Latest articles https://dmtcs.episciences.org/img/episciences_sign_50x50.png https://dmtcs.episciences.org Fri, 29 Nov 2024 00:35:28 +0000 episciences.org https://dmtcs.episciences.org Discrete Mathematics & Theoretical Computer Science Discrete Mathematics & Theoretical Computer Science <![CDATA[Let $G$ be a connected graph and $L(G)$ the set of all integers $k$ such that $G$ contains a spanning tree with exactly $k$ leaves. We show that for a connected graph $G$, the set $L(G)$ is contiguous. It follows from work of Chen, Ren, and Shan that every connected and locally connected $n$-vertex graph -- this includes triangulations -- has a spanning tree with at least $n/2 + 1$ leaves, so by a classic theorem of Whitney and our result, in any plane $4$-connected $n$-vertex triangulation one can find for any integer $k$ which is at least $2$ and at most $n/2 + 1$ a spanning tree with exactly $k$ leaves (and each of these trees can be constructed in polynomial time). We also prove that there exist infinitely many $n$ such that there is a plane $4$-connected $n$-vertex triangulation containing a spanning tree with $2n/3$ leaves, but no spanning tree with more than $2n/3$ leaves.]]> Mon, 18 Nov 2024 08:04:36 +0000 https://doi.org/10.46298/dmtcs.13116 https://doi.org/10.46298/dmtcs.13116 Noguchi, Kenta Zamfirescu, Carol T. Noguchi, Kenta Zamfirescu, Carol T. <![CDATA[Let $G$ be a connected graph and $L(G)$ the set of all integers $k$ such that $G$ contains a spanning tree with exactly $k$ leaves. We show that for a connected graph $G$, the set $L(G)$ is contiguous. It follows from work of Chen, Ren, and Shan that every connected and locally connected $n$-vertex graph -- this includes triangulations -- has a spanning tree with at least $n/2 + 1$ leaves, so by a classic theorem of Whitney and our result, in any plane $4$-connected $n$-vertex triangulation one can find for any integer $k$ which is at least $2$ and at most $n/2 + 1$ a spanning tree with exactly $k$ leaves (and each of these trees can be constructed in polynomial time). We also prove that there exist infinitely many $n$ such that there is a plane $4$-connected $n$-vertex triangulation containing a spanning tree with $2n/3$ leaves, but no spanning tree with more than $2n/3$ leaves.]]> 0 <![CDATA[Cover-Incomparability graphs (C-I graphs) are an interesting class of graphs from posets. A C-I graph is a graph from a poset $P=(V,\le)$ with vertex set $V$, and the edge-set is the union of edge sets of the cover graph and the incomparability graph of the poset. The recognition of the C-I graphs is known to be NP-complete (Maxov\'{a} et al., Order 26(3), 229--236(2009)). In this paper, we prove that chordal graphs having at most two independent simplicial vertices are exactly the chordal graphs which are also C-I graphs. A similar result is obtained for cographs as well. Using the structural results of these graphs, we derive linear time recognition algorithms for chordal graphs and cographs which are C-I graphs.]]> Fri, 15 Nov 2024 21:30:00 +0000 https://doi.org/10.46298/dmtcs.11657 https://doi.org/10.46298/dmtcs.11657 Anil, Arun Changat, Manoj Anil, Arun Changat, Manoj <![CDATA[Cover-Incomparability graphs (C-I graphs) are an interesting class of graphs from posets. A C-I graph is a graph from a poset $P=(V,\le)$ with vertex set $V$, and the edge-set is the union of edge sets of the cover graph and the incomparability graph of the poset. The recognition of the C-I graphs is known to be NP-complete (Maxov\'{a} et al., Order 26(3), 229--236(2009)). In this paper, we prove that chordal graphs having at most two independent simplicial vertices are exactly the chordal graphs which are also C-I graphs. A similar result is obtained for cographs as well. Using the structural results of these graphs, we derive linear time recognition algorithms for chordal graphs and cographs which are C-I graphs.]]> 0 <![CDATA[Lopsp-operations are operations on maps that are applied locally and are guaranteed to preserve all the orientation-preserving symmetries of maps. Well-known examples of such operations are dual, ambo, truncation, and leapfrog. They are described by plane 3-coloured triangulations with specific properties. We developed and implemented a program that can generate all lopsp-operations of a given size by reducing the problem of generating lopsp-operations to generating all plane quadrangulations that are not necessarily simple. We extended the program plantri to generate these quadrangulations.]]> Fri, 15 Nov 2024 21:22:42 +0000 https://doi.org/10.46298/dmtcs.13080 https://doi.org/10.46298/dmtcs.13080 Camp, Heidi Van den McKay, Brendan D. Camp, Heidi Van den McKay, Brendan D. <![CDATA[Lopsp-operations are operations on maps that are applied locally and are guaranteed to preserve all the orientation-preserving symmetries of maps. Well-known examples of such operations are dual, ambo, truncation, and leapfrog. They are described by plane 3-coloured triangulations with specific properties. We developed and implemented a program that can generate all lopsp-operations of a given size by reducing the problem of generating lopsp-operations to generating all plane quadrangulations that are not necessarily simple. We extended the program plantri to generate these quadrangulations.]]> 0 <![CDATA[A subset $S$ of vertices in a graph $G=(V, E)$ is a Dominating Set if each vertex in $V(G)\setminus S$ is adjacent to at least one vertex in $S$. Chellali et al. in 2013, by restricting the number of neighbors in $S$ of a vertex outside $S$, introduced the concept of $[1,j]$-dominating set. A set $D \subseteq V$ of a graph $G = (V, E)$ is called a $[1,j]$-Dominating Set of $G$ if every vertex not in $D$ has at least one neighbor and at most $j$ neighbors in $D$. The Minimum $[1,j]$-Domination problem is the problem of finding the minimum $[1,j]$-dominating set $D$. Given a positive integer $k$ and a graph $G = (V, E)$, the $[1,j]$-Domination Decision problem is to decide whether $G$ has a $[1,j]$-dominating set of cardinality at most $k$. A polynomial-time algorithm was obtained in split graphs for a constant $j$ in contrast to the Dominating Set problem which is NP-hard for split graphs. This result motivates us to investigate the effect of restriction $j$ on the complexity of $[1,j]$-domination problem on various classes of graphs. Although for $j\geq 3$, it has been proved that the minimum of classical domination is equal to minimum $[1,j]$-domination in interval graphs, the complexity of finding the minimum $[1,2]$-domination in interval graphs is still outstanding. In this paper, we propose a polynomial-time algorithm for computing a minimum $[1,2]$-dominating set on interval graphs by a dynamic programming technique. Next, on the negative side, we show that the minimum $[1,2]$-dominating set problem on circle graphs is $NP$-complete.]]> Fri, 15 Nov 2024 20:33:17 +0000 https://doi.org/10.46298/dmtcs.13194 https://doi.org/10.46298/dmtcs.13194 Meybodi, Mohsen Alambardar Poureidi, Abolfazl Meybodi, Mohsen Alambardar Poureidi, Abolfazl <![CDATA[A subset $S$ of vertices in a graph $G=(V, E)$ is a Dominating Set if each vertex in $V(G)\setminus S$ is adjacent to at least one vertex in $S$. Chellali et al. in 2013, by restricting the number of neighbors in $S$ of a vertex outside $S$, introduced the concept of $[1,j]$-dominating set. A set $D \subseteq V$ of a graph $G = (V, E)$ is called a $[1,j]$-Dominating Set of $G$ if every vertex not in $D$ has at least one neighbor and at most $j$ neighbors in $D$. The Minimum $[1,j]$-Domination problem is the problem of finding the minimum $[1,j]$-dominating set $D$. Given a positive integer $k$ and a graph $G = (V, E)$, the $[1,j]$-Domination Decision problem is to decide whether $G$ has a $[1,j]$-dominating set of cardinality at most $k$. A polynomial-time algorithm was obtained in split graphs for a constant $j$ in contrast to the Dominating Set problem which is NP-hard for split graphs. This result motivates us to investigate the effect of restriction $j$ on the complexity of $[1,j]$-domination problem on various classes of graphs. Although for $j\geq 3$, it has been proved that the minimum of classical domination is equal to minimum $[1,j]$-domination in interval graphs, the complexity of finding the minimum $[1,2]$-domination in interval graphs is still outstanding. In this paper, we propose a polynomial-time algorithm for computing a minimum $[1,2]$-dominating set on interval graphs by a dynamic programming technique. Next, on the negative side, we show that the minimum $[1,2]$-dominating set problem on circle graphs is $NP$-complete.]]> 0 <![CDATA[In this work, we study the Biclique-Free Vertex Deletion problem: Given a graph $G$ and integers $k$ and $i \le j$, find a set of at most $k$ vertices that intersects every (not necessarily induced) biclique $K_{i, j}$ in $G$. This is a natural generalization of the Bounded-Degree Deletion problem, wherein one asks whether there is a set of at most $k$ vertices whose deletion results in a graph of a given maximum degree $r$. The two problems coincide when $i = 1$ and $j = r + 1$. We show that Biclique-Free Vertex Deletion is fixed-parameter tractable with respect to $k + d$ for the degeneracy $d$ by developing a $2^{O(d k^2)} \cdot n^{O(1)}$-time algorithm. We also show that it can be solved in $2^{O(f k)} \cdot n^{O(1)}$ time for the feedback vertex number $f$ when $i \ge 2$. In contrast, we find that it is W[1]-hard for the treedepth for any integer $i \ge 1$. Finally, we show that Biclique-Free Vertex Deletion has a polynomial kernel for every $i \ge 1$ when parameterized by the feedback edge number. Previously, for this parameter, its fixed-parameter tractability for $i = 1$ was known (Betzler et al., 2012) but the existence of polynomial kernel was open.]]> Fri, 15 Nov 2024 20:16:36 +0000 https://doi.org/10.46298/dmtcs.13018 https://doi.org/10.46298/dmtcs.13018 Goldmann, Lito Kellerhals, Leon Koana, Tomohiro Goldmann, Lito Kellerhals, Leon Koana, Tomohiro <![CDATA[In this work, we study the Biclique-Free Vertex Deletion problem: Given a graph $G$ and integers $k$ and $i \le j$, find a set of at most $k$ vertices that intersects every (not necessarily induced) biclique $K_{i, j}$ in $G$. This is a natural generalization of the Bounded-Degree Deletion problem, wherein one asks whether there is a set of at most $k$ vertices whose deletion results in a graph of a given maximum degree $r$. The two problems coincide when $i = 1$ and $j = r + 1$. We show that Biclique-Free Vertex Deletion is fixed-parameter tractable with respect to $k + d$ for the degeneracy $d$ by developing a $2^{O(d k^2)} \cdot n^{O(1)}$-time algorithm. We also show that it can be solved in $2^{O(f k)} \cdot n^{O(1)}$ time for the feedback vertex number $f$ when $i \ge 2$. In contrast, we find that it is W[1]-hard for the treedepth for any integer $i \ge 1$. Finally, we show that Biclique-Free Vertex Deletion has a polynomial kernel for every $i \ge 1$ when parameterized by the feedback edge number. Previously, for this parameter, its fixed-parameter tractability for $i = 1$ was known (Betzler et al., 2012) but the existence of polynomial kernel was open.]]> 0 <![CDATA[Complete non-ambiguous trees (CNATs) are combinatorial objects which appear in various contexts.Recently, Chen and Ohlig studied the notion of permutations associated to these objects, and proposed a series of nice conjectures.Most of them were proved by Selig and Zhu, through a connection with the abelian sandpile model.But one conjecture remained open, about the distribution of a natural statistic named determinant.We prove this conjecture, in a bijective way.]]> Fri, 15 Nov 2024 19:28:15 +0000 https://doi.org/10.46298/dmtcs.12850 https://doi.org/10.46298/dmtcs.12850 Aval, Jean-Christophe Aval, Jean-Christophe <![CDATA[Complete non-ambiguous trees (CNATs) are combinatorial objects which appear in various contexts.Recently, Chen and Ohlig studied the notion of permutations associated to these objects, and proposed a series of nice conjectures.Most of them were proved by Selig and Zhu, through a connection with the abelian sandpile model.But one conjecture remained open, about the distribution of a natural statistic named determinant.We prove this conjecture, in a bijective way.]]> 0 <![CDATA[For a graph $G$ and an integer $k\geq 2$, a $\chi'_{k}$-coloring of $G$ is an edge coloring of $G$ such that the subgraph induced by the edges of each color has all degrees congruent to $1 ~ (\mod k)$, and $\chi'_{k}(G)$ is the minimum number of colors in a $\chi'_{k}$-coloring of $G$. In ["The mod $k$ chromatic index of graphs is $O(k)$", J. Graph Theory. 2023; 102: 197-200], Botler, Colucci and Kohayakawa proved that $\chi'_{k}(G)\leq 198k-101$ for every graph $G$. In this paper, we show that $\chi'_{k}(G) \leq 177k-93$.]]> Sun, 10 Nov 2024 15:19:46 +0000 https://doi.org/10.46298/dmtcs.13187 https://doi.org/10.46298/dmtcs.13187 Nweit, Oothan Yang, Daqing Nweit, Oothan Yang, Daqing <![CDATA[For a graph $G$ and an integer $k\geq 2$, a $\chi'_{k}$-coloring of $G$ is an edge coloring of $G$ such that the subgraph induced by the edges of each color has all degrees congruent to $1 ~ (\mod k)$, and $\chi'_{k}(G)$ is the minimum number of colors in a $\chi'_{k}$-coloring of $G$. In ["The mod $k$ chromatic index of graphs is $O(k)$", J. Graph Theory. 2023; 102: 197-200], Botler, Colucci and Kohayakawa proved that $\chi'_{k}(G)\leq 198k-101$ for every graph $G$. In this paper, we show that $\chi'_{k}(G) \leq 177k-93$.]]> 0 <![CDATA[Ammann bars are formed by segments (decorations) on the tiles of a tiling such that forming straight lines with them while tiling forces non-periodicity. Only a few cases are known, starting with Robert Ammann's observations on Penrose tiles, but there is no general explanation or construction. In this article we propose a general method for cut and project tilings based on the notion of subperiods and we illustrate it with an aperiodic set of 36 decorated prototiles related to what we called Cyrenaic tilings.]]> Mon, 04 Nov 2024 13:18:44 +0000 https://doi.org/10.46298/dmtcs.10764 https://doi.org/10.46298/dmtcs.10764 Fernique, Thomas Porrier, Carole Fernique, Thomas Porrier, Carole <![CDATA[Ammann bars are formed by segments (decorations) on the tiles of a tiling such that forming straight lines with them while tiling forces non-periodicity. Only a few cases are known, starting with Robert Ammann's observations on Penrose tiles, but there is no general explanation or construction. In this article we propose a general method for cut and project tilings based on the notion of subperiods and we illustrate it with an aperiodic set of 36 decorated prototiles related to what we called Cyrenaic tilings.]]> 0 <![CDATA[Binary functions are a generalisation of the cocircuit spaces of binary matroids to arbitrary functions. Every rank function is assigned a binary function, and the deletion and contraction operations of binary functions generalise matroid deletion and contraction. We give the excluded minor characterisations for the classes of binary functions with well defined minors, and those with an associated rank function. Within these classes, we also characterise the classes of binary functions corresponding to polymatroids, matroids and binary matroids by their excluded minors. This gives a new proof of Tutte's excluded minor characterisation of binary matroids in the more generalised space of binary functions.]]> Mon, 04 Nov 2024 13:17:38 +0000 https://doi.org/10.46298/dmtcs.12230 https://doi.org/10.46298/dmtcs.12230 Jones, Benjamin R. Jones, Benjamin R. <![CDATA[Binary functions are a generalisation of the cocircuit spaces of binary matroids to arbitrary functions. Every rank function is assigned a binary function, and the deletion and contraction operations of binary functions generalise matroid deletion and contraction. We give the excluded minor characterisations for the classes of binary functions with well defined minors, and those with an associated rank function. Within these classes, we also characterise the classes of binary functions corresponding to polymatroids, matroids and binary matroids by their excluded minors. This gives a new proof of Tutte's excluded minor characterisation of binary matroids in the more generalised space of binary functions.]]> 0 <![CDATA[Bouc (1992) first studied the topological properties of $M_n$, the matching complex of the complete graph of order $n$, in connection with Brown complexes and Quillen complexes. Bj\"{o}rner et al. (1994) showed that $M_n$ is homotopically $(\nu_n-1)$-connected, where $\nu_n=\lfloor{\frac{n+1}{3}}\rfloor-1$, and conjectured that this connectivity bound is sharp. Shareshian and Wachs (2007) settled the conjecture by inductively showing that the $\nu_n$-dimensional homology group of $M_n$ is nontrivial, with Bouc's calculation of $H_1(M_7)$ serving as the pivotal base step. In general, the topology of $M_n$ is not very well-understood, even for a small $n$. In the present article, we look into the topology of $M_n$, and $M_7$ in particular, in the light of discrete Morse theory as developed by Forman (1998). We first construct a gradient vector field on $M_n$ (for $n \ge 5$) that doesn't admit any critical simplices of dimension up to $\nu_n-1$, except one unavoidable $0$-simplex, which also leads to the aforementioned $(\nu_n-1)$-connectedness of $M_n$ in a purely combinatorial way. However, for an efficient homology computation by discrete Morse theoretic techniques, we are required to work with a gradient vector field that admits a low number of critical simplices, and also allows an efficient enumeration of gradient paths. An optimal gradient vector field is one with the least number of critical simplices, but the problem of finding an optimal gradient vector field, in general, is an NP-hard problem (even for $2$-dimensional complexes). We improve the gradient vector field constructed on $M_7$ in particular to a much more efficient (near-optimal) one, and then with the help of this improved gradient vector field, compute the homology groups of $M_7$ in an efficient and algorithmic manner. We also augment this near-optimal gradient vector field to one that we conjecture to be optimal.]]> Mon, 04 Nov 2024 13:17:03 +0000 https://doi.org/10.46298/dmtcs.12887 https://doi.org/10.46298/dmtcs.12887 Mondal, Anupam Mukherjee, Sajal Saha, Kuldeep Mondal, Anupam Mukherjee, Sajal Saha, Kuldeep <![CDATA[Bouc (1992) first studied the topological properties of $M_n$, the matching complex of the complete graph of order $n$, in connection with Brown complexes and Quillen complexes. Bj\"{o}rner et al. (1994) showed that $M_n$ is homotopically $(\nu_n-1)$-connected, where $\nu_n=\lfloor{\frac{n+1}{3}}\rfloor-1$, and conjectured that this connectivity bound is sharp. Shareshian and Wachs (2007) settled the conjecture by inductively showing that the $\nu_n$-dimensional homology group of $M_n$ is nontrivial, with Bouc's calculation of $H_1(M_7)$ serving as the pivotal base step. In general, the topology of $M_n$ is not very well-understood, even for a small $n$. In the present article, we look into the topology of $M_n$, and $M_7$ in particular, in the light of discrete Morse theory as developed by Forman (1998). We first construct a gradient vector field on $M_n$ (for $n \ge 5$) that doesn't admit any critical simplices of dimension up to $\nu_n-1$, except one unavoidable $0$-simplex, which also leads to the aforementioned $(\nu_n-1)$-connectedness of $M_n$ in a purely combinatorial way. However, for an efficient homology computation by discrete Morse theoretic techniques, we are required to work with a gradient vector field that admits a low number of critical simplices, and also allows an efficient enumeration of gradient paths. An optimal gradient vector field is one with the least number of critical simplices, but the problem of finding an optimal gradient vector field, in general, is an NP-hard problem (even for $2$-dimensional complexes). We improve the gradient vector field constructed on $M_7$ in particular to a much more efficient (near-optimal) one, and then with the help of this improved gradient vector field, compute the homology groups of $M_7$ in an efficient and algorithmic manner. We also augment this near-optimal gradient vector field to one that we conjecture to be optimal.]]> 0 <![CDATA[In this paper, we describe minimal string attractors (of size two) of pseudopalindromic prefixes of standard complementary-symmetric Rote sequences. Such a class of Rote sequences forms a subclass of binary generalized pseudostandard sequences, i.e., of sequences obtained when iterating palindromic and antipalindromic closures. When iterating only palindromic closure, palindromic prefixes of standard Sturmian sequences are obtained and their string attractors are of size two. However, already when iterating only antipalindromic closure, antipalindromic prefixes of binary pseudostandard sequences are obtained and we prove that the minimal string attractors are of size three in this case. We conjecture that the pseudopalindromic prefixes of any binary generalized pseudostandard sequence have a minimal string attractor of size at most four.]]> Mon, 04 Nov 2024 13:16:20 +0000 https://doi.org/10.46298/dmtcs.12385 https://doi.org/10.46298/dmtcs.12385 Dvořáková, Lubomíra Hendrychová, Veronika Dvořáková, Lubomíra Hendrychová, Veronika <![CDATA[In this paper, we describe minimal string attractors (of size two) of pseudopalindromic prefixes of standard complementary-symmetric Rote sequences. Such a class of Rote sequences forms a subclass of binary generalized pseudostandard sequences, i.e., of sequences obtained when iterating palindromic and antipalindromic closures. When iterating only palindromic closure, palindromic prefixes of standard Sturmian sequences are obtained and their string attractors are of size two. However, already when iterating only antipalindromic closure, antipalindromic prefixes of binary pseudostandard sequences are obtained and we prove that the minimal string attractors are of size three in this case. We conjecture that the pseudopalindromic prefixes of any binary generalized pseudostandard sequence have a minimal string attractor of size at most four.]]> 0 <![CDATA[In this paper, we study the total displacement statistic of parking functions from the perspective of cooperative game theory. We introduce parking games, which are coalitional cost-sharing games in characteristic function form derived from the total displacement statistic. We show that parking games are supermodular cost-sharing games, indicating that cooperation is difficult (i.e., their core is empty). Next, we study their Shapley value, which formalizes a notion of "fair" cost-sharing and amounts to charging each car for its expected marginal displacement under a random arrival order. Our main contribution is a polynomial-time algorithm to compute the Shapley value of parking games, in contrast with known hardness results on computing the Shapley value of arbitrary games. The algorithm leverages the permutation-invariance of total displacement, combinatorial enumeration, and dynamic programming. We conclude with open questions around an alternative solution concept for supermodular cost-sharing games and connections to other areas in combinatorics.]]> Mon, 04 Nov 2024 13:15:46 +0000 https://doi.org/10.46298/dmtcs.13113 https://doi.org/10.46298/dmtcs.13113 Elder, Jennifer Harris, Pamela E. Kretschmann, Jan Mori, J. Carlos Martínez Elder, Jennifer Harris, Pamela E. Kretschmann, Jan Mori, J. Carlos Martínez <![CDATA[In this paper, we study the total displacement statistic of parking functions from the perspective of cooperative game theory. We introduce parking games, which are coalitional cost-sharing games in characteristic function form derived from the total displacement statistic. We show that parking games are supermodular cost-sharing games, indicating that cooperation is difficult (i.e., their core is empty). Next, we study their Shapley value, which formalizes a notion of "fair" cost-sharing and amounts to charging each car for its expected marginal displacement under a random arrival order. Our main contribution is a polynomial-time algorithm to compute the Shapley value of parking games, in contrast with known hardness results on computing the Shapley value of arbitrary games. The algorithm leverages the permutation-invariance of total displacement, combinatorial enumeration, and dynamic programming. We conclude with open questions around an alternative solution concept for supermodular cost-sharing games and connections to other areas in combinatorics.]]> 0 <![CDATA[Three polynomials are defined for given sets $S$ of $n$ points in general position in the plane: The Voronoi polynomial with coefficients the numbers of vertices of the order-$k$ Voronoi diagrams of $S$, the circle polynomial with coefficients the numbers of circles through three points of $S$ enclosing $k$ points of $S$, and the $E_{\leq k}$ polynomial with coefficients the numbers of (at most $k$)-edges of $S$. We present several formulas for the rectilinear crossing number of $S$ in terms of these polynomials and their roots. We also prove that the roots of the Voronoi polynomial lie on the unit circle if, and only if, $S$ is in convex position. Further, we present bounds on the location of the roots of these polynomials.]]> Mon, 04 Nov 2024 13:14:12 +0000 https://doi.org/10.46298/dmtcs.12443 https://doi.org/10.46298/dmtcs.12443 Claverol, Mercè Heras-Parrilla, Andrea de las Flores-Peñaloza, David Huemer, Clemens Orden, David Claverol, Mercè Heras-Parrilla, Andrea de las Flores-Peñaloza, David Huemer, Clemens Orden, David <![CDATA[Three polynomials are defined for given sets $S$ of $n$ points in general position in the plane: The Voronoi polynomial with coefficients the numbers of vertices of the order-$k$ Voronoi diagrams of $S$, the circle polynomial with coefficients the numbers of circles through three points of $S$ enclosing $k$ points of $S$, and the $E_{\leq k}$ polynomial with coefficients the numbers of (at most $k$)-edges of $S$. We present several formulas for the rectilinear crossing number of $S$ in terms of these polynomials and their roots. We also prove that the roots of the Voronoi polynomial lie on the unit circle if, and only if, $S$ is in convex position. Further, we present bounds on the location of the roots of these polynomials.]]> 0 <![CDATA[Interval parking functions are a generalization of parking functions in which cars have an interval preference for their parking. We generalize this definition to parking functions with $n$ cars and $m\geq n$ parking spots, which we call interval rational parking functions and provide a formula for their enumeration. By specifying an integer parameter $\ell\geq 0$, we then consider the subset of interval rational parking functions in which each car parks at most $\ell$ spots away from their initial preference. We call these $\ell$-interval rational parking functions and provide recursive formulas to enumerate this set for all positive integers $m\geq n$ and $\ell$. We also establish formulas for the number of nondecreasing $\ell$-interval rational parking functions via the outcome map on rational parking functions. We also consider the intersection between $\ell$-interval parking functions and Fubini rankings and show the enumeration of these sets is given by generalized Fibonacci numbers. We conclude by specializing $\ell=1$, and establish that the set of $1$-interval rational parking functions with $n$ cars and $m$ spots are in bijection with the set of barred preferential arrangements of $[n]$ with $m-n$ bars. This readily implies enumerative formulas. Further, in the case where $\ell=1$, we recover the results of Hadaway and Harris that unit interval parking functions are in bijection with the set of Fubini rankings, which are enumerated by the Fubini numbers.]]> Mon, 04 Nov 2024 13:12:09 +0000 https://doi.org/10.46298/dmtcs.12598 https://doi.org/10.46298/dmtcs.12598 Aguilar-Fraga, Tomás Elder, Jennifer Garcia, Rebecca E. Hadaway, Kimberly P. Harris, Pamela E. Harry, Kimberly J. Hogan, Imhotep B. Johnson, Jakeyl Kretschmann, Jan Lawson-Chavanu, Kobe Mori, J. Carlos Martínez Monroe, Casandra D. Quiñonez, Daniel Tolson III, Dirk Williams II, Dwight Anderson Aguilar-Fraga, Tomás Elder, Jennifer Garcia, Rebecca E. Hadaway, Kimberly P. Harris, Pamela E. Harry, Kimberly J. Hogan, Imhotep B. Johnson, Jakeyl Kretschmann, Jan Lawson-Chavanu, Kobe Mori, J. Carlos Martínez Monroe, Casandra D. Quiñonez, Daniel Tolson III, Dirk Williams II, Dwight Anderson <![CDATA[Interval parking functions are a generalization of parking functions in which cars have an interval preference for their parking. We generalize this definition to parking functions with $n$ cars and $m\geq n$ parking spots, which we call interval rational parking functions and provide a formula for their enumeration. By specifying an integer parameter $\ell\geq 0$, we then consider the subset of interval rational parking functions in which each car parks at most $\ell$ spots away from their initial preference. We call these $\ell$-interval rational parking functions and provide recursive formulas to enumerate this set for all positive integers $m\geq n$ and $\ell$. We also establish formulas for the number of nondecreasing $\ell$-interval rational parking functions via the outcome map on rational parking functions. We also consider the intersection between $\ell$-interval parking functions and Fubini rankings and show the enumeration of these sets is given by generalized Fibonacci numbers. We conclude by specializing $\ell=1$, and establish that the set of $1$-interval rational parking functions with $n$ cars and $m$ spots are in bijection with the set of barred preferential arrangements of $[n]$ with $m-n$ bars. This readily implies enumerative formulas. Further, in the case where $\ell=1$, we recover the results of Hadaway and Harris that unit interval parking functions are in bijection with the set of Fubini rankings, which are enumerated by the Fubini numbers.]]> 0 <![CDATA[We consider the class $S_n(1324)$ of permutations of size $n$ that avoid the pattern 1324 and examine the subset $S_n^{a\prec n}(1324)$ of elements for which $a\prec n\prec [a-1]$, $a\ge 1$. This notation means that, when written in one line notation, such a permutation must have $a$ to the left of $n$, and the elements of $\{1,\dots,a-1\}$ must all be to the right of $n$. For $n\ge 2$, we establish a connection between the subset of permutations in $S_n^{1\prec n}(1324)$ having the 1 adjacent to the $n$ (called primitives), and the set of 1324-avoiding dominoes with $n-2$ points. For $a\in\{1,2\}$, we introduce constructive algorithms and give formulas for the enumeration of $S_n^{a\prec n}(1324)$ by the position of $a$ relative to the position of $n$. For $a\ge 3$, we formulate some conjectures for the corresponding generating functions.]]> Mon, 04 Nov 2024 13:10:46 +0000 https://doi.org/10.46298/dmtcs.12629 https://doi.org/10.46298/dmtcs.12629 Gil, Juan B. Lopez, Oscar A. Weiner, Michael D. Gil, Juan B. Lopez, Oscar A. Weiner, Michael D. <![CDATA[We consider the class $S_n(1324)$ of permutations of size $n$ that avoid the pattern 1324 and examine the subset $S_n^{a\prec n}(1324)$ of elements for which $a\prec n\prec [a-1]$, $a\ge 1$. This notation means that, when written in one line notation, such a permutation must have $a$ to the left of $n$, and the elements of $\{1,\dots,a-1\}$ must all be to the right of $n$. For $n\ge 2$, we establish a connection between the subset of permutations in $S_n^{1\prec n}(1324)$ having the 1 adjacent to the $n$ (called primitives), and the set of 1324-avoiding dominoes with $n-2$ points. For $a\in\{1,2\}$, we introduce constructive algorithms and give formulas for the enumeration of $S_n^{a\prec n}(1324)$ by the position of $a$ relative to the position of $n$. For $a\ge 3$, we formulate some conjectures for the corresponding generating functions.]]> 0 <![CDATA[Finding distributions of permutation statistics over pattern-avoiding classes of permutations attracted much attention in the literature. In particular, Bukata et al. found distributions of ascents and descents on permutations avoiding any two patterns of length 3. In this paper, we generalize these results in two different ways: we find explicit formulas for the joint distribution of six statistics (asc, des, lrmax, lrmin, rlmax, rlmin), and also explicit formulas for the joint distribution of four statistics (asc, des, MNA, MND) on these permutations in all cases. The latter result also extends the recent studies by Kitaev and Zhang of the statistics MNA and MND (related to non-overlapping occurrences of ascents and descents) on stack-sortable permutations. All multivariate generating functions in our paper are rational, and we provide combinatorial proofs of five equidistribution results that can be derived from the generating functions.]]> Mon, 04 Nov 2024 13:09:44 +0000 https://doi.org/10.46298/dmtcs.12517 https://doi.org/10.46298/dmtcs.12517 Han, Tian Kitaev, Sergey Han, Tian Kitaev, Sergey <![CDATA[Finding distributions of permutation statistics over pattern-avoiding classes of permutations attracted much attention in the literature. In particular, Bukata et al. found distributions of ascents and descents on permutations avoiding any two patterns of length 3. In this paper, we generalize these results in two different ways: we find explicit formulas for the joint distribution of six statistics (asc, des, lrmax, lrmin, rlmax, rlmin), and also explicit formulas for the joint distribution of four statistics (asc, des, MNA, MND) on these permutations in all cases. The latter result also extends the recent studies by Kitaev and Zhang of the statistics MNA and MND (related to non-overlapping occurrences of ascents and descents) on stack-sortable permutations. All multivariate generating functions in our paper are rational, and we provide combinatorial proofs of five equidistribution results that can be derived from the generating functions.]]> 0 <![CDATA[Shannon Entropy is the preeminent tool for measuring the level of uncertainty (and conversely, information content) in a random variable. In the field of communications, entropy can be used to express the information content of given signals (represented as time series) by considering random variables which sample from specified subsequences. In this paper, we will discuss how an entropy variant, the \textit{permutation entropy} can be used to study and classify radio frequency signals in a noisy environment. The permutation entropy is the entropy of the random variable which samples occurrences of permutation patterns from time series given a fixed window length, making it a function of the distribution of permutation patterns. Since the permutation entropy is a function of the relative order of data, it is (global) amplitude agnostic and thus allows for comparison between signals at different scales. This article is intended to describe a permutation patterns approach to a data driven problem in radio frequency communications research, and includes a primer on all non-permutation pattern specific background. An empirical analysis of the methods herein on radio frequency data is included. No prior knowledge of signals analysis is assumed, and permutation pattern specific notation will be included. This article serves as a self-contained introduction to the relationship between permutation patterns, entropy, and signals analysis for studying radio frequency signals and includes results on a classification task.]]> Mon, 04 Nov 2024 13:08:15 +0000 https://doi.org/10.46298/dmtcs.12644 https://doi.org/10.46298/dmtcs.12644 Kay, Bill Myers, Audun Boydston, Thad Ellwein, Emily Mackenzie, Cameron Alvarez, Iliana Lentz, Erik Kay, Bill Myers, Audun Boydston, Thad Ellwein, Emily Mackenzie, Cameron Alvarez, Iliana Lentz, Erik <![CDATA[Shannon Entropy is the preeminent tool for measuring the level of uncertainty (and conversely, information content) in a random variable. In the field of communications, entropy can be used to express the information content of given signals (represented as time series) by considering random variables which sample from specified subsequences. In this paper, we will discuss how an entropy variant, the \textit{permutation entropy} can be used to study and classify radio frequency signals in a noisy environment. The permutation entropy is the entropy of the random variable which samples occurrences of permutation patterns from time series given a fixed window length, making it a function of the distribution of permutation patterns. Since the permutation entropy is a function of the relative order of data, it is (global) amplitude agnostic and thus allows for comparison between signals at different scales. This article is intended to describe a permutation patterns approach to a data driven problem in radio frequency communications research, and includes a primer on all non-permutation pattern specific background. An empirical analysis of the methods herein on radio frequency data is included. No prior knowledge of signals analysis is assumed, and permutation pattern specific notation will be included. This article serves as a self-contained introduction to the relationship between permutation patterns, entropy, and signals analysis for studying radio frequency signals and includes results on a classification task.]]> 0 <![CDATA[The following problem has been known since the 80s. Let $\Gamma$ be an Abelian group of order $m$ (denoted $|\Gamma|=m$), and let $t$ and $\{m_i\}_{i=1}^{t}$, be positive integers such that $\sum_{i=1}^t m_i=m-1$. Determine when $\Gamma^*=\Gamma\setminus\{0\}$, the set of non-zero elements of $\Gamma$, can be partitioned into disjoint subsets $\{S_i\}_{i=1}^{t}$ such that $|S_i|=m_i$ and $\sum_{s\in S_i}s=0$ for every $1 \leq i \leq t$. Such a subset partition is called a \textit{zero-sum partition}. $|I(\Gamma)|\neq 1$, where $I(\Gamma)$ is the set of involutions in $\Gamma$, is a necessary condition for the existence of zero-sum partitions. In this paper, we show that the additional condition of $m_i\geq 4$ for every $1 \leq i \leq t$, is sufficient. Moreover, we present some applications of zero-sum partitions to magic- and antimagic-type labelings of graphs.]]> Fri, 25 Oct 2024 18:40:16 +0000 https://doi.org/10.46298/dmtcs.12361 https://doi.org/10.46298/dmtcs.12361 Cichacz, Sylwia Suchan, Karol Cichacz, Sylwia Suchan, Karol <![CDATA[The following problem has been known since the 80s. Let $\Gamma$ be an Abelian group of order $m$ (denoted $|\Gamma|=m$), and let $t$ and $\{m_i\}_{i=1}^{t}$, be positive integers such that $\sum_{i=1}^t m_i=m-1$. Determine when $\Gamma^*=\Gamma\setminus\{0\}$, the set of non-zero elements of $\Gamma$, can be partitioned into disjoint subsets $\{S_i\}_{i=1}^{t}$ such that $|S_i|=m_i$ and $\sum_{s\in S_i}s=0$ for every $1 \leq i \leq t$. Such a subset partition is called a \textit{zero-sum partition}. $|I(\Gamma)|\neq 1$, where $I(\Gamma)$ is the set of involutions in $\Gamma$, is a necessary condition for the existence of zero-sum partitions. In this paper, we show that the additional condition of $m_i\geq 4$ for every $1 \leq i \leq t$, is sufficient. Moreover, we present some applications of zero-sum partitions to magic- and antimagic-type labelings of graphs.]]> 0 <![CDATA[A graph $G$ is said to be an $(s, k)$-polar graph if its vertex set admits a partition $(A, B)$ such that $A$ and $B$ induce, respectively, a complete $s$-partite graph and the disjoint union of at most $k$ complete graphs. Polar graphs and monopolar graphs are defined as $(\infty, \infty)$- and $(1, \infty)$-polar graphs, respectively, and unipolar graphs are those graphs with a polar partition $(A, B)$ such that $A$ is a clique. The problems of deciding whether an arbitrary graph is a polar graph or a monopolar graph are known to be NP-complete. In contrast, deciding whether a graph is a unipolar graph can be done in polynomial time. In this work we prove that the three previous problems can be solved in linear time on the classes of $P_4$-sparse and $P_4$-extendible graphs, generalizing analogous results previously known for cographs. Additionally, we provide finite forbidden subgraph characterizations for $(2,2)$-polar graphs on $P_4$-sparse and $P_4$-extendible graphs, also generalizing analogous results recently obtained for the class of cographs.]]> Thu, 10 Oct 2024 13:36:23 +0000 https://doi.org/10.46298/dmtcs.11479 https://doi.org/10.46298/dmtcs.11479 Contreras-Mendoza, Fernando Esteban Hernández-Cruz, César Contreras-Mendoza, Fernando Esteban Hernández-Cruz, César <![CDATA[A graph $G$ is said to be an $(s, k)$-polar graph if its vertex set admits a partition $(A, B)$ such that $A$ and $B$ induce, respectively, a complete $s$-partite graph and the disjoint union of at most $k$ complete graphs. Polar graphs and monopolar graphs are defined as $(\infty, \infty)$- and $(1, \infty)$-polar graphs, respectively, and unipolar graphs are those graphs with a polar partition $(A, B)$ such that $A$ is a clique. The problems of deciding whether an arbitrary graph is a polar graph or a monopolar graph are known to be NP-complete. In contrast, deciding whether a graph is a unipolar graph can be done in polynomial time. In this work we prove that the three previous problems can be solved in linear time on the classes of $P_4$-sparse and $P_4$-extendible graphs, generalizing analogous results previously known for cographs. Additionally, we provide finite forbidden subgraph characterizations for $(2,2)$-polar graphs on $P_4$-sparse and $P_4$-extendible graphs, also generalizing analogous results recently obtained for the class of cographs.]]> 0 <![CDATA[A graph $G$ is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree and an interval $I$, such that each leaf of the tree is a vertex of the graph, and there is an edge $\{ x, y \}$ in $G$ if and only if the weight of the path in the tree connecting $x$ and $y$ lies within the interval $I$. Originating in phylogenetics, PCGs are closely connected to important graph classes like leaf-powers and multi-threshold graphs, widely applied in bioinformatics, especially in understanding evolutionary processes. In this paper we introduce two natural generalizations of the PCG class, namely $k$-OR-PCG and $k$-AND-PCG, which are the classes of graphs that can be expressed as union and intersection, respectively, of $k$ PCGs. These classes can be also described using the concepts of the covering number and the intersection dimension of a graph in relation to the PCG class. We investigate how the classes of OR-PCG and AND-PCG are related to PCGs, $k$-interval-PCGs and other graph classes known in the literature. In particular, we provide upper bounds on the minimum $k$ for which an arbitrary graph $G$ belongs to $k$-interval-PCGs, $k$-OR-PCG or $k$-AND-PCG classes. For particular graph classes we improve these general bounds. Moreover, we show that, for every integer $k$, there exists a bipartite graph that is not in the $k$-interval-PCGs class, proving that there is no finite $k$ for which the $k$-interval-PCG class contains all the graphs. This answers an open question of Ahmed and Rahman from 2017. Finally, using a Ramsey theory argument, we show that for any $k$, there exists graphs that are not in $k$-AND-PCG, and graphs that are not in $k$-OR-PCG.]]> Sun, 06 Oct 2024 16:51:49 +0000 https://doi.org/10.46298/dmtcs.12295 https://doi.org/10.46298/dmtcs.12295 Calamoneri, Tiziana Lafond, Manuel Monti, Angelo Sinaimeri, Blerina Calamoneri, Tiziana Lafond, Manuel Monti, Angelo Sinaimeri, Blerina <![CDATA[A graph $G$ is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree and an interval $I$, such that each leaf of the tree is a vertex of the graph, and there is an edge $\{ x, y \}$ in $G$ if and only if the weight of the path in the tree connecting $x$ and $y$ lies within the interval $I$. Originating in phylogenetics, PCGs are closely connected to important graph classes like leaf-powers and multi-threshold graphs, widely applied in bioinformatics, especially in understanding evolutionary processes. In this paper we introduce two natural generalizations of the PCG class, namely $k$-OR-PCG and $k$-AND-PCG, which are the classes of graphs that can be expressed as union and intersection, respectively, of $k$ PCGs. These classes can be also described using the concepts of the covering number and the intersection dimension of a graph in relation to the PCG class. We investigate how the classes of OR-PCG and AND-PCG are related to PCGs, $k$-interval-PCGs and other graph classes known in the literature. In particular, we provide upper bounds on the minimum $k$ for which an arbitrary graph $G$ belongs to $k$-interval-PCGs, $k$-OR-PCG or $k$-AND-PCG classes. For particular graph classes we improve these general bounds. Moreover, we show that, for every integer $k$, there exists a bipartite graph that is not in the $k$-interval-PCGs class, proving that there is no finite $k$ for which the $k$-interval-PCG class contains all the graphs. This answers an open question of Ahmed and Rahman from 2017. Finally, using a Ramsey theory argument, we show that for any $k$, there exists graphs that are not in $k$-AND-PCG, and graphs that are not in $k$-OR-PCG.]]> 0 <![CDATA[The $[X,Y]$-edge colouring game is played with a set of $k$ colours on a graph $G$ with initially uncoloured edges by two players, Alice (A) and Bob (B). The players move alternately. Player $X\in\{A,B\}$ has the first move. $Y\in\{A,B,-\}$. If $Y\in\{A,B\}$, then only player $Y$ may skip any move, otherwise skipping is not allowed for any player. A move consists of colouring an uncoloured edge with one of the $k$ colours such that adjacent edges have distinct colours. When no more moves are possible, the game ends. If every edge is coloured in the end, Alice wins; otherwise, Bob wins. The $[X,Y]$-game chromatic index $\chi_{[X,Y]}'(G)$ is the smallest nonnegative integer $k$ such that Alice has a winning strategy for the $[X,Y]$-edge colouring game played on $G$ with $k$ colours. The graph $G$ is called line $[X,Y]$-perfect if, for any edge-induced subgraph $H$ of $G$, \[\chi_{[X,Y]}'(H)=\omega(L(H)),\] where $\omega(L(H))$ denotes the clique number of the line graph of $H$. For each of the six possibilities $(X,Y)\in\{A,B\}\times\{A,B,-\}$, we characterise line $[X,Y]$-perfect graphs by forbidden (edge-induced) subgraphs and by explicit structural descriptions, respectively.]]> Sat, 07 Sep 2024 06:12:10 +0000 https://doi.org/10.46298/dmtcs.10971 https://doi.org/10.46298/dmtcs.10971 Andres, Stephan Dominique Fong, Wai Lam Andres, Stephan Dominique Fong, Wai Lam <![CDATA[The $[X,Y]$-edge colouring game is played with a set of $k$ colours on a graph $G$ with initially uncoloured edges by two players, Alice (A) and Bob (B). The players move alternately. Player $X\in\{A,B\}$ has the first move. $Y\in\{A,B,-\}$. If $Y\in\{A,B\}$, then only player $Y$ may skip any move, otherwise skipping is not allowed for any player. A move consists of colouring an uncoloured edge with one of the $k$ colours such that adjacent edges have distinct colours. When no more moves are possible, the game ends. If every edge is coloured in the end, Alice wins; otherwise, Bob wins. The $[X,Y]$-game chromatic index $\chi_{[X,Y]}'(G)$ is the smallest nonnegative integer $k$ such that Alice has a winning strategy for the $[X,Y]$-edge colouring game played on $G$ with $k$ colours. The graph $G$ is called line $[X,Y]$-perfect if, for any edge-induced subgraph $H$ of $G$, \[\chi_{[X,Y]}'(H)=\omega(L(H)),\] where $\omega(L(H))$ denotes the clique number of the line graph of $H$. For each of the six possibilities $(X,Y)\in\{A,B\}\times\{A,B,-\}$, we characterise line $[X,Y]$-perfect graphs by forbidden (edge-induced) subgraphs and by explicit structural descriptions, respectively.]]> 0 <![CDATA[Graph burning is a natural discrete graph algorithm inspired by the spread of social contagion. Despite its simplicity, some open problems remain steadfastly unsolved, notably the burning number conjecture, which says that every connected graph of order $m^2$ has burning number at most $m$. Earlier, we showed that the conjecture also holds for a path forest, which is disconnected, provided each of its paths is sufficiently long. However, finding the least sufficient length for this to hold turns out to be nontrivial. In this note, we present our initial findings and conjectures that associate the problem to some naturally impossibly burnable path forests. It is noteworthy that our problem can be reformulated as a topic concerning sumset partition of integers.]]> Wed, 21 Aug 2024 06:50:18 +0000 https://doi.org/10.46298/dmtcs.12709 https://doi.org/10.46298/dmtcs.12709 Tan, Ta Sheng Teh, Wen Chean Tan, Ta Sheng Teh, Wen Chean <![CDATA[Graph burning is a natural discrete graph algorithm inspired by the spread of social contagion. Despite its simplicity, some open problems remain steadfastly unsolved, notably the burning number conjecture, which says that every connected graph of order $m^2$ has burning number at most $m$. Earlier, we showed that the conjecture also holds for a path forest, which is disconnected, provided each of its paths is sufficiently long. However, finding the least sufficient length for this to hold turns out to be nontrivial. In this note, we present our initial findings and conjectures that associate the problem to some naturally impossibly burnable path forests. It is noteworthy that our problem can be reformulated as a topic concerning sumset partition of integers.]]> 0 <![CDATA[In this paper, we study pattern avoidance in weak ascent sequences, giving some results for patterns of length 3. This is an analogous study to one given by Duncan and Steingr\'imsson (2011) for ascent sequences. More precisely, we provide systematically the generating functions for the number of weak ascent sequences avoiding the patterns $001, 011, 012, 021$, and $102$. Additionally, we establish bijective connections between pattern-avoiding weak ascent sequences and other combinatorial objects, such as compositions, upper triangular 01-matrices, and plane trees.]]> Wed, 21 Aug 2024 06:49:15 +0000 https://doi.org/10.46298/dmtcs.12273 https://doi.org/10.46298/dmtcs.12273 Bényi, Beáta Mansour, Toufik Ramírez, José L. Bényi, Beáta Mansour, Toufik Ramírez, José L. <![CDATA[In this paper, we study pattern avoidance in weak ascent sequences, giving some results for patterns of length 3. This is an analogous study to one given by Duncan and Steingr\'imsson (2011) for ascent sequences. More precisely, we provide systematically the generating functions for the number of weak ascent sequences avoiding the patterns $001, 011, 012, 021$, and $102$. Additionally, we establish bijective connections between pattern-avoiding weak ascent sequences and other combinatorial objects, such as compositions, upper triangular 01-matrices, and plane trees.]]> 0 <![CDATA[We study the following model of disease spread in a social network. At first, all individuals are either infected or healthy. Next, in discrete rounds, the disease spreads in the network from infected to healthy individuals such that a healthy individual gets infected if and only if a sufficient number of its direct neighbors are already infected. We represent the social network as a graph. Inspired by the real-world restrictions in the current epidemic, especially by social and physical distancing requirements, we restrict ourselves to networks that can be represented as geometric intersection graphs. We show that finding a minimal vertex set of initially infected individuals to spread the disease in the whole network is computationally hard, already on unit disk graphs. Hence, to provide some algorithmic results, we focus ourselves on simpler geometric graph classes, such as interval graphs and grid graphs.]]> Wed, 21 Aug 2024 06:48:16 +0000 https://doi.org/10.46298/dmtcs.11591 https://doi.org/10.46298/dmtcs.11591 Dvořák, Michal Knop, Dušan Schierreich, Šimon Dvořák, Michal Knop, Dušan Schierreich, Šimon <![CDATA[We study the following model of disease spread in a social network. At first, all individuals are either infected or healthy. Next, in discrete rounds, the disease spreads in the network from infected to healthy individuals such that a healthy individual gets infected if and only if a sufficient number of its direct neighbors are already infected. We represent the social network as a graph. Inspired by the real-world restrictions in the current epidemic, especially by social and physical distancing requirements, we restrict ourselves to networks that can be represented as geometric intersection graphs. We show that finding a minimal vertex set of initially infected individuals to spread the disease in the whole network is computationally hard, already on unit disk graphs. Hence, to provide some algorithmic results, we focus ourselves on simpler geometric graph classes, such as interval graphs and grid graphs.]]> 0 <![CDATA[We use a recent result of Alin Bostan to prove that the generating functions of two infinite sequences of permutation classes are not algebraic.]]> Wed, 21 Aug 2024 06:46:59 +0000 https://doi.org/10.46298/dmtcs.12539 https://doi.org/10.46298/dmtcs.12539 Bona, Miklos Bona, Miklos <![CDATA[We use a recent result of Alin Bostan to prove that the generating functions of two infinite sequences of permutation classes are not algebraic.]]> 0 <![CDATA[A matroid $M$ is an ordered pair $(E,I)$, where $E$ is a finite set called the ground set and a collection $I\subset 2^{E}$ called the independent sets which satisfy the conditions: (i) $\emptyset \in I$, (ii) $I'\subset I \in I$ implies $I'\in I$, and (iii) $I_1,I_2 \in I$ and $|I_1| < |I_2|$ implies that there is an $e\in I_2$ such that $I_1\cup \{e\} \in I$. The rank $rank(M)$ of a matroid $M$ is the maximum size of an independent set. We say that a matroid $M=(E,I)$ is representable over the reals if there is a map $\varphi \colon E \rightarrow \mathbb{R}^{rank(M)}$ such that $I\in I$ if and only if $\varphi(I)$ forms a linearly independent set. We study the problem of matroid realizability over the reals. Given a matroid $M$, we ask whether there is a set of points in the Euclidean space representing $M$. We show that matroid realizability is $\exists \mathbb R$-complete, already for matroids of rank 3. The complexity class $\exists \mathbb R$ can be defined as the family of algorithmic problems that is polynomial-time is equivalent to determining if a multivariate polynomial with integers coefficients has a real root. Our methods are similar to previous methods from the literature. Yet, the result itself was never pointed out and there is no proof readily available in the language of computer science.]]> Tue, 20 Aug 2024 16:48:54 +0000 https://doi.org/10.46298/dmtcs.10810 https://doi.org/10.46298/dmtcs.10810 Kim, Eun Jung de Mesmay, Arnaud Miltzow, Tillmann Kim, Eun Jung de Mesmay, Arnaud Miltzow, Tillmann <![CDATA[A matroid $M$ is an ordered pair $(E,I)$, where $E$ is a finite set called the ground set and a collection $I\subset 2^{E}$ called the independent sets which satisfy the conditions: (i) $\emptyset \in I$, (ii) $I'\subset I \in I$ implies $I'\in I$, and (iii) $I_1,I_2 \in I$ and $|I_1| < |I_2|$ implies that there is an $e\in I_2$ such that $I_1\cup \{e\} \in I$. The rank $rank(M)$ of a matroid $M$ is the maximum size of an independent set. We say that a matroid $M=(E,I)$ is representable over the reals if there is a map $\varphi \colon E \rightarrow \mathbb{R}^{rank(M)}$ such that $I\in I$ if and only if $\varphi(I)$ forms a linearly independent set. We study the problem of matroid realizability over the reals. Given a matroid $M$, we ask whether there is a set of points in the Euclidean space representing $M$. We show that matroid realizability is $\exists \mathbb R$-complete, already for matroids of rank 3. The complexity class $\exists \mathbb R$ can be defined as the family of algorithmic problems that is polynomial-time is equivalent to determining if a multivariate polynomial with integers coefficients has a real root. Our methods are similar to previous methods from the literature. Yet, the result itself was never pointed out and there is no proof readily available in the language of computer science.]]> 0 <![CDATA[A harmonious coloring of a $k$-uniform hypergraph $H$ is a vertex coloring such that no two vertices in the same edge have the same color, and each $k$-element subset of colors appears on at most one edge. The harmonious number $h(H)$ is the least number of colors needed for such a coloring. The paper contains a new proof of the upper bound $h(H)=O(\sqrt[k]{k!m})$ on the harmonious number of hypergraphs of maximum degree $\Delta$ with $m$ edges. We use the local cut lemma of A. Bernshteyn.]]> Tue, 16 Jul 2024 07:10:02 +0000 https://doi.org/10.46298/dmtcs.11101 https://doi.org/10.46298/dmtcs.11101 Czerwiński, Sebastian Czerwiński, Sebastian <![CDATA[A harmonious coloring of a $k$-uniform hypergraph $H$ is a vertex coloring such that no two vertices in the same edge have the same color, and each $k$-element subset of colors appears on at most one edge. The harmonious number $h(H)$ is the least number of colors needed for such a coloring. The paper contains a new proof of the upper bound $h(H)=O(\sqrt[k]{k!m})$ on the harmonious number of hypergraphs of maximum degree $\Delta$ with $m$ edges. We use the local cut lemma of A. Bernshteyn.]]> 0 <![CDATA[Let u and v be vertices in a connected graph G = (V, E). For any integer k such that 0 ≤ k ≤ dG (u, v), the k-slice Sk (u, v) contains all vertices x on a shortest uv-path such that dG (u, x) = k. The leanness of G is the maximum diameter of a slice. This metric graph invariant has been studied under different names, such as "interval thinness" and "fellow traveler property". Graphs with leanness equal to 0, a.k.a. geodetic graphs, also have received special attention in Graph Theory. The practical computation of leanness in real-life complex networks has been studied recently (Mohammed et al., COMPLEX NETWORKS'21). In this paper, we give a finer-grained complexity analysis of two related problems, namely: deciding whether the leanness of a graph G is at most some small value ℓ; and computing the leanness on specific graph classes. We obtain improved algorithms in some cases, and time complexity lower bounds under plausible hypotheses.]]> Mon, 08 Jul 2024 15:21:39 +0000 https://doi.org/10.46298/dmtcs.12544 https://doi.org/10.46298/dmtcs.12544 Coudert, David Coulomb, Samuel Ducoffe, Guillaume Coudert, David Coulomb, Samuel Ducoffe, Guillaume <![CDATA[Let u and v be vertices in a connected graph G = (V, E). For any integer k such that 0 ≤ k ≤ dG (u, v), the k-slice Sk (u, v) contains all vertices x on a shortest uv-path such that dG (u, x) = k. The leanness of G is the maximum diameter of a slice. This metric graph invariant has been studied under different names, such as "interval thinness" and "fellow traveler property". Graphs with leanness equal to 0, a.k.a. geodetic graphs, also have received special attention in Graph Theory. The practical computation of leanness in real-life complex networks has been studied recently (Mohammed et al., COMPLEX NETWORKS'21). In this paper, we give a finer-grained complexity analysis of two related problems, namely: deciding whether the leanness of a graph G is at most some small value ℓ; and computing the leanness on specific graph classes. We obtain improved algorithms in some cases, and time complexity lower bounds under plausible hypotheses.]]> 0 <![CDATA[We introduce coloring groups, which are permutation groups obtained from a proper edge coloring of a graph. These groups generalize the generalized toggle groups of Striker (which themselves generalize the toggle groups introduced by Cameron and Fon-der-Flaass). We present some general results connecting the structure of a coloring group to the structure of its graph coloring, providing graph-theoretic characterizations of the centralizer and primitivity of a coloring group. We apply these results particularly to generalized toggle groups arising from trees as well as coloring groups arising from the independence posets introduced by Thomas and Williams.]]> Sun, 07 Jul 2024 15:24:33 +0000 https://doi.org/10.46298/dmtcs.12753 https://doi.org/10.46298/dmtcs.12753 Adenbaum, Ben Wilson, Alexander Adenbaum, Ben Wilson, Alexander <![CDATA[We introduce coloring groups, which are permutation groups obtained from a proper edge coloring of a graph. These groups generalize the generalized toggle groups of Striker (which themselves generalize the toggle groups introduced by Cameron and Fon-der-Flaass). We present some general results connecting the structure of a coloring group to the structure of its graph coloring, providing graph-theoretic characterizations of the centralizer and primitivity of a coloring group. We apply these results particularly to generalized toggle groups arising from trees as well as coloring groups arising from the independence posets introduced by Thomas and Williams.]]> 0 <![CDATA[In this paper we prove the following new sufficient condition for a digraph to be Hamiltonian: {\it Let $D$ be a 2-strong digraph of order $n\geq 9$. If $n-1$ vertices of $D$ have degrees at least $n+k$ and the remaining vertex has degree at least $n-k-4$, where $k$ is a non-negative integer, then $D$ is Hamiltonian}. This is an extension of Ghouila-Houri's theorem for 2-strong digraphs and is a generalization of an early result of the author (DAN Arm. SSR (91(2):6-8, 1990). The obtained result is best possible in the sense that for $k=0$ there is a digraph of order $n=8$ (respectively, $n=9$) with the minimum degree $n-4=4$ (respectively, with the minimum $n-5=4$) whose $n-1$ vertices have degrees at least $n-1$, but it is not Hamiltonian. We also give a new sufficient condition for a 3-strong digraph to be Hamiltonian-connected.]]> Mon, 24 Jun 2024 12:52:03 +0000 https://doi.org/10.46298/dmtcs.11560 https://doi.org/10.46298/dmtcs.11560 Darbinyan, Samvel Kh. Darbinyan, Samvel Kh. <![CDATA[In this paper we prove the following new sufficient condition for a digraph to be Hamiltonian: {\it Let $D$ be a 2-strong digraph of order $n\geq 9$. If $n-1$ vertices of $D$ have degrees at least $n+k$ and the remaining vertex has degree at least $n-k-4$, where $k$ is a non-negative integer, then $D$ is Hamiltonian}. This is an extension of Ghouila-Houri's theorem for 2-strong digraphs and is a generalization of an early result of the author (DAN Arm. SSR (91(2):6-8, 1990). The obtained result is best possible in the sense that for $k=0$ there is a digraph of order $n=8$ (respectively, $n=9$) with the minimum degree $n-4=4$ (respectively, with the minimum $n-5=4$) whose $n-1$ vertices have degrees at least $n-1$, but it is not Hamiltonian. We also give a new sufficient condition for a 3-strong digraph to be Hamiltonian-connected.]]> 0 <![CDATA[Given two integers $\ell$ and $p$ as well as $\ell$ graph classes $\mathcal{H}_1,\ldots,\mathcal{H}_\ell$, the problems $\mathsf{GraphPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell,p)$, \break $\mathsf{VertPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell)$, and $\mathsf{EdgePart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell)$ ask, given graph $G$ as input, whether $V(G)$, $V(G)$, $E(G)$ respectively can be partitioned into $\ell$ sets $S_1, \ldots, S_\ell$ such that, for each $i$ between $1$ and $\ell$, $G[S_i] \in \mathcal{H}_i$, $G[S_i] \in \mathcal{H}_i$, $(V(G),S_i) \in \mathcal{H}_i$ respectively. Moreover in $\mathsf{GraphPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell,p)$, we request that the number of edges with endpoints in different sets of the partition is bounded by $p$. We show that if there exist dynamic programming tree-decomposition-based algorithms for recognizing the graph classes $\mathcal{H}_i$, for each $i$, then we can constructively create a dynamic programming tree-decomposition-based algorithms for $\mathsf{GraphPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell,p)$, $\mathsf{VertPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell)$, and $\mathsf{EdgePart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell)$. We apply this approach to known problems. For well-studied problems, like VERTEX COVER and GRAPH $q$-COLORING, we obtain running times that are comparable to those of the best known problem-specific algorithms. For an exotic problem from bioinformatics, called DISPLAYGRAPH, this approach improves the known algorithm parameterized by treewidth.]]> Mon, 24 Jun 2024 07:03:42 +0000 https://doi.org/10.46298/dmtcs.11069 https://doi.org/10.46298/dmtcs.11069 Baste, Julien Baste, Julien <![CDATA[Given two integers $\ell$ and $p$ as well as $\ell$ graph classes $\mathcal{H}_1,\ldots,\mathcal{H}_\ell$, the problems $\mathsf{GraphPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell,p)$, \break $\mathsf{VertPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell)$, and $\mathsf{EdgePart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell)$ ask, given graph $G$ as input, whether $V(G)$, $V(G)$, $E(G)$ respectively can be partitioned into $\ell$ sets $S_1, \ldots, S_\ell$ such that, for each $i$ between $1$ and $\ell$, $G[S_i] \in \mathcal{H}_i$, $G[S_i] \in \mathcal{H}_i$, $(V(G),S_i) \in \mathcal{H}_i$ respectively. Moreover in $\mathsf{GraphPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell,p)$, we request that the number of edges with endpoints in different sets of the partition is bounded by $p$. We show that if there exist dynamic programming tree-decomposition-based algorithms for recognizing the graph classes $\mathcal{H}_i$, for each $i$, then we can constructively create a dynamic programming tree-decomposition-based algorithms for $\mathsf{GraphPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell,p)$, $\mathsf{VertPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell)$, and $\mathsf{EdgePart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell)$. We apply this approach to known problems. For well-studied problems, like VERTEX COVER and GRAPH $q$-COLORING, we obtain running times that are comparable to those of the best known problem-specific algorithms. For an exotic problem from bioinformatics, called DISPLAYGRAPH, this approach improves the known algorithm parameterized by treewidth.]]> 0 <![CDATA[Given a closed simple polygon $P$, we say two points $p,q$ see each other if the segment $pq$ is fully contained in $P$. The art gallery problem seeks a minimum size set $G\subset P$ of guards that sees $P$ completely. The only currently correct algorithm to solve the art gallery problem exactly uses algebraic methods and is attributed to Sharir. As the art gallery problem is ER-complete, it seems unlikely to avoid algebraic methods, without additional assumptions. In this paper, we introduce the notion of vision stability. In order to describe vision stability consider an enhanced guard that can see "around the corner" by an angle of $\delta$ or a diminished guard whose vision is by an angle of $\delta$ "blocked" by reflex vertices. A polygon $P$ has vision stability $\delta$ if the optimal number of enhanced guards to guard $P$ is the same as the optimal number of diminished guards to guard $P$. We will argue that most relevant polygons are vision stable. We describe a one-shot vision stable algorithm that computes an optimal guard set for visionstable polygons using polynomial time and solving one integer program. It guarantees to find the optimal solution for every vision stable polygon. We implemented an iterative visionstable algorithm and show its practical performance is slower, but comparable with other state of the art algorithms. Our iterative algorithm is inspired and follows closely the one-shot algorithm. It delays several steps and only computes them when deemed necessary. Given a chord $c$ of a polygon, we denote by $n(c)$ the number of vertices visible from $c$. The chord-width of a polygon is the maximum $n(c)$ over all possible chords $c$. The set of vision stable polygons admits an FPT algorithm when parametrized by the chord-width. Furthermore, the one-shot algorithm runs in FPT time, when parameterized by the number of reflex vertices.]]> Mon, 24 Jun 2024 07:01:22 +0000 https://doi.org/10.46298/dmtcs.9225 https://doi.org/10.46298/dmtcs.9225 Hengeveld, Simon Miltzow, Tillmann Hengeveld, Simon Miltzow, Tillmann <![CDATA[Given a closed simple polygon $P$, we say two points $p,q$ see each other if the segment $pq$ is fully contained in $P$. The art gallery problem seeks a minimum size set $G\subset P$ of guards that sees $P$ completely. The only currently correct algorithm to solve the art gallery problem exactly uses algebraic methods and is attributed to Sharir. As the art gallery problem is ER-complete, it seems unlikely to avoid algebraic methods, without additional assumptions. In this paper, we introduce the notion of vision stability. In order to describe vision stability consider an enhanced guard that can see "around the corner" by an angle of $\delta$ or a diminished guard whose vision is by an angle of $\delta$ "blocked" by reflex vertices. A polygon $P$ has vision stability $\delta$ if the optimal number of enhanced guards to guard $P$ is the same as the optimal number of diminished guards to guard $P$. We will argue that most relevant polygons are vision stable. We describe a one-shot vision stable algorithm that computes an optimal guard set for visionstable polygons using polynomial time and solving one integer program. It guarantees to find the optimal solution for every vision stable polygon. We implemented an iterative visionstable algorithm and show its practical performance is slower, but comparable with other state of the art algorithms. Our iterative algorithm is inspired and follows closely the one-shot algorithm. It delays several steps and only computes them when deemed necessary. Given a chord $c$ of a polygon, we denote by $n(c)$ the number of vertices visible from $c$. The chord-width of a polygon is the maximum $n(c)$ over all possible chords $c$. The set of vision stable polygons admits an FPT algorithm when parametrized by the chord-width. Furthermore, the one-shot algorithm runs in FPT time, when parameterized by the number of reflex vertices.]]> 0 <![CDATA[An edge $e$ of a matching covered graph $G$ is removable if $G-e$ is also matching covered. Carvalho, Lucchesi, and Murty showed that every brick $G$ different from $K_4$ and $\overline{C_6}$ has at least $\Delta-2$ removable edges, where $\Delta$ is the maximum degree of $G$. In this paper, we generalize the result to irreducible near-bricks, where a graph is irreducible if it contains no single ear of length three or more.]]> Wed, 19 Jun 2024 15:38:38 +0000 https://doi.org/10.46298/dmtcs.11747 https://doi.org/10.46298/dmtcs.11747 Wu, Deyu Zhang, Yipei Wang, Xiumei Wu, Deyu Zhang, Yipei Wang, Xiumei <![CDATA[An edge $e$ of a matching covered graph $G$ is removable if $G-e$ is also matching covered. Carvalho, Lucchesi, and Murty showed that every brick $G$ different from $K_4$ and $\overline{C_6}$ has at least $\Delta-2$ removable edges, where $\Delta$ is the maximum degree of $G$. In this paper, we generalize the result to irreducible near-bricks, where a graph is irreducible if it contains no single ear of length three or more.]]> 0 <![CDATA[A graph on at least ${{k+1}}$ vertices is uniformly $k$-connected if each pair of its vertices is connected by $k$ and not more than $k$ independent paths. We reinvestigate a recent constructive characterization of uniformly $3$-connected graphs and obtain a more detailed result that relates the number of vertices to the operations involved in constructing a respective uniformly $3$-connected graph. Furthermore, we investigate how crossing numbers and treewidths behave under the mentioned constructions. We demonstrate how these results can be utilized to study the structure and properties of uniformly $3$-connected graphs with minimum number of vertices of minimum degree.]]> Fri, 07 Jun 2024 11:10:58 +0000 https://doi.org/10.46298/dmtcs.10407 https://doi.org/10.46298/dmtcs.10407 Göring, Frank Hofmann, Tobias Göring, Frank Hofmann, Tobias <![CDATA[A graph on at least ${{k+1}}$ vertices is uniformly $k$-connected if each pair of its vertices is connected by $k$ and not more than $k$ independent paths. We reinvestigate a recent constructive characterization of uniformly $3$-connected graphs and obtain a more detailed result that relates the number of vertices to the operations involved in constructing a respective uniformly $3$-connected graph. Furthermore, we investigate how crossing numbers and treewidths behave under the mentioned constructions. We demonstrate how these results can be utilized to study the structure and properties of uniformly $3$-connected graphs with minimum number of vertices of minimum degree.]]> 0 <![CDATA[We consider the problem of extending partial edge colorings of iterated cartesian products of even cycles and paths, focusing on the case when the precolored edges satisfy either an Evans-type condition or is a matching. In particular, we prove that if $G=C^d_{2k}$ is the $d$th power of the cartesian product of the even cycle $C_{2k}$ with itself, and at most $2d-1$ edges of $G$ are precolored, then there is a proper $2d$-edge coloring of $G$ that agrees with the partial coloring. We show that the same conclusion holds, without restrictions on the number of precolored edges, if any two precolored edges are at distance at least $4$ from each other. For odd cycles of length at least $5$, we prove that if $G=C^d_{2k+1}$ is the $d$th power of the cartesian product of the odd cycle $C_{2k+1}$ with itself ($k\geq2$), and at most $2d$ edges of $G$ are precolored, then there is a proper $(2d+1)$-edge coloring of $G$ that agrees with the partial coloring. Our results generalize previous ones on precoloring extension of hypercubes [Journal of Graph Theory 95 (2020) 410--444].]]> Tue, 04 Jun 2024 07:37:54 +0000 https://doi.org/10.46298/dmtcs.11377 https://doi.org/10.46298/dmtcs.11377 Casselgren, Carl Johan Granholm, Jonas B. Petros, Fikre B. Casselgren, Carl Johan Granholm, Jonas B. Petros, Fikre B. <![CDATA[We consider the problem of extending partial edge colorings of iterated cartesian products of even cycles and paths, focusing on the case when the precolored edges satisfy either an Evans-type condition or is a matching. In particular, we prove that if $G=C^d_{2k}$ is the $d$th power of the cartesian product of the even cycle $C_{2k}$ with itself, and at most $2d-1$ edges of $G$ are precolored, then there is a proper $2d$-edge coloring of $G$ that agrees with the partial coloring. We show that the same conclusion holds, without restrictions on the number of precolored edges, if any two precolored edges are at distance at least $4$ from each other. For odd cycles of length at least $5$, we prove that if $G=C^d_{2k+1}$ is the $d$th power of the cartesian product of the odd cycle $C_{2k+1}$ with itself ($k\geq2$), and at most $2d$ edges of $G$ are precolored, then there is a proper $(2d+1)$-edge coloring of $G$ that agrees with the partial coloring. Our results generalize previous ones on precoloring extension of hypercubes [Journal of Graph Theory 95 (2020) 410--444].]]> 0 <![CDATA[This paper is devoted to the study of particular geometrically defined intersection classes of graphs. Those were previously studied by Magnant and Martin, who proved that these graphs have arbitrary large chromatic number, while being triangle-free. We give several structural properties of these graphs, and we raise several questions.]]> Fri, 17 May 2024 09:14:11 +0000 https://doi.org/10.46298/dmtcs.10805 https://doi.org/10.46298/dmtcs.10805 Gonçalves, Daniel Limouzy, Vincent Ochem, Pascal Gonçalves, Daniel Limouzy, Vincent Ochem, Pascal <![CDATA[This paper is devoted to the study of particular geometrically defined intersection classes of graphs. Those were previously studied by Magnant and Martin, who proved that these graphs have arbitrary large chromatic number, while being triangle-free. We give several structural properties of these graphs, and we raise several questions.]]> 0 <![CDATA[Let $\pi_n$ be a uniformly chosen random permutation on $[n]$. Using an analysis of the probability that two overlapping consecutive $k$-permutations are order isomorphic, the authors of a recent paper showed that the expected number of distinct consecutive patterns of all lengths $k\in\{1,2,\ldots,n\}$ in $\pi_n$ is $\frac{n^2}{2}(1-o(1))$ as $n\to\infty$. This exhibited the fact that random permutations pack consecutive patterns near-perfectly. We use entirely different methods, namely the Stein-Chen method of Poisson approximation, to reprove and slightly improve their result.]]> Fri, 03 May 2024 08:35:37 +0000 https://doi.org/10.46298/dmtcs.12458 https://doi.org/10.46298/dmtcs.12458 Godbole, Anant Swickheimer, Hannah Godbole, Anant Swickheimer, Hannah <![CDATA[Let $\pi_n$ be a uniformly chosen random permutation on $[n]$. Using an analysis of the probability that two overlapping consecutive $k$-permutations are order isomorphic, the authors of a recent paper showed that the expected number of distinct consecutive patterns of all lengths $k\in\{1,2,\ldots,n\}$ in $\pi_n$ is $\frac{n^2}{2}(1-o(1))$ as $n\to\infty$. This exhibited the fact that random permutations pack consecutive patterns near-perfectly. We use entirely different methods, namely the Stein-Chen method of Poisson approximation, to reprove and slightly improve their result.]]> 0 <![CDATA[We show that the coefficients of the representing polynomial of any monotone Boolean function are the values of the M\"obius function of an atomistic lattice related to this function. Using this we determine the representing polynomial of any Boolean function corresponding to a ST-CONNECTIVITY problem in acyclic quivers (directed acyclic multigraphs). Only monomials corresponding to unions of paths have non-zero coefficients which are $(-1)^D$ where $D$ is an easily computable function of the quiver corresponding to the monomial (it is the number of plane regions in the case of planar graphs). We determine that the number of monomials with non-zero coefficients for the two-dimensional $n \times n$ grid connectivity problem is $2^{\Omega(n^2)}$.]]> Mon, 29 Apr 2024 07:39:18 +0000 https://doi.org/10.46298/dmtcs.9934 https://doi.org/10.46298/dmtcs.9934 Iraids, Jānis Smotrovs, Juris Iraids, Jānis Smotrovs, Juris <![CDATA[We show that the coefficients of the representing polynomial of any monotone Boolean function are the values of the M\"obius function of an atomistic lattice related to this function. Using this we determine the representing polynomial of any Boolean function corresponding to a ST-CONNECTIVITY problem in acyclic quivers (directed acyclic multigraphs). Only monomials corresponding to unions of paths have non-zero coefficients which are $(-1)^D$ where $D$ is an easily computable function of the quiver corresponding to the monomial (it is the number of plane regions in the case of planar graphs). We determine that the number of monomials with non-zero coefficients for the two-dimensional $n \times n$ grid connectivity problem is $2^{\Omega(n^2)}$.]]> 0 <![CDATA[A walk $u_0u_1 \ldots u_{k-1}u_k$ is a \textit{weakly toll walk} if $u_0u_i \in E(G)$ implies $u_i = u_1$ and $u_ju_k\in E(G)$ implies $u_j=u_{k-1}$. A set $S$ of vertices of $G$ is {\it weakly toll convex} if for any two non-adjacent vertices $x,y \in S$ any vertex in a weakly toll walk between $x$ and $y$ is also in $S$. The {\em weakly toll convexity} is the graph convexity space defined over weakly toll convex sets. Many studies are devoted to determine if a graph equipped with a convexity space is a {\em convex geometry}. An \emph{extreme vertex} is an element $x$ of a convex set $S$ such that the set $S\backslash\{x\}$ is also convex. A graph convexity space is said to be a convex geometry if it satisfies the Minkowski-Krein-Milman property, which states that every convex set is the convex hull of its extreme vertices. It is known that chordal, Ptolemaic, weakly polarizable, and interval graphs can be characterized as convex geometries with respect to the monophonic, geodesic, $m^3$, and toll convexities, respectively. Other important classes of graphs can also be characterized in this way. In this paper, we prove that a graph is a convex geometry with respect to the weakly toll convexity if and only if it is a proper interval graph. Furthermore, some well-known graph invariants are studied with respect to the weakly toll convexity.]]> Thu, 18 Apr 2024 08:34:56 +0000 https://doi.org/10.46298/dmtcs.9837 https://doi.org/10.46298/dmtcs.9837 Dourado, Mitre C. Gutierrez, Marisa Protti, Fábio Tondato, Silvia Dourado, Mitre C. Gutierrez, Marisa Protti, Fábio Tondato, Silvia <![CDATA[A walk $u_0u_1 \ldots u_{k-1}u_k$ is a \textit{weakly toll walk} if $u_0u_i \in E(G)$ implies $u_i = u_1$ and $u_ju_k\in E(G)$ implies $u_j=u_{k-1}$. A set $S$ of vertices of $G$ is {\it weakly toll convex} if for any two non-adjacent vertices $x,y \in S$ any vertex in a weakly toll walk between $x$ and $y$ is also in $S$. The {\em weakly toll convexity} is the graph convexity space defined over weakly toll convex sets. Many studies are devoted to determine if a graph equipped with a convexity space is a {\em convex geometry}. An \emph{extreme vertex} is an element $x$ of a convex set $S$ such that the set $S\backslash\{x\}$ is also convex. A graph convexity space is said to be a convex geometry if it satisfies the Minkowski-Krein-Milman property, which states that every convex set is the convex hull of its extreme vertices. It is known that chordal, Ptolemaic, weakly polarizable, and interval graphs can be characterized as convex geometries with respect to the monophonic, geodesic, $m^3$, and toll convexities, respectively. Other important classes of graphs can also be characterized in this way. In this paper, we prove that a graph is a convex geometry with respect to the weakly toll convexity if and only if it is a proper interval graph. Furthermore, some well-known graph invariants are studied with respect to the weakly toll convexity.]]> 0 <![CDATA[We study the Maker-Breaker domination game played by Dominator and Staller on the vertex set of a given graph. Dominator wins when the vertices he has claimed form a dominating set of the graph. Staller wins if she makes it impossible for Dominator to win, or equivalently, she is able to claim some vertex and all its neighbours. Maker-Breaker domination number $\gamma_{MB}(G)$ ($\gamma '_{MB}(G)$) of a graph $G$ is defined to be the minimum number of moves for Dominator to guarantee his winning when he plays first (second). We investigate these two invariants for the Cartesian product of any two graphs. We obtain upper bounds for the Maker-Breaker domination number of the Cartesian product of two arbitrary graphs. Also, we give upper bounds for the Maker-Breaker domination number of the Cartesian product of the complete graph with two vertices and an arbitrary graph. Most importantly, we prove that $\gamma'_{MB}(P_2\square P_n)=n$ for $n\geq 1$, $\gamma_{MB}(P_2\square P_n)$ equals $n$, $n-1$, $n-2$, for $1\leq n\leq 4$, $5\leq n\leq 12$, and $n\geq 13$, respectively. For the disjoint union of $P_2\square P_n$s, we show that $\gamma_{MB}'(\dot\cup_{i=1}^k(P_2\square P_n)_i)=k\cdot n$ ($n\geq 1$), and that $\gamma_{MB}(\dot\cup_{i=1}^k(P_2\square P_n)_i)$ equals $k\cdot n$, $k\cdot n-1$, $k\cdot n-2$ for $1\leq n\leq 4$, $5\leq n\leq 12$, and $n\geq 13$, respectively.]]> Fri, 12 Apr 2024 07:57:46 +0000 https://doi.org/10.46298/dmtcs.10465 https://doi.org/10.46298/dmtcs.10465 Forcan, Jovana Qi, Jiayue Forcan, Jovana Qi, Jiayue <![CDATA[We study the Maker-Breaker domination game played by Dominator and Staller on the vertex set of a given graph. Dominator wins when the vertices he has claimed form a dominating set of the graph. Staller wins if she makes it impossible for Dominator to win, or equivalently, she is able to claim some vertex and all its neighbours. Maker-Breaker domination number $\gamma_{MB}(G)$ ($\gamma '_{MB}(G)$) of a graph $G$ is defined to be the minimum number of moves for Dominator to guarantee his winning when he plays first (second). We investigate these two invariants for the Cartesian product of any two graphs. We obtain upper bounds for the Maker-Breaker domination number of the Cartesian product of two arbitrary graphs. Also, we give upper bounds for the Maker-Breaker domination number of the Cartesian product of the complete graph with two vertices and an arbitrary graph. Most importantly, we prove that $\gamma'_{MB}(P_2\square P_n)=n$ for $n\geq 1$, $\gamma_{MB}(P_2\square P_n)$ equals $n$, $n-1$, $n-2$, for $1\leq n\leq 4$, $5\leq n\leq 12$, and $n\geq 13$, respectively. For the disjoint union of $P_2\square P_n$s, we show that $\gamma_{MB}'(\dot\cup_{i=1}^k(P_2\square P_n)_i)=k\cdot n$ ($n\geq 1$), and that $\gamma_{MB}(\dot\cup_{i=1}^k(P_2\square P_n)_i)$ equals $k\cdot n$, $k\cdot n-1$, $k\cdot n-2$ for $1\leq n\leq 4$, $5\leq n\leq 12$, and $n\geq 13$, respectively.]]> 0 <![CDATA[We explore new connections between complete non-ambiguous trees (CNATs) and permutations. We give a bijection between tree-like tableaux and a specific subset of CNATs. This map is used to establish and solve a recurrence relation for the number of tree-like tableaux of a fixed size without occupied corners, proving a conjecture by Laborde-Zubieta. We end by establishing a row/column swapping operation on CNATs and identify new areas for future research.]]> Wed, 03 Apr 2024 06:22:56 +0000 https://doi.org/10.46298/dmtcs.11169 https://doi.org/10.46298/dmtcs.11169 Chen, Daniel Ohlig, Sebastian Chen, Daniel Ohlig, Sebastian <![CDATA[We explore new connections between complete non-ambiguous trees (CNATs) and permutations. We give a bijection between tree-like tableaux and a specific subset of CNATs. This map is used to establish and solve a recurrence relation for the number of tree-like tableaux of a fixed size without occupied corners, proving a conjecture by Laborde-Zubieta. We end by establishing a row/column swapping operation on CNATs and identify new areas for future research.]]> 0 <![CDATA[We prove that the class of 231-avoiding permutations satisfies a logical limit law, i.e. that for any first-order sentence $\Psi$, in the language of two total orders, the probability $p_{n,\Psi}$ that a uniform random 231-avoiding permutation of size $n$ satisfies $\Psi$ admits a limit as $n$ is large. Moreover, we establish two further results about the behavior and value of $p_{n,\Psi}$: (i) it is either bounded away from $0$, or decays exponentially fast; (ii) the set of possible limits is dense in $[0,1]$. Our tools come mainly from analytic combinatorics and singularity analysis.]]> Tue, 02 Apr 2024 07:53:07 +0000 https://doi.org/10.46298/dmtcs.11751 https://doi.org/10.46298/dmtcs.11751 Albert, Michael Bouvel, Mathilde Féray, Valentin Noy, Marc Albert, Michael Bouvel, Mathilde Féray, Valentin Noy, Marc <![CDATA[We prove that the class of 231-avoiding permutations satisfies a logical limit law, i.e. that for any first-order sentence $\Psi$, in the language of two total orders, the probability $p_{n,\Psi}$ that a uniform random 231-avoiding permutation of size $n$ satisfies $\Psi$ admits a limit as $n$ is large. Moreover, we establish two further results about the behavior and value of $p_{n,\Psi}$: (i) it is either bounded away from $0$, or decays exponentially fast; (ii) the set of possible limits is dense in $[0,1]$. Our tools come mainly from analytic combinatorics and singularity analysis.]]> 0 <![CDATA[The Grundy number of a graph is the maximum number of colours used by the "First-Fit" greedy colouring algorithm over all vertex orderings. Given a vertex ordering $\sigma= v_1,\dots,v_n$, the "First-Fit" greedy colouring algorithm colours the vertices in the order of $\sigma$ by assigning to each vertex the smallest colour unused in its neighbourhood. By restricting this procedure to vertex orderings that are connected, we obtain {\em connected greedy colourings}. For some graphs, all connected greedy colourings use exactly $\chi(G)$ colours; they are called {\em good graphs}. On the opposite, some graphs do not admit any connected greedy colouring using only $\chi(G)$ colours; they are called {\em ugly graphs}. We show that no perfect graph is ugly. We also give simple proofs of this fact for subclasses of perfect graphs (block graphs, comparability graphs), and show that no $K_4$-minor free graph is ugly. Moreover, our proofs are constructive, and imply the existence of polynomial-time algorithms to compute good connected orderings for these graph classes.]]> Tue, 02 Apr 2024 07:48:27 +0000 https://doi.org/10.46298/dmtcs.8715 https://doi.org/10.46298/dmtcs.8715 Beaudou, Laurent Brosse, Caroline Defrain, Oscar Foucaud, Florent Lagoutte, Aurélie Limouzy, Vincent Pastor, Lucas Beaudou, Laurent Brosse, Caroline Defrain, Oscar Foucaud, Florent Lagoutte, Aurélie Limouzy, Vincent Pastor, Lucas <![CDATA[The Grundy number of a graph is the maximum number of colours used by the "First-Fit" greedy colouring algorithm over all vertex orderings. Given a vertex ordering $\sigma= v_1,\dots,v_n$, the "First-Fit" greedy colouring algorithm colours the vertices in the order of $\sigma$ by assigning to each vertex the smallest colour unused in its neighbourhood. By restricting this procedure to vertex orderings that are connected, we obtain {\em connected greedy colourings}. For some graphs, all connected greedy colourings use exactly $\chi(G)$ colours; they are called {\em good graphs}. On the opposite, some graphs do not admit any connected greedy colouring using only $\chi(G)$ colours; they are called {\em ugly graphs}. We show that no perfect graph is ugly. We also give simple proofs of this fact for subclasses of perfect graphs (block graphs, comparability graphs), and show that no $K_4$-minor free graph is ugly. Moreover, our proofs are constructive, and imply the existence of polynomial-time algorithms to compute good connected orderings for these graph classes.]]> 0 <![CDATA[To simplify the analysis of Boolean networks, a reduction in the number of components is often considered. A popular reduction method consists in eliminating components that are not autoregulated, using variable substitution. In this work, we show how this method can be extended, for asynchronous dynamics of Boolean networks, to the elimination of vertices that have a negative autoregulation, and study the effects on the dynamics and interaction structure. For elimination of non-autoregulated variables, the preservation of attractors is in general guaranteed only for fixed points. Here we give sufficient conditions for the preservation of complex attractors. The removal of so called mediator nodes (i.e. vertices with indegree and outdegree one) is often considered, and frequently does not affect the attractor landscape. We clarify that this is not always the case, and in some situations even subtle changes in the interaction structure can lead to a different asymptotic behaviour. Finally, we use properties of the more general elimination method introduced here to give an alternative proof for a bound on the number of attractors of asynchronous Boolean networks in terms of the cardinality of positive feedback vertex sets of the interaction graph.]]> Fri, 22 Mar 2024 15:56:47 +0000 https://doi.org/10.46298/dmtcs.10930 https://doi.org/10.46298/dmtcs.10930 Schwieger, Robert Tonello, Elisa Schwieger, Robert Tonello, Elisa <![CDATA[To simplify the analysis of Boolean networks, a reduction in the number of components is often considered. A popular reduction method consists in eliminating components that are not autoregulated, using variable substitution. In this work, we show how this method can be extended, for asynchronous dynamics of Boolean networks, to the elimination of vertices that have a negative autoregulation, and study the effects on the dynamics and interaction structure. For elimination of non-autoregulated variables, the preservation of attractors is in general guaranteed only for fixed points. Here we give sufficient conditions for the preservation of complex attractors. The removal of so called mediator nodes (i.e. vertices with indegree and outdegree one) is often considered, and frequently does not affect the attractor landscape. We clarify that this is not always the case, and in some situations even subtle changes in the interaction structure can lead to a different asymptotic behaviour. Finally, we use properties of the more general elimination method introduced here to give an alternative proof for a bound on the number of attractors of asynchronous Boolean networks in terms of the cardinality of positive feedback vertex sets of the interaction graph.]]> 0 <![CDATA[In this corrigendum, we give a counterexample to Theorem 5.2 in "On the monophonic rank of a graph" [Discrete Math. Theor. Comput. Sci. 24:2 (2022) #3]. We also present a polynomial-time algorithm for computing the monophonic rank of a starlike graph.]]> Fri, 22 Mar 2024 15:54:12 +0000 https://doi.org/10.46298/dmtcs.11423 https://doi.org/10.46298/dmtcs.11423 Dourado, Mitre C. Ponciano, Vitor S. da Silva, Rômulo L. O. Dourado, Mitre C. Ponciano, Vitor S. da Silva, Rômulo L. O. <![CDATA[In this corrigendum, we give a counterexample to Theorem 5.2 in "On the monophonic rank of a graph" [Discrete Math. Theor. Comput. Sci. 24:2 (2022) #3]. We also present a polynomial-time algorithm for computing the monophonic rank of a starlike graph.]]> 0 <![CDATA[This paper studies the relationship between the modified Foata$\unicode{x2013}$Strehl action (a.k.a. valley-hopping)$\unicode{x2014}$a group action on permutations used to demonstrate the $\gamma$-positivity of the Eulerian polynomials$\unicode{x2014}$and the number of rixed points $\operatorname{rix}$$\unicode{x2014}$a recursively-defined permutation statistic introduced by Lin in the context of an equidistribution problem. We give a linear-time iterative algorithm for computing the set of rixed points, and prove that the $\operatorname{rix}$ statistic is homomesic under valley-hopping. We also demonstrate that a bijection $\Phi$ introduced by Lin and Zeng in the study of the $\operatorname{rix}$ statistic sends orbits of the valley-hopping action to orbits of a cyclic version of valley-hopping, which implies that the number of fixed points $\operatorname{fix}$ is homomesic under cyclic valley-hopping.]]> Tue, 05 Mar 2024 14:23:00 +0000 https://doi.org/10.46298/dmtcs.11553 https://doi.org/10.46298/dmtcs.11553 Lafrenière, Nadia Zhuang, Yan Lafrenière, Nadia Zhuang, Yan <![CDATA[This paper studies the relationship between the modified Foata$\unicode{x2013}$Strehl action (a.k.a. valley-hopping)$\unicode{x2014}$a group action on permutations used to demonstrate the $\gamma$-positivity of the Eulerian polynomials$\unicode{x2014}$and the number of rixed points $\operatorname{rix}$$\unicode{x2014}$a recursively-defined permutation statistic introduced by Lin in the context of an equidistribution problem. We give a linear-time iterative algorithm for computing the set of rixed points, and prove that the $\operatorname{rix}$ statistic is homomesic under valley-hopping. We also demonstrate that a bijection $\Phi$ introduced by Lin and Zeng in the study of the $\operatorname{rix}$ statistic sends orbits of the valley-hopping action to orbits of a cyclic version of valley-hopping, which implies that the number of fixed points $\operatorname{fix}$ is homomesic under cyclic valley-hopping.]]> 0 <![CDATA[In a recent preprint, Matherne, Morales and Selover conjectured that two different representations of unit interval posets are related by the famous zeta map in $q,t$-Catalan combinatorics. This conjecture was proved recently by G\'elinas, Segovia and Thomas using induction. In this short note, we provide a bijective proof of the same conjecture with a reformulation of the zeta map using left-aligned colored trees, first proposed in the study of parabolic Tamari lattices.]]> Fri, 23 Feb 2024 16:38:00 +0000 https://doi.org/10.46298/dmtcs.10837 https://doi.org/10.46298/dmtcs.10837 Fang, Wenjie Fang, Wenjie <![CDATA[In a recent preprint, Matherne, Morales and Selover conjectured that two different representations of unit interval posets are related by the famous zeta map in $q,t$-Catalan combinatorics. This conjecture was proved recently by G\'elinas, Segovia and Thomas using induction. In this short note, we provide a bijective proof of the same conjecture with a reformulation of the zeta map using left-aligned colored trees, first proposed in the study of parabolic Tamari lattices.]]> 0 <![CDATA[Holonomic equations are recursive equations which allow computing efficiently numbers of combinatoric objects. Rémy showed that the holonomic equation associated with binary trees yields an efficient linear random generator of binary trees. I extend this paradigm to Motzkin trees and Schröder trees and show that despite slight differences my algorithm that generates random Schröder trees has linear expected complexity and my algorithm that generates Motzkin trees is in O(n) expected complexity, only if we can implement a specific oracle with a O(1) complexity. For Motzkin trees, I propose a solution which works well for realistic values (up to size ten millions) and yields an efficient algorithm.]]> Fri, 23 Feb 2024 13:48:42 +0000 https://doi.org/10.46298/dmtcs.10952 https://doi.org/10.46298/dmtcs.10952 Lescanne, Pierre Lescanne, Pierre <![CDATA[Holonomic equations are recursive equations which allow computing efficiently numbers of combinatoric objects. Rémy showed that the holonomic equation associated with binary trees yields an efficient linear random generator of binary trees. I extend this paradigm to Motzkin trees and Schröder trees and show that despite slight differences my algorithm that generates random Schröder trees has linear expected complexity and my algorithm that generates Motzkin trees is in O(n) expected complexity, only if we can implement a specific oracle with a O(1) complexity. For Motzkin trees, I propose a solution which works well for realistic values (up to size ten millions) and yields an efficient algorithm.]]> 0 <![CDATA[For the given bipartite graphs $G_1,G_2,\ldots,G_t$, the multicolor bipartite Ramsey number $BR(G_1,G_2,\ldots,G_t)$ is the smallest positive integer $b$ such that any $t$-edge-coloring of $K_{b,b}$ contains a monochromatic subgraph isomorphic to $G_i$, colored with the $i$th color for some $1\leq i\leq t$. We compute the exact values of the bipartite Ramsey numbers $BR(C_8,C_{2n})$ for $n\geq2$.]]> Fri, 16 Feb 2024 13:09:25 +0000 https://doi.org/10.46298/dmtcs.11207 https://doi.org/10.46298/dmtcs.11207 Gholami, Mostafa Rowshan, Yaser Gholami, Mostafa Rowshan, Yaser <![CDATA[For the given bipartite graphs $G_1,G_2,\ldots,G_t$, the multicolor bipartite Ramsey number $BR(G_1,G_2,\ldots,G_t)$ is the smallest positive integer $b$ such that any $t$-edge-coloring of $K_{b,b}$ contains a monochromatic subgraph isomorphic to $G_i$, colored with the $i$th color for some $1\leq i\leq t$. We compute the exact values of the bipartite Ramsey numbers $BR(C_8,C_{2n})$ for $n\geq2$.]]> 0 <![CDATA[The exponential recursive trees model several kinds of networks. At each step of growing of these trees, each node independently attracts a new node with probability p, or fails to do with probability 1 − p. Here, we investigate the number of protected nodes, total path length of protected nodes, and a mean study of the protected node profile of such trees.]]> Thu, 11 Jan 2024 19:09:25 +0000 https://doi.org/10.46298/dmtcs.10524 https://doi.org/10.46298/dmtcs.10524 Javanian, Mehri Aguech, Rafik Javanian, Mehri Aguech, Rafik <![CDATA[The exponential recursive trees model several kinds of networks. At each step of growing of these trees, each node independently attracts a new node with probability p, or fails to do with probability 1 − p. Here, we investigate the number of protected nodes, total path length of protected nodes, and a mean study of the protected node profile of such trees.]]> 0 <![CDATA[Inspired by the split decomposition of graphs and rank-width, we introduce the notion of $r$-splits. We focus on the family of $r$-splits of a graph of order $n$, and we prove that it forms a hypergraph with several properties. We prove that such hypergraphs can be represented using only $\mathcal O(n^{r+1})$ of its hyperedges, despite its potentially exponential number of hyperedges. We also prove that there exist hypergraphs that need at least $\Omega(n^r)$ hyperedges to be represented, using a generalization of set orthogonality.]]> Tue, 02 Jan 2024 14:39:20 +0000 https://doi.org/10.46298/dmtcs.10751 https://doi.org/10.46298/dmtcs.10751 Pitois, François Haddad, Mohammed Seba, Hamida Togni, Olivier Pitois, François Haddad, Mohammed Seba, Hamida Togni, Olivier <![CDATA[Inspired by the split decomposition of graphs and rank-width, we introduce the notion of $r$-splits. We focus on the family of $r$-splits of a graph of order $n$, and we prove that it forms a hypergraph with several properties. We prove that such hypergraphs can be represented using only $\mathcal O(n^{r+1})$ of its hyperedges, despite its potentially exponential number of hyperedges. We also prove that there exist hypergraphs that need at least $\Omega(n^r)$ hyperedges to be represented, using a generalization of set orthogonality.]]> 0 <![CDATA[Symmetric edge polytopes are lattice polytopes associated with finite simple graphs that are of interest in both theory and applications. We investigate the facet structure of symmetric edge polytopes for various models of random graphs. For an Erd\H{o}s-Renyi random graph, we identify a threshold probability at which with high probability the symmetric edge polytope shares many facet-supporting hyperplanes with that of a complete graph. We also investigate the relationship between the average local clustering, also known as the Watts-Strogatz clustering coefficient, and the number of facets for graphs with either a fixed number of edges or a fixed degree sequence. We use well-known Markov Chain Monte Carlo sampling methods to generate empirical evidence that for a fixed degree sequence, higher average local clustering in a connected graph corresponds to higher facet numbers in the associated symmetric edge polytope.]]> Mon, 11 Dec 2023 10:44:04 +0000 https://doi.org/10.46298/dmtcs.9925 https://doi.org/10.46298/dmtcs.9925 Braun, Benjamin Bruegge, Kaitlin Kahle, Matthew Braun, Benjamin Bruegge, Kaitlin Kahle, Matthew <![CDATA[Symmetric edge polytopes are lattice polytopes associated with finite simple graphs that are of interest in both theory and applications. We investigate the facet structure of symmetric edge polytopes for various models of random graphs. For an Erd\H{o}s-Renyi random graph, we identify a threshold probability at which with high probability the symmetric edge polytope shares many facet-supporting hyperplanes with that of a complete graph. We also investigate the relationship between the average local clustering, also known as the Watts-Strogatz clustering coefficient, and the number of facets for graphs with either a fixed number of edges or a fixed degree sequence. We use well-known Markov Chain Monte Carlo sampling methods to generate empirical evidence that for a fixed degree sequence, higher average local clustering in a connected graph corresponds to higher facet numbers in the associated symmetric edge polytope.]]> 0 <![CDATA[Let c be an integer. A c-partite tournament is an orientation of a complete c-partite graph. A c-partite tournament is rich if it is strong, and each partite set has at least two vertices. In 1996, Guo and Volkmann characterized the structure of all rich c-partite tournaments without (c + 1)-cycles, which solved a problem by Bondy. They also put forward a problem that what the structure of rich c-partite tournaments without (c + k)-cycles for some k>1 is. In this paper, we answer the question of Guo and Volkmann for k = 2.]]> Tue, 05 Dec 2023 20:07:01 +0000 https://doi.org/10.46298/dmtcs.9732 https://doi.org/10.46298/dmtcs.9732 Zhang, Jie Wang, Zhilan Yan, Jin Zhang, Jie Wang, Zhilan Yan, Jin <![CDATA[Let c be an integer. A c-partite tournament is an orientation of a complete c-partite graph. A c-partite tournament is rich if it is strong, and each partite set has at least two vertices. In 1996, Guo and Volkmann characterized the structure of all rich c-partite tournaments without (c + 1)-cycles, which solved a problem by Bondy. They also put forward a problem that what the structure of rich c-partite tournaments without (c + k)-cycles for some k>1 is. In this paper, we answer the question of Guo and Volkmann for k = 2.]]> 0 <![CDATA[Mixed graphs have both directed and undirected edges. A mixed cage is a regular mixed graph of given girth with minimum possible order. In this paper mixed cages are studied. Upper bounds are obtained by general construction methods and computer searches.]]> Thu, 30 Nov 2023 08:41:26 +0000 https://doi.org/10.46298/dmtcs.11057 https://doi.org/10.46298/dmtcs.11057 Exoo, Geoffrey Exoo, Geoffrey <![CDATA[Mixed graphs have both directed and undirected edges. A mixed cage is a regular mixed graph of given girth with minimum possible order. In this paper mixed cages are studied. Upper bounds are obtained by general construction methods and computer searches.]]> 0 <![CDATA[Given a set system $\mathcal{X} = \{\mathcal{U},\mathcal{S}\}$, where $\mathcal{U}$ is a set of elements and $\mathcal{S}$ is a set of subsets of $\mathcal{U}$, an exact hitting set $\mathcal{U}'$ is a subset of $\mathcal{U}$ such that each subset in $\mathcal{S}$ contains exactly one element in $\mathcal{U}'$. We refer to a set system as exactly hittable if it has an exact hitting set. In this paper, we study interval graphs which have intersection models that are exactly hittable. We refer to these interval graphs as exactly hittable interval graphs (EHIG). We present a forbidden structure characterization for EHIG. We also show that the class of proper interval graphs is a strict subclass of EHIG. Finally, we give an algorithm that runs in polynomial time to recognize graphs belonging to the class of EHIG.]]> Thu, 30 Nov 2023 08:40:04 +0000 https://doi.org/10.46298/dmtcs.10762 https://doi.org/10.46298/dmtcs.10762 Dhannya, S. M. Narayanaswamy, N. S. Nisha, K. K. Dhannya, S. M. Narayanaswamy, N. S. Nisha, K. K. <![CDATA[Given a set system $\mathcal{X} = \{\mathcal{U},\mathcal{S}\}$, where $\mathcal{U}$ is a set of elements and $\mathcal{S}$ is a set of subsets of $\mathcal{U}$, an exact hitting set $\mathcal{U}'$ is a subset of $\mathcal{U}$ such that each subset in $\mathcal{S}$ contains exactly one element in $\mathcal{U}'$. We refer to a set system as exactly hittable if it has an exact hitting set. In this paper, we study interval graphs which have intersection models that are exactly hittable. We refer to these interval graphs as exactly hittable interval graphs (EHIG). We present a forbidden structure characterization for EHIG. We also show that the class of proper interval graphs is a strict subclass of EHIG. Finally, we give an algorithm that runs in polynomial time to recognize graphs belonging to the class of EHIG.]]> 0 <![CDATA[We define a q-linear path in a hypergraph H as a sequence (e_1,...,e_L) of edges of H such that |e_i ∩ e_i+1 | ∈ [[1, q]] and e_i ∩ e_j = ∅ if |i − j| > 1. In this paper, we study the connected components associated to these paths when q = k − 2 where k is the rank of H. If k = 3 then q = 1 which coincides with the well-known notion of linear path or loose path. We describe the structure of the connected components, using an algorithmic proof which shows that the connected components can be computed in polynomial time. We then mention two consequences of our algorithmic result. The first one is that deciding the winner of the Maker-Breaker game on a hypergraph of rank 3 can be done in polynomial time. The second one is that tractable cases for the NP-complete problem of "Paths Avoiding Forbidden Pairs" in a graph can be deduced from the recognition of a special type of line graph of a hypergraph.]]> Thu, 30 Nov 2023 08:38:57 +0000 https://doi.org/10.46298/dmtcs.10202 https://doi.org/10.46298/dmtcs.10202 Galliot, Florian Gravier, Sylvain Sivignon, Isabelle Galliot, Florian Gravier, Sylvain Sivignon, Isabelle <![CDATA[We define a q-linear path in a hypergraph H as a sequence (e_1,...,e_L) of edges of H such that |e_i ∩ e_i+1 | ∈ [[1, q]] and e_i ∩ e_j = ∅ if |i − j| > 1. In this paper, we study the connected components associated to these paths when q = k − 2 where k is the rank of H. If k = 3 then q = 1 which coincides with the well-known notion of linear path or loose path. We describe the structure of the connected components, using an algorithmic proof which shows that the connected components can be computed in polynomial time. We then mention two consequences of our algorithmic result. The first one is that deciding the winner of the Maker-Breaker game on a hypergraph of rank 3 can be done in polynomial time. The second one is that tractable cases for the NP-complete problem of "Paths Avoiding Forbidden Pairs" in a graph can be deduced from the recognition of a special type of line graph of a hypergraph.]]> 0 <![CDATA[We establish exact values for the $2$-limited broadcast domination number of various grid graphs, in particular $C_m\square C_n$ for $3 \leq m \leq 6$ and all $n\geq m$, $P_m \square C_3$ for all $m \geq 3$, and $P_m \square C_n$ for $4\leq m \leq 5$ and all $n \geq m$. We also produce periodically optimal values for $P_m \square C_4$ and $P_m \square C_6$ for $m \geq 3$, $P_4 \square P_n$ for $n \geq 4$, and $P_5 \square P_n$ for $n \geq 5$. Our method completes an exhaustive case analysis and eliminates cases by combining tools from linear programming with various mathematical proof techniques.]]> Wed, 22 Nov 2023 09:42:44 +0000 https://doi.org/10.46298/dmtcs.11478 https://doi.org/10.46298/dmtcs.11478 Slobodin, Aaron MacGillivray, Gary Myrvold, Wendy Slobodin, Aaron MacGillivray, Gary Myrvold, Wendy <![CDATA[We establish exact values for the $2$-limited broadcast domination number of various grid graphs, in particular $C_m\square C_n$ for $3 \leq m \leq 6$ and all $n\geq m$, $P_m \square C_3$ for all $m \geq 3$, and $P_m \square C_n$ for $4\leq m \leq 5$ and all $n \geq m$. We also produce periodically optimal values for $P_m \square C_4$ and $P_m \square C_6$ for $m \geq 3$, $P_4 \square P_n$ for $n \geq 4$, and $P_5 \square P_n$ for $n \geq 5$. Our method completes an exhaustive case analysis and eliminates cases by combining tools from linear programming with various mathematical proof techniques.]]> 0 <![CDATA[Let $t$ be a positive real number. A graph is called $t$-tough if the removal of any vertex set $S$ that disconnects the graph leaves at most $|S|/t$ components, and all graphs are considered 0-tough. The toughness of a graph is the largest $t$ for which the graph is $t$-tough, whereby the toughness of complete graphs is defined as infinity. A graph is minimally $t$-tough if the toughness of the graph is $t$, and the deletion of any edge from the graph decreases the toughness. In this paper, we investigate the minimum degree and the recognizability of minimally $t$-tough graphs in the classes of chordal graphs, split graphs, claw-free graphs, and $2K_2$-free graphs.]]> Wed, 22 Nov 2023 09:41:35 +0000 https://doi.org/10.46298/dmtcs.10180 https://doi.org/10.46298/dmtcs.10180 Katona, Gyula Y. Varga, Kitti Katona, Gyula Y. Varga, Kitti <![CDATA[Let $t$ be a positive real number. A graph is called $t$-tough if the removal of any vertex set $S$ that disconnects the graph leaves at most $|S|/t$ components, and all graphs are considered 0-tough. The toughness of a graph is the largest $t$ for which the graph is $t$-tough, whereby the toughness of complete graphs is defined as infinity. A graph is minimally $t$-tough if the toughness of the graph is $t$, and the deletion of any edge from the graph decreases the toughness. In this paper, we investigate the minimum degree and the recognizability of minimally $t$-tough graphs in the classes of chordal graphs, split graphs, claw-free graphs, and $2K_2$-free graphs.]]> 0 <![CDATA[We study the enumeration of inversion sequences that avoid the pattern 021 and another pattern of length four. We determine the generating trees for all possible pattern pairs and compute the corresponding generating functions. We introduce the concept of dregular generating trees and conjecture that for any 021-avoiding pattern τ , the generating tree T ({021, τ }) is d-regular for some integer d.]]> Fri, 17 Nov 2023 15:26:30 +0000 https://doi.org/10.46298/dmtcs.10444 https://doi.org/10.46298/dmtcs.10444 Mansour, Toufik Yıldırım, Gökhan Mansour, Toufik Yıldırım, Gökhan <![CDATA[We study the enumeration of inversion sequences that avoid the pattern 021 and another pattern of length four. We determine the generating trees for all possible pattern pairs and compute the corresponding generating functions. We introduce the concept of dregular generating trees and conjecture that for any 021-avoiding pattern τ , the generating tree T ({021, τ }) is d-regular for some integer d.]]> 0 <![CDATA[The bivariate chromatic polynomial $\chi_G(x,y)$ of a graph $G = (V, E)$, introduced by Dohmen-P\"{o}nitz-Tittmann (2003), counts all $x$-colorings of $G$ such that adjacent vertices get different colors if they are $\le y$. We extend this notion to mixed graphs, which have both directed and undirected edges. Our main result is a decomposition formula which expresses $\chi_G(x,y)$ as a sum of bivariate order polynomials (Beck-Farahmand-Karunaratne-Zuniga Ruiz 2020), and a combinatorial reciprocity theorem for $\chi_G(x,y)$.]]> Fri, 17 Nov 2023 15:18:17 +0000 https://doi.org/10.46298/dmtcs.9595 https://doi.org/10.46298/dmtcs.9595 Beck, Matthias Kolhatkar, Sampada Beck, Matthias Kolhatkar, Sampada <![CDATA[The bivariate chromatic polynomial $\chi_G(x,y)$ of a graph $G = (V, E)$, introduced by Dohmen-P\"{o}nitz-Tittmann (2003), counts all $x$-colorings of $G$ such that adjacent vertices get different colors if they are $\le y$. We extend this notion to mixed graphs, which have both directed and undirected edges. Our main result is a decomposition formula which expresses $\chi_G(x,y)$ as a sum of bivariate order polynomials (Beck-Farahmand-Karunaratne-Zuniga Ruiz 2020), and a combinatorial reciprocity theorem for $\chi_G(x,y)$.]]> 0 <![CDATA[We generalize the familiar notion of periodicity in sequences to a new kind of pseudoperiodicity, and we prove some basic results about it. We revisit the results of a 2012 paper of Shevelev and reprove his results in a simpler and more unified manner, and provide a complete answer to one of his previously unresolved questions. We consider finding words with specific pseudoperiod and having the smallest possible critical exponent. Finally, we consider the problem of determining whether a finite word is pseudoperiodic of a given size, and show that it is NP-complete.]]> Mon, 16 Oct 2023 09:07:03 +0000 https://doi.org/10.46298/dmtcs.9919 https://doi.org/10.46298/dmtcs.9919 Meleshko, Joseph Ochem, Pascal Shallit, Jeffrey Shan, Sonja Linghui Meleshko, Joseph Ochem, Pascal Shallit, Jeffrey Shan, Sonja Linghui <![CDATA[We generalize the familiar notion of periodicity in sequences to a new kind of pseudoperiodicity, and we prove some basic results about it. We revisit the results of a 2012 paper of Shevelev and reprove his results in a simpler and more unified manner, and provide a complete answer to one of his previously unresolved questions. We consider finding words with specific pseudoperiod and having the smallest possible critical exponent. Finally, we consider the problem of determining whether a finite word is pseudoperiodic of a given size, and show that it is NP-complete.]]> 0 <![CDATA[Consider the following hat guessing game. A bear sits on each vertex of a graph $G$, and a demon puts on each bear a hat colored by one of $h$ colors. Each bear sees only the hat colors of his neighbors. Based on this information only, each bear has to guess $g$ colors and he guesses correctly if his hat color is included in his guesses. The bears win if at least one bear guesses correctly for any hat arrangement. We introduce a new parameter - fractional hat chromatic number $\hat{\mu}$, arising from the hat guessing game. The parameter $\hat{\mu}$ is related to the hat chromatic number which has been studied before. We present a surprising connection between the hat guessing game and the independence polynomial of graphs. This connection allows us to compute the fractional hat chromatic number of chordal graphs in polynomial time, to bound fractional hat chromatic number by a function of maximum degree of $G$, and to compute the exact value of $\hat{\mu}$ of cliques, paths, and cycles.]]> Mon, 16 Oct 2023 08:51:33 +0000 https://doi.org/10.46298/dmtcs.10802 https://doi.org/10.46298/dmtcs.10802 Blažej, Václav Dvořák, Pavel Opler, Michal Blažej, Václav Dvořák, Pavel Opler, Michal <![CDATA[Consider the following hat guessing game. A bear sits on each vertex of a graph $G$, and a demon puts on each bear a hat colored by one of $h$ colors. Each bear sees only the hat colors of his neighbors. Based on this information only, each bear has to guess $g$ colors and he guesses correctly if his hat color is included in his guesses. The bears win if at least one bear guesses correctly for any hat arrangement. We introduce a new parameter - fractional hat chromatic number $\hat{\mu}$, arising from the hat guessing game. The parameter $\hat{\mu}$ is related to the hat chromatic number which has been studied before. We present a surprising connection between the hat guessing game and the independence polynomial of graphs. This connection allows us to compute the fractional hat chromatic number of chordal graphs in polynomial time, to bound fractional hat chromatic number by a function of maximum degree of $G$, and to compute the exact value of $\hat{\mu}$ of cliques, paths, and cycles.]]> 0 <![CDATA[Homomorphically full graphs are those for which every homomorphic image is isomorphic to a subgraph. We extend the definition of homomorphically full to oriented graphs in two different ways. For the first of these, we show that homomorphically full oriented graphs arise as quasi-transitive orientations of homomorphically full graphs. This in turn yields an efficient recognition and construction algorithms for these homomorphically full oriented graphs. For the second one, we show that the related recognition problem is GI-hard, and that the problem of deciding if a graph admits a homomorphically full orientation is NP-complete. In doing so we show the problem of deciding if two given oriented cliques are isomorphic is GI-complete.]]> Mon, 02 Oct 2023 09:32:12 +0000 https://doi.org/10.46298/dmtcs.9957 https://doi.org/10.46298/dmtcs.9957 Bellitto, Thomas Duffy, Christopher MacGillivray, Gary Bellitto, Thomas Duffy, Christopher MacGillivray, Gary <![CDATA[Homomorphically full graphs are those for which every homomorphic image is isomorphic to a subgraph. We extend the definition of homomorphically full to oriented graphs in two different ways. For the first of these, we show that homomorphically full oriented graphs arise as quasi-transitive orientations of homomorphically full graphs. This in turn yields an efficient recognition and construction algorithms for these homomorphically full oriented graphs. For the second one, we show that the related recognition problem is GI-hard, and that the problem of deciding if a graph admits a homomorphically full orientation is NP-complete. In doing so we show the problem of deciding if two given oriented cliques are isomorphic is GI-complete.]]> 0 <![CDATA[We say that a language $L$ is \emph{constantly growing} if there is a constant $c$ such that for every word $u\in L$ there is a word $v\in L$ with $\vert u\vert<\vert v\vert\leq c+\vert u\vert$. We say that a language $L$ is \emph{geometrically growing} if there is a constant $c$ such that for every word $u\in L$ there is a word $v\in L$ with $\vert u\vert<\vert v\vert\leq c\vert u\vert$. Given two infinite languages $L_1,L_2$, we say that $L_1$ \emph{dissects} $L_2$ if $\vert L_2\setminus L_1\vert=\infty$ and $\vert L_1\cap L_2\vert=\infty$. In 2013, it was shown that for every constantly growing language $L$ there is a regular language $R$ such that $R$ dissects $L$. In the current article we show how to dissect a geometrically growing language by a homomorphic image of intersection of two context-free languages. Consider three alphabets $\Gamma$, $\Sigma$, and $\Theta$ such that $\vert \Sigma\vert=1$ and $\vert \Theta\vert=4$. We prove that there are context-free languages $M_1,M_2\subseteq \Theta^*$, an erasing alphabetical homomorphism $\pi:\Theta^*\rightarrow \Sigma^*$, and a nonerasing alphabetical homomorphism $\varphi : \Gamma^*\rightarrow \Sigma^*$ such that: If $L\subseteq \Gamma^*$ is a geometrically growing language then there is a regular language $R\subseteq \Theta^*$ such that $\varphi^{-1}\left(\pi\left(R\cap M_1\cap M_2\right)\right)$ dissects the language $L$.]]> Mon, 02 Oct 2023 09:31:23 +0000 https://doi.org/10.46298/dmtcs.9063 https://doi.org/10.46298/dmtcs.9063 Rukavicka, Josef Rukavicka, Josef <![CDATA[We say that a language $L$ is \emph{constantly growing} if there is a constant $c$ such that for every word $u\in L$ there is a word $v\in L$ with $\vert u\vert<\vert v\vert\leq c+\vert u\vert$. We say that a language $L$ is \emph{geometrically growing} if there is a constant $c$ such that for every word $u\in L$ there is a word $v\in L$ with $\vert u\vert<\vert v\vert\leq c\vert u\vert$. Given two infinite languages $L_1,L_2$, we say that $L_1$ \emph{dissects} $L_2$ if $\vert L_2\setminus L_1\vert=\infty$ and $\vert L_1\cap L_2\vert=\infty$. In 2013, it was shown that for every constantly growing language $L$ there is a regular language $R$ such that $R$ dissects $L$. In the current article we show how to dissect a geometrically growing language by a homomorphic image of intersection of two context-free languages. Consider three alphabets $\Gamma$, $\Sigma$, and $\Theta$ such that $\vert \Sigma\vert=1$ and $\vert \Theta\vert=4$. We prove that there are context-free languages $M_1,M_2\subseteq \Theta^*$, an erasing alphabetical homomorphism $\pi:\Theta^*\rightarrow \Sigma^*$, and a nonerasing alphabetical homomorphism $\varphi : \Gamma^*\rightarrow \Sigma^*$ such that: If $L\subseteq \Gamma^*$ is a geometrically growing language then there is a regular language $R\subseteq \Theta^*$ such that $\varphi^{-1}\left(\pi\left(R\cap M_1\cap M_2\right)\right)$ dissects the language $L$.]]> 0 <![CDATA[In this paper, we study the problem of gossiping with interference constraint in radio ring networks. Gossiping (or total exchange information) is a protocol where each node in the network has a message and is expected to distribute its own message to every other node in the network. The gossiping problem consists in finding the minimum running time (makespan) of a gossiping protocol and algorithms that attain this makespan. We focus on the case where the transmission network is a ring network. We consider synchronous protocols where it takes one unit of time (step) to transmit a unit-length message. During one step, a node receives at most one message only through one of its two neighbors. We also suppose that, during one step, a node cannot be both a sender and a receiver (half duplex model). Moreover communication is subject to interference constraints. We use a primary node interference model where, if a node receives a message from one of its neighbors, its other neighbor cannot send at the same time. With these assumptions we completely solve the problem for ring networks. We first show lower bounds and then give gossiping algorithms which meet these lower bounds and so are optimal. The number of rounds depends on the congruences of n modulo 12.]]> Mon, 02 Oct 2023 09:30:24 +0000 https://doi.org/10.46298/dmtcs.9399 https://doi.org/10.46298/dmtcs.9399 Bermond, Jean-Claude Kodate, Takako Yu, Joseph Bermond, Jean-Claude Kodate, Takako Yu, Joseph <![CDATA[In this paper, we study the problem of gossiping with interference constraint in radio ring networks. Gossiping (or total exchange information) is a protocol where each node in the network has a message and is expected to distribute its own message to every other node in the network. The gossiping problem consists in finding the minimum running time (makespan) of a gossiping protocol and algorithms that attain this makespan. We focus on the case where the transmission network is a ring network. We consider synchronous protocols where it takes one unit of time (step) to transmit a unit-length message. During one step, a node receives at most one message only through one of its two neighbors. We also suppose that, during one step, a node cannot be both a sender and a receiver (half duplex model). Moreover communication is subject to interference constraints. We use a primary node interference model where, if a node receives a message from one of its neighbors, its other neighbor cannot send at the same time. With these assumptions we completely solve the problem for ring networks. We first show lower bounds and then give gossiping algorithms which meet these lower bounds and so are optimal. The number of rounds depends on the congruences of n modulo 12.]]> 0 <![CDATA[Given a rooted, binary phylogenetic network and a rooted, binary phylogenetic tree, can the tree be embedded into the network? This problem, called \textsc{Tree Containment}, arises when validating networks constructed by phylogenetic inference methods.We present the first algorithm for (rooted) \textsc{Tree Containment} using the treewidth $t$ of the input network $N$ as parameter, showing that the problem can be solved in $2^{O(t^2)}\cdot|N|$ time and space.]]> Mon, 02 Oct 2023 09:29:27 +0000 https://doi.org/10.46298/dmtcs.10116 https://doi.org/10.46298/dmtcs.10116 van Iersel, Leo Jones, Mark Weller, Mathias van Iersel, Leo Jones, Mark Weller, Mathias <![CDATA[Given a rooted, binary phylogenetic network and a rooted, binary phylogenetic tree, can the tree be embedded into the network? This problem, called \textsc{Tree Containment}, arises when validating networks constructed by phylogenetic inference methods.We present the first algorithm for (rooted) \textsc{Tree Containment} using the treewidth $t$ of the input network $N$ as parameter, showing that the problem can be solved in $2^{O(t^2)}\cdot|N|$ time and space.]]> 0 <![CDATA[The analysis of strings of $n$ random variables with geometric distribution has recently attracted renewed interest: Archibald et al. consider the number of distinct adjacent pairs in geometrically distributed words. They obtain the asymptotic ($n\rightarrow\infty$) mean of this number in the cases of different and identical pairs. In this paper we are interested in all asymptotic moments in the identical case, in the asymptotic variance in the different case and in the asymptotic distribution in both cases. We use two approaches: the first one, the probabilistic approach, leads to variances in both cases and to some conjectures on all moments in the identical case and on the distribution in both cases. The second approach, the combinatorial one, relies on multivariate pattern matching techniques, yielding exact formulas for first and second moments. We use such tools as Mellin transforms, Analytic Combinatorics, Markov Chains.]]> Mon, 02 Oct 2023 09:28:16 +0000 https://doi.org/10.46298/dmtcs.9293 https://doi.org/10.46298/dmtcs.9293 Louchard, Guy Schachinger, Werner Ward, Mark Daniel Louchard, Guy Schachinger, Werner Ward, Mark Daniel <![CDATA[The analysis of strings of $n$ random variables with geometric distribution has recently attracted renewed interest: Archibald et al. consider the number of distinct adjacent pairs in geometrically distributed words. They obtain the asymptotic ($n\rightarrow\infty$) mean of this number in the cases of different and identical pairs. In this paper we are interested in all asymptotic moments in the identical case, in the asymptotic variance in the different case and in the asymptotic distribution in both cases. We use two approaches: the first one, the probabilistic approach, leads to variances in both cases and to some conjectures on all moments in the identical case and on the distribution in both cases. The second approach, the combinatorial one, relies on multivariate pattern matching techniques, yielding exact formulas for first and second moments. We use such tools as Mellin transforms, Analytic Combinatorics, Markov Chains.]]> 0 <![CDATA[In the Maker-Breaker domination game played on a graph $G$, Dominator's goal is to select a dominating set and Staller's goal is to claim a closed neighborhood of some vertex. We study the cases when Staller can win the game. If Dominator (resp., Staller) starts the game, then $\gamma_{\rm SMB}(G)$ (resp., $\gamma_{\rm SMB}'(G)$) denotes the minimum number of moves Staller needs to win. For every positive integer $k$, trees $T$ with $\gamma_{\rm SMB}'(T)=k$ are characterized and a general upper bound on $\gamma_{\rm SMB}'$ is proved. Let $S = S(n_1,\dots, n_\ell)$ be the subdivided star obtained from the star with $\ell$ edges by subdividing its edges $n_1-1, \ldots, n_\ell-1$ times, respectively. Then $\gamma_{\rm SMB}'(S)$ is determined in all the cases except when $\ell\ge 4$ and each $n_i$ is even. The simplest formula is obtained when there are at least two odd $n_i$s. If $n_1$ and $n_2$ are the two smallest such numbers, then $\gamma_{\rm SMB}'(S(n_1,\dots, n_\ell))=\lceil \log_2(n_1+n_2+1)\rceil$. For caterpillars, exact formulas for $\gamma_{\rm SMB}$ and for $\gamma_{\rm SMB}'$ are established.]]> Wed, 06 Sep 2023 13:39:28 +0000 https://doi.org/10.46298/dmtcs.10515 https://doi.org/10.46298/dmtcs.10515 Bujtás, Csilla Dokyeesun, Pakanun Klavžar, Sandi Bujtás, Csilla Dokyeesun, Pakanun Klavžar, Sandi <![CDATA[In the Maker-Breaker domination game played on a graph $G$, Dominator's goal is to select a dominating set and Staller's goal is to claim a closed neighborhood of some vertex. We study the cases when Staller can win the game. If Dominator (resp., Staller) starts the game, then $\gamma_{\rm SMB}(G)$ (resp., $\gamma_{\rm SMB}'(G)$) denotes the minimum number of moves Staller needs to win. For every positive integer $k$, trees $T$ with $\gamma_{\rm SMB}'(T)=k$ are characterized and a general upper bound on $\gamma_{\rm SMB}'$ is proved. Let $S = S(n_1,\dots, n_\ell)$ be the subdivided star obtained from the star with $\ell$ edges by subdividing its edges $n_1-1, \ldots, n_\ell-1$ times, respectively. Then $\gamma_{\rm SMB}'(S)$ is determined in all the cases except when $\ell\ge 4$ and each $n_i$ is even. The simplest formula is obtained when there are at least two odd $n_i$s. If $n_1$ and $n_2$ are the two smallest such numbers, then $\gamma_{\rm SMB}'(S(n_1,\dots, n_\ell))=\lceil \log_2(n_1+n_2+1)\rceil$. For caterpillars, exact formulas for $\gamma_{\rm SMB}$ and for $\gamma_{\rm SMB}'$ are established.]]> 0 <![CDATA[The (bitwise) complement $\overline{x}$ of a binary word $x$ is obtained by changing each $0$ in $x$ to $1$ and vice versa. An $\textit{antisquare}$ is a nonempty word of the form $x\, \overline{x}$. In this paper, we study infinite binary words that do not contain arbitrarily large antisquares. For example, we show that the repetition threshold for the language of infinite binary words containing exactly two distinct antisquares is $(5+\sqrt{5})/2$. We also study repetition thresholds for related classes, where "two" in the previous sentence is replaced by a larger number. We say a binary word is $\textit{good}$ if the only antisquares it contains are $01$ and $10$. We characterize the minimal antisquares, that is, those words that are antisquares but all proper factors are good. We determine the growth rate of the number of good words of length $n$ and determine the repetition threshold between polynomial and exponential growth for the number of good words.]]> Wed, 06 Sep 2023 13:37:26 +0000 https://doi.org/10.46298/dmtcs.10063 https://doi.org/10.46298/dmtcs.10063 Baranwal, Aseem Currie, James Mol, Lucas Ochem, Pascal Rampersad, Narad Shallit, Jeffrey Baranwal, Aseem Currie, James Mol, Lucas Ochem, Pascal Rampersad, Narad Shallit, Jeffrey <![CDATA[The (bitwise) complement $\overline{x}$ of a binary word $x$ is obtained by changing each $0$ in $x$ to $1$ and vice versa. An $\textit{antisquare}$ is a nonempty word of the form $x\, \overline{x}$. In this paper, we study infinite binary words that do not contain arbitrarily large antisquares. For example, we show that the repetition threshold for the language of infinite binary words containing exactly two distinct antisquares is $(5+\sqrt{5})/2$. We also study repetition thresholds for related classes, where "two" in the previous sentence is replaced by a larger number. We say a binary word is $\textit{good}$ if the only antisquares it contains are $01$ and $10$. We characterize the minimal antisquares, that is, those words that are antisquares but all proper factors are good. We determine the growth rate of the number of good words of length $n$ and determine the repetition threshold between polynomial and exponential growth for the number of good words.]]> 0 <![CDATA[Regular synchronization languages can be used to define rational relations of finite words, and to characterize subclasses of rational relations, like automatic or recognizable relations. We provide a systematic study of the decidability of uniformization and definability problems for subclasses of rational relations defined in terms of such synchronization languages. We rephrase known results in this setting and complete the picture by adding several new decidability and undecidability results.]]> Wed, 06 Sep 2023 13:35:06 +0000 https://doi.org/10.46298/dmtcs.7460 https://doi.org/10.46298/dmtcs.7460 Löding, Christof Winter, Sarah Löding, Christof Winter, Sarah <![CDATA[Regular synchronization languages can be used to define rational relations of finite words, and to characterize subclasses of rational relations, like automatic or recognizable relations. We provide a systematic study of the decidability of uniformization and definability problems for subclasses of rational relations defined in terms of such synchronization languages. We rephrase known results in this setting and complete the picture by adding several new decidability and undecidability results.]]> 0 <![CDATA[Twin-width is a graph width parameter recently introduced by Bonnet, Kim, Thomass\'{e} & Watrigant. Given two graphs $G$ and $H$ and a graph product $\star$, we address the question: is the twin-width of $G\star H$ bounded by a function of the twin-widths of $G$ and $H$ and their maximum degrees? It is known that a bound of this type holds for strong products (Bonnet, Geniet, Kim, Thomass\'{e} & Watrigant; SODA 2021). We show that bounds of the same form hold for Cartesian, tensor/direct, corona, rooted, replacement, and zig-zag products. For the lexicographical product it is known that the twin-width of the product of two graphs is exactly the maximum of the twin-widths of the individual graphs (Bonnet, Kim, Reinald, Thomass\'{e} & Watrigant; IPEC 2021). In contrast, for the modular product we show that no bound can hold. In addition, we provide examples showing many of our bounds are tight, and give improved bounds for certain classes of graphs.]]> Fri, 09 Jun 2023 13:43:18 +0000 https://doi.org/10.46298/dmtcs.10091 https://doi.org/10.46298/dmtcs.10091 Pettersson, William Sylvester, John Pettersson, William Sylvester, John <![CDATA[Twin-width is a graph width parameter recently introduced by Bonnet, Kim, Thomass\'{e} & Watrigant. Given two graphs $G$ and $H$ and a graph product $\star$, we address the question: is the twin-width of $G\star H$ bounded by a function of the twin-widths of $G$ and $H$ and their maximum degrees? It is known that a bound of this type holds for strong products (Bonnet, Geniet, Kim, Thomass\'{e} & Watrigant; SODA 2021). We show that bounds of the same form hold for Cartesian, tensor/direct, corona, rooted, replacement, and zig-zag products. For the lexicographical product it is known that the twin-width of the product of two graphs is exactly the maximum of the twin-widths of the individual graphs (Bonnet, Kim, Reinald, Thomass\'{e} & Watrigant; IPEC 2021). In contrast, for the modular product we show that no bound can hold. In addition, we provide examples showing many of our bounds are tight, and give improved bounds for certain classes of graphs.]]> 0 <![CDATA[We discuss a notion of convergence for binary trees that is based on subtree sizes. In analogy to recent developments in the theory of graphs, posets and permutations we investigate some general aspects of the topology, such as a characterization of the set of possible limits and its structure as a metric space. For random trees the subtree size topology arises in the context of algorithms for searching and sorting when applied to random input, resulting in a sequence of nested trees. For these we obtain a structural result based on a local version of exchangeability. This in turn leads to a central limit theorem, with possibly mixed asymptotic normality.]]> Tue, 30 May 2023 08:41:14 +0000 https://doi.org/10.46298/dmtcs.10968 https://doi.org/10.46298/dmtcs.10968 Grübel, Rudolf Grübel, Rudolf <![CDATA[We discuss a notion of convergence for binary trees that is based on subtree sizes. In analogy to recent developments in the theory of graphs, posets and permutations we investigate some general aspects of the topology, such as a characterization of the set of possible limits and its structure as a metric space. For random trees the subtree size topology arises in the context of algorithms for searching and sorting when applied to random input, resulting in a sequence of nested trees. For these we obtain a structural result based on a local version of exchangeability. This in turn leads to a central limit theorem, with possibly mixed asymptotic normality.]]> 0 <![CDATA[Gallai's path decomposition conjecture states that if $G$ is a connected graph on $n$ vertices, then the edges of $G$ can be decomposed into at most $\lceil \frac{n }{2} \rceil$ paths. A graph is said to be an odd semi-clique if it can be obtained from a clique on $2k+1$ vertices by deleting at most $k-1$ edges. Bonamy and Perrett asked if the edges of every connected graph $G$ on $n$ vertices can be decomposed into at most $\lfloor \frac{n}{2} \rfloor$ paths unless $G$ is an odd semi-clique. A graph $G$ is said to be 2-degenerate if every subgraph of $G$ has a vertex of degree at most $2$. In this paper, we prove that the edges of any connected 2-degenerate graph $G$ on $n$ vertices can be decomposed into at most $\lfloor \frac{n }{2} \rfloor$ paths unless $G$ is a triangle.]]> Tue, 30 May 2023 08:38:47 +0000 https://doi.org/10.46298/dmtcs.10313 https://doi.org/10.46298/dmtcs.10313 Anto, Nevil Basavaraju, Manu Anto, Nevil Basavaraju, Manu <![CDATA[Gallai's path decomposition conjecture states that if $G$ is a connected graph on $n$ vertices, then the edges of $G$ can be decomposed into at most $\lceil \frac{n }{2} \rceil$ paths. A graph is said to be an odd semi-clique if it can be obtained from a clique on $2k+1$ vertices by deleting at most $k-1$ edges. Bonamy and Perrett asked if the edges of every connected graph $G$ on $n$ vertices can be decomposed into at most $\lfloor \frac{n}{2} \rfloor$ paths unless $G$ is an odd semi-clique. A graph $G$ is said to be 2-degenerate if every subgraph of $G$ has a vertex of degree at most $2$. In this paper, we prove that the edges of any connected 2-degenerate graph $G$ on $n$ vertices can be decomposed into at most $\lfloor \frac{n }{2} \rfloor$ paths unless $G$ is a triangle.]]> 0 <![CDATA[This paper considers the following three Roman domination graph invariants on Kneser graphs: Roman domination, total Roman domination, and signed Roman domination. For Kneser graph $K_{n,k}$, we present exact values for Roman domination number $\gamma_{R}(K_{n,k})$ and total Roman domination number $\gamma_{tR}(K_{n,k})$ proving that for $n\geqslant k(k+1)$, $\gamma_{R}(K_{n,k}) =\gamma_{tR}(K_{n,k}) = 2(k+1)$. For signed Roman domination number $\gamma_{sR}(K_{n,k})$, the new lower and upper bounds for $K_{n,2}$ are provided: we prove that for $n\geqslant 12$, the lower bound is equal to 2, while the upper bound depends on the parity of $n$ and is equal to 3 if $n$ is odd, and equal to $5$ if $n$ is even. For graphs of smaller dimensions, exact values are found by applying exact methods from literature.]]> Fri, 26 May 2023 13:40:48 +0000 https://doi.org/10.46298/dmtcs.10506 https://doi.org/10.46298/dmtcs.10506 Zec, Tatjana Grbić, Milana Zec, Tatjana Grbić, Milana <![CDATA[This paper considers the following three Roman domination graph invariants on Kneser graphs: Roman domination, total Roman domination, and signed Roman domination. For Kneser graph $K_{n,k}$, we present exact values for Roman domination number $\gamma_{R}(K_{n,k})$ and total Roman domination number $\gamma_{tR}(K_{n,k})$ proving that for $n\geqslant k(k+1)$, $\gamma_{R}(K_{n,k}) =\gamma_{tR}(K_{n,k}) = 2(k+1)$. For signed Roman domination number $\gamma_{sR}(K_{n,k})$, the new lower and upper bounds for $K_{n,2}$ are provided: we prove that for $n\geqslant 12$, the lower bound is equal to 2, while the upper bound depends on the parity of $n$ and is equal to 3 if $n$ is odd, and equal to $5$ if $n$ is even. For graphs of smaller dimensions, exact values are found by applying exact methods from literature.]]> 0 <![CDATA[An edge-colored graph is \emph{rainbow }if no two edges of the graph have the same color. An edge-colored graph $G^c$ is called \emph{properly colored} if every two adjacent edges of $G^c$ receive distinct colors in $G^c$. A \emph{strongly edge-colored} graph is a proper edge-colored graph such that every path of length $3$ is rainbow. We call an edge-colored graph $G^c$ \emph{rainbow vertex pair-pancyclic} if any two vertices in $G^c$ are contained in a rainbow cycle of length $\ell$ for each $\ell$ with $3 \leq \ell \leq n$. In this paper, we show that every strongly edge-colored graph $G^c$ of order $n$ with minimum degree $\delta \geq \frac{2n}{3}+1$ is rainbow vertex pair-pancyclicity.]]> Tue, 16 May 2023 13:47:11 +0000 https://doi.org/10.46298/dmtcs.10142 https://doi.org/10.46298/dmtcs.10142 Zhao, Peixue Huang, Fei Zhao, Peixue Huang, Fei <![CDATA[An edge-colored graph is \emph{rainbow }if no two edges of the graph have the same color. An edge-colored graph $G^c$ is called \emph{properly colored} if every two adjacent edges of $G^c$ receive distinct colors in $G^c$. A \emph{strongly edge-colored} graph is a proper edge-colored graph such that every path of length $3$ is rainbow. We call an edge-colored graph $G^c$ \emph{rainbow vertex pair-pancyclic} if any two vertices in $G^c$ are contained in a rainbow cycle of length $\ell$ for each $\ell$ with $3 \leq \ell \leq n$. In this paper, we show that every strongly edge-colored graph $G^c$ of order $n$ with minimum degree $\delta \geq \frac{2n}{3}+1$ is rainbow vertex pair-pancyclicity.]]> 0 <![CDATA[We obtain an explicit formula for the variance of the number of $k$-peaks in a uniformly random permutation. This is then used to obtain an asymptotic formula for the variance of the length of longest $k$-alternating subsequence in random permutations. Also a central limit is proved for the latter statistic.]]> Fri, 05 May 2023 09:51:50 +0000 https://doi.org/10.46298/dmtcs.10296 https://doi.org/10.46298/dmtcs.10296 Çiçeksiz, Altar Demirci, Yunus Emre Işlak, Ümit Çiçeksiz, Altar Demirci, Yunus Emre Işlak, Ümit <![CDATA[We obtain an explicit formula for the variance of the number of $k$-peaks in a uniformly random permutation. This is then used to obtain an asymptotic formula for the variance of the length of longest $k$-alternating subsequence in random permutations. Also a central limit is proved for the latter statistic.]]> 0 <![CDATA[Recently geometric hypergraphs that can be defined by intersections of pseudohalfplanes with a finite point set were defined in a purely combinatorial way. This led to extensions of earlier results about points and halfplanes to pseudohalfplanes, including polychromatic colorings and discrete Helly-type theorems about pseudohalfplanes. Here we continue this line of research and introduce the notion of convex sets of such pseudohalfplane hypergraphs. In this context we prove several results corresponding to classical results about convexity, namely Helly's Theorem, Carath\'eodory's Theorem, Kirchberger's Theorem, Separation Theorem, Radon's Theorem and the Cup-Cap Theorem. These results imply the respective results about pseudoconvex sets in the plane defined using pseudohalfplanes. It turns out that most of our results can be also proved using oriented matroids and topological affine planes (TAPs) but our approach is different from both of them. Compared to oriented matroids, our theory is based on a linear ordering of the vertex set which makes our definitions and proofs quite different and perhaps more elementary. Compared to TAPs, which are continuous objects, our proofs are purely combinatorial and again quite different in flavor. Altogether, we believe that our new approach can further our understanding of these fundamental convexity results.]]> Wed, 03 May 2023 09:26:57 +0000 https://doi.org/10.46298/dmtcs.9255 https://doi.org/10.46298/dmtcs.9255 Keszegh, Balázs Keszegh, Balázs <![CDATA[Recently geometric hypergraphs that can be defined by intersections of pseudohalfplanes with a finite point set were defined in a purely combinatorial way. This led to extensions of earlier results about points and halfplanes to pseudohalfplanes, including polychromatic colorings and discrete Helly-type theorems about pseudohalfplanes. Here we continue this line of research and introduce the notion of convex sets of such pseudohalfplane hypergraphs. In this context we prove several results corresponding to classical results about convexity, namely Helly's Theorem, Carath\'eodory's Theorem, Kirchberger's Theorem, Separation Theorem, Radon's Theorem and the Cup-Cap Theorem. These results imply the respective results about pseudoconvex sets in the plane defined using pseudohalfplanes. It turns out that most of our results can be also proved using oriented matroids and topological affine planes (TAPs) but our approach is different from both of them. Compared to oriented matroids, our theory is based on a linear ordering of the vertex set which makes our definitions and proofs quite different and perhaps more elementary. Compared to TAPs, which are continuous objects, our proofs are purely combinatorial and again quite different in flavor. Altogether, we believe that our new approach can further our understanding of these fundamental convexity results.]]> 0 <![CDATA[An s-branching flow f in a network N = (D, u), where u is the capacity function, is a flow thatreaches every vertex in V(D) from s while loosing exactly one unit of flow in each vertex other thans. Bang-Jensen and Bessy [TCS, 2014] showed that, when every arc has capacity n − 1, a network Nadmits k arc-disjoint s-branching flows if and only if its associated digraph D contains k arc-disjoints-branchings. Thus a classical result by Edmonds stating that a digraph contains k arc-disjoints-branchings if and only if the indegree of every set X ⊆ V (D) \ {s} is at least k also characterizesthe existence of k arc-disjoint s-branching flows in those networks, suggesting that the larger thecapacities are, the closer an s-branching flow is from simply being an s-branching. This observationis further implied by results by Bang-Jensen et al. [DAM, 2016] showing that there is a polynomialalgorithm to find the flows (if they exist) when every arc has capacity n − c, for every fixed c ≥ 1,and that such an algorithm is unlikely to exist for most other choices of the capacities. In this paper,we investigate how a property that is a natural extension of the characterization by Edmonds’ relatesto the existence of k arc-disjoint s-branching flows in networks. Although this property is alwaysnecessary for the existence of the flows, we show that it is not always sufficient and that it is hardto decide if the desired flows exist even if we know beforehand that the network satisfies it. On thepositive side, we show that it guarantees the existence of the desired flows in some particular casesdepending on the choice of the capacity function or on the structure of the underlying graph of D,for example. We remark that, in those positive cases, polynomial time algorithms to find the flowscan be extracted from the constructive proofs.]]> Tue, 02 May 2023 12:51:59 +0000 https://doi.org/10.46298/dmtcs.9302 https://doi.org/10.46298/dmtcs.9302 Carvalho, Cláudio Costa, Jonas Lopes, Raul Maia, Ana Karolinna Nisse, Nicolas Sales, Cláudia, Carvalho, Cláudio Costa, Jonas Lopes, Raul Maia, Ana Karolinna Nisse, Nicolas Sales, Cláudia, <![CDATA[An s-branching flow f in a network N = (D, u), where u is the capacity function, is a flow thatreaches every vertex in V(D) from s while loosing exactly one unit of flow in each vertex other thans. Bang-Jensen and Bessy [TCS, 2014] showed that, when every arc has capacity n − 1, a network Nadmits k arc-disjoint s-branching flows if and only if its associated digraph D contains k arc-disjoints-branchings. Thus a classical result by Edmonds stating that a digraph contains k arc-disjoints-branchings if and only if the indegree of every set X ⊆ V (D) \ {s} is at least k also characterizesthe existence of k arc-disjoint s-branching flows in those networks, suggesting that the larger thecapacities are, the closer an s-branching flow is from simply being an s-branching. This observationis further implied by results by Bang-Jensen et al. [DAM, 2016] showing that there is a polynomialalgorithm to find the flows (if they exist) when every arc has capacity n − c, for every fixed c ≥ 1,and that such an algorithm is unlikely to exist for most other choices of the capacities. In this paper,we investigate how a property that is a natural extension of the characterization by Edmonds’ relatesto the existence of k arc-disjoint s-branching flows in networks. Although this property is alwaysnecessary for the existence of the flows, we show that it is not always sufficient and that it is hardto decide if the desired flows exist even if we know beforehand that the network satisfies it. On thepositive side, we show that it guarantees the existence of the desired flows in some particular casesdepending on the choice of the capacity function or on the structure of the underlying graph of D,for example. We remark that, in those positive cases, polynomial time algorithms to find the flowscan be extracted from the constructive proofs.]]> 0 <![CDATA[In a generalized Tur\'an problem, two graphs $H$ and $F$ are given and the question is the maximum number of copies of $H$ in an $F$-free graph of order $n$. In this paper, we study the number of double stars $S_{k,l}$ in triangle-free graphs. We also study an opposite version of this question: what is the maximum number edges/triangles in graphs with double star type restrictions, which leads us to study two questions related to the extremal number of triangles or edges in graphs with degree-sum constraints over adjacent or non-adjacent vertices.]]> Thu, 20 Apr 2023 08:04:14 +0000 https://doi.org/10.46298/dmtcs.8499 https://doi.org/10.46298/dmtcs.8499 Győri, Ervin Wang, Runze Woolfson, Spencer Győri, Ervin Wang, Runze Woolfson, Spencer <![CDATA[In a generalized Tur\'an problem, two graphs $H$ and $F$ are given and the question is the maximum number of copies of $H$ in an $F$-free graph of order $n$. In this paper, we study the number of double stars $S_{k,l}$ in triangle-free graphs. We also study an opposite version of this question: what is the maximum number edges/triangles in graphs with double star type restrictions, which leads us to study two questions related to the extremal number of triangles or edges in graphs with degree-sum constraints over adjacent or non-adjacent vertices.]]> 0 <![CDATA[We focus on the maximum number of minimal transversals in 3-partite 3-uniform hypergraphs on n vertices. Those hypergraphs (and their minimal transversals) are commonly found in database applications. In this paper we prove that this number grows at least like 1.4977^n and at most like 1.5012^n.]]> Thu, 20 Apr 2023 08:02:06 +0000 https://doi.org/10.46298/dmtcs.7129 https://doi.org/10.46298/dmtcs.7129 Bazin, Alexandre Beaudou, Laurent Kahn, Giacomo Khoshkhah, Kaveh Bazin, Alexandre Beaudou, Laurent Kahn, Giacomo Khoshkhah, Kaveh <![CDATA[We focus on the maximum number of minimal transversals in 3-partite 3-uniform hypergraphs on n vertices. Those hypergraphs (and their minimal transversals) are commonly found in database applications. In this paper we prove that this number grows at least like 1.4977^n and at most like 1.5012^n.]]> 0 <![CDATA[In this paper, we study a variant of graph domination known as $(t, r)$ broadcast domination, first defined in Blessing, Insko, Johnson, and Mauretour in 2015. In this variant, each broadcast provides $t-d$ reception to each vertex a distance $d < t$ from the broadcast. If $d \ge t$ then no reception is provided. A vertex is considered dominated if it receives $r$ total reception from all broadcasts. Our main results provide some upper and lower bounds on the density of a $(t, r)$ dominating pattern of an infinite grid, as well as methods of computing them. Also, when $r \ge 2$ we describe a family of counterexamples to a generalization of Vizing's Conjecture to $(t,r)$ broadcast domination.]]> Mon, 03 Apr 2023 09:38:49 +0000 https://doi.org/10.46298/dmtcs.5732 https://doi.org/10.46298/dmtcs.5732 Shlomi, Tom Shlomi, Tom <![CDATA[In this paper, we study a variant of graph domination known as $(t, r)$ broadcast domination, first defined in Blessing, Insko, Johnson, and Mauretour in 2015. In this variant, each broadcast provides $t-d$ reception to each vertex a distance $d < t$ from the broadcast. If $d \ge t$ then no reception is provided. A vertex is considered dominated if it receives $r$ total reception from all broadcasts. Our main results provide some upper and lower bounds on the density of a $(t, r)$ dominating pattern of an infinite grid, as well as methods of computing them. Also, when $r \ge 2$ we describe a family of counterexamples to a generalization of Vizing's Conjecture to $(t,r)$ broadcast domination.]]> 0 <![CDATA[We show that the counting sequence for permutations avoiding both of the (classical) patterns 1243 and 2134 has the algebraic generating function supplied by Vaclav Kotesovec for sequence A164651 in The On-Line Encyclopedia of Integer Sequences.]]> Thu, 16 Mar 2023 10:21:42 +0000 https://doi.org/10.46298/dmtcs.5287 https://doi.org/10.46298/dmtcs.5287 Callan, David Callan, David <![CDATA[We show that the counting sequence for permutations avoiding both of the (classical) patterns 1243 and 2134 has the algebraic generating function supplied by Vaclav Kotesovec for sequence A164651 in The On-Line Encyclopedia of Integer Sequences.]]> 0 <![CDATA[We examine the classic game of Cops and Robbers played on models of dynamic graphs, that is, graphs evolving over discrete time steps. At each time step, a graph instance is generated as a subgraph of the underlying graph of the model. The cops and the robber take their turns on the current graph instance. The cops win if they can capture the robber at some point in time. Otherwise, the robber wins. In the offline case, the players are fully aware of the evolution sequence, up to some finite time horizon T. We provide a O(n 2k+1 T) algorithm to decide whether a given evolution sequence for an underlying graph with n vertices is k-cop-win via a reduction to a reachability game. In the online case, there is no knowledge of the evolution sequence, and the game might go on forever. Also, each generated instance is required to be connected. We provide a nearly tight characterization for sparse underlying graphs, i.e., with at most linear number of edges. We prove λ + 1 cops suffice to capture the robber in any underlying graph with n − 1 + λ edges. Further, we define a family of underlying graphs with n−1+λ edges where λ−1 cops are necessary (and sufficient) for capture.]]> Thu, 16 Mar 2023 10:17:44 +0000 https://doi.org/10.46298/dmtcs.8784 https://doi.org/10.46298/dmtcs.8784 Balev, Stefan Jiménez Laredo, Juan, Lamprou, Ioannis Pigné, Yoann Sanlaville, Eric Balev, Stefan Jiménez Laredo, Juan, Lamprou, Ioannis Pigné, Yoann Sanlaville, Eric <![CDATA[We examine the classic game of Cops and Robbers played on models of dynamic graphs, that is, graphs evolving over discrete time steps. At each time step, a graph instance is generated as a subgraph of the underlying graph of the model. The cops and the robber take their turns on the current graph instance. The cops win if they can capture the robber at some point in time. Otherwise, the robber wins. In the offline case, the players are fully aware of the evolution sequence, up to some finite time horizon T. We provide a O(n 2k+1 T) algorithm to decide whether a given evolution sequence for an underlying graph with n vertices is k-cop-win via a reduction to a reachability game. In the online case, there is no knowledge of the evolution sequence, and the game might go on forever. Also, each generated instance is required to be connected. We provide a nearly tight characterization for sparse underlying graphs, i.e., with at most linear number of edges. We prove λ + 1 cops suffice to capture the robber in any underlying graph with n − 1 + λ edges. Further, we define a family of underlying graphs with n−1+λ edges where λ−1 cops are necessary (and sufficient) for capture.]]> 0 <![CDATA[Let $\Bigl\langle\matrix{n\cr k}\Bigr\rangle$, $\Bigl\langle\matrix{B_n\cr k}\Bigr\rangle$, and $\Bigl\langle\matrix{D_n\cr k}\Bigr\rangle$ be the Eulerian numbers in the types A, B, and D, respectively -- that is, the number of permutations of n elements with $k$ descents, the number of signed permutations (of $n$ elements) with $k$ type B descents, the number of even signed permutations (of $n$ elements) with $k$ type D descents. Let $S_n(t) = \sum_{k = 0}^{n-1} \Bigl\langle\matrix{n\cr k}\Bigr\rangle t^k$, $B_n(t) = \sum_{k = 0}^n \Bigl\langle\matrix{B_n\cr k}\Bigr\rangle t^k$, and $D_n(t) = \sum_{k = 0}^n \Bigl\langle\matrix{D_n\cr k}\Bigr\rangle t^k$. We give bijective proofs of the identity $$B_n(t^2) = (1 + t)^{n+1}S_n(t) - 2^n tS_n(t^2)$$ and of Stembridge's identity $$D_n(t) = B_n(t) - n2^{n-1}tS_{n-1}(t).$$ These bijective proofs rely on a representation of signed permutations as paths. Using this representation we also establish a bijective correspondence between even signed permutations and pairs $(w, E)$ with $([n], E)$ a threshold graph and $w$ a degree ordering of $([n], E)$, which we use to obtain bijective proofs of enumerative results for threshold graphs.]]> Fri, 10 Mar 2023 14:40:09 +0000 https://doi.org/10.46298/dmtcs.7413 https://doi.org/10.46298/dmtcs.7413 Santocanale, Luigi Santocanale, Luigi <![CDATA[Let $\Bigl\langle\matrix{n\cr k}\Bigr\rangle$, $\Bigl\langle\matrix{B_n\cr k}\Bigr\rangle$, and $\Bigl\langle\matrix{D_n\cr k}\Bigr\rangle$ be the Eulerian numbers in the types A, B, and D, respectively -- that is, the number of permutations of n elements with $k$ descents, the number of signed permutations (of $n$ elements) with $k$ type B descents, the number of even signed permutations (of $n$ elements) with $k$ type D descents. Let $S_n(t) = \sum_{k = 0}^{n-1} \Bigl\langle\matrix{n\cr k}\Bigr\rangle t^k$, $B_n(t) = \sum_{k = 0}^n \Bigl\langle\matrix{B_n\cr k}\Bigr\rangle t^k$, and $D_n(t) = \sum_{k = 0}^n \Bigl\langle\matrix{D_n\cr k}\Bigr\rangle t^k$. We give bijective proofs of the identity $$B_n(t^2) = (1 + t)^{n+1}S_n(t) - 2^n tS_n(t^2)$$ and of Stembridge's identity $$D_n(t) = B_n(t) - n2^{n-1}tS_{n-1}(t).$$ These bijective proofs rely on a representation of signed permutations as paths. Using this representation we also establish a bijective correspondence between even signed permutations and pairs $(w, E)$ with $([n], E)$ a threshold graph and $w$ a degree ordering of $([n], E)$, which we use to obtain bijective proofs of enumerative results for threshold graphs.]]> 0 <![CDATA[We study a family of generalizations of Edge Dominating Set on directed graphs called Directed $(p,q)$-Edge Dominating Set. In this problem an arc $(u,v)$ is said to dominate itself, as well as all arcs which are at distance at most $q$ from $v$, or at distance at most $p$ to $u$. First, we give significantly improved FPT algorithms for the two most important cases of the problem, $(0,1)$-dEDS and $(1,1)$-dEDS (that correspond to versions of Dominating Set on line graphs), as well as polynomial kernels. We also improve the best-known approximation for these cases from logarithmic to constant. In addition, we show that $(p,q)$-dEDS is FPT parameterized by $p+q+tw$, but W-hard parameterized by $tw$ (even if the size of the optimal is added as a second parameter), where $tw$ is the treewidth of the underlying graph of the input. We then go on to focus on the complexity of the problem on tournaments. Here, we provide a complete classification for every possible fixed value of $p,q$, which shows that the problem exhibits a surprising behavior, including cases which are in P; cases which are solvable in quasi-polynomial time but not in P; and a single case $(p=q=1)$ which is NP-hard (under randomized reductions) and cannot be solved in sub-exponential time, under standard assumptions.]]> Fri, 10 Mar 2023 14:36:09 +0000 https://doi.org/10.46298/dmtcs.5378 https://doi.org/10.46298/dmtcs.5378 Belmonte, Rémy Hanaka, Tesshu Katsikarelis, Ioannis Kim, Eun Jung Lampis, Michael Belmonte, Rémy Hanaka, Tesshu Katsikarelis, Ioannis Kim, Eun Jung Lampis, Michael <![CDATA[We study a family of generalizations of Edge Dominating Set on directed graphs called Directed $(p,q)$-Edge Dominating Set. In this problem an arc $(u,v)$ is said to dominate itself, as well as all arcs which are at distance at most $q$ from $v$, or at distance at most $p$ to $u$. First, we give significantly improved FPT algorithms for the two most important cases of the problem, $(0,1)$-dEDS and $(1,1)$-dEDS (that correspond to versions of Dominating Set on line graphs), as well as polynomial kernels. We also improve the best-known approximation for these cases from logarithmic to constant. In addition, we show that $(p,q)$-dEDS is FPT parameterized by $p+q+tw$, but W-hard parameterized by $tw$ (even if the size of the optimal is added as a second parameter), where $tw$ is the treewidth of the underlying graph of the input. We then go on to focus on the complexity of the problem on tournaments. Here, we provide a complete classification for every possible fixed value of $p,q$, which shows that the problem exhibits a surprising behavior, including cases which are in P; cases which are solvable in quasi-polynomial time but not in P; and a single case $(p=q=1)$ which is NP-hard (under randomized reductions) and cannot be solved in sub-exponential time, under standard assumptions.]]> 0 <![CDATA[We study the computational complexity of $c$-Colored $P_\ell$ Deletion and $c$-Colored $C_\ell$ Deletion. In these problems, one is given a $c$-edge-colored graph and wants to destroy all induced $c$-colored paths or cycles, respectively, on $\ell$ vertices by deleting at most $k$ edges. Herein, a path or cycle is $c$-colored if it contains edges of $c$ distinct colors. We show that $c$-Colored $P_\ell$ Deletion and $c$-Colored $C_\ell$ Deletion are NP-hard for each non-trivial combination of $c$ and $\ell$. We then analyze the parameterized complexity of these problems. We extend the notion of neighborhood diversity to edge-colored graphs and show that both problems are fixed-parameter tractable with respect to the colored neighborhood diversity of the input graph. We also provide hardness results to outline the limits of parameterization by the standard parameter solution size $k$. Finally, we consider bicolored input graphs and show a special case of $2$-Colored $P_4$ Deletion that can be solved in polynomial time.]]> Fri, 03 Mar 2023 10:26:24 +0000 https://doi.org/10.46298/dmtcs.7636 https://doi.org/10.46298/dmtcs.7636 Eckstein, Nils Jakob Grüttemeier, Niels Komusiewicz, Christian Sommer, Frank Eckstein, Nils Jakob Grüttemeier, Niels Komusiewicz, Christian Sommer, Frank <![CDATA[We study the computational complexity of $c$-Colored $P_\ell$ Deletion and $c$-Colored $C_\ell$ Deletion. In these problems, one is given a $c$-edge-colored graph and wants to destroy all induced $c$-colored paths or cycles, respectively, on $\ell$ vertices by deleting at most $k$ edges. Herein, a path or cycle is $c$-colored if it contains edges of $c$ distinct colors. We show that $c$-Colored $P_\ell$ Deletion and $c$-Colored $C_\ell$ Deletion are NP-hard for each non-trivial combination of $c$ and $\ell$. We then analyze the parameterized complexity of these problems. We extend the notion of neighborhood diversity to edge-colored graphs and show that both problems are fixed-parameter tractable with respect to the colored neighborhood diversity of the input graph. We also provide hardness results to outline the limits of parameterization by the standard parameter solution size $k$. Finally, we consider bicolored input graphs and show a special case of $2$-Colored $P_4$ Deletion that can be solved in polynomial time.]]> 0 <![CDATA[In this paper we first study $k \times n$ Youden rectangles of small orders. We have enumerated all Youden rectangles for a range of small parameter values, excluding the almost square cases where $k = n-1$, in a large scale computer search. In particular, we verify the previous counts for $(n,k) = (7,3), (7,4)$, and extend this to the cases $(11,5), (11,6), (13,4)$ and $(21,5)$. For small parameter values where no Youden rectangles exist, we also enumerate rectangles where the number of symbols common to two columns is always one of two possible values, differing by 1, which we call \emph{near Youden rectangles}. For all the designs we generate, we calculate the order of the autotopism group and investigate to which degree a certain transformation can yield other row-column designs, namely double arrays, triple arrays and sesqui arrays. Finally, we also investigate certain Latin rectangles with three possible pairwise intersection sizes for the columns and demonstrate that these can give rise to triple and sesqui arrays which cannot be obtained from Youden rectangles, using the transformation mentioned above.]]> Wed, 01 Mar 2023 13:15:44 +0000 https://doi.org/10.46298/dmtcs.6754 https://doi.org/10.46298/dmtcs.6754 Jäger, Gerold Markström, Klas Shcherbak, Denys Öhman, Lars-Daniel Jäger, Gerold Markström, Klas Shcherbak, Denys Öhman, Lars-Daniel <![CDATA[In this paper we first study $k \times n$ Youden rectangles of small orders. We have enumerated all Youden rectangles for a range of small parameter values, excluding the almost square cases where $k = n-1$, in a large scale computer search. In particular, we verify the previous counts for $(n,k) = (7,3), (7,4)$, and extend this to the cases $(11,5), (11,6), (13,4)$ and $(21,5)$. For small parameter values where no Youden rectangles exist, we also enumerate rectangles where the number of symbols common to two columns is always one of two possible values, differing by 1, which we call \emph{near Youden rectangles}. For all the designs we generate, we calculate the order of the autotopism group and investigate to which degree a certain transformation can yield other row-column designs, namely double arrays, triple arrays and sesqui arrays. Finally, we also investigate certain Latin rectangles with three possible pairwise intersection sizes for the columns and demonstrate that these can give rise to triple and sesqui arrays which cannot be obtained from Youden rectangles, using the transformation mentioned above.]]> 0 <![CDATA[Inspired by Lelek's idea from [Disjoint mappings and the span of spaces, Fund. Math. 55 (1964), 199 -- 214], we introduce the novel notion of the span of graphs. Using this, we solve the problem of determining the \emph{maximal safety distance} two players can keep at all times while traversing a graph. Moreover, their moves must be made with respect to certain move rules. For this purpose, we introduce different variants of a span of a given connected graph. All the variants model the maximum safety distance kept by two players in a graph traversal, where the players may only move with accordance to a specific set of rules, and their goal: visit either all vertices, or all edges. For each variant, we show that the solution can be obtained by considering only connected subgraphs of a graph product and the projections to the factors. We characterise graphs in which it is impossible to keep a positive safety distance at all moments in time. Finally, we present a polynomial time algorithm that determines the chosen span variant of a given graph.]]> Wed, 01 Mar 2023 13:14:37 +0000 https://doi.org/10.46298/dmtcs.9859 https://doi.org/10.46298/dmtcs.9859 Banič, Iztok Taranenko, Andrej Banič, Iztok Taranenko, Andrej <![CDATA[Inspired by Lelek's idea from [Disjoint mappings and the span of spaces, Fund. Math. 55 (1964), 199 -- 214], we introduce the novel notion of the span of graphs. Using this, we solve the problem of determining the \emph{maximal safety distance} two players can keep at all times while traversing a graph. Moreover, their moves must be made with respect to certain move rules. For this purpose, we introduce different variants of a span of a given connected graph. All the variants model the maximum safety distance kept by two players in a graph traversal, where the players may only move with accordance to a specific set of rules, and their goal: visit either all vertices, or all edges. For each variant, we show that the solution can be obtained by considering only connected subgraphs of a graph product and the projections to the factors. We characterise graphs in which it is impossible to keep a positive safety distance at all moments in time. Finally, we present a polynomial time algorithm that determines the chosen span variant of a given graph.]]> 0 <![CDATA[The following problem has been known since the 80's. Let $\Gamma$ be an Abelian group of order $m$ (denoted $|\Gamma|=m$), and let $t$ and $m_i$, $1 \leq i \leq t$, be positive integers such that $\sum_{i=1}^t m_i=m-1$. Determine when $\Gamma^*=\Gamma\setminus\{0\}$, the set of non-zero elements of $\Gamma$, can be partitioned into disjoint subsets $S_i$, $1 \leq i \leq t$, such that $|S_i|=m_i$ and $\sum_{s\in S_i}s=0$ for every $i$, $1 \leq i \leq t$. It is easy to check that $m_i\geq 2$ (for every $i$, $1 \leq i \leq t$) and $|I(\Gamma)|\neq 1$ are necessary conditions for the existence of such partitions, where $I(\Gamma)$ is the set of involutions of $\Gamma$. It was proved that the condition $m_i\geq 2$ is sufficient if and only if $|I(\Gamma)|\in\{0,3\}$. For other groups (i.e., for which $|I(\Gamma)|\neq 3$ and $|I(\Gamma)|>1$), only the case of any group $\Gamma$ with $\Gamma\cong(Z_2)^n$ for some positive integer $n$ has been analyzed completely so far, and it was shown independently by several authors that $m_i\geq 3$ is sufficient in this case. Moreover, recently Cichacz and Tuza proved that, if $|\Gamma|$ is large enough and $|I(\Gamma)|>1$, then $m_i\geq 4$ is sufficient. In this paper we generalize this result for every Abelian group of order $2^n$. Namely, we show that the condition $m_i\geq 3$ is sufficient for $\Gamma$ such that $|I(\Gamma)|>1$ and $|\Gamma|=2^n$, for every positive integer $n$. We also present some applications of this result to graph magic- and anti-magic-type labelings.]]> Wed, 01 Mar 2023 13:13:03 +0000 https://doi.org/10.46298/dmtcs.9914 https://doi.org/10.46298/dmtcs.9914 Cichacz, Sylwia Suchan, Karol Cichacz, Sylwia Suchan, Karol <![CDATA[The following problem has been known since the 80's. Let $\Gamma$ be an Abelian group of order $m$ (denoted $|\Gamma|=m$), and let $t$ and $m_i$, $1 \leq i \leq t$, be positive integers such that $\sum_{i=1}^t m_i=m-1$. Determine when $\Gamma^*=\Gamma\setminus\{0\}$, the set of non-zero elements of $\Gamma$, can be partitioned into disjoint subsets $S_i$, $1 \leq i \leq t$, such that $|S_i|=m_i$ and $\sum_{s\in S_i}s=0$ for every $i$, $1 \leq i \leq t$. It is easy to check that $m_i\geq 2$ (for every $i$, $1 \leq i \leq t$) and $|I(\Gamma)|\neq 1$ are necessary conditions for the existence of such partitions, where $I(\Gamma)$ is the set of involutions of $\Gamma$. It was proved that the condition $m_i\geq 2$ is sufficient if and only if $|I(\Gamma)|\in\{0,3\}$. For other groups (i.e., for which $|I(\Gamma)|\neq 3$ and $|I(\Gamma)|>1$), only the case of any group $\Gamma$ with $\Gamma\cong(Z_2)^n$ for some positive integer $n$ has been analyzed completely so far, and it was shown independently by several authors that $m_i\geq 3$ is sufficient in this case. Moreover, recently Cichacz and Tuza proved that, if $|\Gamma|$ is large enough and $|I(\Gamma)|>1$, then $m_i\geq 4$ is sufficient. In this paper we generalize this result for every Abelian group of order $2^n$. Namely, we show that the condition $m_i\geq 3$ is sufficient for $\Gamma$ such that $|I(\Gamma)|>1$ and $|\Gamma|=2^n$, for every positive integer $n$. We also present some applications of this result to graph magic- and anti-magic-type labelings.]]> 0 <![CDATA[Given a boolean predicate $\Pi$ on labeled networks (e.g., proper coloring, leader election, etc.), a self-stabilizing algorithm for $\Pi$ is a distributed algorithm that can start from any initial configuration of the network (i.e., every node has an arbitrary value assigned to each of its variables), and eventually converge to a configuration satisfying $\Pi$. It is known that leader election does not have a deterministic self-stabilizing algorithm using a constant-size register at each node, i.e., for some networks, some of their nodes must have registers whose sizes grow with the size $n$ of the networks. On the other hand, it is also known that leader election can be solved by a deterministic self-stabilizing algorithm using registers of $O(\log \log n)$ bits per node in any $n$-node bounded-degree network. We show that this latter space complexity is optimal. Specifically, we prove that every deterministic self-stabilizing algorithm solving leader election must use $\Omega(\log \log n)$-bit per node registers in some $n$-node networks. In addition, we show that our lower bounds go beyond leader election, and apply to all problems that cannot be solved by anonymous algorithms.]]> Wed, 01 Mar 2023 13:11:38 +0000 https://doi.org/10.46298/dmtcs.9335 https://doi.org/10.46298/dmtcs.9335 Blin, Lélia Feuilloley, Laurent Bouder, Gabriel Le Blin, Lélia Feuilloley, Laurent Bouder, Gabriel Le <![CDATA[Given a boolean predicate $\Pi$ on labeled networks (e.g., proper coloring, leader election, etc.), a self-stabilizing algorithm for $\Pi$ is a distributed algorithm that can start from any initial configuration of the network (i.e., every node has an arbitrary value assigned to each of its variables), and eventually converge to a configuration satisfying $\Pi$. It is known that leader election does not have a deterministic self-stabilizing algorithm using a constant-size register at each node, i.e., for some networks, some of their nodes must have registers whose sizes grow with the size $n$ of the networks. On the other hand, it is also known that leader election can be solved by a deterministic self-stabilizing algorithm using registers of $O(\log \log n)$ bits per node in any $n$-node bounded-degree network. We show that this latter space complexity is optimal. Specifically, we prove that every deterministic self-stabilizing algorithm solving leader election must use $\Omega(\log \log n)$-bit per node registers in some $n$-node networks. In addition, we show that our lower bounds go beyond leader election, and apply to all problems that cannot be solved by anonymous algorithms.]]> 0 <![CDATA[In 2020 Bang-Jensen et. al. generalized the Haj\'os join of two graphs to the class of digraphs and generalized several results for vertex colorings in digraphs. Although, as a consequence of these results, a digraph can be obtained by Haj\'os constructions (directed Haj\'os join and identifying non-adjacent vertices), determining the Haj\'os constructions to obtain the digraph is a complex problem. In particular, Bang-Jensen et al. posed the problem of determining the Haj\'os operations to construct the symmetric 5-cycle from the complete symmetric digraph of order 3 using only Haj\'os constructions. We successfully adapted a rank-based genetic algorithm to solve this problem by the introduction of innovative recombination and mutation operators from graph theory. The Haj\'os Join became the recombination operator and the identification of independent vertices became the mutation operator. In this way, we were able to obtain a sequence of only 16 Haj\'os operations to construct the symmetric cycle of order 5.]]> Wed, 01 Mar 2023 13:10:29 +0000 https://doi.org/10.46298/dmtcs.10189 https://doi.org/10.46298/dmtcs.10189 García-Altamirano, Juan Carlos Olsen, Mika Cervantes-Ojeda, Jorge García-Altamirano, Juan Carlos Olsen, Mika Cervantes-Ojeda, Jorge <![CDATA[In 2020 Bang-Jensen et. al. generalized the Haj\'os join of two graphs to the class of digraphs and generalized several results for vertex colorings in digraphs. Although, as a consequence of these results, a digraph can be obtained by Haj\'os constructions (directed Haj\'os join and identifying non-adjacent vertices), determining the Haj\'os constructions to obtain the digraph is a complex problem. In particular, Bang-Jensen et al. posed the problem of determining the Haj\'os operations to construct the symmetric 5-cycle from the complete symmetric digraph of order 3 using only Haj\'os constructions. We successfully adapted a rank-based genetic algorithm to solve this problem by the introduction of innovative recombination and mutation operators from graph theory. The Haj\'os Join became the recombination operator and the identification of independent vertices became the mutation operator. In this way, we were able to obtain a sequence of only 16 Haj\'os operations to construct the symmetric cycle of order 5.]]> 0 <![CDATA[In 1946, Erd\H{o}s posed the distinct distance problem, which seeks to find the minimum number of distinct distances between pairs of points selected from any configuration of $n$ points in the plane. The problem has since been explored along with many variants, including ones that extend it into higher dimensions. Less studied but no less intriguing is Erd\H{o}s' distinct angle problem, which seeks to find point configurations in the plane that minimize the number of distinct angles. In their recent paper "Distinct Angles in General Position," Fleischmann, Konyagin, Miller, Palsson, Pesikoff, and Wolf use a logarithmic spiral to establish an upper bound of $O(n^2)$ on the minimum number of distinct angles in the plane in general position, which prohibits three points on any line or four on any circle. We consider the question of distinct angles in three dimensions and provide bounds on the minimum number of distinct angles in general position in this setting. We focus on pinned variants of the question, and we examine explicit constructions of point configurations in $\mathbb{R}^3$ which use self-similarity to minimize the number of distinct angles. Furthermore, we study a variant of the distinct angles question regarding distinct angle chains and provide bounds on the minimum number of distinct chains in $\mathbb{R}^2$ and $\mathbb{R}^3$.]]> Mon, 27 Feb 2023 08:39:14 +0000 https://doi.org/10.46298/dmtcs.10037 https://doi.org/10.46298/dmtcs.10037 Ascoli, Ruben Betti, Livia Duke, Jacob Lehmann Liu, Xuyan Milgrim, Wyatt Miller, Steven J. Palsson, Eyvindur A. Acosta, Francisco Romero Iannuzzelli, Santiago Velazquez Ascoli, Ruben Betti, Livia Duke, Jacob Lehmann Liu, Xuyan Milgrim, Wyatt Miller, Steven J. Palsson, Eyvindur A. Acosta, Francisco Romero Iannuzzelli, Santiago Velazquez <![CDATA[In 1946, Erd\H{o}s posed the distinct distance problem, which seeks to find the minimum number of distinct distances between pairs of points selected from any configuration of $n$ points in the plane. The problem has since been explored along with many variants, including ones that extend it into higher dimensions. Less studied but no less intriguing is Erd\H{o}s' distinct angle problem, which seeks to find point configurations in the plane that minimize the number of distinct angles. In their recent paper "Distinct Angles in General Position," Fleischmann, Konyagin, Miller, Palsson, Pesikoff, and Wolf use a logarithmic spiral to establish an upper bound of $O(n^2)$ on the minimum number of distinct angles in the plane in general position, which prohibits three points on any line or four on any circle. We consider the question of distinct angles in three dimensions and provide bounds on the minimum number of distinct angles in general position in this setting. We focus on pinned variants of the question, and we examine explicit constructions of point configurations in $\mathbb{R}^3$ which use self-similarity to minimize the number of distinct angles. Furthermore, we study a variant of the distinct angles question regarding distinct angle chains and provide bounds on the minimum number of distinct chains in $\mathbb{R}^2$ and $\mathbb{R}^3$.]]> 0 <![CDATA[The input to the token swapping problem is a graph with vertices $v_1, v_2, \ldots, v_n$, and $n$ tokens with labels $1, 2, \ldots, n$, one on each vertex. The goal is to get token $i$ to vertex $v_i$ for all $i= 1, \ldots, n$ using a minimum number of swaps, where a swap exchanges the tokens on the endpoints of an edge. We present some results about token swapping on a tree, also known as "sorting with a transposition tree": 1. An optimum swap sequence may need to perform a swap on a leaf vertex that has the correct token (a "happy leaf"), disproving a conjecture of Vaughan. 2. Any algorithm that fixes happy leaves -- as all known approximation algorithms for the problem do -- has approximation factor at least $4/3$. Furthermore, the two best-known 2-approximation algorithms have approximation factor exactly 2. 3. A generalized problem -- weighted coloured token swapping -- is NP-complete on trees, even when they are restricted to be subdivided stars, but solvable in polynomial time on paths and stars. In this version, tokens and vertices have colours, and colours have weights. The goal is to get every token to a vertex of the same colour, and the cost of a swap is the sum of the weights of the two tokens involved.]]> Wed, 18 Jan 2023 12:09:36 +0000 https://doi.org/10.46298/dmtcs.8383 https://doi.org/10.46298/dmtcs.8383 Biniaz, Ahmad Jain, Kshitij Lubiw, Anna Masárová, Zuzana Miltzow, Tillmann Mondal, Debajyoti Naredla, Anurag Murty Tkadlec, Josef Turcotte, Alexi Biniaz, Ahmad Jain, Kshitij Lubiw, Anna Masárová, Zuzana Miltzow, Tillmann Mondal, Debajyoti Naredla, Anurag Murty Tkadlec, Josef Turcotte, Alexi <![CDATA[The input to the token swapping problem is a graph with vertices $v_1, v_2, \ldots, v_n$, and $n$ tokens with labels $1, 2, \ldots, n$, one on each vertex. The goal is to get token $i$ to vertex $v_i$ for all $i= 1, \ldots, n$ using a minimum number of swaps, where a swap exchanges the tokens on the endpoints of an edge. We present some results about token swapping on a tree, also known as "sorting with a transposition tree": 1. An optimum swap sequence may need to perform a swap on a leaf vertex that has the correct token (a "happy leaf"), disproving a conjecture of Vaughan. 2. Any algorithm that fixes happy leaves -- as all known approximation algorithms for the problem do -- has approximation factor at least $4/3$. Furthermore, the two best-known 2-approximation algorithms have approximation factor exactly 2. 3. A generalized problem -- weighted coloured token swapping -- is NP-complete on trees, even when they are restricted to be subdivided stars, but solvable in polynomial time on paths and stars. In this version, tokens and vertices have colours, and colours have weights. The goal is to get every token to a vertex of the same colour, and the cost of a swap is the sum of the weights of the two tokens involved.]]> 0 <![CDATA[In 2010, Kh. Hessami Pilehrood and T. Hessami Pilehrood introduced generating function identities used to obtain series accelerations for values of Dirichlet's $\beta$ function, via the Markov--Wilf--Zeilberger method. Inspired by these past results, together with related results introduced by Chu et al., we introduce a variety of hypergeometric recurrences. We prove these recurrences using the WZ method, and we apply these recurrences to obtain series acceleration identities. We introduce a family of summations generalizing a Ramanujan-type series for $\frac{1}{\pi^2}$ due to Guillera, and a family of summations generalizing an accelerated series for Catalan's constant due to Lupa\c{s}, and many related results.]]> Mon, 02 Jan 2023 09:11:27 +0000 https://doi.org/10.46298/dmtcs.9557 https://doi.org/10.46298/dmtcs.9557 Levrie, Paul Campbell, John Levrie, Paul Campbell, John <![CDATA[In 2010, Kh. Hessami Pilehrood and T. Hessami Pilehrood introduced generating function identities used to obtain series accelerations for values of Dirichlet's $\beta$ function, via the Markov--Wilf--Zeilberger method. Inspired by these past results, together with related results introduced by Chu et al., we introduce a variety of hypergeometric recurrences. We prove these recurrences using the WZ method, and we apply these recurrences to obtain series acceleration identities. We introduce a family of summations generalizing a Ramanujan-type series for $\frac{1}{\pi^2}$ due to Guillera, and a family of summations generalizing an accelerated series for Catalan's constant due to Lupa\c{s}, and many related results.]]> 0 <![CDATA[Let $D$ be an oriented graph. The inversion of a set $X$ of vertices in $D$ consists in reversing the direction of all arcs with both ends in $X$. The inversion number of $D$, denoted by ${\rm inv}(D)$, is the minimum number of inversions needed to make $D$ acyclic. Denoting by $\tau(D)$, $\tau' (D)$, and $\nu(D)$ the cycle transversal number, the cycle arc-transversal number and the cycle packing number of $D$ respectively, one shows that ${\rm inv}(D) \leq \tau' (D)$, ${\rm inv}(D) \leq 2\tau(D)$ and there exists a function $g$ such that ${\rm inv}(D)\leq g(\nu(D))$. We conjecture that for any two oriented graphs $L$ and $R$, ${\rm inv}(L\rightarrow R) ={\rm inv}(L) +{\rm inv}(R)$ where $L\rightarrow R$ is the dijoin of $L$ and $R$. This would imply that the first two inequalities are tight. We prove this conjecture when ${\rm inv}(L)\leq 1$ and ${\rm inv}(R)\leq 2$ and when ${\rm inv}(L) ={\rm inv}(R)=2$ and $L$ and $R$ are strongly connected. We also show that the function $g$ of the third inequality satisfies $g(1)\leq 4$. We then consider the complexity of deciding whether ${\rm inv}(D)\leq k$ for a given oriented graph $D$. We show that it is NP-complete for $k=1$, which together with the above conjecture would imply that it is NP-complete for every $k$. This contrasts with a result of Belkhechine et al. which states that deciding whether ${\rm inv}(T)\leq k$ for a given tournament $T$ is polynomial-time solvable.]]> Wed, 21 Dec 2022 09:14:35 +0000 https://doi.org/10.46298/dmtcs.7474 https://doi.org/10.46298/dmtcs.7474 Bang-Jensen, Jørgen da Silva, Jonas Costa Ferreira Havet, Frédéric Bang-Jensen, Jørgen da Silva, Jonas Costa Ferreira Havet, Frédéric <![CDATA[Let $D$ be an oriented graph. The inversion of a set $X$ of vertices in $D$ consists in reversing the direction of all arcs with both ends in $X$. The inversion number of $D$, denoted by ${\rm inv}(D)$, is the minimum number of inversions needed to make $D$ acyclic. Denoting by $\tau(D)$, $\tau' (D)$, and $\nu(D)$ the cycle transversal number, the cycle arc-transversal number and the cycle packing number of $D$ respectively, one shows that ${\rm inv}(D) \leq \tau' (D)$, ${\rm inv}(D) \leq 2\tau(D)$ and there exists a function $g$ such that ${\rm inv}(D)\leq g(\nu(D))$. We conjecture that for any two oriented graphs $L$ and $R$, ${\rm inv}(L\rightarrow R) ={\rm inv}(L) +{\rm inv}(R)$ where $L\rightarrow R$ is the dijoin of $L$ and $R$. This would imply that the first two inequalities are tight. We prove this conjecture when ${\rm inv}(L)\leq 1$ and ${\rm inv}(R)\leq 2$ and when ${\rm inv}(L) ={\rm inv}(R)=2$ and $L$ and $R$ are strongly connected. We also show that the function $g$ of the third inequality satisfies $g(1)\leq 4$. We then consider the complexity of deciding whether ${\rm inv}(D)\leq k$ for a given oriented graph $D$. We show that it is NP-complete for $k=1$, which together with the above conjecture would imply that it is NP-complete for every $k$. This contrasts with a result of Belkhechine et al. which states that deciding whether ${\rm inv}(T)\leq k$ for a given tournament $T$ is polynomial-time solvable.]]> 0 <![CDATA[We show that permutations avoiding both of the (classical) patterns 4321 and 3241 have the algebraic generating function conjectured by Vladimir Kruchinin.]]> Tue, 13 Dec 2022 08:50:32 +0000 https://doi.org/10.46298/dmtcs.5286 https://doi.org/10.46298/dmtcs.5286 Callan, David Callan, David <![CDATA[We show that permutations avoiding both of the (classical) patterns 4321 and 3241 have the algebraic generating function conjectured by Vladimir Kruchinin.]]> 0 <![CDATA[The average distance of a vertex $v$ of a connected graph $G$ is the arithmetic mean of the distances from $v$ to all other vertices of $G$. The proximity $\pi(G)$ and the remoteness $\rho(G)$ of $G$ are the minimum and the maximum of the average distances of the vertices of $G$, respectively. In this paper, we give upper bounds on the remoteness and proximity for graphs of given order, minimum degree and maximum degree. Our bounds are sharp apart from an additive constant.]]> Wed, 30 Nov 2022 07:46:27 +0000 https://doi.org/10.46298/dmtcs.9432 https://doi.org/10.46298/dmtcs.9432 Dankelmann, Peter Mafunda, Sonwabile Mallu, Sufiyan Dankelmann, Peter Mafunda, Sonwabile Mallu, Sufiyan <![CDATA[The average distance of a vertex $v$ of a connected graph $G$ is the arithmetic mean of the distances from $v$ to all other vertices of $G$. The proximity $\pi(G)$ and the remoteness $\rho(G)$ of $G$ are the minimum and the maximum of the average distances of the vertices of $G$, respectively. In this paper, we give upper bounds on the remoteness and proximity for graphs of given order, minimum degree and maximum degree. Our bounds are sharp apart from an additive constant.]]> 0 <![CDATA[Given an undirected graph $G=(V,E)$ with a nonnegative edge length function and an integer $p$, $0 < p < |V|$, the $p$-centdian problem is to find $p$ vertices (called the {\it centdian set}) of $V$ such that the {\it eccentricity} plus {\it median-distance} is minimized, in which the {\it eccentricity} is the maximum (length) distance of all vertices to their nearest {\it centdian set} and the {\it median-distance} is the total (length) distance of all vertices to their nearest {\it centdian set}. The {\it eccentricity} plus {\it median-distance} is called the {\it centdian-distance}. The purpose of the $p$-centdian problem is to find $p$ open facilities (servers) which satisfy the quality-of-service of the minimum total distance ({\it median-distance}) and the maximum distance ({\it eccentricity}) to their service customers, simultaneously. If we converse the two criteria, that is given the bound of the {\it centdian-distance} and the objective function is to minimize the cardinality of the {\it centdian set}, this problem is called the converse centdian problem. In this paper, we prove the $p$-centdian problem is NP-Complete. Then we design the first non-trivial brute force exact algorithms for the $p$-centdian problem and the converse centdian problem, respectively. Finally, we design two approximation algorithms for both problems.]]> Wed, 30 Nov 2022 07:45:19 +0000 https://doi.org/10.46298/dmtcs.6877 https://doi.org/10.46298/dmtcs.6877 Chen, Yen Hung Chen, Yen Hung <![CDATA[Given an undirected graph $G=(V,E)$ with a nonnegative edge length function and an integer $p$, $0 < p < |V|$, the $p$-centdian problem is to find $p$ vertices (called the {\it centdian set}) of $V$ such that the {\it eccentricity} plus {\it median-distance} is minimized, in which the {\it eccentricity} is the maximum (length) distance of all vertices to their nearest {\it centdian set} and the {\it median-distance} is the total (length) distance of all vertices to their nearest {\it centdian set}. The {\it eccentricity} plus {\it median-distance} is called the {\it centdian-distance}. The purpose of the $p$-centdian problem is to find $p$ open facilities (servers) which satisfy the quality-of-service of the minimum total distance ({\it median-distance}) and the maximum distance ({\it eccentricity}) to their service customers, simultaneously. If we converse the two criteria, that is given the bound of the {\it centdian-distance} and the objective function is to minimize the cardinality of the {\it centdian set}, this problem is called the converse centdian problem. In this paper, we prove the $p$-centdian problem is NP-Complete. Then we design the first non-trivial brute force exact algorithms for the $p$-centdian problem and the converse centdian problem, respectively. Finally, we design two approximation algorithms for both problems.]]> 0 <![CDATA[An equitable coloring of a graph $G=(V,E)$ is a (proper) vertex-coloring of $G$, such that the sizes of any two color classes differ by at most one. In this paper, we consider the equitable coloring problem in block graphs. Recall that the latter are graphs in which each 2-connected component is a complete graph. The problem remains hard in the class of block graphs. In this paper, we present some graph theoretic results relating various parameters. Then we use them in order to trace some algorithmic implications, mainly dealing with the fixed-parameter tractability of the problem.]]> Wed, 30 Nov 2022 07:43:59 +0000 https://doi.org/10.46298/dmtcs.6860 https://doi.org/10.46298/dmtcs.6860 Furmańczyk, Hanna Mkrtchyan, Vahan Furmańczyk, Hanna Mkrtchyan, Vahan <![CDATA[An equitable coloring of a graph $G=(V,E)$ is a (proper) vertex-coloring of $G$, such that the sizes of any two color classes differ by at most one. In this paper, we consider the equitable coloring problem in block graphs. Recall that the latter are graphs in which each 2-connected component is a complete graph. The problem remains hard in the class of block graphs. In this paper, we present some graph theoretic results relating various parameters. Then we use them in order to trace some algorithmic implications, mainly dealing with the fixed-parameter tractability of the problem.]]> 0 <![CDATA[We study the following problem. Given a multiset $M$ of non-negative integers, decide whether there exist and, in the positive case, compute two non-trivial multisets whose Minkowski sum is equal to $M$. The Minkowski sum of two multisets A and B is a multiset containing all possible sums of any element of A and any element of B. This problem was proved to be NP-complete when multisets are replaced by sets. This version of the problem is strictly related to the factorization of boolean polynomials that turns out to be NP-complete as well. When multisets are considered, the problem is equivalent to the factorization of polynomials with non-negative integer coefficients. The computational complexity of both these problems is still unknown. The main contribution of this paper is a heuristic technique for decomposing multisets of non-negative integers. Experimental results show that our heuristic decomposes multisets of hundreds of elements within seconds independently of the magnitude of numbers belonging to the multisets. Our heuristic can be used also for factoring polynomials in N[x]. We show that, when the degree of the polynomials gets larger, our technique is much faster than the state-of-the-art algorithms implemented in commercial software like Mathematica and MatLab.]]> Thu, 03 Nov 2022 08:56:08 +0000 https://doi.org/10.46298/dmtcs.9877 https://doi.org/10.46298/dmtcs.9877 Margara, Luciano Margara, Luciano <![CDATA[We study the following problem. Given a multiset $M$ of non-negative integers, decide whether there exist and, in the positive case, compute two non-trivial multisets whose Minkowski sum is equal to $M$. The Minkowski sum of two multisets A and B is a multiset containing all possible sums of any element of A and any element of B. This problem was proved to be NP-complete when multisets are replaced by sets. This version of the problem is strictly related to the factorization of boolean polynomials that turns out to be NP-complete as well. When multisets are considered, the problem is equivalent to the factorization of polynomials with non-negative integer coefficients. The computational complexity of both these problems is still unknown. The main contribution of this paper is a heuristic technique for decomposing multisets of non-negative integers. Experimental results show that our heuristic decomposes multisets of hundreds of elements within seconds independently of the magnitude of numbers belonging to the multisets. Our heuristic can be used also for factoring polynomials in N[x]. We show that, when the degree of the polynomials gets larger, our technique is much faster than the state-of-the-art algorithms implemented in commercial software like Mathematica and MatLab.]]> 0 <![CDATA[A mixed graph is a set of vertices together with an edge set and an arc set. An $(m,n)$-mixed graph $G$ is a mixed graph whose edges are each assigned one of $m$ colours, and whose arcs are each assigned one of $n$ colours. A \emph{switch} at a vertex $v$ of $G$ permutes the edge colours, the arc colours, and the arc directions of edges and arcs incident with $v$. The group of all allowed switches is $\Gamma$. Let $k \geq 1$ be a fixed integer and $\Gamma$ a fixed permutation group. We consider the problem that takes as input an $(m,n)$-mixed graph $G$ and asks if there a sequence of switches at vertices of $G$ with respect to $\Gamma$ so that the resulting $(m,n)$-mixed graph admits a homomorphism to an $(m,n)$-mixed graph on $k$ vertices. Our main result establishes this problem can be solved in polynomial time for $k \leq 2$, and is NP-hard for $k \geq 3$. This provides a step towards a general dichotomy theorem for the $\Gamma$-switchable homomorphism decision problem.]]> Thu, 03 Nov 2022 08:54:30 +0000 https://doi.org/10.46298/dmtcs.9242 https://doi.org/10.46298/dmtcs.9242 Brewster, Richard C Kidner, Arnott MacGillivray, Gary Brewster, Richard C Kidner, Arnott MacGillivray, Gary <![CDATA[A mixed graph is a set of vertices together with an edge set and an arc set. An $(m,n)$-mixed graph $G$ is a mixed graph whose edges are each assigned one of $m$ colours, and whose arcs are each assigned one of $n$ colours. A \emph{switch} at a vertex $v$ of $G$ permutes the edge colours, the arc colours, and the arc directions of edges and arcs incident with $v$. The group of all allowed switches is $\Gamma$. Let $k \geq 1$ be a fixed integer and $\Gamma$ a fixed permutation group. We consider the problem that takes as input an $(m,n)$-mixed graph $G$ and asks if there a sequence of switches at vertices of $G$ with respect to $\Gamma$ so that the resulting $(m,n)$-mixed graph admits a homomorphism to an $(m,n)$-mixed graph on $k$ vertices. Our main result establishes this problem can be solved in polynomial time for $k \leq 2$, and is NP-hard for $k \geq 3$. This provides a step towards a general dichotomy theorem for the $\Gamma$-switchable homomorphism decision problem.]]> 0 <![CDATA[We enumerate permutations in the two permutation classes $\text{Av}_n(312, 4321)$ and $\text{Av}_n(321, 4123)$ by the number of cycles each permutation admits. We also refine this enumeration with respect to several statistics.]]> Thu, 27 Oct 2022 08:08:29 +0000 https://doi.org/10.46298/dmtcs.6173 https://doi.org/10.46298/dmtcs.6173 Archer, Kassie Archer, Kassie <![CDATA[We enumerate permutations in the two permutation classes $\text{Av}_n(312, 4321)$ and $\text{Av}_n(321, 4123)$ by the number of cycles each permutation admits. We also refine this enumeration with respect to several statistics.]]> 0 <![CDATA[Given a partially order set (poset) $P$, and a pair of families of ideals $\mathcal{I}$ and filters $\mathcal{F}$ in $P$ such that each pair $(I,F)\in \mathcal{I}\times\mathcal{F}$ has a non-empty intersection, the dualization problem over $P$ is to check whether there is an ideal $X$ in $P$ which intersects every member of $\mathcal{F}$ and does not contain any member of $\mathcal{I}$. Equivalently, the problem is to check for a distributive lattice $L=L(P)$, given by the poset $P$ of its set of joint-irreducibles, and two given antichains $\mathcal{A},\mathcal{B}\subseteq L$ such that no $a\in\mathcal{A}$ is dominated by any $b\in\mathcal{B}$, whether $\mathcal{A}$ and $\mathcal{B}$ cover (by domination) the entire lattice. We show that the problem can be solved in quasi-polynomial time in the sizes of $P$, $\mathcal{A}$ and $\mathcal{B}$, thus answering an open question in Babin and Kuznetsov (2017). As an application, we show that minimal infrequent closed sets of attributes in a rational database, with respect to a given implication base of maximum premise size of one, can be enumerated in incremental quasi-polynomial time.]]> Thu, 27 Oct 2022 07:52:30 +0000 https://doi.org/10.46298/dmtcs.6742 https://doi.org/10.46298/dmtcs.6742 Elbassioni, Khaled Elbassioni, Khaled <![CDATA[Given a partially order set (poset) $P$, and a pair of families of ideals $\mathcal{I}$ and filters $\mathcal{F}$ in $P$ such that each pair $(I,F)\in \mathcal{I}\times\mathcal{F}$ has a non-empty intersection, the dualization problem over $P$ is to check whether there is an ideal $X$ in $P$ which intersects every member of $\mathcal{F}$ and does not contain any member of $\mathcal{I}$. Equivalently, the problem is to check for a distributive lattice $L=L(P)$, given by the poset $P$ of its set of joint-irreducibles, and two given antichains $\mathcal{A},\mathcal{B}\subseteq L$ such that no $a\in\mathcal{A}$ is dominated by any $b\in\mathcal{B}$, whether $\mathcal{A}$ and $\mathcal{B}$ cover (by domination) the entire lattice. We show that the problem can be solved in quasi-polynomial time in the sizes of $P$, $\mathcal{A}$ and $\mathcal{B}$, thus answering an open question in Babin and Kuznetsov (2017). As an application, we show that minimal infrequent closed sets of attributes in a rational database, with respect to a given implication base of maximum premise size of one, can be enumerated in incremental quasi-polynomial time.]]> 0 <![CDATA[Dujmovi\'c, Joret, Micek, Morin, Ueckerdt and Wood [J. ACM 2020] proved that for every graph $G$ with Euler genus $g$ there is a graph $H$ with treewidth at most 4 and a path $P$ such that $G\subseteq H \boxtimes P \boxtimes K_{\max\{2g,3\}}$. We improve this result by replacing "4" by "3" and with $H$ planar. We in fact prove a more general result in terms of so-called framed graphs. This implies that every $(g,d)$-map graph is contained in $ H \boxtimes P\boxtimes K_\ell$, for some planar graph $H$ with treewidth $3$, where $\ell=\max\{2g\lfloor \frac{d}{2} \rfloor,d+3\lfloor\frac{d}{2}\rfloor-3\}$. It also implies that every $(g,1)$-planar graph (that is, graphs that can be drawn in a surface of Euler genus $g$ with at most one crossing per edge) is contained in $H\boxtimes P\boxtimes K_{\max\{4g,7\}}$, for some planar graph $H$ with treewidth $3$.]]> Fri, 21 Oct 2022 07:59:07 +0000 https://doi.org/10.46298/dmtcs.8877 https://doi.org/10.46298/dmtcs.8877 Distel, Marc Hickingbotham, Robert Huynh, Tony Wood, David R. Distel, Marc Hickingbotham, Robert Huynh, Tony Wood, David R. <![CDATA[Dujmovi\'c, Joret, Micek, Morin, Ueckerdt and Wood [J. ACM 2020] proved that for every graph $G$ with Euler genus $g$ there is a graph $H$ with treewidth at most 4 and a path $P$ such that $G\subseteq H \boxtimes P \boxtimes K_{\max\{2g,3\}}$. We improve this result by replacing "4" by "3" and with $H$ planar. We in fact prove a more general result in terms of so-called framed graphs. This implies that every $(g,d)$-map graph is contained in $ H \boxtimes P\boxtimes K_\ell$, for some planar graph $H$ with treewidth $3$, where $\ell=\max\{2g\lfloor \frac{d}{2} \rfloor,d+3\lfloor\frac{d}{2}\rfloor-3\}$. It also implies that every $(g,1)$-planar graph (that is, graphs that can be drawn in a surface of Euler genus $g$ with at most one crossing per edge) is contained in $H\boxtimes P\boxtimes K_{\max\{4g,7\}}$, for some planar graph $H$ with treewidth $3$.]]> 0 <![CDATA[We present the Boolean dimension of a graph, we relate it with the notions of inner, geometric and symplectic dimensions, and with the rank and minrank of a graph. We obtain an exact formula for the Boolean dimension of a tree in terms of a certain star decomposition. We relate the Boolean dimension with the inversion index of a tournament.]]> Fri, 23 Sep 2022 09:22:20 +0000 https://doi.org/10.46298/dmtcs.7437 https://doi.org/10.46298/dmtcs.7437 Pouzet, Maurice Si Kaddour, Hamza Thatte, Bhalchandra, Pouzet, Maurice Si Kaddour, Hamza Thatte, Bhalchandra, <![CDATA[We present the Boolean dimension of a graph, we relate it with the notions of inner, geometric and symplectic dimensions, and with the rank and minrank of a graph. We obtain an exact formula for the Boolean dimension of a tree in terms of a certain star decomposition. We relate the Boolean dimension with the inversion index of a tournament.]]> 0 <![CDATA[The ternary relation B(x,y,z) of betweenness states that an element y is between the elements x and z, in some sense depending on the considered structure. In a partially ordered set (N,≤), B(x,y,z):⇔x Sat, 17 Sep 2022 14:51:24 +0000 https://doi.org/10.46298/dmtcs.7288 https://doi.org/10.46298/dmtcs.7288 Courcelle, Bruno Courcelle, Bruno <![CDATA[The ternary relation B(x,y,z) of betweenness states that an element y is between the elements x and z, in some sense depending on the considered structure. In a partially ordered set (N,≤), B(x,y,z):⇔x 0 <![CDATA[A set of vertices $S$ of a graph $G$ is {\em monophonically convex} if every induced path joining two vertices of $S$ is contained in $S$. The {\em monophonic convex hull of $S$}, $\langle S \rangle$, is the smallest monophonically convex set containing $S$. A set $S$ is {\em monophonic convexly independent} if $v \not\in \langle S - \{v\} \rangle$ for every $v \in S$. The {\em monophonic rank} of $G$ is the size of the largest monophonic convexly independent set of $G$. We present a characterization of the monophonic convexly independent sets. Using this result, we show how to determine the monophonic rank of graph classes like bipartite, cactus, triangle-free, and line graphs in polynomial time. Furthermore, we show that this parameter can computed in polynomial time for $1$-starlike graphs, i.e., for split graphs, and that its determination is $\NP$-complete for $k$-starlike graphs for any fixed $k \ge 2$, a subclass of chordal graphs. We also consider this problem on the graphs whose intersection graph of the maximal prime subgraphs is a tree.]]> Thu, 15 Sep 2022 14:27:11 +0000 https://doi.org/10.46298/dmtcs.6835 https://doi.org/10.46298/dmtcs.6835 Dourado, Mitre C. Ponciano, Vitor S. da Silva, Rômulo L. O. Dourado, Mitre C. Ponciano, Vitor S. da Silva, Rômulo L. O. <![CDATA[A set of vertices $S$ of a graph $G$ is {\em monophonically convex} if every induced path joining two vertices of $S$ is contained in $S$. The {\em monophonic convex hull of $S$}, $\langle S \rangle$, is the smallest monophonically convex set containing $S$. A set $S$ is {\em monophonic convexly independent} if $v \not\in \langle S - \{v\} \rangle$ for every $v \in S$. The {\em monophonic rank} of $G$ is the size of the largest monophonic convexly independent set of $G$. We present a characterization of the monophonic convexly independent sets. Using this result, we show how to determine the monophonic rank of graph classes like bipartite, cactus, triangle-free, and line graphs in polynomial time. Furthermore, we show that this parameter can computed in polynomial time for $1$-starlike graphs, i.e., for split graphs, and that its determination is $\NP$-complete for $k$-starlike graphs for any fixed $k \ge 2$, a subclass of chordal graphs. We also consider this problem on the graphs whose intersection graph of the maximal prime subgraphs is a tree.]]> 0 <![CDATA[Motivated by the work on the domination number of directed de Bruijn graphs and some of its generalizations, in this paper we introduce a natural generalization of de Bruijn graphs (directed and undirected), namely $t$-constrained de Bruijn graphs, where $t$ is a positive integer, and then study the domination number of these graphs. Within the definition of $t$-constrained de Bruijn graphs, de Bruijn and Kautz graphs correspond to 1-constrained and 2-constrained de Bruijn graphs, respectively. This generalization inherits many structural properties of de Bruijn graphs and may have similar applications in interconnection networks or bioinformatics. We establish upper and lower bounds for the domination number on $t$-constrained de Bruijn graphs both in the directed and in the undirected case. These bounds are often very close and in some cases we are able to find the exact value.]]> Mon, 22 Aug 2022 10:02:18 +0000 https://doi.org/10.46298/dmtcs.8879 https://doi.org/10.46298/dmtcs.8879 Calamoneri, Tiziana Monti, Angelo Sinaimeri, Blerina Calamoneri, Tiziana Monti, Angelo Sinaimeri, Blerina <![CDATA[Motivated by the work on the domination number of directed de Bruijn graphs and some of its generalizations, in this paper we introduce a natural generalization of de Bruijn graphs (directed and undirected), namely $t$-constrained de Bruijn graphs, where $t$ is a positive integer, and then study the domination number of these graphs. Within the definition of $t$-constrained de Bruijn graphs, de Bruijn and Kautz graphs correspond to 1-constrained and 2-constrained de Bruijn graphs, respectively. This generalization inherits many structural properties of de Bruijn graphs and may have similar applications in interconnection networks or bioinformatics. We establish upper and lower bounds for the domination number on $t$-constrained de Bruijn graphs both in the directed and in the undirected case. These bounds are often very close and in some cases we are able to find the exact value.]]> 0 <![CDATA[Recently, a conjecture due to Hendry was disproved which stated that every Hamiltonian chordal graph is cycle extendible. Here we further explore the conjecture, showing that it fails to hold even when a number of extra conditions are imposed. In particular, we show that Hendry's Conjecture fails for strongly chordal graphs, graphs with high connectivity, and if we relax the definition of "cycle extendible" considerably. We also consider the original conjecture from a subtree intersection model point of view, showing that a result of Abuieda et al is nearly best possible.]]> Mon, 22 Aug 2022 09:57:22 +0000 https://doi.org/10.46298/dmtcs.6700 https://doi.org/10.46298/dmtcs.6700 Lafond, Manuel Seamone, Ben Sherkati, Rezvan Lafond, Manuel Seamone, Ben Sherkati, Rezvan <![CDATA[Recently, a conjecture due to Hendry was disproved which stated that every Hamiltonian chordal graph is cycle extendible. Here we further explore the conjecture, showing that it fails to hold even when a number of extra conditions are imposed. In particular, we show that Hendry's Conjecture fails for strongly chordal graphs, graphs with high connectivity, and if we relax the definition of "cycle extendible" considerably. We also consider the original conjecture from a subtree intersection model point of view, showing that a result of Abuieda et al is nearly best possible.]]> 0 <![CDATA[Tuza famously conjectured in 1981 that in a graph without k+1 edge-disjoint triangles, it suffices to delete at most 2k edges to obtain a triangle-free graph. The conjecture holds for graphs with small treewidth or small maximum average degree, including planar graphs. However, for dense graphs that are neither cliques nor 4-colorable, only asymptotic results are known. Here, we confirm the conjecture for threshold graphs, i.e. graphs that are both split graphs and cographs, and for co-chain graphs with both sides of the same size divisible by 4.]]> Sun, 14 Aug 2022 19:17:06 +0000 https://doi.org/10.46298/dmtcs.7660 https://doi.org/10.46298/dmtcs.7660 Bonamy, Marthe Bożyk, Łukasz Grzesik, Andrzej Hatzel, Meike Masařík, Tomáš Novotná, Jana Okrasa, Karolina Bonamy, Marthe Bożyk, Łukasz Grzesik, Andrzej Hatzel, Meike Masařík, Tomáš Novotná, Jana Okrasa, Karolina <![CDATA[Tuza famously conjectured in 1981 that in a graph without k+1 edge-disjoint triangles, it suffices to delete at most 2k edges to obtain a triangle-free graph. The conjecture holds for graphs with small treewidth or small maximum average degree, including planar graphs. However, for dense graphs that are neither cliques nor 4-colorable, only asymptotic results are known. Here, we confirm the conjecture for threshold graphs, i.e. graphs that are both split graphs and cographs, and for co-chain graphs with both sides of the same size divisible by 4.]]> 0 <![CDATA[We define and study positional marked patterns, permutations $\tau$ where one of elements in $\tau$ is underlined. Given a permutation $\sigma$, we say that $\sigma$ has a $\tau$-match at position $i$ if $\tau$ occurs in $\sigma$ in such a way that $\sigma_i$ plays the role of the underlined element in the occurrence. We let $pmp_\tau(\sigma)$ denote the number of positions $i$ which $\sigma$ has a $\tau$-match. This defines a new class of statistics on permutations, where we study such statistics and prove a number of results. In particular, we prove that two positional marked patterns $1\underline{2}3$ and $1\underline{3}2$ give rise to two statistics that have the same distribution. The equidistibution phenomenon also occurs in other several collections of patterns like $\left \{1\underline{2}3 , 1\underline{3}2 \right \}$, and $\left \{ 1\underline234, 1\underline243, \underline2134, \underline2 1 4 3 \right \}$, as well as two positional marked patterns of any length $n$: $\left \{ 1\underline 2\tau , \underline 21\tau \right \}$.]]> Mon, 08 Aug 2022 16:03:27 +0000 https://doi.org/10.46298/dmtcs.7171 https://doi.org/10.46298/dmtcs.7171 Thamrongpairoj, Sittipong Remmel, Jeffrey B. Thamrongpairoj, Sittipong Remmel, Jeffrey B. <![CDATA[We define and study positional marked patterns, permutations $\tau$ where one of elements in $\tau$ is underlined. Given a permutation $\sigma$, we say that $\sigma$ has a $\tau$-match at position $i$ if $\tau$ occurs in $\sigma$ in such a way that $\sigma_i$ plays the role of the underlined element in the occurrence. We let $pmp_\tau(\sigma)$ denote the number of positions $i$ which $\sigma$ has a $\tau$-match. This defines a new class of statistics on permutations, where we study such statistics and prove a number of results. In particular, we prove that two positional marked patterns $1\underline{2}3$ and $1\underline{3}2$ give rise to two statistics that have the same distribution. The equidistibution phenomenon also occurs in other several collections of patterns like $\left \{1\underline{2}3 , 1\underline{3}2 \right \}$, and $\left \{ 1\underline234, 1\underline243, \underline2134, \underline2 1 4 3 \right \}$, as well as two positional marked patterns of any length $n$: $\left \{ 1\underline 2\tau , \underline 21\tau \right \}$.]]> 0 <![CDATA[The $n$th term of an automatic sequence is the output of a deterministic finite automaton fed with the representation of $n$ in a suitable numeration system. In this paper, instead of considering automatic sequences built on a numeration system with a regular numeration language, we consider those built on languages associated with trees having periodic labeled signatures and, in particular, rational base numeration systems. We obtain two main characterizations of these sequences. The first one is concerned with $r$-block substitutions where $r$ morphisms are applied periodically. In particular, we provide examples of such sequences that are not morphic. The second characterization involves the factors, or subtrees of finite height, of the tree associated with the numeration system and decorated by the terms of the sequence.]]> Tue, 19 Jul 2022 12:30:07 +0000 https://doi.org/10.46298/dmtcs.8455 https://doi.org/10.46298/dmtcs.8455 Rigo, Michel Stipulanti, Manon Rigo, Michel Stipulanti, Manon <![CDATA[The $n$th term of an automatic sequence is the output of a deterministic finite automaton fed with the representation of $n$ in a suitable numeration system. In this paper, instead of considering automatic sequences built on a numeration system with a regular numeration language, we consider those built on languages associated with trees having periodic labeled signatures and, in particular, rational base numeration systems. We obtain two main characterizations of these sequences. The first one is concerned with $r$-block substitutions where $r$ morphisms are applied periodically. In particular, we provide examples of such sequences that are not morphic. The second characterization involves the factors, or subtrees of finite height, of the tree associated with the numeration system and decorated by the terms of the sequence.]]> 0 <![CDATA[For a non-negative integer $s\le |V(G)|-3$, a graph $G$ is $s$-Hamiltonian if the removal of any $k\le s$ vertices results in a Hamiltonian graph. Given a connected simple graph $G$ that is not isomorphic to a path, a cycle, or a $K_{1,3}$, let $\delta(G)$ denote the minimum degree of $G$, let $h_s(G)$ denote the smallest integer $i$ such that the iterated line graph $L^{i}(G)$ is $s$-Hamiltonian, and let $\ell(G)$ denote the length of the longest non-closed path $P$ in which all internal vertices have degree 2 such that $P$ is not both of length 2 and in a $K_3$. For a simple graph $G$, we establish better upper bounds for $h_s(G)$ as follows. \begin{equation*} h_s(G)\le \left\{ \begin{aligned} & \ell(G)+1, &&\mbox{ if }\delta(G)\le 2 \mbox{ and }s=0;\\ & \widetilde d(G)+2+\lceil \lg (s+1)\rceil, &&\mbox{ if }\delta(G)\le 2 \mbox{ and }s\ge 1;\\ & 2+\left\lceil\lg\frac{s+1}{\delta(G)-2}\right\rceil, && \mbox{ if } 3\le\delta(G)\le s+2;\\ & 2, &&{\rm otherwise}, \end{aligned} \right. \end{equation*} where $\widetilde d(G)$ is the smallest integer $i$ such that $\delta(L^i(G))\ge 3$. Consequently, when $s \ge 6$, this new upper bound for the $s$-hamiltonian index implies that $h_s(G) = o(\ell(G)+s+1)$ as $s \to \infty$. This sharpens the result, $h_s(G)\le\ell(G)+s+1$, obtained by Zhang et al. in [Discrete Math., 308 (2008) 4779-4785].]]> Mon, 13 Jun 2022 13:34:01 +0000 https://doi.org/10.46298/dmtcs.8484 https://doi.org/10.46298/dmtcs.8484 Song, Sulin Lei, Lan Shao, Yehong Lai, Hong-Jian Song, Sulin Lei, Lan Shao, Yehong Lai, Hong-Jian <![CDATA[For a non-negative integer $s\le |V(G)|-3$, a graph $G$ is $s$-Hamiltonian if the removal of any $k\le s$ vertices results in a Hamiltonian graph. Given a connected simple graph $G$ that is not isomorphic to a path, a cycle, or a $K_{1,3}$, let $\delta(G)$ denote the minimum degree of $G$, let $h_s(G)$ denote the smallest integer $i$ such that the iterated line graph $L^{i}(G)$ is $s$-Hamiltonian, and let $\ell(G)$ denote the length of the longest non-closed path $P$ in which all internal vertices have degree 2 such that $P$ is not both of length 2 and in a $K_3$. For a simple graph $G$, we establish better upper bounds for $h_s(G)$ as follows. \begin{equation*} h_s(G)\le \left\{ \begin{aligned} & \ell(G)+1, &&\mbox{ if }\delta(G)\le 2 \mbox{ and }s=0;\\ & \widetilde d(G)+2+\lceil \lg (s+1)\rceil, &&\mbox{ if }\delta(G)\le 2 \mbox{ and }s\ge 1;\\ & 2+\left\lceil\lg\frac{s+1}{\delta(G)-2}\right\rceil, && \mbox{ if } 3\le\delta(G)\le s+2;\\ & 2, &&{\rm otherwise}, \end{aligned} \right. \end{equation*} where $\widetilde d(G)$ is the smallest integer $i$ such that $\delta(L^i(G))\ge 3$. Consequently, when $s \ge 6$, this new upper bound for the $s$-hamiltonian index implies that $h_s(G) = o(\ell(G)+s+1)$ as $s \to \infty$. This sharpens the result, $h_s(G)\le\ell(G)+s+1$, obtained by Zhang et al. in [Discrete Math., 308 (2008) 4779-4785].]]> 0 <![CDATA[We study Maker--Breaker total domination game played by two players, Dominator and Staller, on the connected cubic graphs. Staller (playing the role of Maker) wins if she manages to claim an open neighbourhood of a vertex. Dominator wins otherwise (i.e.\ if he can claim a total dominating set of a graph). For certain graphs on $n\geq 6$ vertices, we give the characterization on those which are Dominator's win and those which are Staller's win.]]> Thu, 02 Jun 2022 08:36:32 +0000 https://doi.org/10.46298/dmtcs.8529 https://doi.org/10.46298/dmtcs.8529 Forcan, Jovana Mikalački, Mirjana Forcan, Jovana Mikalački, Mirjana <![CDATA[We study Maker--Breaker total domination game played by two players, Dominator and Staller, on the connected cubic graphs. Staller (playing the role of Maker) wins if she manages to claim an open neighbourhood of a vertex. Dominator wins otherwise (i.e.\ if he can claim a total dominating set of a graph). For certain graphs on $n\geq 6$ vertices, we give the characterization on those which are Dominator's win and those which are Staller's win.]]> 0 <![CDATA[The number of down-steps between pairs of up-steps in $k_t$-Dyck paths, a generalization of Dyck paths consisting of steps $\{(1, k), (1, -1)\}$ such that the path stays (weakly) above the line $y=-t$, is studied. Results are proved bijectively and by means of generating functions, and lead to several interesting identities as well as links to other combinatorial structures. In particular, there is a connection between $k_t$-Dyck paths and perforation patterns for punctured convolutional codes (binary matrices) used in coding theory. Surprisingly, upon restriction to usual Dyck paths this yields a new combinatorial interpretation of Catalan numbers.]]> Tue, 24 May 2022 14:31:23 +0000 https://doi.org/10.46298/dmtcs.7163 https://doi.org/10.46298/dmtcs.7163 Asinowski, Andrei Hackl, Benjamin Selkirk, Sarah J. Asinowski, Andrei Hackl, Benjamin Selkirk, Sarah J. <![CDATA[The number of down-steps between pairs of up-steps in $k_t$-Dyck paths, a generalization of Dyck paths consisting of steps $\{(1, k), (1, -1)\}$ such that the path stays (weakly) above the line $y=-t$, is studied. Results are proved bijectively and by means of generating functions, and lead to several interesting identities as well as links to other combinatorial structures. In particular, there is a connection between $k_t$-Dyck paths and perforation patterns for punctured convolutional codes (binary matrices) used in coding theory. Surprisingly, upon restriction to usual Dyck paths this yields a new combinatorial interpretation of Catalan numbers.]]> 0 <![CDATA[We consider a non-monotone activation process $(X_t)_{t\in\{ 0,1,2,\ldots\}}$ on a graph $G$, where $X_0\subseteq V(G)$, $X_t=\{ u\in V(G):|N_G(u)\cap X_{t-1}|\geq \tau(u)\}$ for every positive integer $t$, and $\tau:V(G)\to \mathbb{Z}$ is a threshold function. The set $X_0$ is a so-called non-monotone target set for $(G,\tau)$ if there is some $t_0$ such that $X_t=V(G)$ for every $t\geq t_0$. Ben-Zwi, Hermelin, Lokshtanov, and Newman [Discrete Optimization 8 (2011) 87-96] asked whether a target set of minimum order can be determined efficiently if $G$ is a tree. We answer their question in the affirmative for threshold functions $\tau$ satisfying $\tau(u)\in \{ 0,1,d_G(u)\}$ for every vertex~$u$. For such restricted threshold functions, we give a characterization of target sets that allows to show that the minimum target set problem remains NP-hard for planar graphs of maximum degree $3$ but is efficiently solvable for graphs of bounded treewidth.]]> Fri, 13 May 2022 12:38:55 +0000 https://doi.org/10.46298/dmtcs.6844 https://doi.org/10.46298/dmtcs.6844 Baste, Julien Ehard, Stefan Rautenbach, Dieter Baste, Julien Ehard, Stefan Rautenbach, Dieter <![CDATA[We consider a non-monotone activation process $(X_t)_{t\in\{ 0,1,2,\ldots\}}$ on a graph $G$, where $X_0\subseteq V(G)$, $X_t=\{ u\in V(G):|N_G(u)\cap X_{t-1}|\geq \tau(u)\}$ for every positive integer $t$, and $\tau:V(G)\to \mathbb{Z}$ is a threshold function. The set $X_0$ is a so-called non-monotone target set for $(G,\tau)$ if there is some $t_0$ such that $X_t=V(G)$ for every $t\geq t_0$. Ben-Zwi, Hermelin, Lokshtanov, and Newman [Discrete Optimization 8 (2011) 87-96] asked whether a target set of minimum order can be determined efficiently if $G$ is a tree. We answer their question in the affirmative for threshold functions $\tau$ satisfying $\tau(u)\in \{ 0,1,d_G(u)\}$ for every vertex~$u$. For such restricted threshold functions, we give a characterization of target sets that allows to show that the minimum target set problem remains NP-hard for planar graphs of maximum degree $3$ but is efficiently solvable for graphs of bounded treewidth.]]> 0 <![CDATA[Layered treewidth and row treewidth are recently introduced graph parameters that have been key ingredients in the solution of several well-known open problems. It follows from the definitions that the layered treewidth of a graph is at most its row treewidth plus 1. Moreover, a minor-closed class has bounded layered treewidth if and only if it has bounded row treewidth. However, it has been open whether row treewidth is bounded by a function of layered treewidth. This paper answers this question in the negative. In particular, for every integer $k$ we describe a graph with layered treewidth 1 and row treewidth $k$. We also prove an analogous result for layered pathwidth and row pathwidth.]]> Fri, 13 May 2022 12:35:08 +0000 https://doi.org/10.46298/dmtcs.7458 https://doi.org/10.46298/dmtcs.7458 Bose, Prosenjit Dujmović, Vida Javarsineh, Mehrnoosh Morin, Pat Wood, David R. Bose, Prosenjit Dujmović, Vida Javarsineh, Mehrnoosh Morin, Pat Wood, David R. <![CDATA[Layered treewidth and row treewidth are recently introduced graph parameters that have been key ingredients in the solution of several well-known open problems. It follows from the definitions that the layered treewidth of a graph is at most its row treewidth plus 1. Moreover, a minor-closed class has bounded layered treewidth if and only if it has bounded row treewidth. However, it has been open whether row treewidth is bounded by a function of layered treewidth. This paper answers this question in the negative. In particular, for every integer $k$ we describe a graph with layered treewidth 1 and row treewidth $k$. We also prove an analogous result for layered pathwidth and row pathwidth.]]> 0 <![CDATA[We study the neighborhood polynomial and the complexity of its computation for chordal graphs. The neighborhood polynomial of a graph is the generating function of subsets of its vertices that have a common neighbor. We introduce a parameter for chordal graphs called anchor width and an algorithm to compute the neighborhood polynomial which runs in polynomial time if the anchor width is polynomially bounded. The anchor width is the maximal number of different sub-cliques of a clique which appear as a common neighborhood. Furthermore we study the anchor width for chordal graphs and some subclasses such as chordal comparability graphs and chordal graphs with bounded leafage. the leafage of a chordal graphs is the minimum number of leaves in the host tree of a subtree representation. We show that the anchor width of a chordal graph is at most $n^{\ell}$ where $\ell$ denotes the leafage. This shows that for some subclasses computing the neighborhood polynomial is possible in polynomial time while it is NP-hard for general chordal graphs.]]> Fri, 06 May 2022 13:09:43 +0000 https://doi.org/10.46298/dmtcs.8388 https://doi.org/10.46298/dmtcs.8388 Bergold, Helena Hochstättler, Winfried Mayer, Uwe Bergold, Helena Hochstättler, Winfried Mayer, Uwe <![CDATA[We study the neighborhood polynomial and the complexity of its computation for chordal graphs. The neighborhood polynomial of a graph is the generating function of subsets of its vertices that have a common neighbor. We introduce a parameter for chordal graphs called anchor width and an algorithm to compute the neighborhood polynomial which runs in polynomial time if the anchor width is polynomially bounded. The anchor width is the maximal number of different sub-cliques of a clique which appear as a common neighborhood. Furthermore we study the anchor width for chordal graphs and some subclasses such as chordal comparability graphs and chordal graphs with bounded leafage. the leafage of a chordal graphs is the minimum number of leaves in the host tree of a subtree representation. We show that the anchor width of a chordal graph is at most $n^{\ell}$ where $\ell$ denotes the leafage. This shows that for some subclasses computing the neighborhood polynomial is possible in polynomial time while it is NP-hard for general chordal graphs.]]> 0 <![CDATA[Given a graph and an integer $k$, it is an NP-complete problem to decide whether there is a dominating set of size at most $k$. In this paper we study this problem for the Kn\"odel Graph on $n$ vertices using elementary number theory techniques. In particular, we show an explicit upper bound for the domination number of the Kn\"odel Graph on $n$ vertices any time that we can find a prime number $p$ dividing $n$ for which $2$ is a primitive root.]]> Fri, 06 May 2022 13:06:07 +0000 https://doi.org/10.46298/dmtcs.7158 https://doi.org/10.46298/dmtcs.7158 Racicot, Jesse Rosso, Giovanni Racicot, Jesse Rosso, Giovanni <![CDATA[Given a graph and an integer $k$, it is an NP-complete problem to decide whether there is a dominating set of size at most $k$. In this paper we study this problem for the Kn\"odel Graph on $n$ vertices using elementary number theory techniques. In particular, we show an explicit upper bound for the domination number of the Kn\"odel Graph on $n$ vertices any time that we can find a prime number $p$ dividing $n$ for which $2$ is a primitive root.]]> 0 <![CDATA[Let $P$ be a set of $n\geq 3$ points in general position in the plane. The edge disjointness graph $D(P)$ of $P$ is the graph whose vertices are all the closed straight line segments with endpoints in $P$, two of which are adjacent in $D(P)$ if and only if they are disjoint. We show that the connectivity of $D(P)$ is at least $\binom{\lfloor\frac{n-2}{2}\rfloor}{2}+\binom{\lceil\frac{n-2}{2}\rceil}{2}$, and that this bound is tight for each $n\geq 3$.]]> Fri, 06 May 2022 13:01:53 +0000 https://doi.org/10.46298/dmtcs.6678 https://doi.org/10.46298/dmtcs.6678 Leaños, J. Ndjatchi, Christophe Ríos-Castro, L. M. Leaños, J. Ndjatchi, Christophe Ríos-Castro, L. M. <![CDATA[Let $P$ be a set of $n\geq 3$ points in general position in the plane. The edge disjointness graph $D(P)$ of $P$ is the graph whose vertices are all the closed straight line segments with endpoints in $P$, two of which are adjacent in $D(P)$ if and only if they are disjoint. We show that the connectivity of $D(P)$ is at least $\binom{\lfloor\frac{n-2}{2}\rfloor}{2}+\binom{\lceil\frac{n-2}{2}\rceil}{2}$, and that this bound is tight for each $n\geq 3$.]]> 0 <![CDATA[The finite models of a universal sentence $\Phi$ in a finite relational signature are the age of a structure if and only if $\Phi$ has the joint embedding property. We prove that the computational problem whether a given universal sentence $\Phi$ has the joint embedding property is undecidable, even if $\Phi$ is additionally Horn and the signature of $\Phi$ only contains relation symbols of arity at most two.]]> Fri, 06 May 2022 12:58:29 +0000 https://doi.org/10.46298/dmtcs.7435 https://doi.org/10.46298/dmtcs.7435 Bodirsky, Manuel Rydval, Jakub Schrottenloher, André Bodirsky, Manuel Rydval, Jakub Schrottenloher, André <![CDATA[The finite models of a universal sentence $\Phi$ in a finite relational signature are the age of a structure if and only if $\Phi$ has the joint embedding property. We prove that the computational problem whether a given universal sentence $\Phi$ has the joint embedding property is undecidable, even if $\Phi$ is additionally Horn and the signature of $\Phi$ only contains relation symbols of arity at most two.]]> 0 <![CDATA[We consider the Erd\H{o}s-P\'osa property for immersions and topological minors in tournaments. We prove that for every simple digraph $H$, $k\in \mathbb{N}$, and tournament $T$, the following statements hold: (i) If in $T$ one cannot find $k$ arc-disjoint immersion copies of $H$, then there exists a set of $\mathcal{O}_H(k^3)$ arcs that intersects all immersion copies of $H$ in $T$. (ii) If in $T$ one cannot find $k$ vertex-disjoint topological minor copies of $H$, then there exists a set of $\mathcal{O}_H(k\log k)$ vertices that intersects all topological minor copies of $H$ in $T$. This improves the results of Raymond [DMTCS '18], who proved similar statements under the assumption that $H$ is strongly connected.]]> Tue, 05 Apr 2022 16:05:27 +0000 https://doi.org/10.46298/dmtcs.7099 https://doi.org/10.46298/dmtcs.7099 Bożyk, Łukasz Pilipczuk, Michał Bożyk, Łukasz Pilipczuk, Michał <![CDATA[We consider the Erd\H{o}s-P\'osa property for immersions and topological minors in tournaments. We prove that for every simple digraph $H$, $k\in \mathbb{N}$, and tournament $T$, the following statements hold: (i) If in $T$ one cannot find $k$ arc-disjoint immersion copies of $H$, then there exists a set of $\mathcal{O}_H(k^3)$ arcs that intersects all immersion copies of $H$ in $T$. (ii) If in $T$ one cannot find $k$ vertex-disjoint topological minor copies of $H$, then there exists a set of $\mathcal{O}_H(k\log k)$ vertices that intersects all topological minor copies of $H$ in $T$. This improves the results of Raymond [DMTCS '18], who proved similar statements under the assumption that $H$ is strongly connected.]]> 0 <![CDATA[An equitable partition into branchings in a digraph is a partition of the arc set into branchings such that the sizes of any two branchings differ at most by one. For a digraph whose arc set can be partitioned into $k$ branchings, there always exists an equitable partition into $k$ branchings. In this paper, we present two extensions of equitable partitions into branchings in digraphs: those into matching forests in mixed graphs; and into $b$-branchings in digraphs. For matching forests, Kir\'{a}ly and Yokoi (2022) considered a tricriteria equitability based on the sizes of the matching forest, and the matching and branching therein. In contrast to this, we introduce a single-criterion equitability based on the number of covered vertices, which is plausible in the light of the delta-matroid structure of matching forests. While the existence of this equitable partition can be derived from a lemma in Kir\'{a}ly and Yokoi, we present its direct and simpler proof. For $b$-branchings, we define an equitability notion based on the size of the $b$-branching and the indegrees of all vertices, and prove that an equitable partition always exists. We then derive the integer decomposition property of the associated polytopes.]]> Thu, 31 Mar 2022 14:27:13 +0000 https://doi.org/10.46298/dmtcs.8719 https://doi.org/10.46298/dmtcs.8719 Takazawa, Kenjiro Takazawa, Kenjiro <![CDATA[An equitable partition into branchings in a digraph is a partition of the arc set into branchings such that the sizes of any two branchings differ at most by one. For a digraph whose arc set can be partitioned into $k$ branchings, there always exists an equitable partition into $k$ branchings. In this paper, we present two extensions of equitable partitions into branchings in digraphs: those into matching forests in mixed graphs; and into $b$-branchings in digraphs. For matching forests, Kir\'{a}ly and Yokoi (2022) considered a tricriteria equitability based on the sizes of the matching forest, and the matching and branching therein. In contrast to this, we introduce a single-criterion equitability based on the number of covered vertices, which is plausible in the light of the delta-matroid structure of matching forests. While the existence of this equitable partition can be derived from a lemma in Kir\'{a}ly and Yokoi, we present its direct and simpler proof. For $b$-branchings, we define an equitability notion based on the size of the $b$-branching and the indegrees of all vertices, and prove that an equitable partition always exists. We then derive the integer decomposition property of the associated polytopes.]]> 0 <![CDATA[A bramble in an undirected graph $G$ is a family of connected subgraphs of $G$ such that for every two subgraphs $H_1$ and $H_2$ in the bramble either $V(H_1) \cap V(H_2) \neq \emptyset$ or there is an edge of $G$ with one endpoint in $V(H_1)$ and the second endpoint in $V(H_2)$. The order of the bramble is the minimum size of a vertex set that intersects all elements of a bramble. Brambles are objects dual to treewidth: As shown by Seymour and Thomas, the maximum order of a bramble in an undirected graph $G$ equals one plus the treewidth of $G$. However, as shown by Grohe and Marx, brambles of high order may necessarily be of exponential size: In a constant-degree $n$-vertex expander a bramble of order $\Omega(n^{1/2+\delta})$ requires size exponential in $\Omega(n^{2\delta})$ for any fixed $\delta \in (0,\frac{1}{2}]$. On the other hand, the combination of results of Grohe and Marx and Chekuri and Chuzhoy shows that a graph of treewidth $k$ admits a bramble of order $\widetilde{\Omega}(k^{1/2})$ and size $\widetilde{\mathcal{O}}(k^{3/2})$. ($\widetilde{\Omega}$ and $\widetilde{\mathcal{O}}$ hide polylogarithmic factors and divisors, respectively.) In this note, we first sharpen the second bound by proving that every graph $G$ of treewidth at least $k$ contains a bramble of order $\widetilde{\Omega}(k^{1/2})$ and congestion $2$, i.e., every vertex of $G$ is contained in at most two elements of the bramble (thus the bramble is of size linear in its order). Second, we provide a tight upper bound for the lower bound of Grohe and Marx: For every $\delta \in (0,\frac{1}{2}]$, every graph $G$ of treewidth at least $k$ contains a bramble of order $\widetilde{\Omega}(k^{1/2+\delta})$ and size $2^{\widetilde{\mathcal{O}}(k^{2\delta})}$.]]> Thu, 31 Mar 2022 14:15:06 +0000 https://doi.org/10.46298/dmtcs.6699 https://doi.org/10.46298/dmtcs.6699 Hatzel, Meike Komosa, Pawel Pilipczuk, Marcin Sorge, Manuel Hatzel, Meike Komosa, Pawel Pilipczuk, Marcin Sorge, Manuel <![CDATA[A bramble in an undirected graph $G$ is a family of connected subgraphs of $G$ such that for every two subgraphs $H_1$ and $H_2$ in the bramble either $V(H_1) \cap V(H_2) \neq \emptyset$ or there is an edge of $G$ with one endpoint in $V(H_1)$ and the second endpoint in $V(H_2)$. The order of the bramble is the minimum size of a vertex set that intersects all elements of a bramble. Brambles are objects dual to treewidth: As shown by Seymour and Thomas, the maximum order of a bramble in an undirected graph $G$ equals one plus the treewidth of $G$. However, as shown by Grohe and Marx, brambles of high order may necessarily be of exponential size: In a constant-degree $n$-vertex expander a bramble of order $\Omega(n^{1/2+\delta})$ requires size exponential in $\Omega(n^{2\delta})$ for any fixed $\delta \in (0,\frac{1}{2}]$. On the other hand, the combination of results of Grohe and Marx and Chekuri and Chuzhoy shows that a graph of treewidth $k$ admits a bramble of order $\widetilde{\Omega}(k^{1/2})$ and size $\widetilde{\mathcal{O}}(k^{3/2})$. ($\widetilde{\Omega}$ and $\widetilde{\mathcal{O}}$ hide polylogarithmic factors and divisors, respectively.) In this note, we first sharpen the second bound by proving that every graph $G$ of treewidth at least $k$ contains a bramble of order $\widetilde{\Omega}(k^{1/2})$ and congestion $2$, i.e., every vertex of $G$ is contained in at most two elements of the bramble (thus the bramble is of size linear in its order). Second, we provide a tight upper bound for the lower bound of Grohe and Marx: For every $\delta \in (0,\frac{1}{2}]$, every graph $G$ of treewidth at least $k$ contains a bramble of order $\widetilde{\Omega}(k^{1/2+\delta})$ and size $2^{\widetilde{\mathcal{O}}(k^{2\delta})}$.]]> 0 <![CDATA[The determining number of a graph $G = (V,E)$ is the minimum cardinality of a set $S\subseteq V$ such that pointwise stabilizer of $S$ under the action of $Aut(G)$ is trivial. In this paper, we provide some improved upper and lower bounds on the determining number of Kneser graphs. Moreover, we provide the exact value of the determining number for some subfamilies of Kneser graphs.]]> Wed, 30 Mar 2022 09:24:23 +0000 https://doi.org/10.46298/dmtcs.7627 https://doi.org/10.46298/dmtcs.7627 Das, Angsuman Dey, Hiranya Kishore Das, Angsuman Dey, Hiranya Kishore <![CDATA[The determining number of a graph $G = (V,E)$ is the minimum cardinality of a set $S\subseteq V$ such that pointwise stabilizer of $S$ under the action of $Aut(G)$ is trivial. In this paper, we provide some improved upper and lower bounds on the determining number of Kneser graphs. Moreover, we provide the exact value of the determining number for some subfamilies of Kneser graphs.]]> 0 <![CDATA[Let $k$ and $n$ be integers such that $1\leq k \leq n-1$, and let $G$ be a simple graph of order $n$. The $k$-token graph $F_k(G)$ of $G$ is the graph whose vertices are the $k$-subsets of $V(G)$, where two vertices are adjacent in $F_k(G)$ whenever their symmetric difference is an edge of $G$. In this paper we show that if $G$ is a tree, then the connectivity of $F_k(G)$ is equal to the minimum degree of $F_k(G)$.]]> Wed, 30 Mar 2022 09:20:29 +0000 https://doi.org/10.46298/dmtcs.7538 https://doi.org/10.46298/dmtcs.7538 Fabila-Monroy, Ruy Leaños, Jesús Trujillo-Negrete, Ana Laura Fabila-Monroy, Ruy Leaños, Jesús Trujillo-Negrete, Ana Laura <![CDATA[Let $k$ and $n$ be integers such that $1\leq k \leq n-1$, and let $G$ be a simple graph of order $n$. The $k$-token graph $F_k(G)$ of $G$ is the graph whose vertices are the $k$-subsets of $V(G)$, where two vertices are adjacent in $F_k(G)$ whenever their symmetric difference is an edge of $G$. In this paper we show that if $G$ is a tree, then the connectivity of $F_k(G)$ is equal to the minimum degree of $F_k(G)$.]]> 0 <![CDATA[This note defines a notion of multiplicity for nodes in a rooted tree and presents an asymptotic calculation of the maximum multiplicity over all leaves in a Bienaym\'e-Galton-Watson tree with critical offspring distribution $\xi$, conditioned on the tree being of size $n$. In particular, we show that if $S_n$ is the maximum multiplicity in a conditional Bienaym\'e-Galton-Watson tree, then $S_n = \Omega(\log n)$ asymptotically in probability and under the further assumption that ${\bf E}\{2^\xi\} < \infty$, we have $S_n = O(\log n)$ asymptotically in probability as well. Explicit formulas are given for the constants in both bounds. We conclude by discussing links with an alternate definition of multiplicity that arises in the root-estimation problem.]]> Wed, 30 Mar 2022 09:17:27 +0000 https://doi.org/10.46298/dmtcs.7515 https://doi.org/10.46298/dmtcs.7515 Brandenberger, Anna M. Devroye, Luc Goh, Marcel K. Zhao, Rosie Y. Brandenberger, Anna M. Devroye, Luc Goh, Marcel K. Zhao, Rosie Y. <![CDATA[This note defines a notion of multiplicity for nodes in a rooted tree and presents an asymptotic calculation of the maximum multiplicity over all leaves in a Bienaym\'e-Galton-Watson tree with critical offspring distribution $\xi$, conditioned on the tree being of size $n$. In particular, we show that if $S_n$ is the maximum multiplicity in a conditional Bienaym\'e-Galton-Watson tree, then $S_n = \Omega(\log n)$ asymptotically in probability and under the further assumption that ${\bf E}\{2^\xi\} < \infty$, we have $S_n = O(\log n)$ asymptotically in probability as well. Explicit formulas are given for the constants in both bounds. We conclude by discussing links with an alternate definition of multiplicity that arises in the root-estimation problem.]]> 0 <![CDATA[Motivated by the study of pattern avoidance in the context of permutations and ordered partitions, we consider the enumeration of weak-ordering chains obtained as leaves of certain restricted rooted trees. A tree of order $n$ is generated by inserting a new variable into each node at every step. A node becomes a leaf either after $n$ steps or when a certain stopping condition is met. In this paper we focus on conditions of size 2 ($x=y$, $x Mon, 21 Mar 2022 14:59:04 +0000 https://doi.org/10.46298/dmtcs.8350 https://doi.org/10.46298/dmtcs.8350 Birmajer, Daniel Gil, Juan B. Kenepp, David S. Weiner, Michael D. Birmajer, Daniel Gil, Juan B. Kenepp, David S. Weiner, Michael D. <![CDATA[Motivated by the study of pattern avoidance in the context of permutations and ordered partitions, we consider the enumeration of weak-ordering chains obtained as leaves of certain restricted rooted trees. A tree of order $n$ is generated by inserting a new variable into each node at every step. A node becomes a leaf either after $n$ steps or when a certain stopping condition is met. In this paper we focus on conditions of size 2 ($x=y$, $x 0 <![CDATA[We assign a relational structure to any finite algebra in a canonical way, using solution sets of equations, and we prove that this relational structure is polymorphism-homogeneous if and only if the algebra itself is polymorphism-homogeneous. We show that polymorphism-homogeneity is also equivalent to the property that algebraic sets (i.e., solution sets of systems of equations) are exactly those sets of tuples that are closed under the centralizer clone of the algebra. Furthermore, we prove that the aforementioned properties hold if and only if the algebra is injective in the category of its finite subpowers. We also consider two additional conditions: a stronger variant for polymorphism-homogeneity and for injectivity, and we describe explicitly the finite semilattices, lattices, Abelian groups and monounary algebras satisfying any one of these three conditions.]]> Mon, 21 Mar 2022 14:54:14 +0000 https://doi.org/10.46298/dmtcs.6904 https://doi.org/10.46298/dmtcs.6904 Tóth, Endre Waldhauser, Tamás Tóth, Endre Waldhauser, Tamás <![CDATA[We assign a relational structure to any finite algebra in a canonical way, using solution sets of equations, and we prove that this relational structure is polymorphism-homogeneous if and only if the algebra itself is polymorphism-homogeneous. We show that polymorphism-homogeneity is also equivalent to the property that algebraic sets (i.e., solution sets of systems of equations) are exactly those sets of tuples that are closed under the centralizer clone of the algebra. Furthermore, we prove that the aforementioned properties hold if and only if the algebra is injective in the category of its finite subpowers. We also consider two additional conditions: a stronger variant for polymorphism-homogeneity and for injectivity, and we describe explicitly the finite semilattices, lattices, Abelian groups and monounary algebras satisfying any one of these three conditions.]]> 0 <![CDATA[Initiated by Davis, Nelson, Petersen and Tenner (2018), the enumerative study of pinnacle sets of permutations has attracted a fair amount of attention recently. In this article, we provide a recurrence that can be used to compute efficiently the number $|\mathfrak{S}_n(P)|$ of permutations of size $n$ with a given pinnacle set $P$, with arithmetic complexity $O(k^4 + k\log n)$ for $P$ of size $k$. A symbolic expression can also be computed in this way for pinnacle sets of fixed size. A weighted sum $q_n(P)$ of $|\mathfrak{S}_n(P)|$ proposed in Davis, Nelson, Petersen and Tenner (2018) seems to have a simple form, and a conjectural form is given recently by Flaque, Novelli and Thibon (2021+). We settle the problem by providing and proving an alternative form of $q_n(P)$, which has a strong combinatorial flavor. We also study admissible orderings of a given pinnacle set, first considered by Rusu (2020) and characterized by Rusu and Tenner (2021), and we give an efficient algorithm for their counting.]]> Fri, 11 Mar 2022 13:00:56 +0000 https://doi.org/10.46298/dmtcs.8321 https://doi.org/10.46298/dmtcs.8321 Fang, Wenjie Fang, Wenjie <![CDATA[Initiated by Davis, Nelson, Petersen and Tenner (2018), the enumerative study of pinnacle sets of permutations has attracted a fair amount of attention recently. In this article, we provide a recurrence that can be used to compute efficiently the number $|\mathfrak{S}_n(P)|$ of permutations of size $n$ with a given pinnacle set $P$, with arithmetic complexity $O(k^4 + k\log n)$ for $P$ of size $k$. A symbolic expression can also be computed in this way for pinnacle sets of fixed size. A weighted sum $q_n(P)$ of $|\mathfrak{S}_n(P)|$ proposed in Davis, Nelson, Petersen and Tenner (2018) seems to have a simple form, and a conjectural form is given recently by Flaque, Novelli and Thibon (2021+). We settle the problem by providing and proving an alternative form of $q_n(P)$, which has a strong combinatorial flavor. We also study admissible orderings of a given pinnacle set, first considered by Rusu (2020) and characterized by Rusu and Tenner (2021), and we give an efficient algorithm for their counting.]]> 0 <![CDATA[The age $\mathcal{A}(G)$ of a graph $G$ (undirected and without loops) is the collection of finite induced subgraphs of $G$, considered up to isomorphy and ordered by embeddability. It is well-quasi-ordered (wqo) for this order if it contains no infinite antichain. A graph is \emph{path-minimal} if it contains finite induced paths of unbounded length and every induced subgraph $G'$ with this property embeds $G$. We construct $2^{\aleph_0}$ path-minimal graphs whose ages are pairwise incomparable with set inclusion and which are wqo. Our construction is based on uniformly recurrent sequences and lexicographical sums of labelled graphs.]]> Tue, 08 Mar 2022 16:07:01 +0000 https://doi.org/10.46298/dmtcs.6915 https://doi.org/10.46298/dmtcs.6915 Pouzet, Maurice Zaguia, Imed Pouzet, Maurice Zaguia, Imed <![CDATA[The age $\mathcal{A}(G)$ of a graph $G$ (undirected and without loops) is the collection of finite induced subgraphs of $G$, considered up to isomorphy and ordered by embeddability. It is well-quasi-ordered (wqo) for this order if it contains no infinite antichain. A graph is \emph{path-minimal} if it contains finite induced paths of unbounded length and every induced subgraph $G'$ with this property embeds $G$. We construct $2^{\aleph_0}$ path-minimal graphs whose ages are pairwise incomparable with set inclusion and which are wqo. Our construction is based on uniformly recurrent sequences and lexicographical sums of labelled graphs.]]> 0 <![CDATA[Let $G=(V(G),E(G))$ be a finite simple undirected graph with vertex set $V(G)$, edge set $E(G)$ and vertex subset $S\subseteq V(G)$. $S$ is termed \emph{open-dominating} if every vertex of $G$ has at least one neighbor in $S$, and \emph{open-independent, open-locating-dominating} (an $OLD_{oind}$-set for short) if no two vertices in $G$ have the same set of neighbors in $S$, and each vertex in $S$ is open-dominated exactly once by $S$. The problem of deciding whether or not $G$ has an $OLD_{oind}$-set has important applications that have been reported elsewhere. As the problem is known to be $\mathcal{NP}$-complete, it appears to be notoriously difficult as we show that its complexity remains the same even for just planar bipartite graphs of maximum degree five and girth six, and also for planar subcubic graphs of girth nine. Also, we present characterizations of both $P_4$-tidy graphs and the complementary prisms of cographs that have an $OLD_{oind}$-set.]]> Mon, 07 Feb 2022 10:59:48 +0000 https://doi.org/10.46298/dmtcs.8440 https://doi.org/10.46298/dmtcs.8440 Cappelle, Márcia R. Coelho, Erika Foulds, Les R. Longo, Humberto J. Cappelle, Márcia R. Coelho, Erika Foulds, Les R. Longo, Humberto J. <![CDATA[Let $G=(V(G),E(G))$ be a finite simple undirected graph with vertex set $V(G)$, edge set $E(G)$ and vertex subset $S\subseteq V(G)$. $S$ is termed \emph{open-dominating} if every vertex of $G$ has at least one neighbor in $S$, and \emph{open-independent, open-locating-dominating} (an $OLD_{oind}$-set for short) if no two vertices in $G$ have the same set of neighbors in $S$, and each vertex in $S$ is open-dominated exactly once by $S$. The problem of deciding whether or not $G$ has an $OLD_{oind}$-set has important applications that have been reported elsewhere. As the problem is known to be $\mathcal{NP}$-complete, it appears to be notoriously difficult as we show that its complexity remains the same even for just planar bipartite graphs of maximum degree five and girth six, and also for planar subcubic graphs of girth nine. Also, we present characterizations of both $P_4$-tidy graphs and the complementary prisms of cographs that have an $OLD_{oind}$-set.]]> 0 <![CDATA[Let asc and desc denote respectively the statistics recording the number of ascents or descents in a sequence having non-negative integer entries. In a recent paper by Andrews and Chern, it was shown that the distribution of asc on the inversion sequence avoidance class $I_n(\geq,\neq,>)$ is the same as that of $n-1-\text{asc}$ on the class $I_n(>,\neq,\geq)$, which confirmed an earlier conjecture of Lin. In this paper, we consider some further enumerative aspects related to this equivalence and, as a consequence, provide an alternative proof of the conjecture. In particular, we find recurrence relations for the joint distribution on $I_n(\geq,\neq,>)$ of asc and desc along with two other parameters, and do the same for $n-1-\text{asc}$ and desc on $I_n(>,\neq,\geq)$. By employing a functional equation approach together with the kernel method, we are able to compute explicitly the generating function for both of the aforementioned joint distributions, which extends (and provides a new proof of) the recent result $|I_n(\geq,\neq,>)|=|I_n(>,\neq,\geq)|$. In both cases, an algorithm is formulated for computing the generating function of the asc distribution on members of each respective class having a fixed number of descents.]]> Mon, 07 Feb 2022 10:56:51 +0000 https://doi.org/10.46298/dmtcs.8330 https://doi.org/10.46298/dmtcs.8330 Mansour, Toufik Shattuck, Mark Mansour, Toufik Shattuck, Mark <![CDATA[Let asc and desc denote respectively the statistics recording the number of ascents or descents in a sequence having non-negative integer entries. In a recent paper by Andrews and Chern, it was shown that the distribution of asc on the inversion sequence avoidance class $I_n(\geq,\neq,>)$ is the same as that of $n-1-\text{asc}$ on the class $I_n(>,\neq,\geq)$, which confirmed an earlier conjecture of Lin. In this paper, we consider some further enumerative aspects related to this equivalence and, as a consequence, provide an alternative proof of the conjecture. In particular, we find recurrence relations for the joint distribution on $I_n(\geq,\neq,>)$ of asc and desc along with two other parameters, and do the same for $n-1-\text{asc}$ and desc on $I_n(>,\neq,\geq)$. By employing a functional equation approach together with the kernel method, we are able to compute explicitly the generating function for both of the aforementioned joint distributions, which extends (and provides a new proof of) the recent result $|I_n(\geq,\neq,>)|=|I_n(>,\neq,\geq)|$. In both cases, an algorithm is formulated for computing the generating function of the asc distribution on members of each respective class having a fixed number of descents.]]> 0 <![CDATA[Hamiltonian cycles in graphs were first studied in the 1850s. Since then, an impressive amount of research has been dedicated to identifying classes of graphs that allow Hamiltonian cycles, and to related questions. The corresponding decision problem, that asks whether a given graph is Hamiltonian (i.\,e.\ admits a Hamiltonian cycle), is one of Karp's famous NP-complete problems. In this paper we study graphs of bounded degree that are \emph{far} from being Hamiltonian, where a graph $G$ on $n$ vertices is \emph{far} from being Hamiltonian, if modifying a constant fraction of $n$ edges is necessary to make $G$ Hamiltonian. We give an explicit deterministic construction of a class of graphs of bounded degree that are locally Hamiltonian, but (globally) far from being Hamiltonian. Here, \emph{locally Hamiltonian} means that every subgraph induced by the neighbourhood of a small vertex set appears in some Hamiltonian graph. More precisely, we obtain graphs which differ in $\Theta(n)$ edges from any Hamiltonian graph, but non-Hamiltonicity cannot be detected in the neighbourhood of $o(n)$ vertices. Our class of graphs yields a class of hard instances for one-sided error property testers with linear query complexity. It is known that any property tester (even with two-sided error) requires a linear number of queries to test Hamiltonicity (Yoshida, Ito, 2010). This is proved via a randomised construction of hard instances. In contrast, our construction is deterministic. So far only very few deterministic constructions of hard instances for property testing are known. We believe that our construction may lead to future insights in graph theory and towards a characterisation of the properties that are testable in the bounded-degree model.]]> Mon, 31 Jan 2022 08:47:41 +0000 https://doi.org/10.46298/dmtcs.7109 https://doi.org/10.46298/dmtcs.7109 Adler, Isolde Köhler, Noleen Adler, Isolde Köhler, Noleen <![CDATA[Hamiltonian cycles in graphs were first studied in the 1850s. Since then, an impressive amount of research has been dedicated to identifying classes of graphs that allow Hamiltonian cycles, and to related questions. The corresponding decision problem, that asks whether a given graph is Hamiltonian (i.\,e.\ admits a Hamiltonian cycle), is one of Karp's famous NP-complete problems. In this paper we study graphs of bounded degree that are \emph{far} from being Hamiltonian, where a graph $G$ on $n$ vertices is \emph{far} from being Hamiltonian, if modifying a constant fraction of $n$ edges is necessary to make $G$ Hamiltonian. We give an explicit deterministic construction of a class of graphs of bounded degree that are locally Hamiltonian, but (globally) far from being Hamiltonian. Here, \emph{locally Hamiltonian} means that every subgraph induced by the neighbourhood of a small vertex set appears in some Hamiltonian graph. More precisely, we obtain graphs which differ in $\Theta(n)$ edges from any Hamiltonian graph, but non-Hamiltonicity cannot be detected in the neighbourhood of $o(n)$ vertices. Our class of graphs yields a class of hard instances for one-sided error property testers with linear query complexity. It is known that any property tester (even with two-sided error) requires a linear number of queries to test Hamiltonicity (Yoshida, Ito, 2010). This is proved via a randomised construction of hard instances. In contrast, our construction is deterministic. So far only very few deterministic constructions of hard instances for property testing are known. We believe that our construction may lead to future insights in graph theory and towards a characterisation of the properties that are testable in the bounded-degree model.]]> 0 <![CDATA[This paper studies three classes of cellular automata from a computational point of view: freezing cellular automata where the state of a cell can only decrease according to some order on states, cellular automata where each cell only makes a bounded number of state changes in any orbit, and finally cellular automata where each orbit converges to some fixed point. Many examples studied in the literature fit into these definitions, in particular the works on cristal growth started by S. Ulam in the 60s. The central question addressed here is how the computational power and computational hardness of basic properties is affected by the constraints of convergence, bounded number of change, or local decreasing of states in each cell. By studying various benchmark problems (short-term prediction, long term reachability, limits) and considering various complexity measures and scales (LOGSPACE vs. PTIME, communication complexity, Turing computability and arithmetical hierarchy) we give a rich and nuanced answer: the overall computational complexity of such cellular automata depends on the class considered (among the three above), the dimension, and the precise problem studied. In particular, we show that all settings can achieve universality in the sense of Blondel-Delvenne-K\r{u}rka, although short term predictability varies from NLOGSPACE to P-complete. Besides, the computability of limit configurations starting from computable initial configurations separates bounded-change from convergent cellular automata in dimension~1, but also dimension~1 versus higher dimensions for freezing cellular automata. Another surprising dimension-sensitive result obtained is that nilpotency becomes decidable in dimension~ 1 for all the three classes, while it stays undecidable even for freezing cellular automata in higher dimension.]]> Mon, 31 Jan 2022 08:44:09 +0000 https://doi.org/10.46298/dmtcs.5734 https://doi.org/10.46298/dmtcs.5734 Ollinger, Nicolas Theyssier, Guillaume Ollinger, Nicolas Theyssier, Guillaume <![CDATA[This paper studies three classes of cellular automata from a computational point of view: freezing cellular automata where the state of a cell can only decrease according to some order on states, cellular automata where each cell only makes a bounded number of state changes in any orbit, and finally cellular automata where each orbit converges to some fixed point. Many examples studied in the literature fit into these definitions, in particular the works on cristal growth started by S. Ulam in the 60s. The central question addressed here is how the computational power and computational hardness of basic properties is affected by the constraints of convergence, bounded number of change, or local decreasing of states in each cell. By studying various benchmark problems (short-term prediction, long term reachability, limits) and considering various complexity measures and scales (LOGSPACE vs. PTIME, communication complexity, Turing computability and arithmetical hierarchy) we give a rich and nuanced answer: the overall computational complexity of such cellular automata depends on the class considered (among the three above), the dimension, and the precise problem studied. In particular, we show that all settings can achieve universality in the sense of Blondel-Delvenne-K\r{u}rka, although short term predictability varies from NLOGSPACE to P-complete. Besides, the computability of limit configurations starting from computable initial configurations separates bounded-change from convergent cellular automata in dimension~1, but also dimension~1 versus higher dimensions for freezing cellular automata. Another surprising dimension-sensitive result obtained is that nilpotency becomes decidable in dimension~ 1 for all the three classes, while it stays undecidable even for freezing cellular automata in higher dimension.]]> 0 <![CDATA[The majorization relation orders the degree sequences of simple graphs into posets called dominance orders. As shown by Ruch and Gutman (1979) and Merris (2002), the degree sequences of threshold and split graphs form upward-closed sets within the dominance orders they belong to, i.e., any degree sequence majorizing a split or threshold sequence must itself be split or threshold, respectively. Motivated by the fact that threshold graphs and split graphs have characterizations in terms of forbidden induced subgraphs, we define a class $\mathcal{F}$ of graphs to be dominance monotone if whenever no realization of $e$ contains an element $\mathcal{F}$ as an induced subgraph, and $d$ majorizes $e$, then no realization of $d$ induces an element of $\mathcal{F}$. We present conditions necessary for a set of graphs to be dominance monotone, and we identify the dominance monotone sets of order at most 3.]]> Thu, 20 Jan 2022 16:56:54 +0000 https://doi.org/10.46298/dmtcs.5666 https://doi.org/10.46298/dmtcs.5666 Barrus, Michael D. Guillaume, Jean A. Barrus, Michael D. Guillaume, Jean A. <![CDATA[The majorization relation orders the degree sequences of simple graphs into posets called dominance orders. As shown by Ruch and Gutman (1979) and Merris (2002), the degree sequences of threshold and split graphs form upward-closed sets within the dominance orders they belong to, i.e., any degree sequence majorizing a split or threshold sequence must itself be split or threshold, respectively. Motivated by the fact that threshold graphs and split graphs have characterizations in terms of forbidden induced subgraphs, we define a class $\mathcal{F}$ of graphs to be dominance monotone if whenever no realization of $e$ contains an element $\mathcal{F}$ as an induced subgraph, and $d$ majorizes $e$, then no realization of $d$ induces an element of $\mathcal{F}$. We present conditions necessary for a set of graphs to be dominance monotone, and we identify the dominance monotone sets of order at most 3.]]> 0 <![CDATA[Let $D$ be a strong balanced digraph on $2a$ vertices. Adamus et al. have proved that $D$ is hamiltonian if $d(u)+d(v)\ge 3a$ whenever $uv\notin A(D)$ and $vu\notin A(D)$. The lower bound $3a$ is tight. In this paper, we shall show that the extremal digraph on this condition is two classes of digraphs that can be clearly characterized. Moreover, we also show that if $d(u)+d(v)\geq 3a-1$ whenever $uv\notin A(D)$ and $vu\notin A(D)$, then $D$ is traceable. The lower bound $3a-1$ is tight.]]> Thu, 20 Jan 2022 16:52:09 +0000 https://doi.org/10.46298/dmtcs.5851 https://doi.org/10.46298/dmtcs.5851 Wang, Ruixia Wu, Linxin Meng, Wei Wang, Ruixia Wu, Linxin Meng, Wei <![CDATA[Let $D$ be a strong balanced digraph on $2a$ vertices. Adamus et al. have proved that $D$ is hamiltonian if $d(u)+d(v)\ge 3a$ whenever $uv\notin A(D)$ and $vu\notin A(D)$. The lower bound $3a$ is tight. In this paper, we shall show that the extremal digraph on this condition is two classes of digraphs that can be clearly characterized. Moreover, we also show that if $d(u)+d(v)\geq 3a-1$ whenever $uv\notin A(D)$ and $vu\notin A(D)$, then $D$ is traceable. The lower bound $3a-1$ is tight.]]> 0 <![CDATA[In Defective Coloring we are given a graph $G$ and two integers $\chi_d$, $\Delta^*$ and are asked if we can $\chi_d$-color $G$ so that the maximum degree induced by any color class is at most $\Delta^*$. We show that this natural generalization of Coloring is much harder on several basic graph classes. In particular, we show that it is NP-hard on split graphs, even when one of the two parameters $\chi_d$, $\Delta^*$ is set to the smallest possible fixed value that does not trivialize the problem ($\chi_d = 2$ or $\Delta^* = 1$). Together with a simple treewidth-based DP algorithm this completely determines the complexity of the problem also on chordal graphs. We then consider the case of cographs and show that, somewhat surprisingly, Defective Coloring turns out to be one of the few natural problems which are NP-hard on this class. We complement this negative result by showing that Defective Coloring is in P for cographs if either $\chi_d$ or $\Delta^*$ is fixed; that it is in P for trivially perfect graphs; and that it admits a sub-exponential time algorithm for cographs when both $\chi_d$ and $\Delta^*$ are unbounded.]]> Thu, 20 Jan 2022 16:47:09 +0000 https://doi.org/10.46298/dmtcs.4926 https://doi.org/10.46298/dmtcs.4926 Belmonte, Rémy Lampis, Michael Mitsou, Valia Belmonte, Rémy Lampis, Michael Mitsou, Valia <![CDATA[In Defective Coloring we are given a graph $G$ and two integers $\chi_d$, $\Delta^*$ and are asked if we can $\chi_d$-color $G$ so that the maximum degree induced by any color class is at most $\Delta^*$. We show that this natural generalization of Coloring is much harder on several basic graph classes. In particular, we show that it is NP-hard on split graphs, even when one of the two parameters $\chi_d$, $\Delta^*$ is set to the smallest possible fixed value that does not trivialize the problem ($\chi_d = 2$ or $\Delta^* = 1$). Together with a simple treewidth-based DP algorithm this completely determines the complexity of the problem also on chordal graphs. We then consider the case of cographs and show that, somewhat surprisingly, Defective Coloring turns out to be one of the few natural problems which are NP-hard on this class. We complement this negative result by showing that Defective Coloring is in P for cographs if either $\chi_d$ or $\Delta^*$ is fixed; that it is in P for trivially perfect graphs; and that it admits a sub-exponential time algorithm for cographs when both $\chi_d$ and $\Delta^*$ are unbounded.]]> 0 <![CDATA[A paired dominating set $P$ is a dominating set with the additional property that $P$ has a perfect matching. While the maximum cardainality of a minimal dominating set in a graph $G$ is called the upper domination number of $G$, denoted by $\Gamma(G)$, the maximum cardinality of a minimal paired dominating set in $G$ is called the upper paired domination number of $G$, denoted by $\Gamma_{pr}(G)$. By Henning and Pradhan (2019), we know that $\Gamma_{pr}(G)\leq 2\Gamma(G)$ for any graph $G$ without isolated vertices. We focus on the graphs satisfying the equality $\Gamma_{pr}(G)= 2\Gamma(G)$. We give characterizations for two special graph classes: bipartite and unicyclic graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ by using the results of Ulatowski (2015). Besides, we study the graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ and a restricted girth. In this context, we provide two characterizations: one for graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ and girth at least 6 and the other for $C_3$-free cactus graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$. We also pose the characterization of the general case of $C_3$-free graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ as an open question.]]> Thu, 16 Dec 2021 13:15:54 +0000 https://doi.org/10.46298/dmtcs.7331 https://doi.org/10.46298/dmtcs.7331 Alizadeh, Hadi Gözüpek, Didem Alizadeh, Hadi Gözüpek, Didem <![CDATA[A paired dominating set $P$ is a dominating set with the additional property that $P$ has a perfect matching. While the maximum cardainality of a minimal dominating set in a graph $G$ is called the upper domination number of $G$, denoted by $\Gamma(G)$, the maximum cardinality of a minimal paired dominating set in $G$ is called the upper paired domination number of $G$, denoted by $\Gamma_{pr}(G)$. By Henning and Pradhan (2019), we know that $\Gamma_{pr}(G)\leq 2\Gamma(G)$ for any graph $G$ without isolated vertices. We focus on the graphs satisfying the equality $\Gamma_{pr}(G)= 2\Gamma(G)$. We give characterizations for two special graph classes: bipartite and unicyclic graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ by using the results of Ulatowski (2015). Besides, we study the graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ and a restricted girth. In this context, we provide two characterizations: one for graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ and girth at least 6 and the other for $C_3$-free cactus graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$. We also pose the characterization of the general case of $C_3$-free graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ as an open question.]]> 0 <![CDATA[Let $H=(V,F)$ be a simple hypergraph without loops. $H$ is called linear if $|f\cap g|\le 1$ for any $f,g\in F$ with $f\not=g$. The $2$-section of $H$, denoted by $[H]_2$, is a graph with $V([H]_2)=V$ and for any $ u,v\in V([H]_2)$, $uv\in E([H]_2)$ if and only if there is $ f\in F$ such that $u,v\in f$. The treewidth of a graph is an important invariant in structural and algorithmic graph theory. In this paper, we consider the treewidth of the $2$-section of a linear hypergraph. We will use the minimum degree, maximum degree, anti-rank and average rank of a linear hypergraph to determine the upper and lower bounds of the treewidth of its $2$-section. Since for any graph $G$, there is a linear hypergraph $H$ such that $[H]_2\cong G$, we provide a method to estimate the bound of treewidth of graph by the parameters of the hypergraph.]]> Thu, 09 Dec 2021 15:59:54 +0000 https://doi.org/10.46298/dmtcs.6499 https://doi.org/10.46298/dmtcs.6499 Liu, Ke Lu, Mei Liu, Ke Lu, Mei <![CDATA[Let $H=(V,F)$ be a simple hypergraph without loops. $H$ is called linear if $|f\cap g|\le 1$ for any $f,g\in F$ with $f\not=g$. The $2$-section of $H$, denoted by $[H]_2$, is a graph with $V([H]_2)=V$ and for any $ u,v\in V([H]_2)$, $uv\in E([H]_2)$ if and only if there is $ f\in F$ such that $u,v\in f$. The treewidth of a graph is an important invariant in structural and algorithmic graph theory. In this paper, we consider the treewidth of the $2$-section of a linear hypergraph. We will use the minimum degree, maximum degree, anti-rank and average rank of a linear hypergraph to determine the upper and lower bounds of the treewidth of its $2$-section. Since for any graph $G$, there is a linear hypergraph $H$ such that $[H]_2\cong G$, we provide a method to estimate the bound of treewidth of graph by the parameters of the hypergraph.]]> 0 <![CDATA[Fici, Restivo, Silva, and Zamboni define a $k$-antipower to be a word composed of $k$ pairwise distinct, concatenated words of equal length. Berger and Defant conjecture that for any sufficiently well-behaved aperiodic morphic word $w$, there exists a constant $c$ such that for any $k$ and any index $i$, a $k$-antipower with block length at most $ck$ starts at the $i$th position of $w$. They prove their conjecture in the case of binary words, and we extend their result to alphabets of arbitrary finite size and characterize those words for which the result does not hold. We also prove their conjecture in the specific case of the Fibonacci word.]]> Thu, 09 Dec 2021 15:53:04 +0000 https://doi.org/10.46298/dmtcs.7134 https://doi.org/10.46298/dmtcs.7134 Garg, Swapnil Garg, Swapnil <![CDATA[Fici, Restivo, Silva, and Zamboni define a $k$-antipower to be a word composed of $k$ pairwise distinct, concatenated words of equal length. Berger and Defant conjecture that for any sufficiently well-behaved aperiodic morphic word $w$, there exists a constant $c$ such that for any $k$ and any index $i$, a $k$-antipower with block length at most $ck$ starts at the $i$th position of $w$. They prove their conjecture in the case of binary words, and we extend their result to alphabets of arbitrary finite size and characterize those words for which the result does not hold. We also prove their conjecture in the specific case of the Fibonacci word.]]> 0 <![CDATA[A graph $G$ is $k$-$weighted-list-antimagic$ if for any vertex weighting $\omega\colon V(G)\to\mathbb{R}$ and any list assignment $L\colon E(G)\to2^{\mathbb{R}}$ with $|L(e)|\geq |E(G)|+k$ there exists an edge labeling $f$ such that $f(e)\in L(e)$ for all $e\in E(G)$, labels of edges are pairwise distinct, and the sum of the labels on edges incident to a vertex plus the weight of that vertex is distinct from the sum at every other vertex. In this paper we prove that every graph on $n$ vertices having no $K_1$ or $K_2$ component is $\lfloor{\frac{4n}{3}}\rfloor$-weighted-list-antimagic. An oriented graph $G$ is $k$-$oriented-antimagic$ if there exists an injective edge labeling from $E(G)$ into $\{1,\dotsc,|E(G)|+k\}$ such that the sum of the labels on edges incident to and oriented toward a vertex minus the sum of the labels on edges incident to and oriented away from that vertex is distinct from the difference of sums at every other vertex. We prove that every graph on $n$ vertices with no $K_1$ component admits an orientation that is $\lfloor{\frac{2n}{3}}\rfloor$-oriented-antimagic.]]> Thu, 02 Dec 2021 15:23:48 +0000 https://doi.org/10.46298/dmtcs.5631 https://doi.org/10.46298/dmtcs.5631 Berikkyzy, Zhanar Brandt, Axel Jahanbekam, Sogol Larsen, Victor Rorabaugh, Danny Berikkyzy, Zhanar Brandt, Axel Jahanbekam, Sogol Larsen, Victor Rorabaugh, Danny <![CDATA[A graph $G$ is $k$-$weighted-list-antimagic$ if for any vertex weighting $\omega\colon V(G)\to\mathbb{R}$ and any list assignment $L\colon E(G)\to2^{\mathbb{R}}$ with $|L(e)|\geq |E(G)|+k$ there exists an edge labeling $f$ such that $f(e)\in L(e)$ for all $e\in E(G)$, labels of edges are pairwise distinct, and the sum of the labels on edges incident to a vertex plus the weight of that vertex is distinct from the sum at every other vertex. In this paper we prove that every graph on $n$ vertices having no $K_1$ or $K_2$ component is $\lfloor{\frac{4n}{3}}\rfloor$-weighted-list-antimagic. An oriented graph $G$ is $k$-$oriented-antimagic$ if there exists an injective edge labeling from $E(G)$ into $\{1,\dotsc,|E(G)|+k\}$ such that the sum of the labels on edges incident to and oriented toward a vertex minus the sum of the labels on edges incident to and oriented away from that vertex is distinct from the difference of sums at every other vertex. We prove that every graph on $n$ vertices with no $K_1$ component admits an orientation that is $\lfloor{\frac{2n}{3}}\rfloor$-oriented-antimagic.]]> 0 <![CDATA[We are given $n$ balls and an unknown coloring of them with two colors. Our goal is to find a ball that belongs to the larger color class, or show that the color classes have the same size. We can ask sets of $k$ balls as queries, and the problem has different variants, according to what the answers to the queries can be. These questions has attracted several researchers, but the focus of most research was the adaptive version, where queries are decided sequentially, after learning the answer to the previous query. Here we study the non-adaptive version, where all the queries have to be asked at the same time.]]> Fri, 26 Nov 2021 10:00:41 +0000 https://doi.org/10.46298/dmtcs.7084 https://doi.org/10.46298/dmtcs.7084 Gerbner, Dániel Vizer, Máté Gerbner, Dániel Vizer, Máté <![CDATA[We are given $n$ balls and an unknown coloring of them with two colors. Our goal is to find a ball that belongs to the larger color class, or show that the color classes have the same size. We can ask sets of $k$ balls as queries, and the problem has different variants, according to what the answers to the queries can be. These questions has attracted several researchers, but the focus of most research was the adaptive version, where queries are decided sequentially, after learning the answer to the previous query. Here we study the non-adaptive version, where all the queries have to be asked at the same time.]]> 0 <![CDATA[Boolean nested canalizing functions (NCFs) have important applications in molecular regulatory networks, engineering and computer science. In this paper, we study their certificate complexity. For both Boolean values $b\in\{0,1\}$, we obtain a formula for $b$-certificate complexity and consequently, we develop a direct proof of the certificate complexity formula of an NCF. Symmetry is another interesting property of Boolean functions and we significantly simplify the proofs of some recent theorems about partial symmetry of NCFs. We also describe the algebraic normal form of $s$-symmetric NCFs. We obtain the general formula of the cardinality of the set of $n$-variable $s$-symmetric Boolean NCFs for $s=1,\dots,n$. In particular, we enumerate the strongly asymmetric Boolean NCFs.]]> Fri, 26 Nov 2021 09:50:34 +0000 https://doi.org/10.46298/dmtcs.6191 https://doi.org/10.46298/dmtcs.6191 Li, Yuan Ingram, Frank Zhang, Huaming Li, Yuan Ingram, Frank Zhang, Huaming <![CDATA[Boolean nested canalizing functions (NCFs) have important applications in molecular regulatory networks, engineering and computer science. In this paper, we study their certificate complexity. For both Boolean values $b\in\{0,1\}$, we obtain a formula for $b$-certificate complexity and consequently, we develop a direct proof of the certificate complexity formula of an NCF. Symmetry is another interesting property of Boolean functions and we significantly simplify the proofs of some recent theorems about partial symmetry of NCFs. We also describe the algebraic normal form of $s$-symmetric NCFs. We obtain the general formula of the cardinality of the set of $n$-variable $s$-symmetric Boolean NCFs for $s=1,\dots,n$. In particular, we enumerate the strongly asymmetric Boolean NCFs.]]> 0 <![CDATA[When can we compute the diameter of a graph in quasi linear time? We address this question for the class of {\em split graphs}, that we observe to be the hardest instances for deciding whether the diameter is at most two. We stress that although the diameter of a non-complete split graph can only be either $2$ or $3$, under the Strong Exponential-Time Hypothesis (SETH) we cannot compute the diameter of an $n$-vertex $m$-edge split graph in less than quadratic time -- in the size $n+m$ of the input. Therefore it is worth to study the complexity of diameter computation on {\em subclasses} of split graphs, in order to better understand the complexity border. Specifically, we consider the split graphs with bounded {\em clique-interval number} and their complements, with the former being a natural variation of the concept of interval number for split graphs that we introduce in this paper. We first discuss the relations between the clique-interval number and other graph invariants such as the classic interval number of graphs, the treewidth, the {\em VC-dimension} and the {\em stabbing number} of a related hypergraph. Then, in part based on these above relations, we almost completely settle the complexity of diameter computation on these subclasses of split graphs: - For the $k$-clique-interval split graphs, we can compute their diameter in truly subquadratic time if $k={\cal O}(1)$, and even in quasi linear time if $k=o(\log{n})$ and in addition a corresponding ordering of the vertices in the clique is given. However, under SETH this cannot be done in truly subquadratic time for any $k = \omega(\log{n})$. - For the {\em complements} of $k$-clique-interval split graphs, we can compute their diameter in truly subquadratic time if $k={\cal O}(1)$, and even in time ${\cal O}(km)$ if a corresponding ordering of the vertices in the stable set is given. Again this latter result is optimal under SETH up to polylogarithmic factors. Our findings raise the question whether a $k$-clique interval ordering can always be computed in quasi linear time. We prove that it is the case for $k=1$ and for some subclasses such as bounded-treewidth split graphs, threshold graphs and comparability split graphs. Finally, we prove that some important subclasses of split graphs -- including the ones mentioned above -- have a bounded clique-interval number.]]> Mon, 15 Nov 2021 10:16:07 +0000 https://doi.org/10.46298/dmtcs.6422 https://doi.org/10.46298/dmtcs.6422 Ducoffe, Guillaume Habib, Michel Viennot, Laurent Ducoffe, Guillaume Habib, Michel Viennot, Laurent <![CDATA[When can we compute the diameter of a graph in quasi linear time? We address this question for the class of {\em split graphs}, that we observe to be the hardest instances for deciding whether the diameter is at most two. We stress that although the diameter of a non-complete split graph can only be either $2$ or $3$, under the Strong Exponential-Time Hypothesis (SETH) we cannot compute the diameter of an $n$-vertex $m$-edge split graph in less than quadratic time -- in the size $n+m$ of the input. Therefore it is worth to study the complexity of diameter computation on {\em subclasses} of split graphs, in order to better understand the complexity border. Specifically, we consider the split graphs with bounded {\em clique-interval number} and their complements, with the former being a natural variation of the concept of interval number for split graphs that we introduce in this paper. We first discuss the relations between the clique-interval number and other graph invariants such as the classic interval number of graphs, the treewidth, the {\em VC-dimension} and the {\em stabbing number} of a related hypergraph. Then, in part based on these above relations, we almost completely settle the complexity of diameter computation on these subclasses of split graphs: - For the $k$-clique-interval split graphs, we can compute their diameter in truly subquadratic time if $k={\cal O}(1)$, and even in quasi linear time if $k=o(\log{n})$ and in addition a corresponding ordering of the vertices in the clique is given. However, under SETH this cannot be done in truly subquadratic time for any $k = \omega(\log{n})$. - For the {\em complements} of $k$-clique-interval split graphs, we can compute their diameter in truly subquadratic time if $k={\cal O}(1)$, and even in time ${\cal O}(km)$ if a corresponding ordering of the vertices in the stable set is given. Again this latter result is optimal under SETH up to polylogarithmic factors. Our findings raise the question whether a $k$-clique interval ordering can always be computed in quasi linear time. We prove that it is the case for $k=1$ and for some subclasses such as bounded-treewidth split graphs, threshold graphs and comparability split graphs. Finally, we prove that some important subclasses of split graphs -- including the ones mentioned above -- have a bounded clique-interval number.]]> 0 <![CDATA[In this paper, we prove a collection of results on graphical indices. We determine the extremal graphs attaining the maximal generalized Wiener index (e.g. the hyper-Wiener index) among all graphs with given matching number or independence number. This generalizes some work of Dankelmann, as well as some work of Chung. We also show alternative proofs for two recents results on maximizing the Wiener index and external Wiener index by deriving it from earlier results. We end with proving two conjectures. We prove that the maximum for the difference of the Wiener index and the eccentricity is attained by the path if the order $n$ is at least $9$ and that the maximum weighted Szeged index of graphs of given order is attained by the balanced complete bipartite graphs.]]> Mon, 15 Nov 2021 10:10:33 +0000 https://doi.org/10.46298/dmtcs.6896 https://doi.org/10.46298/dmtcs.6896 Cambie, Stijn Cambie, Stijn <![CDATA[In this paper, we prove a collection of results on graphical indices. We determine the extremal graphs attaining the maximal generalized Wiener index (e.g. the hyper-Wiener index) among all graphs with given matching number or independence number. This generalizes some work of Dankelmann, as well as some work of Chung. We also show alternative proofs for two recents results on maximizing the Wiener index and external Wiener index by deriving it from earlier results. We end with proving two conjectures. We prove that the maximum for the difference of the Wiener index and the eccentricity is attained by the path if the order $n$ is at least $9$ and that the maximum weighted Szeged index of graphs of given order is attained by the balanced complete bipartite graphs.]]> 0 <![CDATA[In this article we present theoretical and computational results on the existence of polyhedral embeddings of graphs. The emphasis is on cubic graphs. We also describe an efficient algorithm to compute all polyhedral embeddings of a given cubic graph and constructions for cubic graphs with some special properties of their polyhedral embeddings. Some key results are that even cubic graphs with a polyhedral embedding on the torus can also have polyhedral embeddings in arbitrarily high genus, in fact in a genus {\em close} to the theoretical maximum for that number of vertices, and that there is no bound on the number of genera in which a cubic graph can have a polyhedral embedding. While these results suggest a large variety of polyhedral embeddings, computations for up to 28 vertices suggest that by far most of the cubic graphs do not have a polyhedral embedding in any genus and that the ratio of these graphs is increasing with the number of vertices.]]> Fri, 05 Nov 2021 15:40:54 +0000 https://doi.org/10.46298/dmtcs.6729 https://doi.org/10.46298/dmtcs.6729 Brinkmann, Gunnar Tucker, Thomas Van Cleemput, Nico Brinkmann, Gunnar Tucker, Thomas Van Cleemput, Nico <![CDATA[In this article we present theoretical and computational results on the existence of polyhedral embeddings of graphs. The emphasis is on cubic graphs. We also describe an efficient algorithm to compute all polyhedral embeddings of a given cubic graph and constructions for cubic graphs with some special properties of their polyhedral embeddings. Some key results are that even cubic graphs with a polyhedral embedding on the torus can also have polyhedral embeddings in arbitrarily high genus, in fact in a genus {\em close} to the theoretical maximum for that number of vertices, and that there is no bound on the number of genera in which a cubic graph can have a polyhedral embedding. While these results suggest a large variety of polyhedral embeddings, computations for up to 28 vertices suggest that by far most of the cubic graphs do not have a polyhedral embedding in any genus and that the ratio of these graphs is increasing with the number of vertices.]]> 0 <![CDATA[A function on an algebra is congruence preserving if, for any congruence, it maps pairs of congruent elements onto pairs of congruent elements. We show that on the algebra of binary trees whose leaves are labeled by letters of an alphabet containing at least three letters, a function is congruence preserving if and only if it is polynomial.]]> Thu, 04 Nov 2021 08:44:33 +0000 https://doi.org/10.46298/dmtcs.6890 https://doi.org/10.46298/dmtcs.6890 Arnold, Andre Cegielski, Patrick Grigorieff, Serge Guessarian, Irene Arnold, Andre Cegielski, Patrick Grigorieff, Serge Guessarian, Irene <![CDATA[A function on an algebra is congruence preserving if, for any congruence, it maps pairs of congruent elements onto pairs of congruent elements. We show that on the algebra of binary trees whose leaves are labeled by letters of an alphabet containing at least three letters, a function is congruence preserving if and only if it is polynomial.]]> 0 <![CDATA[A distributed graph algorithm is basically an algorithm where every node of a graph can look at its neighborhood at some distance in the graph and chose its output. As distributed environment are subject to faults, an important issue is to be able to check that the output is correct, or in general that the network is in proper configuration with respect to some predicate. One would like this checking to be very local, to avoid using too much resources. Unfortunately most predicates cannot be checked this way, and that is where certification comes into play. Local certification (also known as proof-labeling schemes, locally checkable proofs or distributed verification) consists in assigning labels to the nodes, that certify that the configuration is correct. There are several point of view on this topic: it can be seen as a part of self-stabilizing algorithms, as labeling problem, or as a non-deterministic distributed decision. This paper is an introduction to the domain of local certification, giving an overview of the history, the techniques and the current research directions.]]> Wed, 15 Sep 2021 12:18:21 +0000 https://doi.org/10.46298/dmtcs.6280 https://doi.org/10.46298/dmtcs.6280 Feuilloley, Laurent Feuilloley, Laurent <![CDATA[A distributed graph algorithm is basically an algorithm where every node of a graph can look at its neighborhood at some distance in the graph and chose its output. As distributed environment are subject to faults, an important issue is to be able to check that the output is correct, or in general that the network is in proper configuration with respect to some predicate. One would like this checking to be very local, to avoid using too much resources. Unfortunately most predicates cannot be checked this way, and that is where certification comes into play. Local certification (also known as proof-labeling schemes, locally checkable proofs or distributed verification) consists in assigning labels to the nodes, that certify that the configuration is correct. There are several point of view on this topic: it can be seen as a part of self-stabilizing algorithms, as labeling problem, or as a non-deterministic distributed decision. This paper is an introduction to the domain of local certification, giving an overview of the history, the techniques and the current research directions.]]> 0 <![CDATA[As a step towards resolving a question of Ru\v{s}kuc on the decidability of joint embedding for hereditary classes of permutations, which may be viewed as structures in a language of 2 linear orders, we show the corresponding problem is undecidable for hereditary classes of structures in a language of 3 linear orders.]]> Tue, 14 Sep 2021 08:40:59 +0000 https://doi.org/10.46298/dmtcs.6165 https://doi.org/10.46298/dmtcs.6165 Braunfeld, Samuel Braunfeld, Samuel <![CDATA[As a step towards resolving a question of Ru\v{s}kuc on the decidability of joint embedding for hereditary classes of permutations, which may be viewed as structures in a language of 2 linear orders, we show the corresponding problem is undecidable for hereditary classes of structures in a language of 3 linear orders.]]> 0 <![CDATA[In the $d$-dimensional hypercube bin packing problem, a given list of $d$-dimensional hypercubes must be packed into the smallest number of hypercube bins. Epstein and van Stee [SIAM J. Comput. 35 (2005)] showed that the asymptotic performance ratio $\rho$ of the online bounded space variant is $\Omega(\log d)$ and $O(d/\log d)$, and conjectured that it is $\Theta(\log d)$. We show that $\rho$ is in fact $\Theta(d/\log d)$, using probabilistic arguments.]]> Tue, 14 Sep 2021 08:37:20 +0000 https://doi.org/10.46298/dmtcs.8325 https://doi.org/10.46298/dmtcs.8325 Kohayakawa, Yoshiharu Miyazawa, Flávio Keidi Wakabayashi, Yoshiko Kohayakawa, Yoshiharu Miyazawa, Flávio Keidi Wakabayashi, Yoshiko <![CDATA[In the $d$-dimensional hypercube bin packing problem, a given list of $d$-dimensional hypercubes must be packed into the smallest number of hypercube bins. Epstein and van Stee [SIAM J. Comput. 35 (2005)] showed that the asymptotic performance ratio $\rho$ of the online bounded space variant is $\Omega(\log d)$ and $O(d/\log d)$, and conjectured that it is $\Theta(\log d)$. We show that $\rho$ is in fact $\Theta(d/\log d)$, using probabilistic arguments.]]> 0 <![CDATA[A subset $A \subset \mathbb R^2$ is said to avoid distance $1$ if: $\forall x,y \in A, \left\| x-y \right\|_2 \neq 1.$ In this paper we study the number $m_1(\mathbb R^2)$ which is the supremum of the upper densities of measurable sets avoiding distance 1 in the Euclidean plane. Intuitively, $m_1(\mathbb R^2)$ represents the highest proportion of the plane that can be filled by a set avoiding distance 1. This parameter is related to the fractional chromatic number $\chi_f(\mathbb R^2)$ of the plane. We establish that $m_1(\mathbb R^2) \leq 0.25647$ and $\chi_f(\mathbb R^2) \geq 3.8991$.]]> Tue, 31 Aug 2021 16:35:13 +0000 https://doi.org/10.46298/dmtcs.5153 https://doi.org/10.46298/dmtcs.5153 Bellitto, Thomas Pêcher, Arnaud Sédillot, Antoine Bellitto, Thomas Pêcher, Arnaud Sédillot, Antoine <![CDATA[A subset $A \subset \mathbb R^2$ is said to avoid distance $1$ if: $\forall x,y \in A, \left\| x-y \right\|_2 \neq 1.$ In this paper we study the number $m_1(\mathbb R^2)$ which is the supremum of the upper densities of measurable sets avoiding distance 1 in the Euclidean plane. Intuitively, $m_1(\mathbb R^2)$ represents the highest proportion of the plane that can be filled by a set avoiding distance 1. This parameter is related to the fractional chromatic number $\chi_f(\mathbb R^2)$ of the plane. We establish that $m_1(\mathbb R^2) \leq 0.25647$ and $\chi_f(\mathbb R^2) \geq 3.8991$.]]> 0 <![CDATA[Defant, Engen, and Miller defined a permutation to be uniquely sorted if it has exactly one preimage under West's stack-sorting map. We enumerate classes of uniquely sorted permutations that avoid a pattern of length three and a pattern of length four by establishing bijections between these classes and various lattice paths. This allows us to prove nine conjectures of Defant.]]> Tue, 31 Aug 2021 07:30:13 +0000 https://doi.org/10.46298/dmtcs.6494 https://doi.org/10.46298/dmtcs.6494 Mularczyk, Hanna Mularczyk, Hanna <![CDATA[Defant, Engen, and Miller defined a permutation to be uniquely sorted if it has exactly one preimage under West's stack-sorting map. We enumerate classes of uniquely sorted permutations that avoid a pattern of length three and a pattern of length four by establishing bijections between these classes and various lattice paths. This allows us to prove nine conjectures of Defant.]]> 0 <![CDATA[We show that, with the exception of the words $a^2ba^2$ and $b^2ab^2$, all (finite or infinite) binary patterns in the Prouhet-Thue-Morse sequence can actually be found in that sequence as segments (up to exchange of letters in the infinite case). This result was previously attributed to unpublished work by D. Guaiana and may also be derived from publications of A. Shur only available in Russian. We also identify the (finitely many) finite binary patterns that appear non trivially, in the sense that they are obtained by applying an endomorphism that does not map the set of all segments of the sequence into itself.]]> Mon, 30 Aug 2021 12:58:00 +0000 https://doi.org/10.46298/dmtcs.5460 https://doi.org/10.46298/dmtcs.5460 Almeida, Jorge Klíma, Ondřej Almeida, Jorge Klíma, Ondřej <![CDATA[We show that, with the exception of the words $a^2ba^2$ and $b^2ab^2$, all (finite or infinite) binary patterns in the Prouhet-Thue-Morse sequence can actually be found in that sequence as segments (up to exchange of letters in the infinite case). This result was previously attributed to unpublished work by D. Guaiana and may also be derived from publications of A. Shur only available in Russian. We also identify the (finitely many) finite binary patterns that appear non trivially, in the sense that they are obtained by applying an endomorphism that does not map the set of all segments of the sequence into itself.]]> 0 <![CDATA[An outer-1-planar graph is a graph admitting a drawing in the plane so that all vertices appear in the outer region of the drawing and every edge crosses at most one other edge. This paper establishes the local structure of outer-1-planar graphs by proving that each outer-1-planar graph contains one of the seventeen fixed configurations, and the list of those configurations is minimal in the sense that for each fixed configuration there exist outer-1-planar graphs containing this configuration that do not contain any of another sixteen configurations. There are two interesting applications of this structural theorem. First of all, we conclude that every (resp. maximal) outer-1-planar graph of minimum degree at least 2 has an edge with the sum of the degrees of its two end-vertices being at most 9 (resp. 7), and this upper bound is sharp. On the other hand, we show that the list 3-dynamic chromatic number of every outer-1-planar graph is at most 6, and this upper bound is best possible.]]> Fri, 27 Aug 2021 06:57:03 +0000 https://doi.org/10.46298/dmtcs.5860 https://doi.org/10.46298/dmtcs.5860 Li, Yan Zhang, Xin Li, Yan Zhang, Xin <![CDATA[An outer-1-planar graph is a graph admitting a drawing in the plane so that all vertices appear in the outer region of the drawing and every edge crosses at most one other edge. This paper establishes the local structure of outer-1-planar graphs by proving that each outer-1-planar graph contains one of the seventeen fixed configurations, and the list of those configurations is minimal in the sense that for each fixed configuration there exist outer-1-planar graphs containing this configuration that do not contain any of another sixteen configurations. There are two interesting applications of this structural theorem. First of all, we conclude that every (resp. maximal) outer-1-planar graph of minimum degree at least 2 has an edge with the sum of the degrees of its two end-vertices being at most 9 (resp. 7), and this upper bound is sharp. On the other hand, we show that the list 3-dynamic chromatic number of every outer-1-planar graph is at most 6, and this upper bound is best possible.]]> 0 <![CDATA[The Hausdorff distance is a relatively new measure of similarity of graphs. The notion of the Hausdorff distance considers a special kind of a common subgraph of the compared graphs and depends on the structural properties outside of the common subgraph. There was no known efficient algorithm for the problem of determining the Hausdorff distance between two trees, and in this paper we present a polynomial-time algorithm for it. The algorithm is recursive and it utilizes the divide and conquer technique. As a subtask it also uses the procedure that is based on the well known graph algorithm of finding the maximum bipartite matching.]]> Thu, 19 Aug 2021 15:29:10 +0000 https://doi.org/10.46298/dmtcs.6952 https://doi.org/10.46298/dmtcs.6952 Kelenc, Aleksander Kelenc, Aleksander <![CDATA[The Hausdorff distance is a relatively new measure of similarity of graphs. The notion of the Hausdorff distance considers a special kind of a common subgraph of the compared graphs and depends on the structural properties outside of the common subgraph. There was no known efficient algorithm for the problem of determining the Hausdorff distance between two trees, and in this paper we present a polynomial-time algorithm for it. The algorithm is recursive and it utilizes the divide and conquer technique. As a subtask it also uses the procedure that is based on the well known graph algorithm of finding the maximum bipartite matching.]]> 0 <![CDATA[A family S of convex sets in the plane defines a hypergraph H = (S, E) as follows. Every subfamily S' of S defines a hyperedge of H if and only if there exists a halfspace h that fully contains S' , and no other set of S is fully contained in h. In this case, we say that h realizes S'. We say a set S is shattered, if all its subsets are realized. The VC-dimension of a hypergraph H is the size of the largest shattered set. We show that the VC-dimension for pairwise disjoint convex sets in the plane is bounded by 3, and this is tight. In contrast, we show the VC-dimension of convex sets in the plane (not necessarily disjoint) is unbounded. We provide a quadratic lower bound in the number of pairs of intersecting sets in a shattered family of convex sets in the plane. We also show that the VC-dimension is unbounded for pairwise disjoint convex sets in R^d , for d > 2. We focus on, possibly intersecting, segments in the plane and determine that the VC-dimension is always at most 5. And this is tight, as we construct a set of five segments that can be shattered. We give two exemplary applications. One for a geometric set cover problem and one for a range-query data structure problem, to motivate our findings.]]> Thu, 19 Aug 2021 15:24:12 +0000 https://doi.org/10.46298/dmtcs.6631 https://doi.org/10.46298/dmtcs.6631 Grelier, Nicolas Ilchi, Saeed Gh. Miltzow, Tillmann Smorodinsky, Shakhar Grelier, Nicolas Ilchi, Saeed Gh. Miltzow, Tillmann Smorodinsky, Shakhar <![CDATA[A family S of convex sets in the plane defines a hypergraph H = (S, E) as follows. Every subfamily S' of S defines a hyperedge of H if and only if there exists a halfspace h that fully contains S' , and no other set of S is fully contained in h. In this case, we say that h realizes S'. We say a set S is shattered, if all its subsets are realized. The VC-dimension of a hypergraph H is the size of the largest shattered set. We show that the VC-dimension for pairwise disjoint convex sets in the plane is bounded by 3, and this is tight. In contrast, we show the VC-dimension of convex sets in the plane (not necessarily disjoint) is unbounded. We provide a quadratic lower bound in the number of pairs of intersecting sets in a shattered family of convex sets in the plane. We also show that the VC-dimension is unbounded for pairwise disjoint convex sets in R^d , for d > 2. We focus on, possibly intersecting, segments in the plane and determine that the VC-dimension is always at most 5. And this is tight, as we construct a set of five segments that can be shattered. We give two exemplary applications. One for a geometric set cover problem and one for a range-query data structure problem, to motivate our findings.]]> 0 <![CDATA[Two permutation classes, the X-class and subpermutations of the increasing oscillation are shown to exhibit an exponential Wilf-collapse. This means that the number of distinct enumerations of principal subclasses of each of these classes grows much more slowly than the class itself whereas a priori, based only on symmetries of the class, there is no reason to expect this. The underlying cause of the collapse in both cases is the ability to apply some form of local symmetry which, combined with a greedy algorithm for detecting patterns in these classes, yields a Wilf-collapse.]]> Thu, 19 Aug 2021 15:18:29 +0000 https://doi.org/10.46298/dmtcs.5986 https://doi.org/10.46298/dmtcs.5986 Albert, Michael Jelínek, Vít Opler, Michal Albert, Michael Jelínek, Vít Opler, Michal <![CDATA[Two permutation classes, the X-class and subpermutations of the increasing oscillation are shown to exhibit an exponential Wilf-collapse. This means that the number of distinct enumerations of principal subclasses of each of these classes grows much more slowly than the class itself whereas a priori, based only on symmetries of the class, there is no reason to expect this. The underlying cause of the collapse in both cases is the ability to apply some form of local symmetry which, combined with a greedy algorithm for detecting patterns in these classes, yields a Wilf-collapse.]]> 0 <![CDATA[The dynamics of certain combinatorial actions and their liftings to actions at the piecewise-linear and birational level have been studied lately with an eye towards questions of periodicity, orbit structure, and invariants. One key property enjoyed by the rowmotion operator on certain finite partially-ordered sets is homomesy, where the average value of a statistic is the same for all orbits. To prove refined versions of homomesy in the product of two chain posets, J. Propp and the second author used an equivariant bijection discovered (less formally) by R. Stanley and H. Thomas. We explore the lifting of this "Stanley--Thomas word" to the piecewise-linear, birational, and noncommutative realms. Although the map is no longer a bijection, so cannot be used to prove periodicity directly, it still gives enough information to prove the homomesy at the piecewise-linear and birational levels (a result previously shown by D. Grinberg, S. Hopkins, and S. Okada). Even at the noncommutative level, the Stanley--Thomas word of a poset labeling rotates cyclically with the lifting of antichain rowmotion. Along the way we give some formulas for noncommutative antichain rowmotion that we hope will be first steps towards proving the conjectured periodicity at this level.]]> Wed, 18 Aug 2021 07:18:26 +0000 https://doi.org/10.46298/dmtcs.6633 https://doi.org/10.46298/dmtcs.6633 Joseph, Michael Roby, Tom Joseph, Michael Roby, Tom <![CDATA[The dynamics of certain combinatorial actions and their liftings to actions at the piecewise-linear and birational level have been studied lately with an eye towards questions of periodicity, orbit structure, and invariants. One key property enjoyed by the rowmotion operator on certain finite partially-ordered sets is homomesy, where the average value of a statistic is the same for all orbits. To prove refined versions of homomesy in the product of two chain posets, J. Propp and the second author used an equivariant bijection discovered (less formally) by R. Stanley and H. Thomas. We explore the lifting of this "Stanley--Thomas word" to the piecewise-linear, birational, and noncommutative realms. Although the map is no longer a bijection, so cannot be used to prove periodicity directly, it still gives enough information to prove the homomesy at the piecewise-linear and birational levels (a result previously shown by D. Grinberg, S. Hopkins, and S. Okada). Even at the noncommutative level, the Stanley--Thomas word of a poset labeling rotates cyclically with the lifting of antichain rowmotion. Along the way we give some formulas for noncommutative antichain rowmotion that we hope will be first steps towards proving the conjectured periodicity at this level.]]> 0 <![CDATA[In this paper, we enumerate Dumont permutations of the fourth kind avoiding or containing certain permutations of length 4. We also conjecture a Wilf-equivalence of two 4-letter patterns on Dumont permutations of the first kind.]]> Tue, 06 Jul 2021 12:35:41 +0000 https://doi.org/10.46298/dmtcs.6174 https://doi.org/10.46298/dmtcs.6174 Burstein, Alexander Jones, Opel Burstein, Alexander Jones, Opel <![CDATA[In this paper, we enumerate Dumont permutations of the fourth kind avoiding or containing certain permutations of length 4. We also conjecture a Wilf-equivalence of two 4-letter patterns on Dumont permutations of the first kind.]]> 0 <![CDATA[A dominating set $D$ of a graph $G$ without isolated vertices is called semipaired dominating set if $D$ can be partitioned into $2$-element subsets such that the vertices in each set are at distance at most $2$. The semipaired domination number, denoted by $\gamma_{pr2}(G)$ is the minimum cardinality of a semipaired dominating set of $G$. Given a graph $G$ with no isolated vertices, the \textsc{Minimum Semipaired Domination} problem is to find a semipaired dominating set of $G$ of cardinality $\gamma_{pr2}(G)$. The decision version of the \textsc{Minimum Semipaired Domination} problem is already known to be NP-complete for chordal graphs, an important graph class. In this paper, we show that the decision version of the \textsc{Minimum Semipaired Domination} problem remains NP-complete for split graphs, a subclass of chordal graphs. On the positive side, we propose a linear-time algorithm to compute a minimum cardinality semipaired dominating set of block graphs. In addition, we prove that the \textsc{Minimum Semipaired Domination} problem is APX-complete for graphs with maximum degree $3$.]]> Tue, 06 Jul 2021 12:30:51 +0000 https://doi.org/10.46298/dmtcs.6782 https://doi.org/10.46298/dmtcs.6782 Henning, Michael A. Pandey, Arti Tripathi, Vikash Henning, Michael A. Pandey, Arti Tripathi, Vikash <![CDATA[A dominating set $D$ of a graph $G$ without isolated vertices is called semipaired dominating set if $D$ can be partitioned into $2$-element subsets such that the vertices in each set are at distance at most $2$. The semipaired domination number, denoted by $\gamma_{pr2}(G)$ is the minimum cardinality of a semipaired dominating set of $G$. Given a graph $G$ with no isolated vertices, the \textsc{Minimum Semipaired Domination} problem is to find a semipaired dominating set of $G$ of cardinality $\gamma_{pr2}(G)$. The decision version of the \textsc{Minimum Semipaired Domination} problem is already known to be NP-complete for chordal graphs, an important graph class. In this paper, we show that the decision version of the \textsc{Minimum Semipaired Domination} problem remains NP-complete for split graphs, a subclass of chordal graphs. On the positive side, we propose a linear-time algorithm to compute a minimum cardinality semipaired dominating set of block graphs. In addition, we prove that the \textsc{Minimum Semipaired Domination} problem is APX-complete for graphs with maximum degree $3$.]]> 0 <![CDATA[A skew shape is the difference of two top-left justified Ferrers shapes sharing the same top-left corner. We study integer fillings of skew shapes. As our first main result, we show that for a specific hereditary class of skew shapes, which we call D-free shapes, the fillings that avoid a north-east chain of size $k$ are in bijection with fillings that avoid a south-east chain of the same size. Since Ferrers shapes are a subclass of D-free shapes, this result can be seen as a generalization of previous analogous results for Ferrers shapes. As our second main result, we construct a bijection between 01-fillings of an arbitrary skew shape that avoid a south-east chain of size 2, and the 01-fillings of the same shape that simultaneously avoid a north-east chain of size 2 and a particular non-square subfilling. This generalizes a previous result for transversal fillings.]]> Fri, 18 Jun 2021 12:31:44 +0000 https://doi.org/10.46298/dmtcs.6171 https://doi.org/10.46298/dmtcs.6171 Jelínek, Vít Karpilovskij, Mark Jelínek, Vít Karpilovskij, Mark <![CDATA[A skew shape is the difference of two top-left justified Ferrers shapes sharing the same top-left corner. We study integer fillings of skew shapes. As our first main result, we show that for a specific hereditary class of skew shapes, which we call D-free shapes, the fillings that avoid a north-east chain of size $k$ are in bijection with fillings that avoid a south-east chain of the same size. Since Ferrers shapes are a subclass of D-free shapes, this result can be seen as a generalization of previous analogous results for Ferrers shapes. As our second main result, we construct a bijection between 01-fillings of an arbitrary skew shape that avoid a south-east chain of size 2, and the 01-fillings of the same shape that simultaneously avoid a north-east chain of size 2 and a particular non-square subfilling. This generalizes a previous result for transversal fillings.]]> 0 <![CDATA[We consider weighted tree automata (wta) over strong bimonoids and their initial algebra semantics and their run semantics. There are wta for which these semantics are different; however, for bottom-up deterministic wta and for wta over semirings, the difference vanishes. A wta is crisp-deterministic if it is bottom-up deterministic and each transition is weighted by one of the unit elements of the strong bimonoid. We prove that the class of weighted tree languages recognized by crisp-deterministic wta is the same as the class of recognizable step mappings. Moreover, we investigate the following two crisp-determinization problems: for a given wta ${\cal A}$, (a) does there exist a crisp-deterministic wta which computes the initial algebra semantics of ${\cal A}$ and (b) does there exist a crisp-deterministic wta which computes the run semantics of ${\cal A}$? We show that the finiteness of the Nerode algebra ${\cal N}({\cal A})$ of ${\cal A}$ implies a positive answer for (a), and that the finite order property of ${\cal A}$ implies a positive answer for (b). We show a sufficient condition which guarantees the finiteness of ${\cal N}({\cal A})$ and a sufficient condition which guarantees the finite order property of ${\cal A}$. Also, we provide an algorithm for the construction of the crisp-deterministic wta according to (a) if ${\cal N}({\cal A})$ is finite, and similarly for (b) if ${\cal A}$ has finite order property. We prove that it is undecidable whether an arbitrary wta ${\cal A}$ is crisp-determinizable. We also prove that both, the finiteness of ${\cal N}({\cal A})$ and the finite order property of ${\cal A}$ are undecidable.]]> Tue, 15 Jun 2021 07:40:34 +0000 https://doi.org/10.46298/dmtcs.5943 https://doi.org/10.46298/dmtcs.5943 Fülöp, Zoltán Kószó, Dávid Vogler, Heiko Fülöp, Zoltán Kószó, Dávid Vogler, Heiko <![CDATA[We consider weighted tree automata (wta) over strong bimonoids and their initial algebra semantics and their run semantics. There are wta for which these semantics are different; however, for bottom-up deterministic wta and for wta over semirings, the difference vanishes. A wta is crisp-deterministic if it is bottom-up deterministic and each transition is weighted by one of the unit elements of the strong bimonoid. We prove that the class of weighted tree languages recognized by crisp-deterministic wta is the same as the class of recognizable step mappings. Moreover, we investigate the following two crisp-determinization problems: for a given wta ${\cal A}$, (a) does there exist a crisp-deterministic wta which computes the initial algebra semantics of ${\cal A}$ and (b) does there exist a crisp-deterministic wta which computes the run semantics of ${\cal A}$? We show that the finiteness of the Nerode algebra ${\cal N}({\cal A})$ of ${\cal A}$ implies a positive answer for (a), and that the finite order property of ${\cal A}$ implies a positive answer for (b). We show a sufficient condition which guarantees the finiteness of ${\cal N}({\cal A})$ and a sufficient condition which guarantees the finite order property of ${\cal A}$. Also, we provide an algorithm for the construction of the crisp-deterministic wta according to (a) if ${\cal N}({\cal A})$ is finite, and similarly for (b) if ${\cal A}$ has finite order property. We prove that it is undecidable whether an arbitrary wta ${\cal A}$ is crisp-determinizable. We also prove that both, the finiteness of ${\cal N}({\cal A})$ and the finite order property of ${\cal A}$ are undecidable.]]> 0 <![CDATA[Edmonds, Lov\'asz, and Pulleyblank showed that if a matching covered graph has a nontrivial tight cut, then it also has a nontrivial ELP-cut. Carvalho et al. gave a stronger conjecture: if a matching covered graph has a nontrivial tight cut $C$, then it also has a nontrivial ELP-cut that does not cross $C$. Chen, et al gave a proof of the conjecture. This note is inspired by the paper of Carvalho et al. We give a simplified proof of the conjecture, and prove the following result which is slightly stronger than the conjecture: if a nontrivial tight cut $C$ of a matching covered graph $G$ is not an ELP-cut, then there is a sequence $G_1=G, G_2,\ldots,G_r, r\geq2$ of matching covered graphs, such that for $i=1, 2,\ldots, r-1$, $G_i$ has an ELP-cut $C_i$, and $G_{i+1}$ is a $C_i$-contraction of $G_i$, and $C$ is a $2$-separation cut of $G_r$.]]> Mon, 14 Jun 2021 09:51:56 +0000 https://doi.org/10.46298/dmtcs.6013 https://doi.org/10.46298/dmtcs.6013 Zhao, Xiao Chen, Sheng Zhao, Xiao Chen, Sheng <![CDATA[Edmonds, Lov\'asz, and Pulleyblank showed that if a matching covered graph has a nontrivial tight cut, then it also has a nontrivial ELP-cut. Carvalho et al. gave a stronger conjecture: if a matching covered graph has a nontrivial tight cut $C$, then it also has a nontrivial ELP-cut that does not cross $C$. Chen, et al gave a proof of the conjecture. This note is inspired by the paper of Carvalho et al. We give a simplified proof of the conjecture, and prove the following result which is slightly stronger than the conjecture: if a nontrivial tight cut $C$ of a matching covered graph $G$ is not an ELP-cut, then there is a sequence $G_1=G, G_2,\ldots,G_r, r\geq2$ of matching covered graphs, such that for $i=1, 2,\ldots, r-1$, $G_i$ has an ELP-cut $C_i$, and $G_{i+1}$ is a $C_i$-contraction of $G_i$, and $C$ is a $2$-separation cut of $G_r$.]]> 0 <![CDATA[We introduce and study the Bicolored $P_3$ Deletion problem defined as follows. The input is a graph $G=(V,E)$ where the edge set $E$ is partitioned into a set $E_r$ of red edges and a set $E_b$ of blue edges. The question is whether we can delete at most $k$ edges such that $G$ does not contain a bicolored $P_3$ as an induced subgraph. Here, a bicolored $P_3$ is a path on three vertices with one blue and one red edge. We show that Bicolored $P_3$ Deletion is NP-hard and cannot be solved in $2^{o(|V|+|E|)}$ time on bounded-degree graphs if the ETH is true. Then, we show that Bicolored $P_3$ Deletion is polynomial-time solvable when $G$ does not contain a bicolored $K_3$, that is, a triangle with edges of both colors. Moreover, we provide a polynomial-time algorithm for the case that $G$ contains no blue $P_3$, red $P_3$, blue $K_3$, and red $K_3$. Finally, we show that Bicolored $P_3$ Deletion can be solved in $ O(1.84^k\cdot |V| \cdot |E|)$ time and that it admits a kernel with $ O(k\Delta\min(k,\Delta))$ vertices, where $\Delta$ is the maximum degree of $G$.]]> Tue, 08 Jun 2021 08:23:01 +0000 https://doi.org/10.46298/dmtcs.6108 https://doi.org/10.46298/dmtcs.6108 Grüttemeier, Niels Komusiewicz, Christian Schestag, Jannik Sommer, Frank Grüttemeier, Niels Komusiewicz, Christian Schestag, Jannik Sommer, Frank <![CDATA[We introduce and study the Bicolored $P_3$ Deletion problem defined as follows. The input is a graph $G=(V,E)$ where the edge set $E$ is partitioned into a set $E_r$ of red edges and a set $E_b$ of blue edges. The question is whether we can delete at most $k$ edges such that $G$ does not contain a bicolored $P_3$ as an induced subgraph. Here, a bicolored $P_3$ is a path on three vertices with one blue and one red edge. We show that Bicolored $P_3$ Deletion is NP-hard and cannot be solved in $2^{o(|V|+|E|)}$ time on bounded-degree graphs if the ETH is true. Then, we show that Bicolored $P_3$ Deletion is polynomial-time solvable when $G$ does not contain a bicolored $K_3$, that is, a triangle with edges of both colors. Moreover, we provide a polynomial-time algorithm for the case that $G$ contains no blue $P_3$, red $P_3$, blue $K_3$, and red $K_3$. Finally, we show that Bicolored $P_3$ Deletion can be solved in $ O(1.84^k\cdot |V| \cdot |E|)$ time and that it admits a kernel with $ O(k\Delta\min(k,\Delta))$ vertices, where $\Delta$ is the maximum degree of $G$.]]> 0 <![CDATA[Fix a finite undirected graph $\Gamma$ and a vertex $v$ of $\Gamma$. Let $E$ be the set of edges of $\Gamma$. We call a subset $F$ of $E$ pandemic if each edge of $\Gamma$ has at least one endpoint that can be connected to $v$ by an $F$-path (i.e., a path using edges from $F$ only). In 1984, Elser showed that the sum of $\left(-1\right)^{\left| F\right|}$ over all pandemic subsets $F$ of $E$ is $0$ if $E\neq \varnothing$. We give a simple proof of this result via a sign-reversing involution, and discuss variants, generalizations and refinements, revealing connections to abstract convexity (the notion of an antimatroid) and discrete Morse theory.]]> Thu, 03 Jun 2021 09:41:13 +0000 https://doi.org/10.46298/dmtcs.7012 https://doi.org/10.46298/dmtcs.7012 Grinberg, Darij Grinberg, Darij <![CDATA[Fix a finite undirected graph $\Gamma$ and a vertex $v$ of $\Gamma$. Let $E$ be the set of edges of $\Gamma$. We call a subset $F$ of $E$ pandemic if each edge of $\Gamma$ has at least one endpoint that can be connected to $v$ by an $F$-path (i.e., a path using edges from $F$ only). In 1984, Elser showed that the sum of $\left(-1\right)^{\left| F\right|}$ over all pandemic subsets $F$ of $E$ is $0$ if $E\neq \varnothing$. We give a simple proof of this result via a sign-reversing involution, and discuss variants, generalizations and refinements, revealing connections to abstract convexity (the notion of an antimatroid) and discrete Morse theory.]]> 0 <![CDATA[Binary relations derived from labeled rooted trees play an import role in mathematical biology as formal models of evolutionary relationships. The (symmetrized) Fitch relation formalizes xenology as the pairs of genes separated by at least one horizontal transfer event. As a natural generalization, we consider symmetrized Fitch maps, that is, symmetric maps $\varepsilon$ that assign a subset of colors to each pair of vertices in $X$ and that can be explained by a tree $T$ with edges that are labeled with subsets of colors in the sense that the color $m$ appears in $\varepsilon(x,y)$ if and only if $m$ appears in a label along the unique path between $x$ and $y$ in $T$. We first give an alternative characterization of the monochromatic case and then give a characterization of symmetrized Fitch maps in terms of compatibility of a certain set of quartets. We show that recognition of symmetrized Fitch maps is NP-complete. In the restricted case where $|\varepsilon(x,y)|\leq 1$ the problem becomes polynomial, since such maps coincide with class of monochromatic Fitch maps whose graph-representations form precisely the class of complete multi-partite graphs.]]> Thu, 03 Jun 2021 09:37:26 +0000 https://doi.org/10.46298/dmtcs.6040 https://doi.org/10.46298/dmtcs.6040 Hellmuth, Marc Seemann, Carsten R. Stadler, Peter F. Hellmuth, Marc Seemann, Carsten R. Stadler, Peter F. <![CDATA[Binary relations derived from labeled rooted trees play an import role in mathematical biology as formal models of evolutionary relationships. The (symmetrized) Fitch relation formalizes xenology as the pairs of genes separated by at least one horizontal transfer event. As a natural generalization, we consider symmetrized Fitch maps, that is, symmetric maps $\varepsilon$ that assign a subset of colors to each pair of vertices in $X$ and that can be explained by a tree $T$ with edges that are labeled with subsets of colors in the sense that the color $m$ appears in $\varepsilon(x,y)$ if and only if $m$ appears in a label along the unique path between $x$ and $y$ in $T$. We first give an alternative characterization of the monochromatic case and then give a characterization of symmetrized Fitch maps in terms of compatibility of a certain set of quartets. We show that recognition of symmetrized Fitch maps is NP-complete. In the restricted case where $|\varepsilon(x,y)|\leq 1$ the problem becomes polynomial, since such maps coincide with class of monochromatic Fitch maps whose graph-representations form precisely the class of complete multi-partite graphs.]]> 0 <![CDATA[Let $G$ be a connected graph of order $n$.The Wiener index $W(G)$ of $G$ is the sum of the distances between all unordered pairs of vertices of $G$. In this paper we show that the well-known upper bound $\big( \frac{n}{\delta+1}+2\big) {n \choose 2}$ on the Wiener index of a graph of order $n$ and minimum degree $\delta$ [M. Kouider, P. Winkler, Mean distance and minimum degree. J. Graph Theory 25 no. 1 (1997)] can be improved significantly if the graph contains also a vertex of large degree. Specifically, we give the asymptotically sharp bound $W(G) \leq {n-\Delta+\delta \choose 2} \frac{n+2\Delta}{\delta+1}+ 2n(n-1)$ on the Wiener index of a graph $G$ of order $n$, minimum degree $\delta$ and maximum degree $\Delta$. We prove a similar result for triangle-free graphs, and we determine a bound on the Wiener index of $C_4$-free graphs of given order, minimum and maximum degree and show that it is, in some sense, best possible.]]> Thu, 03 Jun 2021 09:31:33 +0000 https://doi.org/10.46298/dmtcs.6956 https://doi.org/10.46298/dmtcs.6956 Dankelmann, Peter Alochukwu, Alex Dankelmann, Peter Alochukwu, Alex <![CDATA[Let $G$ be a connected graph of order $n$.The Wiener index $W(G)$ of $G$ is the sum of the distances between all unordered pairs of vertices of $G$. In this paper we show that the well-known upper bound $\big( \frac{n}{\delta+1}+2\big) {n \choose 2}$ on the Wiener index of a graph of order $n$ and minimum degree $\delta$ [M. Kouider, P. Winkler, Mean distance and minimum degree. J. Graph Theory 25 no. 1 (1997)] can be improved significantly if the graph contains also a vertex of large degree. Specifically, we give the asymptotically sharp bound $W(G) \leq {n-\Delta+\delta \choose 2} \frac{n+2\Delta}{\delta+1}+ 2n(n-1)$ on the Wiener index of a graph $G$ of order $n$, minimum degree $\delta$ and maximum degree $\Delta$. We prove a similar result for triangle-free graphs, and we determine a bound on the Wiener index of $C_4$-free graphs of given order, minimum and maximum degree and show that it is, in some sense, best possible.]]> 0 <![CDATA[This paper introduces a notion of equivalence for higher-dimensional automata, called weak equivalence. Weak equivalence focuses mainly on a traditional trace language and a new homology language, which captures the overall independence structure of an HDA. It is shown that weak equivalence is compatible with both the tensor product and the coproduct of HDAs and that, under certain conditions, HDAs may be reduced to weakly equivalent smaller ones by merging and collapsing cubes.]]> Tue, 18 May 2021 09:47:50 +0000 https://doi.org/10.46298/dmtcs.5884 https://doi.org/10.46298/dmtcs.5884 Kahl, Thomas Kahl, Thomas <![CDATA[This paper introduces a notion of equivalence for higher-dimensional automata, called weak equivalence. Weak equivalence focuses mainly on a traditional trace language and a new homology language, which captures the overall independence structure of an HDA. It is shown that weak equivalence is compatible with both the tensor product and the coproduct of HDAs and that, under certain conditions, HDAs may be reduced to weakly equivalent smaller ones by merging and collapsing cubes.]]> 0 <![CDATA[Flip-sort is a natural sorting procedure which raises fascinating combinatorial questions. It finds its roots in the seminal work of Knuth on stack-based sorting algorithms and leads to many links with permutation patterns. We present several structural, enumerative, and algorithmic results on permutations that need few (resp. many) iterations of this procedure to be sorted. In particular, we give the shape of the permutations after one iteration, and characterize several families of permutations related to the best and worst cases of flip-sort. En passant, we also give some links between pop-stack sorting, automata, and lattice paths, and introduce several tactics of bijective proofs which have their own interest.]]> Fri, 30 Apr 2021 08:26:14 +0000 https://doi.org/10.46298/dmtcs.6196 https://doi.org/10.46298/dmtcs.6196 Asinowski, Andrei Banderier, Cyril Hackl, Benjamin Asinowski, Andrei Banderier, Cyril Hackl, Benjamin <![CDATA[Flip-sort is a natural sorting procedure which raises fascinating combinatorial questions. It finds its roots in the seminal work of Knuth on stack-based sorting algorithms and leads to many links with permutation patterns. We present several structural, enumerative, and algorithmic results on permutations that need few (resp. many) iterations of this procedure to be sorted. In particular, we give the shape of the permutations after one iteration, and characterize several families of permutations related to the best and worst cases of flip-sort. En passant, we also give some links between pop-stack sorting, automata, and lattice paths, and introduce several tactics of bijective proofs which have their own interest.]]> 0 <![CDATA[A mixed dominating set is a collection of vertices and edges that dominates all vertices and edges of a graph. We study the complexity of exact and parameterized algorithms for \textsc{Mixed Dominating Set}, resolving some open questions. In particular, we settle the problem's complexity parameterized by treewidth and pathwidth by giving an algorithm running in time $O^*(5^{tw})$ (improving the current best $O^*(6^{tw})$), as well as a lower bound showing that our algorithm cannot be improved under the Strong Exponential Time Hypothesis (SETH), even if parameterized by pathwidth (improving a lower bound of $O^*((2 - \varepsilon)^{pw})$). Furthermore, by using a simple but so far overlooked observation on the structure of minimal solutions, we obtain branching algorithms which improve both the best known FPT algorithm for this problem, from $O^*(4.172^k)$ to $O^*(3.510^k)$, and the best known exponential-time exact algorithm, from $O^*(2^n)$ and exponential space, to $O^*(1.912^n)$ and polynomial space.]]> Fri, 30 Apr 2021 08:23:11 +0000 https://doi.org/10.46298/dmtcs.6824 https://doi.org/10.46298/dmtcs.6824 Dublois, Louis Lampis, Michael Paschos, Vangelis Th. Dublois, Louis Lampis, Michael Paschos, Vangelis Th. <![CDATA[A mixed dominating set is a collection of vertices and edges that dominates all vertices and edges of a graph. We study the complexity of exact and parameterized algorithms for \textsc{Mixed Dominating Set}, resolving some open questions. In particular, we settle the problem's complexity parameterized by treewidth and pathwidth by giving an algorithm running in time $O^*(5^{tw})$ (improving the current best $O^*(6^{tw})$), as well as a lower bound showing that our algorithm cannot be improved under the Strong Exponential Time Hypothesis (SETH), even if parameterized by pathwidth (improving a lower bound of $O^*((2 - \varepsilon)^{pw})$). Furthermore, by using a simple but so far overlooked observation on the structure of minimal solutions, we obtain branching algorithms which improve both the best known FPT algorithm for this problem, from $O^*(4.172^k)$ to $O^*(3.510^k)$, and the best known exponential-time exact algorithm, from $O^*(2^n)$ and exponential space, to $O^*(1.912^n)$ and polynomial space.]]> 0 <![CDATA[Let $\lambda$ be a partition with no more than $n$ parts. Let $\beta$ be a weakly increasing $n$-tuple with entries from $\{ 1, ... , n \}$. The flagged Schur function in the variables $x_1, ... , x_n$ that is indexed by $\lambda$ and $\beta$ has been defined to be the sum of the content weight monomials for the semistandard Young tableaux of shape $\lambda$ whose values are row-wise bounded by the entries of $\beta$. Gessel and Viennot gave a determinant expression for the flagged Schur function indexed by $\lambda$ and $\beta$; this could be done since the pair $(\lambda, \beta)$ satisfied their "nonpermutable" condition for the sequence of terminals of an $n$-tuple of lattice paths that they used to model the tableaux. We generalize flagged Schur functions by dropping the requirement that $\beta$ be weakly increasing. Then for each $\lambda$ we give a condition on the entries of $\beta$ for the pair $(\lambda, \beta)$ to be nonpermutable that is both necessary and sufficient. When the parts of $\lambda$ are not distinct there will be multiple row bound $n$-tuples $\beta$ that will produce the same set of tableaux. We accordingly group the bounding $\beta$ into equivalence classes and identify the most efficient $\beta$ in each class for the determinant computation. We recently showed that many other sets of objects that are indexed by $n$ and $\lambda$ are enumerated by the number of these efficient $n$-tuples. We called these counts "parabolic Catalan numbers". It is noted that the $GL(n)$ Demazure characters (key polynomials) indexed by 312-avoiding permutations can also be expressed with these determinants.]]> Fri, 23 Apr 2021 09:42:21 +0000 https://doi.org/10.46298/dmtcs.6632 https://doi.org/10.46298/dmtcs.6632 Proctor, Robert A. Willis, Matthew J. Proctor, Robert A. Willis, Matthew J. <![CDATA[Let $\lambda$ be a partition with no more than $n$ parts. Let $\beta$ be a weakly increasing $n$-tuple with entries from $\{ 1, ... , n \}$. The flagged Schur function in the variables $x_1, ... , x_n$ that is indexed by $\lambda$ and $\beta$ has been defined to be the sum of the content weight monomials for the semistandard Young tableaux of shape $\lambda$ whose values are row-wise bounded by the entries of $\beta$. Gessel and Viennot gave a determinant expression for the flagged Schur function indexed by $\lambda$ and $\beta$; this could be done since the pair $(\lambda, \beta)$ satisfied their "nonpermutable" condition for the sequence of terminals of an $n$-tuple of lattice paths that they used to model the tableaux. We generalize flagged Schur functions by dropping the requirement that $\beta$ be weakly increasing. Then for each $\lambda$ we give a condition on the entries of $\beta$ for the pair $(\lambda, \beta)$ to be nonpermutable that is both necessary and sufficient. When the parts of $\lambda$ are not distinct there will be multiple row bound $n$-tuples $\beta$ that will produce the same set of tableaux. We accordingly group the bounding $\beta$ into equivalence classes and identify the most efficient $\beta$ in each class for the determinant computation. We recently showed that many other sets of objects that are indexed by $n$ and $\lambda$ are enumerated by the number of these efficient $n$-tuples. We called these counts "parabolic Catalan numbers". It is noted that the $GL(n)$ Demazure characters (key polynomials) indexed by 312-avoiding permutations can also be expressed with these determinants.]]> 0 <![CDATA[Catalan words are particular growth-restricted words counted by the eponymous integer sequence. In this article we consider Catalan words avoiding a pair of patterns of length 3, pursuing the recent initiating work of the first and last authors and of S. Kirgizov where (among other things) the enumeration of Catalan words avoiding a patterns of length 3 is completed. More precisely, we explore systematically the structural properties of the sets of words under consideration and give enumerating results by means of recursive decomposition, constructive bijections or bivariate generating functions with respect to the length and descent number. Some of the obtained enumerating sequences are known, and thus the corresponding results establish new combinatorial interpretations for them.]]> Fri, 16 Apr 2021 07:43:30 +0000 https://doi.org/10.46298/dmtcs.6002 https://doi.org/10.46298/dmtcs.6002 Baril, Jean-Luc Khalil, Carine Vajnovszki, Vincent Baril, Jean-Luc Khalil, Carine Vajnovszki, Vincent <![CDATA[Catalan words are particular growth-restricted words counted by the eponymous integer sequence. In this article we consider Catalan words avoiding a pair of patterns of length 3, pursuing the recent initiating work of the first and last authors and of S. Kirgizov where (among other things) the enumeration of Catalan words avoiding a patterns of length 3 is completed. More precisely, we explore systematically the structural properties of the sets of words under consideration and give enumerating results by means of recursive decomposition, constructive bijections or bivariate generating functions with respect to the length and descent number. Some of the obtained enumerating sequences are known, and thus the corresponding results establish new combinatorial interpretations for them.]]> 0 <![CDATA[We give three applications of a recently-proven "Decomposition Lemma," which allows one to count preimages of certain sets of permutations under West's stack-sorting map $s$. We first enumerate the permutation class $s^{-1}(\text{Av}(231,321))=\text{Av}(2341,3241,45231)$, finding a new example of an unbalanced Wilf equivalence. This result is equivalent to the enumeration of permutations sortable by ${\bf B}\circ s$, where ${\bf B}$ is the bubble sort map. We then prove that the sets $s^{-1}(\text{Av}(231,312))$, $s^{-1}(\text{Av}(132,231))=\text{Av}(2341,1342,\underline{32}41,\underline{31}42)$, and $s^{-1}(\text{Av}(132,312))=\text{Av}(1342,3142,3412,34\underline{21})$ are counted by the so-called "Boolean-Catalan numbers," settling a conjecture of the current author and another conjecture of Hossain. This completes the enumerations of all sets of the form $s^{-1}(\text{Av}(\tau^{(1)},\ldots,\tau^{(r)}))$ for $\{\tau^{(1)},\ldots,\tau^{(r)}\}\subseteq S_3$ with the exception of the set $\{321\}$. We also find an explicit formula for $|s^{-1}(\text{Av}_{n,k}(231,312,321))|$, where $\text{Av}_{n,k}(231,312,321)$ is the set of permutations in $\text{Av}_n(231,312,321)$ with $k$ descents. This allows us to prove a conjectured identity involving Catalan numbers and order ideals in Young's lattice.]]> Fri, 09 Apr 2021 15:03:26 +0000 https://doi.org/10.46298/dmtcs.6709 https://doi.org/10.46298/dmtcs.6709 Defant, Colin Defant, Colin <![CDATA[We give three applications of a recently-proven "Decomposition Lemma," which allows one to count preimages of certain sets of permutations under West's stack-sorting map $s$. We first enumerate the permutation class $s^{-1}(\text{Av}(231,321))=\text{Av}(2341,3241,45231)$, finding a new example of an unbalanced Wilf equivalence. This result is equivalent to the enumeration of permutations sortable by ${\bf B}\circ s$, where ${\bf B}$ is the bubble sort map. We then prove that the sets $s^{-1}(\text{Av}(231,312))$, $s^{-1}(\text{Av}(132,231))=\text{Av}(2341,1342,\underline{32}41,\underline{31}42)$, and $s^{-1}(\text{Av}(132,312))=\text{Av}(1342,3142,3412,34\underline{21})$ are counted by the so-called "Boolean-Catalan numbers," settling a conjecture of the current author and another conjecture of Hossain. This completes the enumerations of all sets of the form $s^{-1}(\text{Av}(\tau^{(1)},\ldots,\tau^{(r)}))$ for $\{\tau^{(1)},\ldots,\tau^{(r)}\}\subseteq S_3$ with the exception of the set $\{321\}$. We also find an explicit formula for $|s^{-1}(\text{Av}_{n,k}(231,312,321))|$, where $\text{Av}_{n,k}(231,312,321)$ is the set of permutations in $\text{Av}_n(231,312,321)$ with $k$ descents. This allows us to prove a conjectured identity involving Catalan numbers and order ideals in Young's lattice.]]> 0 <![CDATA[In this paper, we study the staircase encoding of permutations, which maps a permutation to a staircase grid with cells filled with permutations. We consider many cases, where restricted to a permutation class, the staircase encoding becomes a bijection to its image. We describe the image of those restrictions using independent sets of graphs weighted with permutations. We derive the generating function for the independent sets and then for their weighted counterparts. The bijections we establish provide the enumeration of permutation classes. We use our results to uncover some unbalanced Wilf-equivalences of permutation classes and outline how to do random sampling in the permutation classes. In particular, we cover the classes $\mathrm{Av}(2314,3124)$, $\mathrm{Av}(2413,3142)$, $\mathrm{Av}(2413,3124)$, $\mathrm{Av}(2413,2134)$ and $\mathrm{Av}(2314,2143)$, as well as many subclasses.]]> Mon, 29 Mar 2021 09:02:41 +0000 https://doi.org/10.46298/dmtcs.5995 https://doi.org/10.46298/dmtcs.5995 Bean, Christian Nadeau, Émile Ulfarsson, Henning Bean, Christian Nadeau, Émile Ulfarsson, Henning <![CDATA[In this paper, we study the staircase encoding of permutations, which maps a permutation to a staircase grid with cells filled with permutations. We consider many cases, where restricted to a permutation class, the staircase encoding becomes a bijection to its image. We describe the image of those restrictions using independent sets of graphs weighted with permutations. We derive the generating function for the independent sets and then for their weighted counterparts. The bijections we establish provide the enumeration of permutation classes. We use our results to uncover some unbalanced Wilf-equivalences of permutation classes and outline how to do random sampling in the permutation classes. In particular, we cover the classes $\mathrm{Av}(2314,3124)$, $\mathrm{Av}(2413,3142)$, $\mathrm{Av}(2413,3124)$, $\mathrm{Av}(2413,2134)$ and $\mathrm{Av}(2314,2143)$, as well as many subclasses.]]> 0 <![CDATA[We introduce a new boundedness condition for affine permutations, motivated by the fruitful concept of periodic boundary conditions in statistical physics. We study pattern avoidance in bounded affine permutations. In particular, we show that if $\tau$ is one of the finite increasing oscillations, then every $\tau$-avoiding affine permutation satisfies the boundedness condition. We also explore the enumeration of pattern-avoiding affine permutations that can be decomposed into blocks, using analytic methods to relate their exact and asymptotic enumeration to that of the underlying ordinary permutations. Finally, we perform exact and asymptotic enumeration of the set of all bounded affine permutations of size $n$. A companion paper will focus on avoidance of monotone decreasing patterns in bounded affine permutations.]]> Mon, 29 Mar 2021 08:57:54 +0000 https://doi.org/10.46298/dmtcs.6178 https://doi.org/10.46298/dmtcs.6178 Madras, Neal Troyka, Justin M. Madras, Neal Troyka, Justin M. <![CDATA[We introduce a new boundedness condition for affine permutations, motivated by the fruitful concept of periodic boundary conditions in statistical physics. We study pattern avoidance in bounded affine permutations. In particular, we show that if $\tau$ is one of the finite increasing oscillations, then every $\tau$-avoiding affine permutation satisfies the boundedness condition. We also explore the enumeration of pattern-avoiding affine permutations that can be decomposed into blocks, using analytic methods to relate their exact and asymptotic enumeration to that of the underlying ordinary permutations. Finally, we perform exact and asymptotic enumeration of the set of all bounded affine permutations of size $n$. A companion paper will focus on avoidance of monotone decreasing patterns in bounded affine permutations.]]> 0 <![CDATA[We consider non-trivial homomorphisms to reflexive oriented graphs in which some pair of adjacent vertices have the same image. Using a notion of convexity for oriented graphs, we study those oriented graphs that do not admit such homomorphisms. We fully classify those oriented graphs with tree-width $2$ that do not admit such homomorphisms and show that it is NP-complete to decide if a graph admits an orientation that does not admit such homomorphisms. We prove analogous results for $2$-edge-coloured graphs. We apply our results on oriented graphs to provide a new tool in the study of chromatic number of orientations of planar graphs -- a long-standing open problem.]]> Mon, 29 Mar 2021 08:53:54 +0000 https://doi.org/10.46298/dmtcs.6773 https://doi.org/10.46298/dmtcs.6773 Duffy, Christopher Shan, Sonja Linghui Duffy, Christopher Shan, Sonja Linghui <![CDATA[We consider non-trivial homomorphisms to reflexive oriented graphs in which some pair of adjacent vertices have the same image. Using a notion of convexity for oriented graphs, we study those oriented graphs that do not admit such homomorphisms. We fully classify those oriented graphs with tree-width $2$ that do not admit such homomorphisms and show that it is NP-complete to decide if a graph admits an orientation that does not admit such homomorphisms. We prove analogous results for $2$-edge-coloured graphs. We apply our results on oriented graphs to provide a new tool in the study of chromatic number of orientations of planar graphs -- a long-standing open problem.]]> 0 <![CDATA[The forbidden number $\mathrm{forb}(m,F)$, which denotes the maximum number of unique columns in an $m$-rowed $(0,1)$-matrix with no submatrix that is a row and column permutation of $F$, has been widely studied in extremal set theory. Recently, this function was extended to $r$-matrices, whose entries lie in $\{0,1,\dots,r-1\}$. The combinatorics of the generalized forbidden number is less well-studied. In this paper, we provide exact bounds for many $(0,1)$-matrices $F$, including all $2$-rowed matrices when $r > 3$. We also prove a stability result for the $2\times 2$ identity matrix. Along the way, we expose some interesting qualitative differences between the cases $r=2$, $r = 3$, and $r > 3$.]]> Tue, 23 Mar 2021 09:42:29 +0000 https://doi.org/10.46298/dmtcs.6613 https://doi.org/10.46298/dmtcs.6613 Dillon, Travis Sali, Attila Dillon, Travis Sali, Attila <![CDATA[The forbidden number $\mathrm{forb}(m,F)$, which denotes the maximum number of unique columns in an $m$-rowed $(0,1)$-matrix with no submatrix that is a row and column permutation of $F$, has been widely studied in extremal set theory. Recently, this function was extended to $r$-matrices, whose entries lie in $\{0,1,\dots,r-1\}$. The combinatorics of the generalized forbidden number is less well-studied. In this paper, we provide exact bounds for many $(0,1)$-matrices $F$, including all $2$-rowed matrices when $r > 3$. We also prove a stability result for the $2\times 2$ identity matrix. Along the way, we expose some interesting qualitative differences between the cases $r=2$, $r = 3$, and $r > 3$.]]> 0 <![CDATA[Let $G$ be a a connected graph. The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices. We provide asymptotic formulae for the maximum Wiener index of simple triangulations and quadrangulations with given connectivity, as the order increases, and make conjectures for the extremal triangulations and quadrangulations based on computational evidence. If $\overline{\sigma}(v)$ denotes the arithmetic mean of the distances from $v$ to all other vertices of $G$, then the remoteness of $G$ is defined as the largest value of $\overline{\sigma}(v)$ over all vertices $v$ of $G$. We give sharp upper bounds on the remoteness of simple triangulations and quadrangulations of given order and connectivity.]]> Mon, 08 Mar 2021 08:48:59 +0000 https://doi.org/10.46298/dmtcs.6473 https://doi.org/10.46298/dmtcs.6473 Czabarka, Éva Dankelmann, Peter Olsen, Trevor Székely, László A. Czabarka, Éva Dankelmann, Peter Olsen, Trevor Székely, László A. <![CDATA[Let $G$ be a a connected graph. The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices. We provide asymptotic formulae for the maximum Wiener index of simple triangulations and quadrangulations with given connectivity, as the order increases, and make conjectures for the extremal triangulations and quadrangulations based on computational evidence. If $\overline{\sigma}(v)$ denotes the arithmetic mean of the distances from $v$ to all other vertices of $G$, then the remoteness of $G$ is defined as the largest value of $\overline{\sigma}(v)$ over all vertices $v$ of $G$. We give sharp upper bounds on the remoteness of simple triangulations and quadrangulations of given order and connectivity.]]> 0 <![CDATA[Recently, Yamazaki et al. provided an algorithm that enumerates all non-isomorphic interval graphs on $n$ vertices with an $O(n^4)$ time delay. In this paper, we improve their algorithm and achieve $O(n^3 \log n)$ time delay. We also extend the catalog of these graphs providing a list of all non-isomorphic interval graphs for all $n$ up to $15$.]]> Mon, 08 Mar 2021 08:45:41 +0000 https://doi.org/10.46298/dmtcs.6164 https://doi.org/10.46298/dmtcs.6164 Mikos, Patryk Mikos, Patryk <![CDATA[Recently, Yamazaki et al. provided an algorithm that enumerates all non-isomorphic interval graphs on $n$ vertices with an $O(n^4)$ time delay. In this paper, we improve their algorithm and achieve $O(n^3 \log n)$ time delay. We also extend the catalog of these graphs providing a list of all non-isomorphic interval graphs for all $n$ up to $15$.]]> 0 <![CDATA[Recently, Fici, Restivo, Silva, and Zamboni introduced the notion of a $k$-anti-power, which is defined as a word of the form $w^{(1)} w^{(2)} \cdots w^{(k)}$, where $w^{(1)}, w^{(2)}, \ldots, w^{(k)}$ are distinct words of the same length. For an infinite word $w$ and a positive integer $k$, define $AP_j(w,k)$ to be the set of all integers $m$ such that $w_{j+1} w_{j+2} \cdots w_{j+km}$ is a $k$-anti-power, where $w_i$ denotes the $i$-th letter of $w$. Define also $\mathcal{F}_j(k) = (2 \mathbb{Z}^+ - 1) \cap AP_j(\mathbf{t},k)$, where $\mathbf{t}$ denotes the Thue-Morse word. For all $k \in \mathbb{Z}^+$, $\gamma_j(k) = \min (AP_j(\mathbf{t},k))$ is a well-defined positive integer, and for $k \in \mathbb{Z}^+$ sufficiently large, $\Gamma_j(k) = \sup ((2 \mathbb{Z}^+ -1) \setminus \mathcal{F}_j(k))$ is a well-defined odd positive integer. In his 2018 paper, Defant shows that $\gamma_0(k)$ and $\Gamma_0(k)$ grow linearly in $k$. We generalize Defant's methods to prove that $\gamma_j(k)$ and $\Gamma_j(k)$ grow linearly in $k$ for any nonnegative integer $j$. In particular, we show that $\displaystyle 1/10 \leq \liminf_{k \rightarrow \infty} (\gamma_j(k)/k) \leq 9/10$ and $\displaystyle 1/5 \leq \limsup_{k \rightarrow \infty} (\gamma_j(k)/k) \leq 3/2$. Additionally, we show that $\displaystyle \liminf_{k \rightarrow \infty} (\Gamma_j(k)/k) = 3/2$ and $\displaystyle \limsup_{k \rightarrow \infty} (\Gamma_j(k)/k) = 3$.]]> Thu, 25 Feb 2021 09:28:02 +0000 https://doi.org/10.46298/dmtcs.5483 https://doi.org/10.46298/dmtcs.5483 Gaetz, Marisa Gaetz, Marisa <![CDATA[Recently, Fici, Restivo, Silva, and Zamboni introduced the notion of a $k$-anti-power, which is defined as a word of the form $w^{(1)} w^{(2)} \cdots w^{(k)}$, where $w^{(1)}, w^{(2)}, \ldots, w^{(k)}$ are distinct words of the same length. For an infinite word $w$ and a positive integer $k$, define $AP_j(w,k)$ to be the set of all integers $m$ such that $w_{j+1} w_{j+2} \cdots w_{j+km}$ is a $k$-anti-power, where $w_i$ denotes the $i$-th letter of $w$. Define also $\mathcal{F}_j(k) = (2 \mathbb{Z}^+ - 1) \cap AP_j(\mathbf{t},k)$, where $\mathbf{t}$ denotes the Thue-Morse word. For all $k \in \mathbb{Z}^+$, $\gamma_j(k) = \min (AP_j(\mathbf{t},k))$ is a well-defined positive integer, and for $k \in \mathbb{Z}^+$ sufficiently large, $\Gamma_j(k) = \sup ((2 \mathbb{Z}^+ -1) \setminus \mathcal{F}_j(k))$ is a well-defined odd positive integer. In his 2018 paper, Defant shows that $\gamma_0(k)$ and $\Gamma_0(k)$ grow linearly in $k$. We generalize Defant's methods to prove that $\gamma_j(k)$ and $\Gamma_j(k)$ grow linearly in $k$ for any nonnegative integer $j$. In particular, we show that $\displaystyle 1/10 \leq \liminf_{k \rightarrow \infty} (\gamma_j(k)/k) \leq 9/10$ and $\displaystyle 1/5 \leq \limsup_{k \rightarrow \infty} (\gamma_j(k)/k) \leq 3/2$. Additionally, we show that $\displaystyle \liminf_{k \rightarrow \infty} (\Gamma_j(k)/k) = 3/2$ and $\displaystyle \limsup_{k \rightarrow \infty} (\Gamma_j(k)/k) = 3$.]]> 0 <![CDATA[In a recent work, Bensmail, Blanc, Cohen, Havet and Rocha, motivated by applications for TDMA scheduling problems, have introduced the notion of BMRN*-colouring of digraphs, which is a type of arc-colouring with particular colouring constraints. In particular, they gave a special focus to planar digraphs. They notably proved that every planar digraph can be 8-BMRN*-coloured, while there exist planar digraphs for which 7 colours are needed in a BMRN*-colouring. They also proved that the problem of deciding whether a planar digraph can be 3-BMRN*-coloured is NP-hard. In this work, we pursue these investigations on planar digraphs, in particular by answering some of the questions left open by the authors in that seminal work. We exhibit planar digraphs needing 8 colours to be BMRN*-coloured, thus showing that the upper bound of Bensmail, Blanc, Cohen, Havet and Rocha cannot be decreased in general. We also generalize their complexity result by showing that the problem of deciding whether a planar digraph can be k-BMRN*-coloured is NP-hard for every k ∈ {3,...,6}. Finally, we investigate the connection between the girth of a planar digraphs and the least number of colours in its BMRN*-colourings.]]> Thu, 25 Feb 2021 09:24:16 +0000 https://doi.org/10.46298/dmtcs.5798 https://doi.org/10.46298/dmtcs.5798 Bensmail, Julien Fioravantes, Foivos Bensmail, Julien Fioravantes, Foivos <![CDATA[In a recent work, Bensmail, Blanc, Cohen, Havet and Rocha, motivated by applications for TDMA scheduling problems, have introduced the notion of BMRN*-colouring of digraphs, which is a type of arc-colouring with particular colouring constraints. In particular, they gave a special focus to planar digraphs. They notably proved that every planar digraph can be 8-BMRN*-coloured, while there exist planar digraphs for which 7 colours are needed in a BMRN*-colouring. They also proved that the problem of deciding whether a planar digraph can be 3-BMRN*-coloured is NP-hard. In this work, we pursue these investigations on planar digraphs, in particular by answering some of the questions left open by the authors in that seminal work. We exhibit planar digraphs needing 8 colours to be BMRN*-coloured, thus showing that the upper bound of Bensmail, Blanc, Cohen, Havet and Rocha cannot be decreased in general. We also generalize their complexity result by showing that the problem of deciding whether a planar digraph can be k-BMRN*-coloured is NP-hard for every k ∈ {3,...,6}. Finally, we investigate the connection between the girth of a planar digraphs and the least number of colours in its BMRN*-colourings.]]> 0 <![CDATA[In this paper we present an average-case analysis of closed lambda terms with restricted values of De Bruijn indices in the model where each occurrence of a variable contributes one to the size. Given a fixed integer k, a lambda term in which all De Bruijn indices are bounded by k has the following shape: It starts with k De Bruijn levels, forming the so-called hat of the term, to which some number of k-colored Motzkin trees are attached. By means of analytic combinatorics, we show that the size of this hat is constant on average and that the average number of De Bruijn levels of k-colored Motzkin trees of size n is asymptotically Θ(√ n). Combining these two facts, we conclude that the maximal non-empty De Bruijn level in a lambda term with restrictions on De Bruijn indices and of size n is, on average, also of order √ n. On this basis, we provide the average unary profile of such lambda terms.]]> Fri, 12 Feb 2021 09:04:37 +0000 https://doi.org/10.46298/dmtcs.5836 https://doi.org/10.46298/dmtcs.5836 Grygiel, Katarzyna Larcher, Isabella Grygiel, Katarzyna Larcher, Isabella <![CDATA[In this paper we present an average-case analysis of closed lambda terms with restricted values of De Bruijn indices in the model where each occurrence of a variable contributes one to the size. Given a fixed integer k, a lambda term in which all De Bruijn indices are bounded by k has the following shape: It starts with k De Bruijn levels, forming the so-called hat of the term, to which some number of k-colored Motzkin trees are attached. By means of analytic combinatorics, we show that the size of this hat is constant on average and that the average number of De Bruijn levels of k-colored Motzkin trees of size n is asymptotically Θ(√ n). Combining these two facts, we conclude that the maximal non-empty De Bruijn level in a lambda term with restrictions on De Bruijn indices and of size n is, on average, also of order √ n. On this basis, we provide the average unary profile of such lambda terms.]]> 0 <![CDATA[A sequence of geometric random variables of length $n$ is a sequence of $n$ independent and identically distributed geometric random variables ($\Gamma_1, \Gamma_2, \dots, \Gamma_n$) where $\mathbb{P}(\Gamma_j=i)=pq^{i-1}$ for $1~\leq~j~\leq~n$ with $p+q=1.$ We study the number of distinct adjacent two letter patterns in such sequences. Initially we directly count the number of distinct pairs in words of short length. Because of the rapid growth of the number of word patterns we change our approach to this problem by obtaining an expression for the expected number of distinct pairs in words of length $n$. We also obtain the asymptotics for the expected number as $n \to \infty$.]]> Thu, 28 Jan 2021 14:22:47 +0000 https://doi.org/10.23638/DMTCS-22-4-10 https://doi.org/10.23638/DMTCS-22-4-10 Archibald, Margaret Blecher, Aubrey Brennan, Charlotte Knopfmacher, Arnold Wagner, Stephan Ward, Mark Archibald, Margaret Blecher, Aubrey Brennan, Charlotte Knopfmacher, Arnold Wagner, Stephan Ward, Mark <![CDATA[A sequence of geometric random variables of length $n$ is a sequence of $n$ independent and identically distributed geometric random variables ($\Gamma_1, \Gamma_2, \dots, \Gamma_n$) where $\mathbb{P}(\Gamma_j=i)=pq^{i-1}$ for $1~\leq~j~\leq~n$ with $p+q=1.$ We study the number of distinct adjacent two letter patterns in such sequences. Initially we directly count the number of distinct pairs in words of short length. Because of the rapid growth of the number of word patterns we change our approach to this problem by obtaining an expression for the expected number of distinct pairs in words of length $n$. We also obtain the asymptotics for the expected number as $n \to \infty$.]]> 0 <![CDATA[In this paper we address the problem of understanding Concurrency Theory from a combinatorial point of view. We are interested in quantitative results and algorithmic tools to refine our understanding of the classical combinatorial explosion phenomenon arising in concurrency. This paper is essentially focusing on the the notion of synchronization from the point of view of combinatorics. As a first step, we address the quantitative problem of counting the number of executions of simple processes interacting with synchronization barriers. We elaborate a systematic decomposition of processes that produces a symbolic integral formula to solve the problem. Based on this procedure, we develop a generic algorithm to generate process executions uniformly at random. For some interesting sub-classes of processes we propose very efficient counting and random sampling algorithms. All these algorithms have one important characteristic in common: they work on the control graph of processes and thus do not require the explicit construction of the state-space.]]> Mon, 25 Jan 2021 13:58:42 +0000 https://doi.org/10.46298/dmtcs.5820 https://doi.org/10.46298/dmtcs.5820 Bodini, OLivier Dien, Matthieu Genitrini, Antoine Peschanski, Frédéric Bodini, OLivier Dien, Matthieu Genitrini, Antoine Peschanski, Frédéric <![CDATA[In this paper we address the problem of understanding Concurrency Theory from a combinatorial point of view. We are interested in quantitative results and algorithmic tools to refine our understanding of the classical combinatorial explosion phenomenon arising in concurrency. This paper is essentially focusing on the the notion of synchronization from the point of view of combinatorics. As a first step, we address the quantitative problem of counting the number of executions of simple processes interacting with synchronization barriers. We elaborate a systematic decomposition of processes that produces a symbolic integral formula to solve the problem. Based on this procedure, we develop a generic algorithm to generate process executions uniformly at random. For some interesting sub-classes of processes we propose very efficient counting and random sampling algorithms. All these algorithms have one important characteristic in common: they work on the control graph of processes and thus do not require the explicit construction of the state-space.]]> 0 <![CDATA[Y. Manoussakis (J. Graph Theory 16, 1992, 51-59) proposed the following conjecture. \noindent\textbf{Conjecture}. {\it Let $D$ be a 2-strongly connected digraph of order $n$ such that for all distinct pairs of non-adjacent vertices $x$, $y$ and $w$, $z$, we have $d(x)+d(y)+d(w)+d(z)\geq 4n-3$. Then $D$ is Hamiltonian.} In this paper, we confirm this conjecture. Moreover, we prove that if a digraph $D$ satisfies the conditions of this conjecture and has a pair of non-adjacent vertices $\{x,y\}$ such that $d(x)+d(y)\leq 2n-4$, then $D$ contains cycles of all lengths $3, 4, \ldots , n$.]]> Mon, 18 Jan 2021 12:23:33 +0000 https://doi.org/10.23638/DMTCS-22-4-12 https://doi.org/10.23638/DMTCS-22-4-12 Darbinyan, Samvel Kh. Darbinyan, Samvel Kh. <![CDATA[Y. Manoussakis (J. Graph Theory 16, 1992, 51-59) proposed the following conjecture. \noindent\textbf{Conjecture}. {\it Let $D$ be a 2-strongly connected digraph of order $n$ such that for all distinct pairs of non-adjacent vertices $x$, $y$ and $w$, $z$, we have $d(x)+d(y)+d(w)+d(z)\geq 4n-3$. Then $D$ is Hamiltonian.} In this paper, we confirm this conjecture. Moreover, we prove that if a digraph $D$ satisfies the conditions of this conjecture and has a pair of non-adjacent vertices $\{x,y\}$ such that $d(x)+d(y)\leq 2n-4$, then $D$ contains cycles of all lengths $3, 4, \ldots , n$.]]> 0 <![CDATA[We provide a pair of ribbon graphs that have the same rotor routing and Bernardi sandpile torsors, but different topological genus. This resolves a question posed by M. Chan [Cha]. We also show that if we are given a graph, but not its ribbon structure, along with the rotor routing sandpile torsors, we are able to determine the ribbon graph's genus.]]> Thu, 07 Jan 2021 09:16:32 +0000 https://doi.org/10.46298/dmtcs.6176 https://doi.org/10.46298/dmtcs.6176 McDonough, Alex McDonough, Alex <![CDATA[We provide a pair of ribbon graphs that have the same rotor routing and Bernardi sandpile torsors, but different topological genus. This resolves a question posed by M. Chan [Cha]. We also show that if we are given a graph, but not its ribbon structure, along with the rotor routing sandpile torsors, we are able to determine the ribbon graph's genus.]]> 0 <![CDATA[Neumann-Lara and Škrekovski conjectured that every planar digraph is 2-colourable. We show that this conjecture is equivalent to the more general statement that all oriented K_5-minor-free graphs are 2-colourable.]]> Tue, 05 Jan 2021 08:49:23 +0000 https://doi.org/10.23638/DMTCS-22-4-11 https://doi.org/10.23638/DMTCS-22-4-11 Steiner, Raphael Steiner, Raphael <![CDATA[Neumann-Lara and Škrekovski conjectured that every planar digraph is 2-colourable. We show that this conjecture is equivalent to the more general statement that all oriented K_5-minor-free graphs are 2-colourable.]]> 0 <![CDATA[A graph $G$ is a cocomparability graph if there exists an acyclic transitive orientation of the edges of its complement graph $\overline{G}$. LBFS$^{+}$ is a variant of the generic Lexicographic Breadth First Search (LBFS), which uses a specific tie-breaking mechanism. Starting with some ordering $\sigma_{0}$ of $G$, let $\{\sigma_{i}\}_{i\geq 1}$ be the sequence of orderings such that $\sigma_{i}=$LBFS$^{+}(G, \sigma_{i-1})$. The LexCycle($G$) is defined as the maximum length of a cycle of vertex orderings of $G$ obtained via such a sequence of LBFS$^{+}$ sweeps. Dusart and Habib conjectured in 2017 that LexCycle($G$)=2 if $G$ is a cocomparability graph and proved it holds for interval graphs. In this paper, we show that LexCycle($G$)=2 if $G$ is a $\overline{P_{2}\cup P_{3}}$-free cocomparability graph, where a $\overline{P_{2}\cup P_{3}}$ is the graph whose complement is the disjoint union of $P_{2}$ and $P_{3}$. As corollaries, it's applicable for diamond-free cocomparability graphs, cocomparability graphs with girth at least 4, as well as interval graphs.]]> Mon, 28 Dec 2020 09:08:45 +0000 https://doi.org/10.23638/DMTCS-22-4-13 https://doi.org/10.23638/DMTCS-22-4-13 Gao, Xiao-Lu Xu, Shou-Jun Gao, Xiao-Lu Xu, Shou-Jun <![CDATA[A graph $G$ is a cocomparability graph if there exists an acyclic transitive orientation of the edges of its complement graph $\overline{G}$. LBFS$^{+}$ is a variant of the generic Lexicographic Breadth First Search (LBFS), which uses a specific tie-breaking mechanism. Starting with some ordering $\sigma_{0}$ of $G$, let $\{\sigma_{i}\}_{i\geq 1}$ be the sequence of orderings such that $\sigma_{i}=$LBFS$^{+}(G, \sigma_{i-1})$. The LexCycle($G$) is defined as the maximum length of a cycle of vertex orderings of $G$ obtained via such a sequence of LBFS$^{+}$ sweeps. Dusart and Habib conjectured in 2017 that LexCycle($G$)=2 if $G$ is a cocomparability graph and proved it holds for interval graphs. In this paper, we show that LexCycle($G$)=2 if $G$ is a $\overline{P_{2}\cup P_{3}}$-free cocomparability graph, where a $\overline{P_{2}\cup P_{3}}$ is the graph whose complement is the disjoint union of $P_{2}$ and $P_{3}$. As corollaries, it's applicable for diamond-free cocomparability graphs, cocomparability graphs with girth at least 4, as well as interval graphs.]]> 0 <![CDATA[Given a graph $G$ and an integer $p$, a coloring $f : V(G) \to \mathbb{N}$ is \emph{$p$-centered} if for every connected subgraph $H$ of $G$, either $f$ uses more than $p$ colors on $H$ or there is a color that appears exactly once in $H$. The notion of $p$-centered colorings plays a central role in the theory of sparse graphs. In this note we show two lower bounds on the number of colors required in a $p$-centered coloring. First, we consider monotone classes of graphs whose shallow minors have average degree bounded polynomially in the radius, or equivalently (by a result of Dvo\v{r}\'ak and Norin), admitting strongly sublinear separators. We construct such a class such that $p$-centered colorings require a number of colors super-polynomial in $p$. This is in contrast with a recent result of Pilipczuk and Siebertz, who established a polynomial upper bound in the special case of graphs excluding a fixed minor. Second, we consider graphs of maximum degree $\Delta$. D\k{e}bski, Felsner, Micek, and Schr\"{o}der recently proved that these graphs have $p$-centered colorings with $O(\Delta^{2-1/p} p)$ colors. We show that there are graphs of maximum degree $\Delta$ that require $\Omega(\Delta^{2-1/p} p \ln^{-1/p}\Delta)$ colors in any $p$-centered coloring, thus matching their upper bound up to a logarithmic factor.]]> Wed, 11 Nov 2020 09:20:55 +0000 https://doi.org/10.23638/DMTCS-22-4-9 https://doi.org/10.23638/DMTCS-22-4-9 Dubois, Loïc Joret, Gwenaël Perarnau, Guillem Pilipczuk, Marcin Pitois, François Dubois, Loïc Joret, Gwenaël Perarnau, Guillem Pilipczuk, Marcin Pitois, François <![CDATA[Given a graph $G$ and an integer $p$, a coloring $f : V(G) \to \mathbb{N}$ is \emph{$p$-centered} if for every connected subgraph $H$ of $G$, either $f$ uses more than $p$ colors on $H$ or there is a color that appears exactly once in $H$. The notion of $p$-centered colorings plays a central role in the theory of sparse graphs. In this note we show two lower bounds on the number of colors required in a $p$-centered coloring. First, we consider monotone classes of graphs whose shallow minors have average degree bounded polynomially in the radius, or equivalently (by a result of Dvo\v{r}\'ak and Norin), admitting strongly sublinear separators. We construct such a class such that $p$-centered colorings require a number of colors super-polynomial in $p$. This is in contrast with a recent result of Pilipczuk and Siebertz, who established a polynomial upper bound in the special case of graphs excluding a fixed minor. Second, we consider graphs of maximum degree $\Delta$. D\k{e}bski, Felsner, Micek, and Schr\"{o}der recently proved that these graphs have $p$-centered colorings with $O(\Delta^{2-1/p} p)$ colors. We show that there are graphs of maximum degree $\Delta$ that require $\Omega(\Delta^{2-1/p} p \ln^{-1/p}\Delta)$ colors in any $p$-centered coloring, thus matching their upper bound up to a logarithmic factor.]]> 0 <![CDATA[Lov{\'a}sz showed that a matching covered graph $G$ has an ear decomposition starting with an arbitrary edge of $G$. Let $G$ be a graph which has a perfect matching. We call $G$ cycle-nice if for each even cycle $C$ of $G$, $G-V(C)$ has a perfect matching. If $G$ is a cycle-nice matching covered graph, then $G$ has ear decompositions starting with an arbitrary even cycle of $G$. In this paper, we characterize cycle-nice claw-free plane graphs. We show that the only cycle-nice simple 3-connected claw-free plane graphs are $K_4$, $W_5$ and $\overline C_6$. Furthermore, every cycle-nice 2-connected claw-free plane graph can be obtained from a graph in the family ${\cal F}$ by a sequence of three types of operations, where ${\cal F}$ consists of even cycles, a diamond, $K_4$, and $\overline C_6$.]]> Mon, 12 Oct 2020 07:36:00 +0000 https://doi.org/10.23638/DMTCS-22-4-6 https://doi.org/10.23638/DMTCS-22-4-6 Zhang, Shanshan Wang, Xiumei Yuan, Jinjiang Zhang, Shanshan Wang, Xiumei Yuan, Jinjiang <![CDATA[Lov{\'a}sz showed that a matching covered graph $G$ has an ear decomposition starting with an arbitrary edge of $G$. Let $G$ be a graph which has a perfect matching. We call $G$ cycle-nice if for each even cycle $C$ of $G$, $G-V(C)$ has a perfect matching. If $G$ is a cycle-nice matching covered graph, then $G$ has ear decompositions starting with an arbitrary even cycle of $G$. In this paper, we characterize cycle-nice claw-free plane graphs. We show that the only cycle-nice simple 3-connected claw-free plane graphs are $K_4$, $W_5$ and $\overline C_6$. Furthermore, every cycle-nice 2-connected claw-free plane graph can be obtained from a graph in the family ${\cal F}$ by a sequence of three types of operations, where ${\cal F}$ consists of even cycles, a diamond, $K_4$, and $\overline C_6$.]]> 0 <![CDATA[We consider the convex hull $P_{\varphi}(G)$ of all satisfying assignments of a given MSO formula $\varphi$ on a given graph $G$. We show that there exists an extended formulation of the polytope $P_{\varphi}(G)$ that can be described by $f(|\varphi|,\tau)\cdot n$ inequalities, where $n$ is the number of vertices in $G$, $\tau$ is the treewidth of $G$ and $f$ is a computable function depending only on $\varphi$ and $\tau.$ In other words, we prove that the extension complexity of $P_{\varphi}(G)$ is linear in the size of the graph $G$, with a constant depending on the treewidth of $G$ and the formula $\varphi$. This provides a very general yet very simple meta-theorem about the extension complexity of polytopes related to a wide class of problems and graphs. As a corollary of our main result, we obtain an analogous result % for the weaker MSO$_1$ logic on the wider class of graphs of bounded cliquewidth. Furthermore, we study our main geometric tool which we term the glued product of polytopes. While the glued product of polytopes has been known since the '90s, we are the first to show that it preserves decomposability and boundedness of treewidth of the constraint matrix. This implies that our extension of $P_\varphi(G)$ is decomposable and has a constraint matrix of bounded treewidth; so far only few classes of polytopes are known to be decomposable. These properties make our extension useful in the construction of algorithms.]]> Thu, 01 Oct 2020 15:42:02 +0000 https://doi.org/10.23638/DMTCS-22-4-8 https://doi.org/10.23638/DMTCS-22-4-8 Kolman, Petr Koutecký, Martin Tiwary, Hans Raj Kolman, Petr Koutecký, Martin Tiwary, Hans Raj <![CDATA[We consider the convex hull $P_{\varphi}(G)$ of all satisfying assignments of a given MSO formula $\varphi$ on a given graph $G$. We show that there exists an extended formulation of the polytope $P_{\varphi}(G)$ that can be described by $f(|\varphi|,\tau)\cdot n$ inequalities, where $n$ is the number of vertices in $G$, $\tau$ is the treewidth of $G$ and $f$ is a computable function depending only on $\varphi$ and $\tau.$ In other words, we prove that the extension complexity of $P_{\varphi}(G)$ is linear in the size of the graph $G$, with a constant depending on the treewidth of $G$ and the formula $\varphi$. This provides a very general yet very simple meta-theorem about the extension complexity of polytopes related to a wide class of problems and graphs. As a corollary of our main result, we obtain an analogous result % for the weaker MSO$_1$ logic on the wider class of graphs of bounded cliquewidth. Furthermore, we study our main geometric tool which we term the glued product of polytopes. While the glued product of polytopes has been known since the '90s, we are the first to show that it preserves decomposability and boundedness of treewidth of the constraint matrix. This implies that our extension of $P_\varphi(G)$ is decomposable and has a constraint matrix of bounded treewidth; so far only few classes of polytopes are known to be decomposable. These properties make our extension useful in the construction of algorithms.]]> 0 <![CDATA[This work is a contribution to the study of rewrite games. Positions are finite words, and the possible moves are defined by a finite number of local rewriting rules. We introduce and investigate taking-and-merging games, that is, where each rule is of the form a^k->epsilon. We give sufficient conditions for a game to be such that the losing positions (resp. the positions with a given Grundy value) form a regular language or a context-free language. We formulate several related open questions in parallel with the famous conjecture of Guy about the periodicity of the Grundy function of octal games. Finally we show that more general rewrite games quickly lead to undecidable problems. Namely, it is undecidable whether there exists a winning position in a given regular language, even if we restrict to games where each move strictly reduces the length of the current position. We formulate several related open questions in parallel with the famous conjecture of Guy about the periodicity of the Grundy function of octal games.]]> Wed, 23 Sep 2020 11:47:19 +0000 https://doi.org/10.23638/DMTCS-22-4-5 https://doi.org/10.23638/DMTCS-22-4-5 Duchêne, Eric Marsault, Victor Parreau, Aline Rigo, Michel Duchêne, Eric Marsault, Victor Parreau, Aline Rigo, Michel <![CDATA[This work is a contribution to the study of rewrite games. Positions are finite words, and the possible moves are defined by a finite number of local rewriting rules. We introduce and investigate taking-and-merging games, that is, where each rule is of the form a^k->epsilon. We give sufficient conditions for a game to be such that the losing positions (resp. the positions with a given Grundy value) form a regular language or a context-free language. We formulate several related open questions in parallel with the famous conjecture of Guy about the periodicity of the Grundy function of octal games. Finally we show that more general rewrite games quickly lead to undecidable problems. Namely, it is undecidable whether there exists a winning position in a given regular language, even if we restrict to games where each move strictly reduces the length of the current position. We formulate several related open questions in parallel with the famous conjecture of Guy about the periodicity of the Grundy function of octal games.]]> 0 <![CDATA[In the (binary) Distinct Vectors problem we are given a binary matrix A with pairwise different rows and want to select at most k columns such that, restricting the matrix to these columns, all rows are still pairwise different. A result by Froese et al. [JCSS] implies a 2^2^(O(k)) * poly(|A|)-time brute-force algorithm for Distinct Vectors. We show that this running time bound is essentially optimal by showing that there is a constant c such that the existence of an algorithm solving Distinct Vectors with running time 2^(O(2^(ck))) * poly(|A|) would contradict the Exponential Time Hypothesis.]]> Fri, 18 Sep 2020 13:34:05 +0000 https://doi.org/10.23638/DMTCS-22-4-7 https://doi.org/10.23638/DMTCS-22-4-7 Pilipczuk, Marcin Sorge, Manuel Pilipczuk, Marcin Sorge, Manuel <![CDATA[In the (binary) Distinct Vectors problem we are given a binary matrix A with pairwise different rows and want to select at most k columns such that, restricting the matrix to these columns, all rows are still pairwise different. A result by Froese et al. [JCSS] implies a 2^2^(O(k)) * poly(|A|)-time brute-force algorithm for Distinct Vectors. We show that this running time bound is essentially optimal by showing that there is a constant c such that the existence of an algorithm solving Distinct Vectors with running time 2^(O(2^(ck))) * poly(|A|) would contradict the Exponential Time Hypothesis.]]> 0 <![CDATA[The packing number of a graph $G$ is the maximum number of closed neighborhoods of vertices in $G$ with pairwise empty intersections. Similarly, the open packing number of $G$ is the maximum number of open neighborhoods in $G$ with pairwise empty intersections. We consider the packing and open packing numbers on graph products. In particular we give a complete solution with respect to some properties of factors in the case of lexicographic and rooted products. For Cartesian, strong and direct products, we present several lower and upper bounds on these parameters.]]> Thu, 27 Aug 2020 07:18:21 +0000 https://doi.org/10.23638/DMTCS-22-4-1 https://doi.org/10.23638/DMTCS-22-4-1 Mojdeh, Doost Ali Peterin, Iztok Samadi, Babak Yero, Ismael G. Mojdeh, Doost Ali Peterin, Iztok Samadi, Babak Yero, Ismael G. <![CDATA[The packing number of a graph $G$ is the maximum number of closed neighborhoods of vertices in $G$ with pairwise empty intersections. Similarly, the open packing number of $G$ is the maximum number of open neighborhoods in $G$ with pairwise empty intersections. We consider the packing and open packing numbers on graph products. In particular we give a complete solution with respect to some properties of factors in the case of lexicographic and rooted products. For Cartesian, strong and direct products, we present several lower and upper bounds on these parameters.]]> 0 <![CDATA[Assume that two robots are located at the centre of a unit disk. Their goal is to evacuate from the disk through an exit at an unknown location on the boundary of the disk. At any time the robots can move anywhere they choose on the disk, independently of each other, with maximum speed $1$. The robots can cooperate by exchanging information whenever they meet. We study algorithms for the two robots to minimize the evacuation time: the time when both robots reach the exit. In [CGGKMP14] the authors gave an algorithm defining trajectories for the two robots yielding evacuation time at most $5.740$ and also proved that any algorithm has evacuation time at least $3+ \frac{\pi}{4} + \sqrt{2} \approx 5.199$. We improve both the upper and lower bound on the evacuation time of a unit disk. Namely, we present a new non-trivial algorithm whose evacuation time is at most $5.628$ and show that any algorithm has evacuation time at least $3+ \frac{\pi}{6} + \sqrt{3} \approx 5.255$. To achieve the upper bound, we designed an algorithm which proposes a forced meeting between the two robots, even if the exit has not been found by either of them. We also show that such a strategy is provably optimal for a related problem of searching for an exit placed at the vertices of a regular hexagon.]]> Thu, 27 Aug 2020 07:13:53 +0000 https://doi.org/10.23638/DMTCS-22-4-4 https://doi.org/10.23638/DMTCS-22-4-4 Czyzowicz, Jurek Georgiou, Konstantinos Kranakis, Evangelos Narayanan, Lata Opatrny, Jarda Vogtenhuber, Birgit Czyzowicz, Jurek Georgiou, Konstantinos Kranakis, Evangelos Narayanan, Lata Opatrny, Jarda Vogtenhuber, Birgit <![CDATA[Assume that two robots are located at the centre of a unit disk. Their goal is to evacuate from the disk through an exit at an unknown location on the boundary of the disk. At any time the robots can move anywhere they choose on the disk, independently of each other, with maximum speed $1$. The robots can cooperate by exchanging information whenever they meet. We study algorithms for the two robots to minimize the evacuation time: the time when both robots reach the exit. In [CGGKMP14] the authors gave an algorithm defining trajectories for the two robots yielding evacuation time at most $5.740$ and also proved that any algorithm has evacuation time at least $3+ \frac{\pi}{4} + \sqrt{2} \approx 5.199$. We improve both the upper and lower bound on the evacuation time of a unit disk. Namely, we present a new non-trivial algorithm whose evacuation time is at most $5.628$ and show that any algorithm has evacuation time at least $3+ \frac{\pi}{6} + \sqrt{3} \approx 5.255$. To achieve the upper bound, we designed an algorithm which proposes a forced meeting between the two robots, even if the exit has not been found by either of them. We also show that such a strategy is provably optimal for a related problem of searching for an exit placed at the vertices of a regular hexagon.]]> 0 <![CDATA[We introduce MSO graph storage types, and call a storage type MSO-expressible if it is isomorphic to some MSO graph storage type. An MSO graph storage type has MSO-definable sets of graphs as storage configurations and as storage transformations. We consider sequential automata with MSO graph storage and associate with each such automaton a string language (in the usual way) and a graph language; a graph is accepted by the automaton if it represents a correct sequence of storage configurations for a given input string. For each MSO graph storage type, we define an MSO logic which is a subset of the usual MSO logic on graphs. We prove a B\"uchi-Elgot-Trakhtenbrot theorem, both for the string case and the graph case. Moreover, we prove that (i) each MSO graph transduction can be used as storage transformation in an MSO graph storage type, (ii) every automatic storage type is MSO-expressible, and (iii) the pushdown operator on storage types preserves the property of MSO-expressibility. Thus, the iterated pushdown storage types are MSO-expressible.]]> Thu, 27 Aug 2020 07:12:06 +0000 https://doi.org/10.23638/DMTCS-22-4-3 https://doi.org/10.23638/DMTCS-22-4-3 Engelfriet, Joost Vogler, Heiko Engelfriet, Joost Vogler, Heiko <![CDATA[We introduce MSO graph storage types, and call a storage type MSO-expressible if it is isomorphic to some MSO graph storage type. An MSO graph storage type has MSO-definable sets of graphs as storage configurations and as storage transformations. We consider sequential automata with MSO graph storage and associate with each such automaton a string language (in the usual way) and a graph language; a graph is accepted by the automaton if it represents a correct sequence of storage configurations for a given input string. For each MSO graph storage type, we define an MSO logic which is a subset of the usual MSO logic on graphs. We prove a B\"uchi-Elgot-Trakhtenbrot theorem, both for the string case and the graph case. Moreover, we prove that (i) each MSO graph transduction can be used as storage transformation in an MSO graph storage type, (ii) every automatic storage type is MSO-expressible, and (iii) the pushdown operator on storage types preserves the property of MSO-expressibility. Thus, the iterated pushdown storage types are MSO-expressible.]]> 0 <![CDATA[We consider nondeterministic higher-order recursion schemes as recognizers of languages of finite words or finite trees. We propose a type system that allows to solve the simultaneous-unboundedness problem (SUP) for schemes, which asks, given a set of letters A and a scheme G, whether it is the case that for every number n the scheme accepts a word (a tree) in which every letter from A appears at least n times. Using this type system we prove that SUP is (m-1)-EXPTIME-complete for word-recognizing schemes of order m, and m-EXPTIME-complete for tree-recognizing schemes of order m. Moreover, we establish the reflection property for SUP: out of an input scheme G one can create its enhanced version that recognizes the same language but is aware of the answer to SUP.]]> Tue, 18 Aug 2020 07:20:48 +0000 https://doi.org/10.23638/DMTCS-22-4-2 https://doi.org/10.23638/DMTCS-22-4-2 Parys, Paweł Parys, Paweł <![CDATA[We consider nondeterministic higher-order recursion schemes as recognizers of languages of finite words or finite trees. We propose a type system that allows to solve the simultaneous-unboundedness problem (SUP) for schemes, which asks, given a set of letters A and a scheme G, whether it is the case that for every number n the scheme accepts a word (a tree) in which every letter from A appears at least n times. Using this type system we prove that SUP is (m-1)-EXPTIME-complete for word-recognizing schemes of order m, and m-EXPTIME-complete for tree-recognizing schemes of order m. Moreover, we establish the reflection property for SUP: out of an input scheme G one can create its enhanced version that recognizes the same language but is aware of the answer to SUP.]]> 0 <![CDATA[The purpose of this paper is to study a statistic that is used to compare the similarity between two strings, which is first introduced by Michael Steele in 1982. It was proposed as an alternative to the length of the longest common subsequences, for which the variance problem is still open. Our results include moment asymptotics and distributional asymptotics for Steele's statistic and a variation of it in random words and random permutations.]]> Fri, 10 Jul 2020 10:03:55 +0000 https://doi.org/10.23638/DMTCS-22-1-18 https://doi.org/10.23638/DMTCS-22-1-18 Işlak, Ümit Özdemir, Alperen Y. Işlak, Ümit Özdemir, Alperen Y. <![CDATA[The purpose of this paper is to study a statistic that is used to compare the similarity between two strings, which is first introduced by Michael Steele in 1982. It was proposed as an alternative to the length of the longest common subsequences, for which the variance problem is still open. Our results include moment asymptotics and distributional asymptotics for Steele's statistic and a variation of it in random words and random permutations.]]> 0 <![CDATA[A rearrangement operation makes a small graph-theoretical change to a phylogenetic network to transform it into another one. For unrooted phylogenetic trees and networks, popular rearrangement operations are tree bisection and reconnection (TBR) and prune and regraft (PR) (called subtree prune and regraft (SPR) on trees). Each of these operations induces a metric on the sets of phylogenetic trees and networks. The TBR-distance between two unrooted phylogenetic trees $T$ and $T'$ can be characterised by a maximum agreement forest, that is, a forest with a minimum number of components that covers both $T$ and $T'$ in a certain way. This characterisation has facilitated the development of fixed-parameter tractable algorithms and approximation algorithms. Here, we introduce maximum agreement graphs as a generalisations of maximum agreement forests for phylogenetic networks. While the agreement distance -- the metric induced by maximum agreement graphs -- does not characterise the TBR-distance of two networks, we show that it still provides constant-factor bounds on the TBR-distance. We find similar results for PR in terms of maximum endpoint agreement graphs.]]> Thu, 09 Jul 2020 08:30:40 +0000 https://doi.org/10.23638/DMTCS-22-1-22 https://doi.org/10.23638/DMTCS-22-1-22 Klawitter, Jonathan Klawitter, Jonathan <![CDATA[A rearrangement operation makes a small graph-theoretical change to a phylogenetic network to transform it into another one. For unrooted phylogenetic trees and networks, popular rearrangement operations are tree bisection and reconnection (TBR) and prune and regraft (PR) (called subtree prune and regraft (SPR) on trees). Each of these operations induces a metric on the sets of phylogenetic trees and networks. The TBR-distance between two unrooted phylogenetic trees $T$ and $T'$ can be characterised by a maximum agreement forest, that is, a forest with a minimum number of components that covers both $T$ and $T'$ in a certain way. This characterisation has facilitated the development of fixed-parameter tractable algorithms and approximation algorithms. Here, we introduce maximum agreement graphs as a generalisations of maximum agreement forests for phylogenetic networks. While the agreement distance -- the metric induced by maximum agreement graphs -- does not characterise the TBR-distance of two networks, we show that it still provides constant-factor bounds on the TBR-distance. We find similar results for PR in terms of maximum endpoint agreement graphs.]]> 0 <![CDATA[The enumeration of inversion sequences avoiding a single pattern was initiated by Corteel--Martinez--Savage--Weselcouch and Mansour--Shattuck independently. Their work has sparked various investigations of generalized patterns in inversion sequences, including patterns of relation triples by Martinez and Savage, consecutive patterns by Auli and Elizalde, and vincular patterns by Lin and Yan. In this paper, we carried out the systematic study of inversion sequences avoiding two patterns of length $3$. Our enumerative results establish further connections to the OEIS sequences and some classical combinatorial objects, such as restricted permutations, weighted ordered trees and set partitions. Since patterns of relation triples are some special multiple patterns of length $3$, our results complement the work by Martinez and Savage. In particular, one of their conjectures regarding the enumeration of $(021,120)$-avoiding inversion sequences is solved.]]> Mon, 29 Jun 2020 13:38:31 +0000 https://doi.org/10.23638/DMTCS-22-1-23 https://doi.org/10.23638/DMTCS-22-1-23 Yan, Chunyan Lin, Zhicong Yan, Chunyan Lin, Zhicong <![CDATA[The enumeration of inversion sequences avoiding a single pattern was initiated by Corteel--Martinez--Savage--Weselcouch and Mansour--Shattuck independently. Their work has sparked various investigations of generalized patterns in inversion sequences, including patterns of relation triples by Martinez and Savage, consecutive patterns by Auli and Elizalde, and vincular patterns by Lin and Yan. In this paper, we carried out the systematic study of inversion sequences avoiding two patterns of length $3$. Our enumerative results establish further connections to the OEIS sequences and some classical combinatorial objects, such as restricted permutations, weighted ordered trees and set partitions. Since patterns of relation triples are some special multiple patterns of length $3$, our results complement the work by Martinez and Savage. In particular, one of their conjectures regarding the enumeration of $(021,120)$-avoiding inversion sequences is solved.]]> 0 <![CDATA[We study the dissection of a square into congruent convex polygons. Yuan \emph{et al.} [Dissecting the square into five congruent parts, Discrete Math. \textbf{339} (2016) 288-298] asked whether, if the number of tiles is a prime number $\geq 3$, it is true that the tile must be a rectangle. We conjecture that the same conclusion still holds even if the number of tiles is an odd number $\geq 3$. Our conjecture has been confirmed for triangles in earlier works. We prove that the conjecture holds if either the tile is a convex $q$-gon with $q\geq 6$ or it is a right-angle trapezoid.]]> Mon, 29 Jun 2020 13:34:40 +0000 https://doi.org/10.23638/DMTCS-22-1-21 https://doi.org/10.23638/DMTCS-22-1-21 Rao, Hui Ren, Lei Wang, Yang Rao, Hui Ren, Lei Wang, Yang <![CDATA[We study the dissection of a square into congruent convex polygons. Yuan \emph{et al.} [Dissecting the square into five congruent parts, Discrete Math. \textbf{339} (2016) 288-298] asked whether, if the number of tiles is a prime number $\geq 3$, it is true that the tile must be a rectangle. We conjecture that the same conclusion still holds even if the number of tiles is an odd number $\geq 3$. Our conjecture has been confirmed for triangles in earlier works. We prove that the conjecture holds if either the tile is a convex $q$-gon with $q\geq 6$ or it is a right-angle trapezoid.]]> 0 <![CDATA[We investigate the partitioning of partial orders into a minimal number of heapable subsets. We prove a characterization result reminiscent of the proof of Dilworth's theorem, which yields as a byproduct a flow-based algorithm for computing such a minimal decomposition. On the other hand, in the particular case of sets and sequences of intervals we prove that this minimal decomposition can be computed by a simple greedy-type algorithm. The paper ends with a couple of open problems related to the analog of the Ulam-Hammersley problem for decompositions of sets and sequences of random intervals into heapable sets.]]> Mon, 29 Jun 2020 13:29:19 +0000 https://doi.org/10.23638/DMTCS-22-1-17 https://doi.org/10.23638/DMTCS-22-1-17 Balogh, János Bonchiş, Cosmin Diniş, Diana Istrate, Gabriel Todinca, Ioan Balogh, János Bonchiş, Cosmin Diniş, Diana Istrate, Gabriel Todinca, Ioan <![CDATA[We investigate the partitioning of partial orders into a minimal number of heapable subsets. We prove a characterization result reminiscent of the proof of Dilworth's theorem, which yields as a byproduct a flow-based algorithm for computing such a minimal decomposition. On the other hand, in the particular case of sets and sequences of intervals we prove that this minimal decomposition can be computed by a simple greedy-type algorithm. The paper ends with a couple of open problems related to the analog of the Ulam-Hammersley problem for decompositions of sets and sequences of random intervals into heapable sets.]]> 0 <![CDATA[We determine the critical exponent and the recurrence function of complementary symmetric Rote sequences. The formulae are expressed in terms of the continued fraction expansions associated with the S-adic representations of the corresponding standard Sturmian sequences. The results are based on a thorough study of return words to bispecial factors of Sturmian sequences. Using the formula for the critical exponent, we describe all complementary symmetric Rote sequences with the critical exponent less than or equal to 3, and we show that there are uncountably many complementary symmetric Rote sequences with the critical exponent less than the critical exponent of the Fibonacci sequence. Our study is motivated by a~conjecture on sequences rich in palindromes formulated by Baranwal and Shallit. Its recent solution by Curie, Mol, and Rampersad uses two particular complementary symmetric Rote sequences.]]> Sat, 06 Jun 2020 11:59:49 +0000 https://doi.org/10.23638/DMTCS-22-1-20 https://doi.org/10.23638/DMTCS-22-1-20 Dvořáková, Lubomíra Medková, Kateřina Pelantová, Edita Dvořáková, Lubomíra Medková, Kateřina Pelantová, Edita <![CDATA[We determine the critical exponent and the recurrence function of complementary symmetric Rote sequences. The formulae are expressed in terms of the continued fraction expansions associated with the S-adic representations of the corresponding standard Sturmian sequences. The results are based on a thorough study of return words to bispecial factors of Sturmian sequences. Using the formula for the critical exponent, we describe all complementary symmetric Rote sequences with the critical exponent less than or equal to 3, and we show that there are uncountably many complementary symmetric Rote sequences with the critical exponent less than the critical exponent of the Fibonacci sequence. Our study is motivated by a~conjecture on sequences rich in palindromes formulated by Baranwal and Shallit. Its recent solution by Curie, Mol, and Rampersad uses two particular complementary symmetric Rote sequences.]]> 0 <![CDATA[Golumbic, Lipshteyn, and Stern defined in 2009 the class of EPG graphs, the intersection graph class of edge paths on a grid. An EPG graph $G$ is a graph that admits a representation where its vertices correspond to paths in a grid $Q$, such that two vertices of $G$ are adjacent if and only if their corresponding paths in $Q$ have a common edge. If the paths in the representation have at most $k$ bends, we say that it is a $B_k$-EPG representation. A collection $C$ of sets satisfies the Helly property when every sub-collection of $C$ that is pairwise intersecting has at least one common element. In this paper, we show that given a graph $G$ and an integer $k$, the problem of determining whether $G$ admits a $B_k$-EPG representation whose edge-intersections of paths satisfy the Helly property, so-called Helly-$B_k$-EPG representation, is in NP, for every $k$ bounded by a polynomial function of $|V(G)|$. Moreover, we show that the problem of recognizing Helly-$B_1$-EPG graphs is NP-complete, and it remains NP-complete even when restricted to 2-apex and 3-degenerate graphs.]]> Thu, 04 Jun 2020 14:56:38 +0000 https://doi.org/10.23638/DMTCS-22-1-19 https://doi.org/10.23638/DMTCS-22-1-19 Bornstein, Claudson F. Golumbic, Martin Charles Santos, Tanilson D. Souza, Uéverton S. Szwarcfiter, Jayme L. Bornstein, Claudson F. Golumbic, Martin Charles Santos, Tanilson D. Souza, Uéverton S. Szwarcfiter, Jayme L. <![CDATA[Golumbic, Lipshteyn, and Stern defined in 2009 the class of EPG graphs, the intersection graph class of edge paths on a grid. An EPG graph $G$ is a graph that admits a representation where its vertices correspond to paths in a grid $Q$, such that two vertices of $G$ are adjacent if and only if their corresponding paths in $Q$ have a common edge. If the paths in the representation have at most $k$ bends, we say that it is a $B_k$-EPG representation. A collection $C$ of sets satisfies the Helly property when every sub-collection of $C$ that is pairwise intersecting has at least one common element. In this paper, we show that given a graph $G$ and an integer $k$, the problem of determining whether $G$ admits a $B_k$-EPG representation whose edge-intersections of paths satisfy the Helly property, so-called Helly-$B_k$-EPG representation, is in NP, for every $k$ bounded by a polynomial function of $|V(G)|$. Moreover, we show that the problem of recognizing Helly-$B_1$-EPG graphs is NP-complete, and it remains NP-complete even when restricted to 2-apex and 3-degenerate graphs.]]> 0 <![CDATA[Finding a solution to a Constraint Satisfaction Problem (CSP) is known to be an NP-hard task. This has motivatedthe multitude of works that have been devoted to developing techniques that simplify CSP instances before or duringtheir resolution.The present work proposes rigidly enforced schemes for simplifying binary CSPs that allow the narrowing of valuedomains, either via value merging or via value suppression. The proposed schemes can be viewed as parametrizedgeneralizations of two widely studied CSP simplification techniques, namely, value merging and neighbourhoodsubstitutability. Besides, we show that both schemes may be strengthened in order to allow variable elimination,which may result in more significant simplifications. This work contributes also to the theory of tractable CSPs byidentifying a new tractable class of binary CSP.]]> Thu, 04 Jun 2020 14:38:55 +0000 https://doi.org/10.23638/DMTCS-22-1-10 https://doi.org/10.23638/DMTCS-22-1-10 Naanaa, Wady Naanaa, Wady <![CDATA[Finding a solution to a Constraint Satisfaction Problem (CSP) is known to be an NP-hard task. This has motivatedthe multitude of works that have been devoted to developing techniques that simplify CSP instances before or duringtheir resolution.The present work proposes rigidly enforced schemes for simplifying binary CSPs that allow the narrowing of valuedomains, either via value merging or via value suppression. The proposed schemes can be viewed as parametrizedgeneralizations of two widely studied CSP simplification techniques, namely, value merging and neighbourhoodsubstitutability. Besides, we show that both schemes may be strengthened in order to allow variable elimination,which may result in more significant simplifications. This work contributes also to the theory of tractable CSPs byidentifying a new tractable class of binary CSP.]]> 0 <![CDATA[Let $G=(X,Y;E)$ be a bipartite graph, where $X$ and $Y$ are color classes and $E$ is the set of edges of $G$. Lov\'asz and Plummer \cite{LoPl86} asked whether one can decide in polynomial time that a given bipartite graph $G=(X,Y; E)$ admits a 1-anti-factor, that is subset $F$ of $E$ such that $d_F(v)=1$ for all $v\in X$ and $d_F(v)\neq 1$ for all $v\in Y$. Cornu\'ejols \cite{CHP} answered this question in the affirmative. Yu and Liu \cite{YL09} asked whether, for a given integer $k\geq 3$, every $k$-regular bipartite graph contains a 1-anti-factor. This paper answers this question in the affirmative.]]> Thu, 04 Jun 2020 14:33:38 +0000 https://doi.org/10.23638/DMTCS-22-1-16 https://doi.org/10.23638/DMTCS-22-1-16 Lu, Hongliang Wang, Wei Yan, Juan Lu, Hongliang Wang, Wei Yan, Juan <![CDATA[Let $G=(X,Y;E)$ be a bipartite graph, where $X$ and $Y$ are color classes and $E$ is the set of edges of $G$. Lov\'asz and Plummer \cite{LoPl86} asked whether one can decide in polynomial time that a given bipartite graph $G=(X,Y; E)$ admits a 1-anti-factor, that is subset $F$ of $E$ such that $d_F(v)=1$ for all $v\in X$ and $d_F(v)\neq 1$ for all $v\in Y$. Cornu\'ejols \cite{CHP} answered this question in the affirmative. Yu and Liu \cite{YL09} asked whether, for a given integer $k\geq 3$, every $k$-regular bipartite graph contains a 1-anti-factor. This paper answers this question in the affirmative.]]> 0 <![CDATA[A formal inverse of a given automatic sequence (the sequence of coefficients of the composition inverse of its associated formal power series) is also automatic. The comparison of properties of the original sequence and its formal inverse is an interesting problem. Such an analysis has been done before for the Thue{Morse sequence. In this paper, we describe arithmetic properties of formal inverses of the generalized Thue-Morse sequences and formal inverses of two modifications of the Rudin{Shapiro sequence. In each case, we give the recurrence relations and the automaton, then we analyze the lengths of strings of consecutive identical letters as well as the frequencies of letters. We also compare the obtained results with the original sequences.]]> Mon, 25 May 2020 08:03:35 +0000 https://doi.org/10.23638/DMTCS-22-1-15 https://doi.org/10.23638/DMTCS-22-1-15 Merta, Łukasz Merta, Łukasz <![CDATA[A formal inverse of a given automatic sequence (the sequence of coefficients of the composition inverse of its associated formal power series) is also automatic. The comparison of properties of the original sequence and its formal inverse is an interesting problem. Such an analysis has been done before for the Thue{Morse sequence. In this paper, we describe arithmetic properties of formal inverses of the generalized Thue-Morse sequences and formal inverses of two modifications of the Rudin{Shapiro sequence. In each case, we give the recurrence relations and the automaton, then we analyze the lengths of strings of consecutive identical letters as well as the frequencies of letters. We also compare the obtained results with the original sequences.]]> 0 <![CDATA[Let $S_{a,b}$ denote the sequence of leading digits of $a^n$ in base $b$. It is well known that if $a$ is not a rational power of $b$, then the sequence $S_{a,b}$ satisfies Benford's Law; that is, digit $d$ occurs in $S_{a,b}$ with frequency $\log_{b}(1+1/d)$, for $d=1,2,\dots,b-1$. In this paper, we investigate the \emph{complexity} of such sequences. We focus mainly on the \emph{block complexity}, $p_{a,b}(n)$, defined as the number of distinct blocks of length $n$ appearing in $S_{a,b}$. In our main result we determine $p_{a,b}(n)$ for all squarefree bases $b\ge 5$ and all rational numbers $a>0$ that are not integral powers of $b$. In particular, we show that, for all such pairs $(a,b)$, the complexity function $p_{a,b}(n)$ is \emph{affine}, i.e., satisfies $p_{a,b}(n)=c_{a,b} n + d_{a,b}$ for all $n\ge1$, with coefficients $c_{a,b}\ge1$ and $d_{a,b}\ge0$, given explicitly in terms of $a$ and $b$. We also show that the requirement that $b$ be squarefree cannot be dropped: If $b$ is not squarefree, then there exist integers $a$ with $1 Thu, 30 Apr 2020 09:13:09 +0000 https://doi.org/10.23638/DMTCS-22-1-14 https://doi.org/10.23638/DMTCS-22-1-14 He, Xinwei Hildebrand, A. J. Li, Yuchen Zhang, Yunyi He, Xinwei Hildebrand, A. J. Li, Yuchen Zhang, Yunyi <![CDATA[Let $S_{a,b}$ denote the sequence of leading digits of $a^n$ in base $b$. It is well known that if $a$ is not a rational power of $b$, then the sequence $S_{a,b}$ satisfies Benford's Law; that is, digit $d$ occurs in $S_{a,b}$ with frequency $\log_{b}(1+1/d)$, for $d=1,2,\dots,b-1$. In this paper, we investigate the \emph{complexity} of such sequences. We focus mainly on the \emph{block complexity}, $p_{a,b}(n)$, defined as the number of distinct blocks of length $n$ appearing in $S_{a,b}$. In our main result we determine $p_{a,b}(n)$ for all squarefree bases $b\ge 5$ and all rational numbers $a>0$ that are not integral powers of $b$. In particular, we show that, for all such pairs $(a,b)$, the complexity function $p_{a,b}(n)$ is \emph{affine}, i.e., satisfies $p_{a,b}(n)=c_{a,b} n + d_{a,b}$ for all $n\ge1$, with coefficients $c_{a,b}\ge1$ and $d_{a,b}\ge0$, given explicitly in terms of $a$ and $b$. We also show that the requirement that $b$ be squarefree cannot be dropped: If $b$ is not squarefree, then there exist integers $a$ with $1 0 <![CDATA[It is known that 2-state binary and 3-state unary probabilistic finite automata and 2-state unary quantum finite automata recognize uncountably many languages with cutpoints. These results have been obtained by associating each recognized language with a cutpoint and then by using the fact that there are uncountably many cutpoints. In this note, we prove the same results for fixed cutpoints: each recognized language is associated with an automaton (i.e., algorithm), and the proofs use the fact that there are uncountably many automata. For each case, we present a new construction.]]> Thu, 30 Apr 2020 08:51:36 +0000 https://doi.org/10.23638/DMTCS-22-1-13 https://doi.org/10.23638/DMTCS-22-1-13 Naumovs, Aleksejs Dimitrijevs, Maksims Yakaryılmaz, Abuzer Naumovs, Aleksejs Dimitrijevs, Maksims Yakaryılmaz, Abuzer <![CDATA[It is known that 2-state binary and 3-state unary probabilistic finite automata and 2-state unary quantum finite automata recognize uncountably many languages with cutpoints. These results have been obtained by associating each recognized language with a cutpoint and then by using the fact that there are uncountably many cutpoints. In this note, we prove the same results for fixed cutpoints: each recognized language is associated with an automaton (i.e., algorithm), and the proofs use the fact that there are uncountably many automata. For each case, we present a new construction.]]> 0 <![CDATA[Let v be a grid path made of north and east steps. The lattice TAM(v), based on all grid paths weakly above the grid path v sharing the same endpoints as v, was introduced by Pre ́ville-Ratelle and Viennot (2014) and corresponds to the usual Tamari lattice in the case v = (NE)n. They showed that TAM(v) is isomorphic to the dual of TAM(←−v ), where ←−v is the reverse of v with N and E exchanged. Our main contribution is a bijection from intervals in TAM(v) to non-separable planar maps. It follows that the number of intervals in TAM(v) over all v of length n is 2(3n+3)! (n+2)!(2n+3)! . This formula was first obtained by Tutte(1963) for non-separable planar maps.]]> Wed, 22 Apr 2020 21:33:39 +0000 https://doi.org/10.46298/dmtcs.6421 https://doi.org/10.46298/dmtcs.6421 Fang, Wenjie Préville-Ratelle, Louis-François Fang, Wenjie Préville-Ratelle, Louis-François <![CDATA[Let v be a grid path made of north and east steps. The lattice TAM(v), based on all grid paths weakly above the grid path v sharing the same endpoints as v, was introduced by Pre ́ville-Ratelle and Viennot (2014) and corresponds to the usual Tamari lattice in the case v = (NE)n. They showed that TAM(v) is isomorphic to the dual of TAM(←−v ), where ←−v is the reverse of v with N and E exchanged. Our main contribution is a bijection from intervals in TAM(v) to non-separable planar maps. It follows that the number of intervals in TAM(v) over all v of length n is 2(3n+3)! (n+2)!(2n+3)! . This formula was first obtained by Tutte(1963) for non-separable planar maps.]]> 0 <![CDATA[Corner polyhedra were introduced by Eppstein and Mumford (2014) as the set of simply connected 3D polyhedra such that all vertices have non negative integer coordinates, edges are parallel to the coordinate axes and all vertices but one can be seen from infinity in the direction (1, 1, 1). These authors gave a remarkable characterization of the set of corner polyhedra graphs, that is graphs that can be skeleton of a corner polyhedron: as planar maps, they are the duals of some particular bipartite triangulations, which we call hereafter corner triangulations.In this paper we count corner polyhedral graphs by determining the generating function of the corner triangulations with respect to the number of vertices: we obtain an explicit rational expression for it in terms of the Catalan gen- erating function. We first show that this result can be derived using Tutte's classical compositional approach. Then, in order to explain the occurrence of the Catalan series we give a direct algebraic decomposition of corner triangu- lations: in particular we exhibit a family of almond triangulations that admit a recursive decomposition structurally equivalent to the decomposition of binary trees. Finally we sketch a direct bijection between binary trees and almond triangulations. Our combinatorial analysis yields a simpler alternative to the algorithm of Eppstein and Mumford for endowing a corner polyhedral graph with the cycle cover structure needed to realize it as a polyhedral graph.]]> Wed, 22 Apr 2020 21:33:38 +0000 https://doi.org/10.46298/dmtcs.6420 https://doi.org/10.46298/dmtcs.6420 Dervieux, Clement Poulalhon, Dominique Schaeffer, Gilles Dervieux, Clement Poulalhon, Dominique Schaeffer, Gilles <![CDATA[Corner polyhedra were introduced by Eppstein and Mumford (2014) as the set of simply connected 3D polyhedra such that all vertices have non negative integer coordinates, edges are parallel to the coordinate axes and all vertices but one can be seen from infinity in the direction (1, 1, 1). These authors gave a remarkable characterization of the set of corner polyhedra graphs, that is graphs that can be skeleton of a corner polyhedron: as planar maps, they are the duals of some particular bipartite triangulations, which we call hereafter corner triangulations.In this paper we count corner polyhedral graphs by determining the generating function of the corner triangulations with respect to the number of vertices: we obtain an explicit rational expression for it in terms of the Catalan gen- erating function. We first show that this result can be derived using Tutte's classical compositional approach. Then, in order to explain the occurrence of the Catalan series we give a direct algebraic decomposition of corner triangu- lations: in particular we exhibit a family of almond triangulations that admit a recursive decomposition structurally equivalent to the decomposition of binary trees. Finally we sketch a direct bijection between binary trees and almond triangulations. Our combinatorial analysis yields a simpler alternative to the algorithm of Eppstein and Mumford for endowing a corner polyhedral graph with the cycle cover structure needed to realize it as a polyhedral graph.]]> 0 <![CDATA[We present a new, explicit sum formula for symmetric Macdonald polynomials Pλ and show that they can be written as a trace over a product of (infinite dimensional) matrices. These matrices satisfy the Zamolodchikov– Faddeev (ZF) algebra. We construct solutions of the ZF algebra from a rank-reduced version of the Yang–Baxter algebra. As a corollary, we find that the normalization of the stationary measure of the multi-species asymmetric exclusion process is a Macdonald polynomial with all variables set equal to one.]]> Wed, 22 Apr 2020 21:33:38 +0000 https://doi.org/10.46298/dmtcs.6419 https://doi.org/10.46298/dmtcs.6419 Cantini, Luigi De Gier, Jan Wheeler, Michael Cantini, Luigi De Gier, Jan Wheeler, Michael <![CDATA[We present a new, explicit sum formula for symmetric Macdonald polynomials Pλ and show that they can be written as a trace over a product of (infinite dimensional) matrices. These matrices satisfy the Zamolodchikov– Faddeev (ZF) algebra. We construct solutions of the ZF algebra from a rank-reduced version of the Yang–Baxter algebra. As a corollary, we find that the normalization of the stationary measure of the multi-species asymmetric exclusion process is a Macdonald polynomial with all variables set equal to one.]]> 0 <![CDATA[We prove a conjecture of J.-C. Novelli, J.-Y. Thibon, and L. K. Williams (2010) about an equivalence of two triples of statistics on permutations. To prove this conjecture, we construct a bijection through different combinatorial objects, starting with a Catalan based object related to the PASEP.]]> Wed, 22 Apr 2020 21:33:37 +0000 https://doi.org/10.46298/dmtcs.6418 https://doi.org/10.46298/dmtcs.6418 Nunge, Arthur Nunge, Arthur <![CDATA[We prove a conjecture of J.-C. Novelli, J.-Y. Thibon, and L. K. Williams (2010) about an equivalence of two triples of statistics on permutations. To prove this conjecture, we construct a bijection through different combinatorial objects, starting with a Catalan based object related to the PASEP.]]> 0 <![CDATA[It has been shown by Pittel and Romik that the random surface associated with a large rectangular Youngtableau converges to a deterministic limit. We study the fluctuations from this limit along the edges of the rectangle.We show that in the corner, these fluctuations are gaussian whereas, away from the corner and when the rectangle isa square, the fluctuations are given by the Tracy-Widom distribution. Our method is based on a connection with theJacobi ensemble.]]> Wed, 22 Apr 2020 21:33:36 +0000 https://doi.org/10.46298/dmtcs.6417 https://doi.org/10.46298/dmtcs.6417 Marchal, Philippe Marchal, Philippe <![CDATA[It has been shown by Pittel and Romik that the random surface associated with a large rectangular Youngtableau converges to a deterministic limit. We study the fluctuations from this limit along the edges of the rectangle.We show that in the corner, these fluctuations are gaussian whereas, away from the corner and when the rectangle isa square, the fluctuations are given by the Tracy-Widom distribution. Our method is based on a connection with theJacobi ensemble.]]> 0 <![CDATA[In the 1970s, Tutte developed a clever algebraic approach, based on certain " invariants " , to solve a functional equation that arises in the enumeration of properly colored triangulations. The enumeration of plane lattice walks confined to the first quadrant is governed by similar equations, and has led in the past decade to a rich collection of attractive results dealing with the nature (algebraic, D-finite or not) of the associated generating function, depending on the set of allowed steps. We first adapt Tutte's approach to prove (or reprove) the algebraicity of all quadrant models known or conjectured to be algebraic (with one small exception). This includes Gessel's famous model, and the first proof ever found for one model with weighted steps. To be applicable, the method requires the existence of two rational functions called invariant and decoupling function respectively. When they exist, algebraicity comes out (almost) automatically. Then, we move to an analytic viewpoint which has already proved very powerful, leading in particular to integral expressions of the generating function in the non-D-finite cases, as well as to proofs of non-D-finiteness. We develop in this context a weaker notion of invariant. Now all quadrant models have invariants, and for those that have in addition a decoupling function, we obtain integral-free expressions of the generating function, and a proof that this series is differentially algebraic (that is, satisfies a non-linear differential equation).]]> Wed, 22 Apr 2020 21:33:36 +0000 https://doi.org/10.46298/dmtcs.6416 https://doi.org/10.46298/dmtcs.6416 Bernardi, Olivier Bousquet-Mélou, Mireille Raschel, Kilian Bernardi, Olivier Bousquet-Mélou, Mireille Raschel, Kilian <![CDATA[In the 1970s, Tutte developed a clever algebraic approach, based on certain " invariants " , to solve a functional equation that arises in the enumeration of properly colored triangulations. The enumeration of plane lattice walks confined to the first quadrant is governed by similar equations, and has led in the past decade to a rich collection of attractive results dealing with the nature (algebraic, D-finite or not) of the associated generating function, depending on the set of allowed steps. We first adapt Tutte's approach to prove (or reprove) the algebraicity of all quadrant models known or conjectured to be algebraic (with one small exception). This includes Gessel's famous model, and the first proof ever found for one model with weighted steps. To be applicable, the method requires the existence of two rational functions called invariant and decoupling function respectively. When they exist, algebraicity comes out (almost) automatically. Then, we move to an analytic viewpoint which has already proved very powerful, leading in particular to integral expressions of the generating function in the non-D-finite cases, as well as to proofs of non-D-finiteness. We develop in this context a weaker notion of invariant. Now all quadrant models have invariants, and for those that have in addition a decoupling function, we obtain integral-free expressions of the generating function, and a proof that this series is differentially algebraic (that is, satisfies a non-linear differential equation).]]> 0 <![CDATA[We continue the investigations of lattice walks in the three-dimensional lattice restricted to the positive octant. We separate models which clearly have a D-finite generating function from models for which there is no reason to expect that their generating function is D-finite, and we isolate a small set of models whose nature remains unclear and requires further investigation. For these, we give some experimental results about their asymptotic behaviour, based on the inspection of a large number of initial terms. At least for some of them, the guessed asymptotic form seems to tip the balance towards non-D-finiteness.]]> Wed, 22 Apr 2020 21:33:35 +0000 https://doi.org/10.46298/dmtcs.6415 https://doi.org/10.46298/dmtcs.6415 Bacher, Axel Kauers, Manuel Yatchak, Rika Bacher, Axel Kauers, Manuel Yatchak, Rika <![CDATA[We continue the investigations of lattice walks in the three-dimensional lattice restricted to the positive octant. We separate models which clearly have a D-finite generating function from models for which there is no reason to expect that their generating function is D-finite, and we isolate a small set of models whose nature remains unclear and requires further investigation. For these, we give some experimental results about their asymptotic behaviour, based on the inspection of a large number of initial terms. At least for some of them, the guessed asymptotic form seems to tip the balance towards non-D-finiteness.]]> 0 <![CDATA[We present a new definition of non-ambiguous trees (NATs) as labelled binary trees. We thus get a differ- ential equation whose solution can be described combinatorially. This yield a new formula for the number of NATs. We also obtain q-versions of our formula. And we generalize NATs to higher dimension.]]> Wed, 22 Apr 2020 21:33:34 +0000 https://doi.org/10.46298/dmtcs.6414 https://doi.org/10.46298/dmtcs.6414 Aval, Jean-Christophe Boussicault, Adrien Delcroix-Oger, Bérénice Hivert, Florent Laborde-Zubieta, Patxi Aval, Jean-Christophe Boussicault, Adrien Delcroix-Oger, Bérénice Hivert, Florent Laborde-Zubieta, Patxi <![CDATA[We present a new definition of non-ambiguous trees (NATs) as labelled binary trees. We thus get a differ- ential equation whose solution can be described combinatorially. This yield a new formula for the number of NATs. We also obtain q-versions of our formula. And we generalize NATs to higher dimension.]]> 0 <![CDATA[A q-integral over an order polytope coming from a poset is interpreted as a generating function of linear extensions of the poset. As an application, theq-beta integral and aq-analog of Dirichlet’s integral are computed. A combinatorial interpretation of aq-Selberg integral is also obtained.]]> Wed, 22 Apr 2020 21:33:34 +0000 https://doi.org/10.46298/dmtcs.6413 https://doi.org/10.46298/dmtcs.6413 Kim, Jang Soo Kim, Jang Soo <![CDATA[A q-integral over an order polytope coming from a poset is interpreted as a generating function of linear extensions of the poset. As an application, theq-beta integral and aq-analog of Dirichlet’s integral are computed. A combinatorial interpretation of aq-Selberg integral is also obtained.]]> 0 <![CDATA[We describe a bijective proof of Macdonald's reduced word identity using pipe dreams and Little's bumping algorithm. The proof extends to a principal specialization of the identity due to Fomin and Stanley. Our bijective tools also allow us to address a problem posed by Fomin and Kirillov from 1997, using work of Wachs, Lenart and Serrano- Stump.]]> Wed, 22 Apr 2020 21:33:33 +0000 https://doi.org/10.46298/dmtcs.6412 https://doi.org/10.46298/dmtcs.6412 Billey, Sara, Holroyd, Alexander Young, Benjamin, Billey, Sara, Holroyd, Alexander Young, Benjamin, <![CDATA[We describe a bijective proof of Macdonald's reduced word identity using pipe dreams and Little's bumping algorithm. The proof extends to a principal specialization of the identity due to Fomin and Stanley. Our bijective tools also allow us to address a problem posed by Fomin and Kirillov from 1997, using work of Wachs, Lenart and Serrano- Stump.]]> 0 <![CDATA[In this extended abstract, we give a complete description and enumeration of smooth and rationally smooth Schubert varieties in finite type. In particular, we show that rationally smooth Schubert varieties are in bijection with a new combinatorial data structure called staircase diagrams.]]> Wed, 22 Apr 2020 21:33:32 +0000 https://doi.org/10.46298/dmtcs.6411 https://doi.org/10.46298/dmtcs.6411 Richmond, Edward Slofstra, William Richmond, Edward Slofstra, William <![CDATA[In this extended abstract, we give a complete description and enumeration of smooth and rationally smooth Schubert varieties in finite type. In particular, we show that rationally smooth Schubert varieties are in bijection with a new combinatorial data structure called staircase diagrams.]]> 0 <![CDATA[We describe a combinatorial formula for the coefficients when the dual immaculate quasisymmetric func- tions are decomposed into Young quasisymmetric Schur functions. We prove this using an analogue of Schensted insertion. We also provide a Remmel-Whitney style rule to generate these coefficients algorithmically.]]> Wed, 22 Apr 2020 21:33:31 +0000 https://doi.org/10.46298/dmtcs.6410 https://doi.org/10.46298/dmtcs.6410 Allen, Edward, Hallam, Joshua Mason, Sarah, Allen, Edward, Hallam, Joshua Mason, Sarah, <![CDATA[We describe a combinatorial formula for the coefficients when the dual immaculate quasisymmetric func- tions are decomposed into Young quasisymmetric Schur functions. We prove this using an analogue of Schensted insertion. We also provide a Remmel-Whitney style rule to generate these coefficients algorithmically.]]> 0 <![CDATA[We introduce a new family of complete lattices, arising from a digraph together with a valuation on its vertices and generalizing a previous construction of the author. We then apply this to the study of two long-standing conjectures of Dyer, and we provide a description of the Tamari lattice with this theory.]]> Wed, 22 Apr 2020 21:33:29 +0000 https://doi.org/10.46298/dmtcs.6409 https://doi.org/10.46298/dmtcs.6409 Viard, Francois Viard, Francois <![CDATA[We introduce a new family of complete lattices, arising from a digraph together with a valuation on its vertices and generalizing a previous construction of the author. We then apply this to the study of two long-standing conjectures of Dyer, and we provide a description of the Tamari lattice with this theory.]]> 0 <![CDATA[A classical result by Schoenberg (1942) identifies all real-valued functions that preserve positive semidefi- niteness (psd) when applied entrywise to matrices of arbitrary dimension. Schoenberg's work has continued to attract significant interest, including renewed recent attention due to applications in high-dimensional statistics. However, despite a great deal of effort in the area, an effective characterization of entrywise functions preserving positivity in a fixed dimension remains elusive to date. As a first step, we characterize new classes of polynomials preserving pos- itivity in fixed dimension. The proof of our main result is representation theoretic, and employs Schur polynomials. An alternate, variational approach also leads to several interesting consequences including (a) a hitherto unexplored Schubert cell-type stratification of the cone of psd matrices, (b) new connections between generalized Rayleigh quo- tients of Hadamard powers and Schur polynomials, and (c) a description of the joint kernels of Hadamard powers.]]> Wed, 22 Apr 2020 21:33:29 +0000 https://doi.org/10.46298/dmtcs.6408 https://doi.org/10.46298/dmtcs.6408 Belton, Alexander Guillot, Dominique Khare, Apoorva Putinar, Mihai Belton, Alexander Guillot, Dominique Khare, Apoorva Putinar, Mihai <![CDATA[A classical result by Schoenberg (1942) identifies all real-valued functions that preserve positive semidefi- niteness (psd) when applied entrywise to matrices of arbitrary dimension. Schoenberg's work has continued to attract significant interest, including renewed recent attention due to applications in high-dimensional statistics. However, despite a great deal of effort in the area, an effective characterization of entrywise functions preserving positivity in a fixed dimension remains elusive to date. As a first step, we characterize new classes of polynomials preserving pos- itivity in fixed dimension. The proof of our main result is representation theoretic, and employs Schur polynomials. An alternate, variational approach also leads to several interesting consequences including (a) a hitherto unexplored Schubert cell-type stratification of the cone of psd matrices, (b) new connections between generalized Rayleigh quo- tients of Hadamard powers and Schur polynomials, and (c) a description of the joint kernels of Hadamard powers.]]> 0 <![CDATA[The involution walk is a random walk on the symmetric group generated by involutions with a number of 2-cycles sampled from the binomial distribution with parameter p. This is a parallelization of the lazy transposition walk onthesymmetricgroup.Theinvolutionwalkisshowninthispapertomixfor1 ≤p≤1fixed,nsufficientlylarge 2 in between log1/p(n) steps and log2/(1+p)(n) steps. The paper introduces a new technique for finding eigenvalues of random walks on the symmetric group generated by many conjugacy classes using the character polynomial for the characters of the representations of the symmetric group. This is especially efficient at calculating the large eigenvalues. The smaller eigenvalues are handled by developing monotonicity relations that also give after sufficient time the likelihood order, the order from most likely to least likely state. The walk was introduced to study a conjecture about a random walk on the unitary group from the information theory of back holes.]]> Wed, 22 Apr 2020 21:33:28 +0000 https://doi.org/10.46298/dmtcs.6407 https://doi.org/10.46298/dmtcs.6407 Bernstein, Megan Bernstein, Megan <![CDATA[The involution walk is a random walk on the symmetric group generated by involutions with a number of 2-cycles sampled from the binomial distribution with parameter p. This is a parallelization of the lazy transposition walk onthesymmetricgroup.Theinvolutionwalkisshowninthispapertomixfor1 ≤p≤1fixed,nsufficientlylarge 2 in between log1/p(n) steps and log2/(1+p)(n) steps. The paper introduces a new technique for finding eigenvalues of random walks on the symmetric group generated by many conjugacy classes using the character polynomial for the characters of the representations of the symmetric group. This is especially efficient at calculating the large eigenvalues. The smaller eigenvalues are handled by developing monotonicity relations that also give after sufficient time the likelihood order, the order from most likely to least likely state. The walk was introduced to study a conjecture about a random walk on the unitary group from the information theory of back holes.]]> 0 <![CDATA[Introduced by Kawanaka in order to find the unipotent representations of finite groups of Lie type, gener- alized Gelfand–Graev characters have remained somewhat mysterious. Even in the case of the finite general linear groups, the combinatorics of their decompositions has not been worked out. This paper re-interprets Kawanaka's def- inition in type A in a way that gives far more flexibility in computations. We use these alternate constructions to show how to obtain generalized Gelfand–Graev representations directly from the maximal unipotent subgroups. We also explicitly decompose the corresponding generalized Gelfand–Graev characters in terms of unipotent representations, thereby recovering the Kostka–Foulkes polynomials as multiplicities.]]> Wed, 22 Apr 2020 21:33:27 +0000 https://doi.org/10.46298/dmtcs.6406 https://doi.org/10.46298/dmtcs.6406 Andrews, Scott Thiem, Nathaniel Andrews, Scott Thiem, Nathaniel <![CDATA[Introduced by Kawanaka in order to find the unipotent representations of finite groups of Lie type, gener- alized Gelfand–Graev characters have remained somewhat mysterious. Even in the case of the finite general linear groups, the combinatorics of their decompositions has not been worked out. This paper re-interprets Kawanaka's def- inition in type A in a way that gives far more flexibility in computations. We use these alternate constructions to show how to obtain generalized Gelfand–Graev representations directly from the maximal unipotent subgroups. We also explicitly decompose the corresponding generalized Gelfand–Graev characters in terms of unipotent representations, thereby recovering the Kostka–Foulkes polynomials as multiplicities.]]> 0 <![CDATA[Start with a permutation matrix π and consider all matrices that can be obtained from π by taking downward row operations and rightward column operations; the closure of this set gives the matrix Schubert variety Xπ. We characterize when the ideal defining Xπ is toric (with respect to a 2n − 1-dimensional torus) and study the associated polytope of its projectivization. We construct regular triangulations of these polytopes which we show are geometric realizations of a family of subword complexes. We also show that these complexes can be realized geometrically via regular triangulations of root polytopes. This implies that a family of β-Grothendieck polynomials are special cases of reduced forms in the subdivision algebra of root polytopes. We also write the volume and Ehrhart series of root polytopes in terms of β-Grothendieck polynomials. Subword complexes were introduced by Knutson and Miller in 2004, who showed that they are homeomorphic to balls or spheres and raised the question of their polytopal realizations.]]> Wed, 22 Apr 2020 21:33:27 +0000 https://doi.org/10.46298/dmtcs.6405 https://doi.org/10.46298/dmtcs.6405 Escobar, Laura Mészáros, Karola Escobar, Laura Mészáros, Karola <![CDATA[Start with a permutation matrix π and consider all matrices that can be obtained from π by taking downward row operations and rightward column operations; the closure of this set gives the matrix Schubert variety Xπ. We characterize when the ideal defining Xπ is toric (with respect to a 2n − 1-dimensional torus) and study the associated polytope of its projectivization. We construct regular triangulations of these polytopes which we show are geometric realizations of a family of subword complexes. We also show that these complexes can be realized geometrically via regular triangulations of root polytopes. This implies that a family of β-Grothendieck polynomials are special cases of reduced forms in the subdivision algebra of root polytopes. We also write the volume and Ehrhart series of root polytopes in terms of β-Grothendieck polynomials. Subword complexes were introduced by Knutson and Miller in 2004, who showed that they are homeomorphic to balls or spheres and raised the question of their polytopal realizations.]]> 0 <![CDATA[The Plücker relations which define the Grassmann manifolds as projective varieties are well known. Grass-mann manifolds are examples of minuscule flag manifolds. We study the generalized Plücker relations for minuscule flag manifolds independent of Lie type. To do this we combinatorially model the Plücker coordinates based on Wild-berger’s construction of minuscule Lie algebra representations; it uses the colored partially ordered sets known asminuscule posets. We obtain, uniformly across Lie type, descriptions of the Plücker relations of “extreme weight”. We show that these are “supported” by “double-tailed diamond” sublattices of minuscule lattices. From this, we obtain a complete set of Plücker relations for the exceptional minuscule flag manifolds. These Plücker relations are straightening laws for their coordinate rings.]]> Wed, 22 Apr 2020 21:33:26 +0000 https://doi.org/10.46298/dmtcs.6404 https://doi.org/10.46298/dmtcs.6404 Lax, David C Lax, David C <![CDATA[The Plücker relations which define the Grassmann manifolds as projective varieties are well known. Grass-mann manifolds are examples of minuscule flag manifolds. We study the generalized Plücker relations for minuscule flag manifolds independent of Lie type. To do this we combinatorially model the Plücker coordinates based on Wild-berger’s construction of minuscule Lie algebra representations; it uses the colored partially ordered sets known asminuscule posets. We obtain, uniformly across Lie type, descriptions of the Plücker relations of “extreme weight”. We show that these are “supported” by “double-tailed diamond” sublattices of minuscule lattices. From this, we obtain a complete set of Plücker relations for the exceptional minuscule flag manifolds. These Plücker relations are straightening laws for their coordinate rings.]]> 0 <![CDATA[There are two reasonable ways to put a cluster structure on a positroid variety. In one, the initial seed is a set of Plu ̈cker coordinates. In the other, the initial seed consists of certain monomials in the edge weights of a plabic graph. We will describe an automorphism of the positroid variety, the twist, which takes one to the other. For the big positroid cell, this was already done by Marsh and Scott; we generalize their results to all positroid varieties. This also provides an inversion of the boundary measurement map which is more general than Talaska's, in that it works for all reduced plabic graphs rather than just Le-diagrams. This is the analogue for positroid varieties of the twist map of Berenstein, Fomin and Zelevinsky for double Bruhat cells. Our construction involved the combinatorics of dimer configurations on bipartite planar graphs.]]> Wed, 22 Apr 2020 21:33:25 +0000 https://doi.org/10.46298/dmtcs.6403 https://doi.org/10.46298/dmtcs.6403 Muller, Greg Speyer, David E. Muller, Greg Speyer, David E. <![CDATA[There are two reasonable ways to put a cluster structure on a positroid variety. In one, the initial seed is a set of Plu ̈cker coordinates. In the other, the initial seed consists of certain monomials in the edge weights of a plabic graph. We will describe an automorphism of the positroid variety, the twist, which takes one to the other. For the big positroid cell, this was already done by Marsh and Scott; we generalize their results to all positroid varieties. This also provides an inversion of the boundary measurement map which is more general than Talaska's, in that it works for all reduced plabic graphs rather than just Le-diagrams. This is the analogue for positroid varieties of the twist map of Berenstein, Fomin and Zelevinsky for double Bruhat cells. Our construction involved the combinatorics of dimer configurations on bipartite planar graphs.]]> 0 <![CDATA[We study the motion of a robotic arm inside a rectangular tunnel of width 2. We prove that the configuration space S of all possible positions of the robot is a CAT(0) cubical complex. Before this work, very few families of robots were known to have CAT(0) configuration spaces. This property allows us to move the arm optimally from one position to another.]]> Wed, 22 Apr 2020 21:33:25 +0000 https://doi.org/10.46298/dmtcs.6402 https://doi.org/10.46298/dmtcs.6402 Ardila, Federico Bastidas, Hanner Ceballos, Cesar Guo, John Ardila, Federico Bastidas, Hanner Ceballos, Cesar Guo, John <![CDATA[We study the motion of a robotic arm inside a rectangular tunnel of width 2. We prove that the configuration space S of all possible positions of the robot is a CAT(0) cubical complex. Before this work, very few families of robots were known to have CAT(0) configuration spaces. This property allows us to move the arm optimally from one position to another.]]> 0 <![CDATA[Generalizing the connection between the classes of the sylvester congruence and the binary trees, we show that the classes of the congruence of the weak order on Sn defined as the transitive closure of the rewriting rule UacV1b1 ···VkbkW ≡k UcaV1b1 ···VkbkW, for letters a < b1,...,bk < c and words U,V1,...,Vk,W on [n], are in bijection with acyclic k-triangulations of the (n + 2k)-gon, or equivalently with acyclic pipe dreams for the permutation (1,...,k,n + k,...,k + 1,n + k + 1,...,n + 2k). It enables us to transport the known lattice and Hopf algebra structures from the congruence classes of ≡k to these acyclic pipe dreams, and to describe the product and coproduct of this algebra in terms of pipe dreams. Moreover, it shows that the fan obtained by coarsening the Coxeter fan according to the classes of ≡k is the normal fan of the corresponding brick polytope]]> Wed, 22 Apr 2020 21:33:24 +0000 https://doi.org/10.46298/dmtcs.6401 https://doi.org/10.46298/dmtcs.6401 Pilaud, Vincent Pilaud, Vincent <![CDATA[Generalizing the connection between the classes of the sylvester congruence and the binary trees, we show that the classes of the congruence of the weak order on Sn defined as the transitive closure of the rewriting rule UacV1b1 ···VkbkW ≡k UcaV1b1 ···VkbkW, for letters a < b1,...,bk < c and words U,V1,...,Vk,W on [n], are in bijection with acyclic k-triangulations of the (n + 2k)-gon, or equivalently with acyclic pipe dreams for the permutation (1,...,k,n + k,...,k + 1,n + k + 1,...,n + 2k). It enables us to transport the known lattice and Hopf algebra structures from the congruence classes of ≡k to these acyclic pipe dreams, and to describe the product and coproduct of this algebra in terms of pipe dreams. Moreover, it shows that the fan obtained by coarsening the Coxeter fan according to the classes of ≡k is the normal fan of the corresponding brick polytope]]> 0 <![CDATA[Graph associahedra are polytopes realizing the nested complex N(G) on connected subgraphs of a graph G.While all known explicit constructions produce polytopes with the same normal fan, the great variety of fan realizationsof classical associahedra and the analogy between finite type cluster complexes and nested complexes incitedus to transpose S. Fomin and A. Zelevinsky's construction of compatibility fans for generalized associahedra (2003)to graph associahedra. Using a compatibility degree, we construct one fan realization of N(G) for each of its facets.Specifying G to paths and cycles, we recover a construction by F. Santos for classical associahedra (2011) and extendF. Chapoton, S. Fomin and A. Zelevinsky's construction (2002) for type B and C generalized associahedra.]]> Wed, 22 Apr 2020 21:33:23 +0000 https://doi.org/10.46298/dmtcs.6400 https://doi.org/10.46298/dmtcs.6400 Manneville, Thibault Pilaud, Vincent Manneville, Thibault Pilaud, Vincent <![CDATA[Graph associahedra are polytopes realizing the nested complex N(G) on connected subgraphs of a graph G.While all known explicit constructions produce polytopes with the same normal fan, the great variety of fan realizationsof classical associahedra and the analogy between finite type cluster complexes and nested complexes incitedus to transpose S. Fomin and A. Zelevinsky's construction of compatibility fans for generalized associahedra (2003)to graph associahedra. Using a compatibility degree, we construct one fan realization of N(G) for each of its facets.Specifying G to paths and cycles, we recover a construction by F. Santos for classical associahedra (2011) and extendF. Chapoton, S. Fomin and A. Zelevinsky's construction (2002) for type B and C generalized associahedra.]]> 0 <![CDATA[We investigate a poset structure that extends the weak order on a finite Coxeter group W to the set of all faces of the permutahedron of W. We call this order the facial weak order. We first provide two alternative characterizations of this poset: a first one, geometric, that generalizes the notion of inversion sets of roots, and a second one, combinatorial, that uses comparisons of the minimal and maximal length representatives of the cosets. These characterizations are then used to show that the facial weak order is in fact a lattice, generalizing a well-known result of A. Bjo ̈rner for the classical weak order. Finally, we show that any lattice congruence of the classical weak order induces a lattice congruence of the facial weak order, and we give a geometric interpretation of its classes.]]> Wed, 22 Apr 2020 21:33:23 +0000 https://doi.org/10.46298/dmtcs.6399 https://doi.org/10.46298/dmtcs.6399 Dermenjian, Aram Hohlweg, Christophe Pilaud, Vincent Dermenjian, Aram Hohlweg, Christophe Pilaud, Vincent <![CDATA[We investigate a poset structure that extends the weak order on a finite Coxeter group W to the set of all faces of the permutahedron of W. We call this order the facial weak order. We first provide two alternative characterizations of this poset: a first one, geometric, that generalizes the notion of inversion sets of roots, and a second one, combinatorial, that uses comparisons of the minimal and maximal length representatives of the cosets. These characterizations are then used to show that the facial weak order is in fact a lattice, generalizing a well-known result of A. Bjo ̈rner for the classical weak order. Finally, we show that any lattice congruence of the classical weak order induces a lattice congruence of the facial weak order, and we give a geometric interpretation of its classes.]]> 0 <![CDATA[We give a different presentation of a recent bijection due to Chapuy and Dołe ̨ga for nonorientable bipartite quadrangulations and we extend it to the case of nonorientable general maps. This can be seen as a Bouttier–Di Francesco–Guitter-like generalization of the Cori–Vauquelin–Schaeffer bijection in the context of general nonori- entable surfaces. In the particular case of triangulations, the encoding objects take a particularly simple form and we recover a famous asymptotic enumeration formula found by Gao.]]> Wed, 22 Apr 2020 21:33:22 +0000 https://doi.org/10.46298/dmtcs.6398 https://doi.org/10.46298/dmtcs.6398 Bettinelli, Jérémie Bettinelli, Jérémie <![CDATA[We give a different presentation of a recent bijection due to Chapuy and Dołe ̨ga for nonorientable bipartite quadrangulations and we extend it to the case of nonorientable general maps. This can be seen as a Bouttier–Di Francesco–Guitter-like generalization of the Cori–Vauquelin–Schaeffer bijection in the context of general nonori- entable surfaces. In the particular case of triangulations, the encoding objects take a particularly simple form and we recover a famous asymptotic enumeration formula found by Gao.]]> 0 <![CDATA[There are three main constructions of supercharacter theories for a group G. The first, defined by Diaconis and Isaacs, comes from the action of a group A via automorphisms on our given group G. Another general way to construct a supercharacter theory for G, defined by Diaconis and Isaacs, uses the action of a group A of automor- phisms of the cyclotomic field Q[ζ|G|]. The third, defined by Hendrickson, is combining a supercharacter theories of a normal subgroup N of G with a supercharacter theory of G/N . In this paper we construct a supercharacter theory from an arbitrary set of normal subgroups of G. We show that when we consider the set of all normal subgroups of G, the corresponding supercharacter theory is related to a partition of G given by certain values on the central primitive idempotents. Also, we show the supercharacter theories that we construct can not be obtained via automorphisms or a single normal subgroup.]]> Wed, 22 Apr 2020 21:33:22 +0000 https://doi.org/10.46298/dmtcs.6397 https://doi.org/10.46298/dmtcs.6397 Aliniaeifard, Farid Aliniaeifard, Farid <![CDATA[There are three main constructions of supercharacter theories for a group G. The first, defined by Diaconis and Isaacs, comes from the action of a group A via automorphisms on our given group G. Another general way to construct a supercharacter theory for G, defined by Diaconis and Isaacs, uses the action of a group A of automor- phisms of the cyclotomic field Q[ζ|G|]. The third, defined by Hendrickson, is combining a supercharacter theories of a normal subgroup N of G with a supercharacter theory of G/N . In this paper we construct a supercharacter theory from an arbitrary set of normal subgroups of G. We show that when we consider the set of all normal subgroups of G, the corresponding supercharacter theory is related to a partition of G given by certain values on the central primitive idempotents. Also, we show the supercharacter theories that we construct can not be obtained via automorphisms or a single normal subgroup.]]> 0 <![CDATA[In his study of Kazhdan-Lusztig cells in affine type A, Shi has introduced an affine analog of Robinson- Schensted correspondence. We generalize the Matrix-Ball Construction of Viennot and Fulton to give a more combi- natorial realization of Shi's algorithm. As a biproduct, we also give a way to realize the affine correspondence via the usual Robinson-Schensted bumping algorithm. Next, inspired by Honeywill, we extend the algorithm to a bijection between extended affine symmetric group and triples (P, Q, ρ) where P and Q are tabloids and ρ is a dominant weight. The weights ρ get a natural interpretation in terms of the Affine Matrix-Ball Construction. Finally, we prove that fibers of the inverse map possess a Weyl group symmetry, explaining the dominance condition on weights.]]> Wed, 22 Apr 2020 21:33:21 +0000 https://doi.org/10.46298/dmtcs.6396 https://doi.org/10.46298/dmtcs.6396 Chmutov, Michael Pylyavskyy, Pavlo Yudovina, Elena Chmutov, Michael Pylyavskyy, Pavlo Yudovina, Elena <![CDATA[In his study of Kazhdan-Lusztig cells in affine type A, Shi has introduced an affine analog of Robinson- Schensted correspondence. We generalize the Matrix-Ball Construction of Viennot and Fulton to give a more combi- natorial realization of Shi's algorithm. As a biproduct, we also give a way to realize the affine correspondence via the usual Robinson-Schensted bumping algorithm. Next, inspired by Honeywill, we extend the algorithm to a bijection between extended affine symmetric group and triples (P, Q, ρ) where P and Q are tabloids and ρ is a dominant weight. The weights ρ get a natural interpretation in terms of the Affine Matrix-Ball Construction. Finally, we prove that fibers of the inverse map possess a Weyl group symmetry, explaining the dominance condition on weights.]]> 0 <![CDATA[We provide bijections between the cluster variables (and clusters) in two families of cluster algebras which have received considerable attention. These cluster algebras are the ones associated with certain Grassmannians of k-planes, and those associated with certain spaces of decorated SLk-local systems in the disk in the work of Fock and Goncharov. When k is 3, this bijection can be described explicitly using the combinatorics of Kuperberg's basis of non-elliptic webs. Using our bijection and symmetries of these cluster algebras, we provide evidence for conjectures of Fomin and Pylyavskyy concerning cluster variables in Grassmannians of 3-planes. We also prove their conjecture that there are infinitely many indecomposable nonarborizable webs in the Grassmannian of 3-planes in 9-dimensional space.]]> Wed, 22 Apr 2020 21:33:20 +0000 https://doi.org/10.46298/dmtcs.6395 https://doi.org/10.46298/dmtcs.6395 Fraser, Chris Fraser, Chris <![CDATA[We provide bijections between the cluster variables (and clusters) in two families of cluster algebras which have received considerable attention. These cluster algebras are the ones associated with certain Grassmannians of k-planes, and those associated with certain spaces of decorated SLk-local systems in the disk in the work of Fock and Goncharov. When k is 3, this bijection can be described explicitly using the combinatorics of Kuperberg's basis of non-elliptic webs. Using our bijection and symmetries of these cluster algebras, we provide evidence for conjectures of Fomin and Pylyavskyy concerning cluster variables in Grassmannians of 3-planes. We also prove their conjecture that there are infinitely many indecomposable nonarborizable webs in the Grassmannian of 3-planes in 9-dimensional space.]]> 0 <![CDATA[We construct a type A(1) n−1 affine geometric crystal structure on the Grassmannian Gr(k, n). The tropicalization of this structure recovers the combinatorics of crystal operators on semistandard Young tableaux of rectangular shape (with n − k rows), including the affine crystal operator e 0. In particular, the promotion operation on these tableaux essentially corresponds to cyclically shifting the Plu ̈cker coordinates of the Grassmannian.]]> Wed, 22 Apr 2020 21:33:19 +0000 https://doi.org/10.46298/dmtcs.6393 https://doi.org/10.46298/dmtcs.6393 Frieden, Gabriel Frieden, Gabriel <![CDATA[We construct a type A(1) n−1 affine geometric crystal structure on the Grassmannian Gr(k, n). The tropicalization of this structure recovers the combinatorics of crystal operators on semistandard Young tableaux of rectangular shape (with n − k rows), including the affine crystal operator e 0. In particular, the promotion operation on these tableaux essentially corresponds to cyclically shifting the Plu ̈cker coordinates of the Grassmannian.]]> 0 <![CDATA[We introduce a new concept of resonance on discrete dynamical systems. Our main result is an equivariant bijection between plane partitions in a box under rowmotion and increasing tableaux under K-promotion, using a generalization of the equivariance of promotion and rowmotion [J. Striker–N. Williams '12] to higher dimensional lattices. This theorem implies new results for K-promotion and new proofs of previous results on plane partitions.]]> Wed, 22 Apr 2020 21:33:19 +0000 https://doi.org/10.46298/dmtcs.6394 https://doi.org/10.46298/dmtcs.6394 Dilks, Kevin Pechenik, Oliver Striker, Jessica Dilks, Kevin Pechenik, Oliver Striker, Jessica <![CDATA[We introduce a new concept of resonance on discrete dynamical systems. Our main result is an equivariant bijection between plane partitions in a box under rowmotion and increasing tableaux under K-promotion, using a generalization of the equivariance of promotion and rowmotion [J. Striker–N. Williams '12] to higher dimensional lattices. This theorem implies new results for K-promotion and new proofs of previous results on plane partitions.]]> 0 <![CDATA[The positroid decomposition of the Grassmannian refines the well-known Schubert decomposition, and has a rich combinatorial structure. There are a number of interesting combinatorial posets which index positroid varieties,just as Young diagrams index Schubert varieties. In addition, Postnikov’s boundary measurement map gives a family of parametrizations for each positroid variety. The domain of each parametrization is the space of edge weights of a weighted planar network. The positroid stratification of the Grassmannian provides an elementary example of Lusztig’s theory of total non negativity for partial flag varieties, and has remarkable applications to particle physics.We generalize the combinatorics of positroid varieties to the Lagrangian Grassmannian, the moduli space of maximal isotropic subspaces with respect to a symplectic form]]> Wed, 22 Apr 2020 21:33:18 +0000 https://doi.org/10.46298/dmtcs.6392 https://doi.org/10.46298/dmtcs.6392 Karpman, Rachel Karpman, Rachel <![CDATA[The positroid decomposition of the Grassmannian refines the well-known Schubert decomposition, and has a rich combinatorial structure. There are a number of interesting combinatorial posets which index positroid varieties,just as Young diagrams index Schubert varieties. In addition, Postnikov’s boundary measurement map gives a family of parametrizations for each positroid variety. The domain of each parametrization is the space of edge weights of a weighted planar network. The positroid stratification of the Grassmannian provides an elementary example of Lusztig’s theory of total non negativity for partial flag varieties, and has remarkable applications to particle physics.We generalize the combinatorics of positroid varieties to the Lagrangian Grassmannian, the moduli space of maximal isotropic subspaces with respect to a symplectic form]]> 0 <![CDATA[We introduce the difference operator for functions defined on strict partitions and prove a polynomiality property for a summation involving the bar length (hook length) and content statistics. As an application, several new hook-content formulas for strict partitions are derived.]]> Wed, 22 Apr 2020 21:33:17 +0000 https://doi.org/10.46298/dmtcs.6391 https://doi.org/10.46298/dmtcs.6391 Han, Guo-Niu Xiong, Huan Han, Guo-Niu Xiong, Huan <![CDATA[We introduce the difference operator for functions defined on strict partitions and prove a polynomiality property for a summation involving the bar length (hook length) and content statistics. As an application, several new hook-content formulas for strict partitions are derived.]]> 0 <![CDATA[We consider the enumeration of walks on the two-dimensional non-negative integer lattice with steps defined by a finite set S ⊆ {±1, 0}2 . Up to isomorphism there are 79 unique two-dimensional models to consider, and previous work in this area has used the kernel method, along with a rigorous computer algebra approach, to show that 23 of the 79 models admit D-finite generating functions. In 2009, Bostan and Kauers used Pade ́-Hermite approximants to guess differential equations which these 23 generating functions satisfy, in the process guessing asymptotics of their coefficient sequences. In this article we provide, for the first time, a complete rigorous verification of these guesses. Our technique is to use the kernel method to express 19 of the 23 generating functions as diagonals of tri-variate rational functions and apply the methods of analytic combinatorics in several variables (the remaining 4 models have algebraic generating functions and can thus be handled by univariate techniques). This approach also shows the link between combinatorial properties of the models and features of its asymptotics such as asymptotic and polynomial growth factors. In addition, we give expressions for the number of walks returning to the x-axis, the y-axis, and the origin, proving recently conjectured asymptotics of Bostan, Chyzak, van Hoeij, Kauers, and Pech.]]> Wed, 22 Apr 2020 21:33:16 +0000 https://doi.org/10.46298/dmtcs.6390 https://doi.org/10.46298/dmtcs.6390 Melczer, Stephen Wilson, Mark C. Melczer, Stephen Wilson, Mark C. <![CDATA[We consider the enumeration of walks on the two-dimensional non-negative integer lattice with steps defined by a finite set S ⊆ {±1, 0}2 . Up to isomorphism there are 79 unique two-dimensional models to consider, and previous work in this area has used the kernel method, along with a rigorous computer algebra approach, to show that 23 of the 79 models admit D-finite generating functions. In 2009, Bostan and Kauers used Pade ́-Hermite approximants to guess differential equations which these 23 generating functions satisfy, in the process guessing asymptotics of their coefficient sequences. In this article we provide, for the first time, a complete rigorous verification of these guesses. Our technique is to use the kernel method to express 19 of the 23 generating functions as diagonals of tri-variate rational functions and apply the methods of analytic combinatorics in several variables (the remaining 4 models have algebraic generating functions and can thus be handled by univariate techniques). This approach also shows the link between combinatorial properties of the models and features of its asymptotics such as asymptotic and polynomial growth factors. In addition, we give expressions for the number of walks returning to the x-axis, the y-axis, and the origin, proving recently conjectured asymptotics of Bostan, Chyzak, van Hoeij, Kauers, and Pech.]]> 0 <![CDATA[Plabic graphs are combinatorial objects used to study the totally nonnegative Grassmannian. Faces of plabic graphs are labeled by k-element sets of positive integers, and a collection of such k-element sets are the face labels of a plabic graph if that collection forms a maximal weakly separated collection. There are moves that one can apply to plabic graphs, and thus to maximal weakly separated collections, analogous to mutations of seeds in cluster algebras. In this short note, we show if two maximal weakly separated collections can be mutated from one to another, then one can do so while freezing the face labels they have in common. In particular, this provides a new, and we think simpler, proof of Postnikov's result that any two reduced plabic graphs with the same decorated permutations can be mutated to each other.]]> Wed, 22 Apr 2020 21:33:16 +0000 https://doi.org/10.46298/dmtcs.6389 https://doi.org/10.46298/dmtcs.6389 Oh, Suho Speyer, David Oh, Suho Speyer, David <![CDATA[Plabic graphs are combinatorial objects used to study the totally nonnegative Grassmannian. Faces of plabic graphs are labeled by k-element sets of positive integers, and a collection of such k-element sets are the face labels of a plabic graph if that collection forms a maximal weakly separated collection. There are moves that one can apply to plabic graphs, and thus to maximal weakly separated collections, analogous to mutations of seeds in cluster algebras. In this short note, we show if two maximal weakly separated collections can be mutated from one to another, then one can do so while freezing the face labels they have in common. In particular, this provides a new, and we think simpler, proof of Postnikov's result that any two reduced plabic graphs with the same decorated permutations can be mutated to each other.]]> 0 <![CDATA[We use Khovanov-Lauda-Rouquier (KLR) algebras to categorify a crystal isomorphism between a funda-mental crystal and the tensor product of a Kirillov-Reshetikhin crystal and another fundamental crystal, all in affine type. The nodes of the Kirillov-Reshetikhin crystal correspond to a family of “trivial” modules. The nodes of the fun-damental crystal correspond to simple modules of the corresponding cyclotomic KLR algebra. The crystal operators correspond to socle of restriction and behave compatibly with the rule for tensor product of crystal graphs.]]> Wed, 22 Apr 2020 21:33:15 +0000 https://doi.org/10.46298/dmtcs.6388 https://doi.org/10.46298/dmtcs.6388 Kvinge, Henry Vazirani, Monica Kvinge, Henry Vazirani, Monica <![CDATA[We use Khovanov-Lauda-Rouquier (KLR) algebras to categorify a crystal isomorphism between a funda-mental crystal and the tensor product of a Kirillov-Reshetikhin crystal and another fundamental crystal, all in affine type. The nodes of the Kirillov-Reshetikhin crystal correspond to a family of “trivial” modules. The nodes of the fun-damental crystal correspond to simple modules of the corresponding cyclotomic KLR algebra. The crystal operators correspond to socle of restriction and behave compatibly with the rule for tensor product of crystal graphs.]]> 0 <![CDATA[We study the relationship between rational slope Dyck paths and invariant subsets in Z, extending the work of the first two authors in the relatively prime case. We also find a bijection between (dn, dm)–Dyck paths and d-tuples of (n, m)-Dyck paths endowed with certain gluing data. These are first steps towards understanding the relationship between the rational slope Catalan combinatorics in non relatively prime case and the geometry of affine Springer fibers and representation theory.]]> Wed, 22 Apr 2020 21:33:14 +0000 https://doi.org/10.46298/dmtcs.6387 https://doi.org/10.46298/dmtcs.6387 Gorsky, Eugene Mazin, Mikhail Vazirani, Monica Gorsky, Eugene Mazin, Mikhail Vazirani, Monica <![CDATA[We study the relationship between rational slope Dyck paths and invariant subsets in Z, extending the work of the first two authors in the relatively prime case. We also find a bijection between (dn, dm)–Dyck paths and d-tuples of (n, m)-Dyck paths endowed with certain gluing data. These are first steps towards understanding the relationship between the rational slope Catalan combinatorics in non relatively prime case and the geometry of affine Springer fibers and representation theory.]]> 0 <![CDATA[The Schubert polynomials lift the Schur basis of symmetric polynomials into a basis for Z[x1; x2; : : :]. We suggest the prism tableau model for these polynomials. A novel aspect of this alternative to earlier results is that it directly invokes semistandard tableaux; it does so as part of a colored tableau amalgam. In the Grassmannian case, a prism tableau with colors ignored is a semistandard Young tableau. Our arguments are developed from the Gr¨obner geometry of matrix Schubert varieties.]]> Wed, 22 Apr 2020 21:33:13 +0000 https://doi.org/10.46298/dmtcs.6386 https://doi.org/10.46298/dmtcs.6386 Weigandt, Anna Yong, Alexander Weigandt, Anna Yong, Alexander <![CDATA[The Schubert polynomials lift the Schur basis of symmetric polynomials into a basis for Z[x1; x2; : : :]. We suggest the prism tableau model for these polynomials. A novel aspect of this alternative to earlier results is that it directly invokes semistandard tableaux; it does so as part of a colored tableau amalgam. In the Grassmannian case, a prism tableau with colors ignored is a semistandard Young tableau. Our arguments are developed from the Gr¨obner geometry of matrix Schubert varieties.]]> 0 <![CDATA[Based on results by Brugallé and Mikhalkin, Fomin and Mikhalkin give formulas for computing classical Severi degrees Nd,δ using long-edge graphs. In 2012, Block, Colley and Kennedy considered the logarithmic versionof a special function associated to long-edge graphs which appeared in Fomin-Mikhalkin’s formula, and conjecturedit to be linear. They have since proved their conjecture. At the same time, motivated by their conjecture, we considera special multivariate function associated to long-edge graphs that generalizes their function. The main result of thispaper is that the multivariate function we define is always linear.The first application of our linearity result is that by applying it to classical Severi degrees, we recover quadraticity of Qd,δ and a bound δ for the threshold of polynomiality ofNd,δ.Next, in joint work with Osserman, we apply thelinearity result to a special family of toric surfaces and obtain universal polynomial results having connections to the Göttsche-Yau-Zaslow formula. As a result, we provide combinatorial formulas for the two unidentified power series B1(q) and B2(q) appearing in the Göttsche-Yau-Zaslow formula.The proof of our linearity result is completely combinatorial. We defineτ-graphs which generalize long-edge graphs,and a closely related family of combinatorial objects we call (τ,n)-words. By introducing height functions and aconcept of irreducibility, we describe ways to decompose certain families of (τ,n)-words into irreducible words,which leads to the desired results.]]> Wed, 22 Apr 2020 21:33:13 +0000 https://doi.org/10.46298/dmtcs.6385 https://doi.org/10.46298/dmtcs.6385 Liu, Fu Liu, Fu <![CDATA[Based on results by Brugallé and Mikhalkin, Fomin and Mikhalkin give formulas for computing classical Severi degrees Nd,δ using long-edge graphs. In 2012, Block, Colley and Kennedy considered the logarithmic versionof a special function associated to long-edge graphs which appeared in Fomin-Mikhalkin’s formula, and conjecturedit to be linear. They have since proved their conjecture. At the same time, motivated by their conjecture, we considera special multivariate function associated to long-edge graphs that generalizes their function. The main result of thispaper is that the multivariate function we define is always linear.The first application of our linearity result is that by applying it to classical Severi degrees, we recover quadraticity of Qd,δ and a bound δ for the threshold of polynomiality ofNd,δ.Next, in joint work with Osserman, we apply thelinearity result to a special family of toric surfaces and obtain universal polynomial results having connections to the Göttsche-Yau-Zaslow formula. As a result, we provide combinatorial formulas for the two unidentified power series B1(q) and B2(q) appearing in the Göttsche-Yau-Zaslow formula.The proof of our linearity result is completely combinatorial. We defineτ-graphs which generalize long-edge graphs,and a closely related family of combinatorial objects we call (τ,n)-words. By introducing height functions and aconcept of irreducibility, we describe ways to decompose certain families of (τ,n)-words into irreducible words,which leads to the desired results.]]> 0 <![CDATA[We conjecture two combinatorial interpretations for the symmetric function ∆eken, where ∆f is an eigenoperator for the modified Macdonald polynomials defined by Bergeron, Garsia, Haiman, and Tesler. Both interpretations can be seen as generalizations of the Shuffle Conjecture, a statement originally conjectured by Haglund, Haiman, Remmel, Loehr, and Ulyanov and recently proved by Carlsson and Mellit. We show how previous work of the second and third authors on Tesler matrices and ordered set partitions can be used to verify several cases of our conjectures. Furthermore, we use a reciprocity identity and LLT polynomials to prove another case. Finally, we show how our conjectures inspire 4-variable generalizations of the Catalan numbers, extending work of Garsia, Haiman, and the first author.]]> Wed, 22 Apr 2020 21:33:12 +0000 https://doi.org/10.46298/dmtcs.6384 https://doi.org/10.46298/dmtcs.6384 Haglund, James Remmel, Jeffrey B. Wilson, Andrew Timothy Haglund, James Remmel, Jeffrey B. Wilson, Andrew Timothy <![CDATA[We conjecture two combinatorial interpretations for the symmetric function ∆eken, where ∆f is an eigenoperator for the modified Macdonald polynomials defined by Bergeron, Garsia, Haiman, and Tesler. Both interpretations can be seen as generalizations of the Shuffle Conjecture, a statement originally conjectured by Haglund, Haiman, Remmel, Loehr, and Ulyanov and recently proved by Carlsson and Mellit. We show how previous work of the second and third authors on Tesler matrices and ordered set partitions can be used to verify several cases of our conjectures. Furthermore, we use a reciprocity identity and LLT polynomials to prove another case. Finally, we show how our conjectures inspire 4-variable generalizations of the Catalan numbers, extending work of Garsia, Haiman, and the first author.]]> 0 <![CDATA[In general, the existence of a maximal green sequence is not mutation invariant. In this paper we show that it is in fact mutation invariant for cluster quivers associated to most marked surfaces. We develop a procedure to find maximal green sequences for cluster quivers associated to an arbitrary triangulation of closed higher genus marked surfaces with at least two punctures. As a corollary, it follows that any triangulation of a marked surface with at least one boundary component has a maximal green sequence.]]> Wed, 22 Apr 2020 21:33:11 +0000 https://doi.org/10.46298/dmtcs.6383 https://doi.org/10.46298/dmtcs.6383 Mills, Matthew R. Mills, Matthew R. <![CDATA[In general, the existence of a maximal green sequence is not mutation invariant. In this paper we show that it is in fact mutation invariant for cluster quivers associated to most marked surfaces. We develop a procedure to find maximal green sequences for cluster quivers associated to an arbitrary triangulation of closed higher genus marked surfaces with at least two punctures. As a corollary, it follows that any triangulation of a marked surface with at least one boundary component has a maximal green sequence.]]> 0 <![CDATA[We consider GLn (Fq)-analogues of certain factorization problems in the symmetric group Sn: ratherthan counting factorizations of the long cycle(1,2, . . . , n) given the number of cycles of each factor, we countfactorizations of a regular elliptic element given the fixed space dimension of each factor. We show that, as in Sn, the generating function counting these factorizations has attractive coefficients after an appropriate change of basis.Our work generalizes several recent results on factorizations in GLn (Fq) and also uses a character-based approach.We end with an asymptotic application and some questions.]]> Wed, 22 Apr 2020 21:33:10 +0000 https://doi.org/10.46298/dmtcs.6382 https://doi.org/10.46298/dmtcs.6382 Lewis, Joel Brewster Morales, Alejandro H. Lewis, Joel Brewster Morales, Alejandro H. <![CDATA[We consider GLn (Fq)-analogues of certain factorization problems in the symmetric group Sn: ratherthan counting factorizations of the long cycle(1,2, . . . , n) given the number of cycles of each factor, we countfactorizations of a regular elliptic element given the fixed space dimension of each factor. We show that, as in Sn, the generating function counting these factorizations has attractive coefficients after an appropriate change of basis.Our work generalizes several recent results on factorizations in GLn (Fq) and also uses a character-based approach.We end with an asymptotic application and some questions.]]> 0 <![CDATA[The consecutive pattern poset is the infinite partially ordered set of all permutations where σ ≤ τ if τ has a subsequence of adjacent entries in the same relative order as the entries of σ. We study the structure of the intervals in this poset from topological, poset-theoretic, and enumerative perspectives. In particular, we prove that all intervals are rank-unimodal and strongly Sperner, and we characterize disconnected and shellable intervals. We also show that most intervals are not shellable and have Mo ̈bius function equal to zero.]]> Wed, 22 Apr 2020 21:33:09 +0000 https://doi.org/10.46298/dmtcs.6380 https://doi.org/10.46298/dmtcs.6380 Elizalde, Sergi McNamara, Peter R. W. Elizalde, Sergi McNamara, Peter R. W. <![CDATA[The consecutive pattern poset is the infinite partially ordered set of all permutations where σ ≤ τ if τ has a subsequence of adjacent entries in the same relative order as the entries of σ. We study the structure of the intervals in this poset from topological, poset-theoretic, and enumerative perspectives. In particular, we prove that all intervals are rank-unimodal and strongly Sperner, and we characterize disconnected and shellable intervals. We also show that most intervals are not shellable and have Mo ̈bius function equal to zero.]]> 0 <![CDATA[We establish a combinatorial connection between the real geometry and the K-theory of complex Schubert curves Spλ‚q, which are one-dimensional Schubert problems defined with respect to flags osculating the rational normal curve. In a previous paper, the second author showed that the real geometry of these curves is described by the orbits of a map ω on skew tableaux, defined as the commutator of jeu de taquin rectification and promotion. In particular, the real locus of the Schubert curve is naturally a covering space of RP1, with ω as the monodromy operator.We provide a fast, local algorithm for computing ω without rectifying the skew tableau, and show that certain steps in our algorithm are in bijective correspondence with Pechenik and Yong's genomic tableaux, which enumerate the K-theoretic Littlewood-Richardson coefficient associated to the Schubert curve. Using this bijection, we give purely combinatorial proofs of several numerical results involving the K-theory and real geometry of Spλ‚q.]]> Wed, 22 Apr 2020 21:33:09 +0000 https://doi.org/10.46298/dmtcs.6381 https://doi.org/10.46298/dmtcs.6381 Gillespie, Maria Monks Levinson, Jake Gillespie, Maria Monks Levinson, Jake <![CDATA[We establish a combinatorial connection between the real geometry and the K-theory of complex Schubert curves Spλ‚q, which are one-dimensional Schubert problems defined with respect to flags osculating the rational normal curve. In a previous paper, the second author showed that the real geometry of these curves is described by the orbits of a map ω on skew tableaux, defined as the commutator of jeu de taquin rectification and promotion. In particular, the real locus of the Schubert curve is naturally a covering space of RP1, with ω as the monodromy operator.We provide a fast, local algorithm for computing ω without rectifying the skew tableau, and show that certain steps in our algorithm are in bijective correspondence with Pechenik and Yong's genomic tableaux, which enumerate the K-theoretic Littlewood-Richardson coefficient associated to the Schubert curve. Using this bijection, we give purely combinatorial proofs of several numerical results involving the K-theory and real geometry of Spλ‚q.]]> 0 <![CDATA[Given a tree embedded in a disk, we define two lattices - the oriented flip graph of noncrossing arcs and the lattice of noncrossing tree partitions. When the interior vertices of the tree have degree 3, the oriented flip graph is equivalent to the oriented exchange graph of a type A cluster algebra. Our main result is an isomorphism between the shard intersection order of the oriented flip graph and the lattice of noncrossing tree partitions. As a consequence, we deduce a simple characterization of c-matrices of type A cluster algebras.]]> Wed, 22 Apr 2020 21:33:08 +0000 https://doi.org/10.46298/dmtcs.6379 https://doi.org/10.46298/dmtcs.6379 Garver, Alexander McConville, Thomas Garver, Alexander McConville, Thomas <![CDATA[Given a tree embedded in a disk, we define two lattices - the oriented flip graph of noncrossing arcs and the lattice of noncrossing tree partitions. When the interior vertices of the tree have degree 3, the oriented flip graph is equivalent to the oriented exchange graph of a type A cluster algebra. Our main result is an isomorphism between the shard intersection order of the oriented flip graph and the lattice of noncrossing tree partitions. As a consequence, we deduce a simple characterization of c-matrices of type A cluster algebras.]]> 0 <![CDATA[We introduce n(n − 1)/2 natural involutions (“toggles”) on the set S of noncrossing partitions π of size n, along with certain composite operations obtained by composing these involutions. We show that for many operations T of this kind, a surprisingly large family of functions f on S (including the function that sends π to the number of blocks of π) exhibits the homomesy phenomenon: the average of f over the elements of a T -orbit is the same for all T -orbits. Our methods apply more broadly to toggle operations on independent sets of certain graphs.]]> Wed, 22 Apr 2020 21:33:06 +0000 https://doi.org/10.46298/dmtcs.6378 https://doi.org/10.46298/dmtcs.6378 Einstein, David Farber, Miriam Gunawan, Emily Joseph, Michael Macauley, Matthew Propp, James Rubinstein-Salzedo, Simon Einstein, David Farber, Miriam Gunawan, Emily Joseph, Michael Macauley, Matthew Propp, James Rubinstein-Salzedo, Simon <![CDATA[We introduce n(n − 1)/2 natural involutions (“toggles”) on the set S of noncrossing partitions π of size n, along with certain composite operations obtained by composing these involutions. We show that for many operations T of this kind, a surprisingly large family of functions f on S (including the function that sends π to the number of blocks of π) exhibits the homomesy phenomenon: the average of f over the elements of a T -orbit is the same for all T -orbits. Our methods apply more broadly to toggle operations on independent sets of certain graphs.]]> 0 <![CDATA[Lusztig's theory of PBW bases gives a way to realize the crystal B(∞) for any simple complex Lie algebra where the underlying set consists of Kostant partitions. In fact, there are many different such realizations, one for each reduced expression for the longest element of the Weyl group. There is an algorithm to calculate the actions of the crystal operators, but it can be quite complicated. For ADE types, we give conditions on the reduced expression which ensure that the corresponding crystal operators are given by simple combinatorial bracketing rules. We then give at least one reduced expression satisfying our conditions in every type except E8, and discuss the resulting combinatorics. Finally, we describe the relationship with more standard tableaux combinatorics in types A and D.]]> Wed, 22 Apr 2020 21:33:05 +0000 https://doi.org/10.46298/dmtcs.6377 https://doi.org/10.46298/dmtcs.6377 Salisbury, Ben Schultze, Adam Tingley, Peter Salisbury, Ben Schultze, Adam Tingley, Peter <![CDATA[Lusztig's theory of PBW bases gives a way to realize the crystal B(∞) for any simple complex Lie algebra where the underlying set consists of Kostant partitions. In fact, there are many different such realizations, one for each reduced expression for the longest element of the Weyl group. There is an algorithm to calculate the actions of the crystal operators, but it can be quite complicated. For ADE types, we give conditions on the reduced expression which ensure that the corresponding crystal operators are given by simple combinatorial bracketing rules. We then give at least one reduced expression satisfying our conditions in every type except E8, and discuss the resulting combinatorics. Finally, we describe the relationship with more standard tableaux combinatorics in types A and D.]]> 0 <![CDATA[The poset P of all permutations ordered by pattern containment is a fundamental object of study in the field of permutation patterns. This poset has a very rich and complex topology and an understanding of its Möbius function has proved particularly elusive, although results have been slowly emerging in the last few years. Using a variety of topological techniques we present a two term formula for the Mo ̈bius function of intervals in P. The first term in this formula is, up to sign, the number of so called normal occurrences of one permutation in another. Our definition of normal occurrences is similar to those that have appeared in several variations in the literature on the Möbius function of this and other posets, but simpler than most of them. The second term in the formula is (still) complicated, but we conjecture that it equals zero for a significant proportion of intervals. We present some cases where the second term vanishes and others where it is nonzero. Computing the Möbius function recursively from its definition has exponential complexity, whereas the computation of the first term in our formula is polynomial and the exponential part is isolated to the second term, which seems to often vanish. This is thus the first polynomial time formula for the Möbius function of what appears to be a large proportion of all intervals of P.]]> Wed, 22 Apr 2020 21:33:05 +0000 https://doi.org/10.46298/dmtcs.6376 https://doi.org/10.46298/dmtcs.6376 Smith, Jason P Smith, Jason P <![CDATA[The poset P of all permutations ordered by pattern containment is a fundamental object of study in the field of permutation patterns. This poset has a very rich and complex topology and an understanding of its Möbius function has proved particularly elusive, although results have been slowly emerging in the last few years. Using a variety of topological techniques we present a two term formula for the Mo ̈bius function of intervals in P. The first term in this formula is, up to sign, the number of so called normal occurrences of one permutation in another. Our definition of normal occurrences is similar to those that have appeared in several variations in the literature on the Möbius function of this and other posets, but simpler than most of them. The second term in the formula is (still) complicated, but we conjecture that it equals zero for a significant proportion of intervals. We present some cases where the second term vanishes and others where it is nonzero. Computing the Möbius function recursively from its definition has exponential complexity, whereas the computation of the first term in our formula is polynomial and the exponential part is isolated to the second term, which seems to often vanish. This is thus the first polynomial time formula for the Möbius function of what appears to be a large proportion of all intervals of P.]]> 0 <![CDATA[Amoebas are projections of complex algebraic varieties in the algebraic torus under a Log-absolute value map, which have connections to various mathematical subjects. While amoebas of hypersurfaces have been inten- sively studied during the last years, the non-hypersurface case is barely understood so far. We investigate intersections of amoebas of n hypersurfaces in (C∗)n, which are genuine supersets of amoebas given by non-hypersurface vari- eties. Our main results are amoeba analogs of Bernstein's Theorem and Be ́zout's Theorem providing an upper bound for the number of connected components of such intersections. Moreover, we show that the order map for hypersur- face amoebas can be generalized in a natural way to intersections of amoebas. We show that, analogous to the case of amoebas of hypersurfaces, the restriction of this generalized order map to a single connected component is still 1-to-1.]]> Wed, 22 Apr 2020 21:33:04 +0000 https://doi.org/10.46298/dmtcs.6375 https://doi.org/10.46298/dmtcs.6375 Juhnke-Kubitzke, Martina De Wolff, Timo Juhnke-Kubitzke, Martina De Wolff, Timo <![CDATA[Amoebas are projections of complex algebraic varieties in the algebraic torus under a Log-absolute value map, which have connections to various mathematical subjects. While amoebas of hypersurfaces have been inten- sively studied during the last years, the non-hypersurface case is barely understood so far. We investigate intersections of amoebas of n hypersurfaces in (C∗)n, which are genuine supersets of amoebas given by non-hypersurface vari- eties. Our main results are amoeba analogs of Bernstein's Theorem and Be ́zout's Theorem providing an upper bound for the number of connected components of such intersections. Moreover, we show that the order map for hypersur- face amoebas can be generalized in a natural way to intersections of amoebas. We show that, analogous to the case of amoebas of hypersurfaces, the restriction of this generalized order map to a single connected component is still 1-to-1.]]> 0 <![CDATA[The dual stable Grothendieck polynomials are a deformation of the Schur functions, originating in the study of the K-theory of the Grassmannian. We generalize these polynomials by introducing a countable family of additional parameters such that the generalization still defines symmetric functions. We outline two self-contained proofs of this fact, one of which constructs a family of involutions on the set of reverse plane partitions generalizing the Bender-Knuth involutions on semistandard tableaux, whereas the other classifies the structure of reverse plane partitions with entries 1 and 2.]]> Wed, 22 Apr 2020 21:33:03 +0000 https://doi.org/10.46298/dmtcs.6374 https://doi.org/10.46298/dmtcs.6374 Galashin, Pavel Grinberg, Darij Liu, Gaku Galashin, Pavel Grinberg, Darij Liu, Gaku <![CDATA[The dual stable Grothendieck polynomials are a deformation of the Schur functions, originating in the study of the K-theory of the Grassmannian. We generalize these polynomials by introducing a countable family of additional parameters such that the generalization still defines symmetric functions. We outline two self-contained proofs of this fact, one of which constructs a family of involutions on the set of reverse plane partitions generalizing the Bender-Knuth involutions on semistandard tableaux, whereas the other classifies the structure of reverse plane partitions with entries 1 and 2.]]> 0 <![CDATA[Characterizing sets of permutations whose associated quasisymmetric function is symmetric and Schur- positive is a long-standing problem in algebraic combinatorics. In this paper we present a general method to construct Schur-positive sets and multisets, based on geometric grid classes and the product operation. Our approach produces many new instances of Schur-positive sets, and provides a broad framework that explains the existence of known such sets that until now were sporadic cases.]]> Wed, 22 Apr 2020 21:33:02 +0000 https://doi.org/10.46298/dmtcs.6373 https://doi.org/10.46298/dmtcs.6373 Elizalde, Sergi Roichman, Yuval Elizalde, Sergi Roichman, Yuval <![CDATA[Characterizing sets of permutations whose associated quasisymmetric function is symmetric and Schur- positive is a long-standing problem in algebraic combinatorics. In this paper we present a general method to construct Schur-positive sets and multisets, based on geometric grid classes and the product operation. Our approach produces many new instances of Schur-positive sets, and provides a broad framework that explains the existence of known such sets that until now were sporadic cases.]]> 0 <![CDATA[We present a general diagrammatic approach to the construction of efficient algorithms for computingthe Fourier transform of a function on a finite group. By extending work which connects Bratteli diagrams to theconstruction of Fast Fourier Transform algorithms we make explicit use of the path algebra connection and work inthe setting of quivers. In this setting the complexity of an algorithm for computing a Fourier transform reduces to pathcounting in the Bratelli diagram, and we generalize Stanley's work on differential posets to provide such counts. Ourmethods give improved upper bounds for computing the Fourier transform for the general linear groups over finitefields, the classical Weyl groups, and homogeneous spaces of finite groups.]]> Wed, 22 Apr 2020 21:33:01 +0000 https://doi.org/10.46298/dmtcs.6372 https://doi.org/10.46298/dmtcs.6372 Maslan, David Rockmore, Daniel N. Wolff, Sarah Maslan, David Rockmore, Daniel N. Wolff, Sarah <![CDATA[We present a general diagrammatic approach to the construction of efficient algorithms for computingthe Fourier transform of a function on a finite group. By extending work which connects Bratteli diagrams to theconstruction of Fast Fourier Transform algorithms we make explicit use of the path algebra connection and work inthe setting of quivers. In this setting the complexity of an algorithm for computing a Fourier transform reduces to pathcounting in the Bratelli diagram, and we generalize Stanley's work on differential posets to provide such counts. Ourmethods give improved upper bounds for computing the Fourier transform for the general linear groups over finitefields, the classical Weyl groups, and homogeneous spaces of finite groups.]]> 0 <![CDATA[We present two results, the first on the distribution of the roots of a polynomial over the ring of integers modulo n and the second on the distribution of the roots of the Sylvester resultant of two multivariate polynomials. The second result has application to polynomial GCD computation and solving polynomial diophantine equations.]]> Wed, 22 Apr 2020 21:33:00 +0000 https://doi.org/10.46298/dmtcs.6371 https://doi.org/10.46298/dmtcs.6371 Monagan, Michael Tuncer, Baris Monagan, Michael Tuncer, Baris <![CDATA[We present two results, the first on the distribution of the roots of a polynomial over the ring of integers modulo n and the second on the distribution of the roots of the Sylvester resultant of two multivariate polynomials. The second result has application to polynomial GCD computation and solving polynomial diophantine equations.]]> 0 <![CDATA[We generalize the definition of Yang-Baxter basis of type A Hecke algebra introduced by A.Lascoux, B.Leclerc and J.Y.Thibon (Letters in Math. Phys., 40 (1997), 75–90) to all the Lie types and prove their duality. As an application we give a solution to Casselman's problem on Iwahori fixed vectors of principal series representation of p-adic groups.]]> Wed, 22 Apr 2020 21:32:59 +0000 https://doi.org/10.46298/dmtcs.6370 https://doi.org/10.46298/dmtcs.6370 Nakasuji, Maki Naruse, Hiroshi Nakasuji, Maki Naruse, Hiroshi <![CDATA[We generalize the definition of Yang-Baxter basis of type A Hecke algebra introduced by A.Lascoux, B.Leclerc and J.Y.Thibon (Letters in Math. Phys., 40 (1997), 75–90) to all the Lie types and prove their duality. As an application we give a solution to Casselman's problem on Iwahori fixed vectors of principal series representation of p-adic groups.]]> 0 <![CDATA[this is an extended abstract of the full version. We study n-vertex d-dimensional polytopes with at most one nonsimplex facet with, say, d + s vertices, called almost simplicial polytopes. We provide tight lower and upper bounds for the face numbers of these polytopes as functions of d, n and s, thus generalizing the classical Lower Bound Theorem by Barnette and Upper Bound Theorem by McMullen, which treat the case s = 0. We characterize the minimizers and provide examples of maximizers, for any d.]]> Wed, 22 Apr 2020 21:32:58 +0000 https://doi.org/10.46298/dmtcs.6369 https://doi.org/10.46298/dmtcs.6369 Nevo, Eran Pineda-Villavicencio, Guillermo Ugon, Julien Yost, David Nevo, Eran Pineda-Villavicencio, Guillermo Ugon, Julien Yost, David <![CDATA[this is an extended abstract of the full version. We study n-vertex d-dimensional polytopes with at most one nonsimplex facet with, say, d + s vertices, called almost simplicial polytopes. We provide tight lower and upper bounds for the face numbers of these polytopes as functions of d, n and s, thus generalizing the classical Lower Bound Theorem by Barnette and Upper Bound Theorem by McMullen, which treat the case s = 0. We characterize the minimizers and provide examples of maximizers, for any d.]]> 0 <![CDATA[We enumerate the connected graphs that contain a linear number of edges with respect to the number of vertices. So far, only the first term of the asymptotics was known. Using analytic combinatorics, i.e. generating function manipulations, we derive the complete asymptotic expansion.]]> Wed, 22 Apr 2020 21:32:57 +0000 https://doi.org/10.46298/dmtcs.6368 https://doi.org/10.46298/dmtcs.6368 De Panafieu, Élie De Panafieu, Élie <![CDATA[We enumerate the connected graphs that contain a linear number of edges with respect to the number of vertices. So far, only the first term of the asymptotics was known. Using analytic combinatorics, i.e. generating function manipulations, we derive the complete asymptotic expansion.]]> 0 <![CDATA[The geometric Littlewood-Richardson (LR) rule is a combinatorial algorithm for computing LR coefficients derived from degenerating the Richardson variety into a union of Schubert varieties in the Grassmannian. Such rules were first given by Vakil and later generalized by Coskun. In this paper we give a noncommutative version of the geometric LR rule. As a consequence, we establish a geometric explanation for the positivity of noncommutative LR coefficients in certain cases.]]> Wed, 22 Apr 2020 21:32:56 +0000 https://doi.org/10.46298/dmtcs.6367 https://doi.org/10.46298/dmtcs.6367 Richmond, Edward Tewari, Vasu Van Willigenburg, Stephanie Richmond, Edward Tewari, Vasu Van Willigenburg, Stephanie <![CDATA[The geometric Littlewood-Richardson (LR) rule is a combinatorial algorithm for computing LR coefficients derived from degenerating the Richardson variety into a union of Schubert varieties in the Grassmannian. Such rules were first given by Vakil and later generalized by Coskun. In this paper we give a noncommutative version of the geometric LR rule. As a consequence, we establish a geometric explanation for the positivity of noncommutative LR coefficients in certain cases.]]> 0 <![CDATA[We consider families of quasisymmetric functions with the property that if a symmetric function f is a positive sum of functions in one of these families, then f is necessarily a positive sum of Schur functions. Furthermore, in each of the families studied, we give a combinatorial description of the Schur coefficients of f. We organize six such families into a poset, where functions in higher families in the poset are always positive integer sums of functions in each of the lower families.]]> Wed, 22 Apr 2020 21:32:55 +0000 https://doi.org/10.46298/dmtcs.6366 https://doi.org/10.46298/dmtcs.6366 Roberts, Austin Roberts, Austin <![CDATA[We consider families of quasisymmetric functions with the property that if a symmetric function f is a positive sum of functions in one of these families, then f is necessarily a positive sum of Schur functions. Furthermore, in each of the families studied, we give a combinatorial description of the Schur coefficients of f. We organize six such families into a poset, where functions in higher families in the poset are always positive integer sums of functions in each of the lower families.]]> 0 <![CDATA[Motivated by a question of Duval and Reiner about higher Laplacians of simplicial complexes, we describe various relaxations of the defining axioms of matroid theory to obtain larger classes of simplicial complexes that contain pure shifted simplicial complexes. The resulting classes retain some of the matroid properties and allow us to classify matroid properties according to the relevant axioms needed to prove them. We illustrate this by discussing Tutte polynomials. Furthermore, we extend a conjecture of Stanley on h-vectors and provide evidence to show that the extension is better suited than matroids to study the conjecture.]]> Wed, 22 Apr 2020 21:32:55 +0000 https://doi.org/10.46298/dmtcs.6365 https://doi.org/10.46298/dmtcs.6365 Samper, Jose ́ Alejandro Samper, Jose ́ Alejandro <![CDATA[Motivated by a question of Duval and Reiner about higher Laplacians of simplicial complexes, we describe various relaxations of the defining axioms of matroid theory to obtain larger classes of simplicial complexes that contain pure shifted simplicial complexes. The resulting classes retain some of the matroid properties and allow us to classify matroid properties according to the relevant axioms needed to prove them. We illustrate this by discussing Tutte polynomials. Furthermore, we extend a conjecture of Stanley on h-vectors and provide evidence to show that the extension is better suited than matroids to study the conjecture.]]> 0 <![CDATA[Let gcd(a, b) = 1. J. Olsson and D. Stanton proved that the maximum number of boxes in a simultaneous (a, b)-core is (a2 −1)(b2 −1) 24, and showed that this maximum is achieved by a unique core. P. Johnson combined Ehrhart theory with the polynomial method to prove D. Armstrong's conjecture that the expected number of boxes in a simultaneous (a, b)-core is (a−1)(b−1)(a+b+1) 24. We apply P. Johnson's method to compute the variance and third moment. By extending the definitions of “simultaneous cores” and “number of boxes” to affine Weyl groups, we give uniform generalizations of these formulae to simply-laced affine types. We further explain the appearance of the number 24 using the “strange formula” of H. Freudenthal and H. de Vries.]]> Wed, 22 Apr 2020 21:32:54 +0000 https://doi.org/10.46298/dmtcs.6364 https://doi.org/10.46298/dmtcs.6364 Thiel, Marko Williams, Nathan Thiel, Marko Williams, Nathan <![CDATA[Let gcd(a, b) = 1. J. Olsson and D. Stanton proved that the maximum number of boxes in a simultaneous (a, b)-core is (a2 −1)(b2 −1) 24, and showed that this maximum is achieved by a unique core. P. Johnson combined Ehrhart theory with the polynomial method to prove D. Armstrong's conjecture that the expected number of boxes in a simultaneous (a, b)-core is (a−1)(b−1)(a+b+1) 24. We apply P. Johnson's method to compute the variance and third moment. By extending the definitions of “simultaneous cores” and “number of boxes” to affine Weyl groups, we give uniform generalizations of these formulae to simply-laced affine types. We further explain the appearance of the number 24 using the “strange formula” of H. Freudenthal and H. de Vries.]]> 0 <![CDATA[We prove that the noncrossing partition lattices associated with the complex reflection groups G(d, d, n) for d, n ≥ 2 admit a decomposition into saturated chains that are symmetric about the middle ranks. A consequence of this result is that these lattices have the strong Sperner property, which asserts that the cardinality of the union of the k largest antichains does not exceed the sum of the k largest ranks for all k ≤ n. Subsequently, we use a computer to complete the proof that any noncrossing partition lattice associated with a well-generated complex reflection group is strongly Sperner, thus affirmatively answering a special case of a question of D. Armstrong. This was previously established only for the Coxeter groups of type A and B.]]> Wed, 22 Apr 2020 21:32:53 +0000 https://doi.org/10.46298/dmtcs.6363 https://doi.org/10.46298/dmtcs.6363 Mühle, Henri Mühle, Henri <![CDATA[We prove that the noncrossing partition lattices associated with the complex reflection groups G(d, d, n) for d, n ≥ 2 admit a decomposition into saturated chains that are symmetric about the middle ranks. A consequence of this result is that these lattices have the strong Sperner property, which asserts that the cardinality of the union of the k largest antichains does not exceed the sum of the k largest ranks for all k ≤ n. Subsequently, we use a computer to complete the proof that any noncrossing partition lattice associated with a well-generated complex reflection group is strongly Sperner, thus affirmatively answering a special case of a question of D. Armstrong. This was previously established only for the Coxeter groups of type A and B.]]> 0 <![CDATA[In this work, we construct elliptic analogues of the rook numbers and file numbers by attaching elliptic weights to the cells in a board. We show that our elliptic rook and file numbers satisfy elliptic extensions of corre- sponding factorization theorems which in the classical case were established by Goldman, Joichi and White and by Garsia and Remmel in the file number case. This factorization theorem can be used to define elliptic analogues of various kinds of Stirling numbers of the first and second kind as well as Abel numbers. We also give analogous results for matchings of graphs, elliptically extending the result of Haglund and Remmel.]]> Wed, 22 Apr 2020 21:32:53 +0000 https://doi.org/10.46298/dmtcs.6362 https://doi.org/10.46298/dmtcs.6362 Schlosser, Michael J. Yoo, Meesue Schlosser, Michael J. Yoo, Meesue <![CDATA[In this work, we construct elliptic analogues of the rook numbers and file numbers by attaching elliptic weights to the cells in a board. We show that our elliptic rook and file numbers satisfy elliptic extensions of corre- sponding factorization theorems which in the classical case were established by Goldman, Joichi and White and by Garsia and Remmel in the file number case. This factorization theorem can be used to define elliptic analogues of various kinds of Stirling numbers of the first and second kind as well as Abel numbers. We also give analogous results for matchings of graphs, elliptically extending the result of Haglund and Remmel.]]> 0 <![CDATA[The problem of computing products of Schubert classes in the cohomology ring can be formulated as theproblem of expanding skew Schur polynomial into the basis of ordinary Schur polynomials. We reformulate theproblem of computing the structure constants of the Grothendieck ring of a Grassmannian variety with respect to itsbasis of Schubert structure sheaves in a similar way; we address the problem of expanding the generating functions forskew reverse-plane partitions into the basis of polynomials which are Hall-dual to stable Grothendieck polynomials. From this point of view, we produce a chain of bijections leading to Buch’s K-theoretic Littlewood-Richardson rule.]]> Wed, 22 Apr 2020 21:32:52 +0000 https://doi.org/10.46298/dmtcs.6361 https://doi.org/10.46298/dmtcs.6361 Li, Huilan Morse, Jennifer Shields, Pat Li, Huilan Morse, Jennifer Shields, Pat <![CDATA[The problem of computing products of Schubert classes in the cohomology ring can be formulated as theproblem of expanding skew Schur polynomial into the basis of ordinary Schur polynomials. We reformulate theproblem of computing the structure constants of the Grothendieck ring of a Grassmannian variety with respect to itsbasis of Schubert structure sheaves in a similar way; we address the problem of expanding the generating functions forskew reverse-plane partitions into the basis of polynomials which are Hall-dual to stable Grothendieck polynomials. From this point of view, we produce a chain of bijections leading to Buch’s K-theoretic Littlewood-Richardson rule.]]> 0 <![CDATA[For a finite subgroup G of the special unitary group SU2, we study the centralizer algebra Zk(G) = EndG(V⊗k) of G acting on the k-fold tensor product of its defining representation V = C2. The McKay corre- spondence relates the representation theory of these groups to an associated affine Dynkin diagram, and we use this connection to study the structure and representation theory of Zk(G) via the combinatorics of the Dynkin diagram. When G equals the binary tetrahedral, octahedral, or icosahedral group, we exhibit remarkable connections between Zk (G) and the Martin-Jones set partition algebras.]]> Wed, 22 Apr 2020 21:32:51 +0000 https://doi.org/10.46298/dmtcs.6360 https://doi.org/10.46298/dmtcs.6360 Benkart, Georgia Halverson, Tom Benkart, Georgia Halverson, Tom <![CDATA[For a finite subgroup G of the special unitary group SU2, we study the centralizer algebra Zk(G) = EndG(V⊗k) of G acting on the k-fold tensor product of its defining representation V = C2. The McKay corre- spondence relates the representation theory of these groups to an associated affine Dynkin diagram, and we use this connection to study the structure and representation theory of Zk(G) via the combinatorics of the Dynkin diagram. When G equals the binary tetrahedral, octahedral, or icosahedral group, we exhibit remarkable connections between Zk (G) and the Martin-Jones set partition algebras.]]> 0 <![CDATA[We introduce a Hopf algebra structure of subword complexes, including both finite and infinite types. We present an explicit cancellation free formula for the antipode using acyclic orientations of certain graphs, and show that this Hopf algebra induces a natural non-trivial sub-Hopf algebra on c-clusters in the theory of cluster algebras.]]> Wed, 22 Apr 2020 21:32:51 +0000 https://doi.org/10.46298/dmtcs.6359 https://doi.org/10.46298/dmtcs.6359 Bergeron, Nantel Ceballos, Cesar Bergeron, Nantel Ceballos, Cesar <![CDATA[We introduce a Hopf algebra structure of subword complexes, including both finite and infinite types. We present an explicit cancellation free formula for the antipode using acyclic orientations of certain graphs, and show that this Hopf algebra induces a natural non-trivial sub-Hopf algebra on c-clusters in the theory of cluster algebras.]]> 0 <![CDATA[We introduce and study a special class of ideals over the semiring of tropical polynomials, which we calltropical ideals, with the goal of developing a useful and solid algebraic foundation for tropical geometry. We exploretheir rich combinatorial structure, and prove that they satisfy numerous properties analogous to classical ideals.]]> Wed, 22 Apr 2020 21:32:50 +0000 https://doi.org/10.46298/dmtcs.6358 https://doi.org/10.46298/dmtcs.6358 Maclagan, Diane Rincón, Felipe Maclagan, Diane Rincón, Felipe <![CDATA[We introduce and study a special class of ideals over the semiring of tropical polynomials, which we calltropical ideals, with the goal of developing a useful and solid algebraic foundation for tropical geometry. We exploretheir rich combinatorial structure, and prove that they satisfy numerous properties analogous to classical ideals.]]> 0 <![CDATA[We provide a new succession rule (i.e. generating tree) associated with Schröder numbers, that interpolates between the known succession rules for Catalan and Baxter numbers. We define Schröder and Baxter generalizations of parallelogram polyominoes (called slicings) which grow according to these succession rules. We also exhibit Schröder subclasses of Baxter classes, namely a Schröder subset of triples of non-intersecting lattice paths, and a new Schröder subset of Baxter permutations.]]> Wed, 22 Apr 2020 21:32:49 +0000 https://doi.org/10.46298/dmtcs.6357 https://doi.org/10.46298/dmtcs.6357 Bouvel, Mathilde Guerrini, Veronica Rinaldi, Simone Bouvel, Mathilde Guerrini, Veronica Rinaldi, Simone <![CDATA[We provide a new succession rule (i.e. generating tree) associated with Schröder numbers, that interpolates between the known succession rules for Catalan and Baxter numbers. We define Schröder and Baxter generalizations of parallelogram polyominoes (called slicings) which grow according to these succession rules. We also exhibit Schröder subclasses of Baxter classes, namely a Schröder subset of triples of non-intersecting lattice paths, and a new Schröder subset of Baxter permutations.]]> 0 <![CDATA[The totally nonnegative Grassmannian Gr≥0 k,n is the set of k-dimensional subspaces V of Rn whose nonzero Plucker coordinates all have the same sign. In their study of scattering amplitudes in N = 4 supersym- metric Yang-Mills theory, Arkani-Hamed and Trnka (2013) considered the image (called an amplituhedron) of Gr≥0 k,n under a linear map Z : Rn → Rr, where k ≤ r and the r × r minors of Z are all positive. One reason they required this positivity condition is to ensure that the map Gr≥0 k,n → Grk,r induced by Z is well defined, i.e. it takes everynelement of Gr≥0 k,n to a k-dimensional subspace of Rr. Lam (2015) gave a sufficient condition for the induced map Gr≥0 k,n → Grk,r to be well defined, in which case he called the image a Grassmann polytope. (In the case k = 1, Grassmann polytopes are just polytopes, and amplituhedra are cyclic polytopes.) We give a necessary and sufficient condition for the induced map Gr≥0 k,n → Grk,r to be well defined, in terms of sign variation. Using previous work we presented at FPSAC 2015, we obtain an equivalent condition in terms of the r × r minors of Z (assuming Z has rank r).]]> Wed, 22 Apr 2020 21:32:49 +0000 https://doi.org/10.46298/dmtcs.6356 https://doi.org/10.46298/dmtcs.6356 Karp, Steven N. Karp, Steven N. <![CDATA[The totally nonnegative Grassmannian Gr≥0 k,n is the set of k-dimensional subspaces V of Rn whose nonzero Plucker coordinates all have the same sign. In their study of scattering amplitudes in N = 4 supersym- metric Yang-Mills theory, Arkani-Hamed and Trnka (2013) considered the image (called an amplituhedron) of Gr≥0 k,n under a linear map Z : Rn → Rr, where k ≤ r and the r × r minors of Z are all positive. One reason they required this positivity condition is to ensure that the map Gr≥0 k,n → Grk,r induced by Z is well defined, i.e. it takes everynelement of Gr≥0 k,n to a k-dimensional subspace of Rr. Lam (2015) gave a sufficient condition for the induced map Gr≥0 k,n → Grk,r to be well defined, in which case he called the image a Grassmann polytope. (In the case k = 1, Grassmann polytopes are just polytopes, and amplituhedra are cyclic polytopes.) We give a necessary and sufficient condition for the induced map Gr≥0 k,n → Grk,r to be well defined, in terms of sign variation. Using previous work we presented at FPSAC 2015, we obtain an equivalent condition in terms of the r × r minors of Z (assuming Z has rank r).]]> 0 <![CDATA[The celebrated hook-length formula gives a product formula for the number of standard Young tableaux of a straight shape. In 2014, Naruse announced a more general formula for the number of standard Young tableaux of skew shapes as a positive sum over excited diagrams of products of hook-lengths. We give two q-analogues of Naruse's formula for the skew Schur functions and for counting reverse plane partitions of skew shapes. We also apply our results to border strip shapes and their generalizations.]]> Wed, 22 Apr 2020 21:32:48 +0000 https://doi.org/10.46298/dmtcs.6354 https://doi.org/10.46298/dmtcs.6354 Morales, Alejandro H. Pak, Igor Panova, Greta Morales, Alejandro H. Pak, Igor Panova, Greta <![CDATA[The celebrated hook-length formula gives a product formula for the number of standard Young tableaux of a straight shape. In 2014, Naruse announced a more general formula for the number of standard Young tableaux of skew shapes as a positive sum over excited diagrams of products of hook-lengths. We give two q-analogues of Naruse's formula for the skew Schur functions and for counting reverse plane partitions of skew shapes. We also apply our results to border strip shapes and their generalizations.]]> 0 <![CDATA[We prove that the external activity complex Act<(M) of a matroid is shellable. In fact, we show that every linear extension of Las Vergnas's external/internal order Wed, 22 Apr 2020 21:32:48 +0000 https://doi.org/10.46298/dmtcs.6355 https://doi.org/10.46298/dmtcs.6355 Ardila, Federico Castillo, Federico Samper, Jose, Ardila, Federico Castillo, Federico Samper, Jose, <![CDATA[We prove that the external activity complex Act<(M) of a matroid is shellable. In fact, we show that every linear extension of Las Vergnas's external/internal order 0 <![CDATA[For any Coxeter system (W, S) of rank n, we introduce an abstract boolean complex (simplicial poset) of dimension 2n − 1 which contains the Coxeter complex as a relative subcomplex. Faces are indexed by triples (J,w,K), where J and K are subsets of the set S of simple generators, and w is a minimal length representative for the double parabolic coset WJ wWK . There is exactly one maximal face for each element of the group W . The complex is shellable and thin, which implies the complex is a sphere for the finite Coxeter groups. In this case, a natural refinement of the h-polynomial is given by the “two-sided” W -Eulerian polynomial, i.e., the generating function for the joint distribution of left and right descents in W .]]> Wed, 22 Apr 2020 21:32:47 +0000 https://doi.org/10.46298/dmtcs.6353 https://doi.org/10.46298/dmtcs.6353 Petersen, T. Kyle Petersen, T. Kyle <![CDATA[For any Coxeter system (W, S) of rank n, we introduce an abstract boolean complex (simplicial poset) of dimension 2n − 1 which contains the Coxeter complex as a relative subcomplex. Faces are indexed by triples (J,w,K), where J and K are subsets of the set S of simple generators, and w is a minimal length representative for the double parabolic coset WJ wWK . There is exactly one maximal face for each element of the group W . The complex is shellable and thin, which implies the complex is a sphere for the finite Coxeter groups. In this case, a natural refinement of the h-polynomial is given by the “two-sided” W -Eulerian polynomial, i.e., the generating function for the joint distribution of left and right descents in W .]]> 0 <![CDATA[We show that the density μ of the Smith normal form (SNF) of a random integer matrix exists and equals a product of densities μps of SNF over Z/psZ with p a prime and s some positive integer. Our approach is to connect the SNF of a matrix with the greatest common divisors (gcds) of certain polynomials of matrix entries, and develop the theory of multi-gcd distribution of polynomial values at a random integer vector. We also derive a formula for μps and determine the density μ for several interesting types of sets.]]> Wed, 22 Apr 2020 21:32:46 +0000 https://doi.org/10.46298/dmtcs.6352 https://doi.org/10.46298/dmtcs.6352 Wang, Yinghui Stanley, Richard P. Wang, Yinghui Stanley, Richard P. <![CDATA[We show that the density μ of the Smith normal form (SNF) of a random integer matrix exists and equals a product of densities μps of SNF over Z/psZ with p a prime and s some positive integer. Our approach is to connect the SNF of a matrix with the greatest common divisors (gcds) of certain polynomials of matrix entries, and develop the theory of multi-gcd distribution of polynomial values at a random integer vector. We also derive a formula for μps and determine the density μ for several interesting types of sets.]]> 0 <![CDATA[The main objects of noncrossing Catalan combinatorics associated to a finite Coxeter system are noncross- ing partitions, sortable elements, and cluster complexes. The first and the third of these have known Fuss–Catalan generalizations. We provide new viewpoints for these, introduce a corresponding generalization of sortable elements as elements in the positive Artin monoid, and show how this perspective ties together all three generalizations.]]> Wed, 22 Apr 2020 21:32:45 +0000 https://doi.org/10.46298/dmtcs.6351 https://doi.org/10.46298/dmtcs.6351 Stump, Christian Thomas, Hugh Williams, Nathan Stump, Christian Thomas, Hugh Williams, Nathan <![CDATA[The main objects of noncrossing Catalan combinatorics associated to a finite Coxeter system are noncross- ing partitions, sortable elements, and cluster complexes. The first and the third of these have known Fuss–Catalan generalizations. We provide new viewpoints for these, introduce a corresponding generalization of sortable elements as elements in the positive Artin monoid, and show how this perspective ties together all three generalizations.]]> 0 <![CDATA[We present some results on the freeness or non freeness of some tridendriform algebras. In particular, we give a combinatorial proof of the freeness of WQSym, an algebra based on packed words, result already known with an algebraic proof. Then, we prove the non-freeness of an another tridendriform algebra, PQSym, a conjecture remained open. The method of these proofs is generalizable, in particular it has been used to prove the freeness of the dendriform algebra FQSym and the quadrialgebra of 2-permutations.]]> Wed, 22 Apr 2020 21:32:45 +0000 https://doi.org/10.46298/dmtcs.6350 https://doi.org/10.46298/dmtcs.6350 Vong, Vincent Vong, Vincent <![CDATA[We present some results on the freeness or non freeness of some tridendriform algebras. In particular, we give a combinatorial proof of the freeness of WQSym, an algebra based on packed words, result already known with an algebraic proof. Then, we prove the non-freeness of an another tridendriform algebra, PQSym, a conjecture remained open. The method of these proofs is generalizable, in particular it has been used to prove the freeness of the dendriform algebra FQSym and the quadrialgebra of 2-permutations.]]> 0 <![CDATA[We define two refinements of the skew length statistic on simultaneous core partitions. The first one relies on hook lengths and is used to prove a refined version of the theorem stating that the skew length is invariant under conjugation of the core. The second one is equivalent to a generalisation of Shi tableaux to the rational level of Catalan combinatorics. We prove that the rational Shi tableau is injective. Moreover we present a uniform definition of the rational Shi tableau for Weyl groups and conjecture injectivity in the general case.]]> Wed, 22 Apr 2020 21:32:44 +0000 https://doi.org/10.46298/dmtcs.6349 https://doi.org/10.46298/dmtcs.6349 Sulzgruber, Robin Sulzgruber, Robin <![CDATA[We define two refinements of the skew length statistic on simultaneous core partitions. The first one relies on hook lengths and is used to prove a refined version of the theorem stating that the skew length is invariant under conjugation of the core. The second one is equivalent to a generalisation of Shi tableaux to the rational level of Catalan combinatorics. We prove that the rational Shi tableau is injective. Moreover we present a uniform definition of the rational Shi tableau for Weyl groups and conjecture injectivity in the general case.]]> 0 <![CDATA[Tanglegrams are a class of graphs arising in computer science and in biological research on cospeciation and coevolution. They are formed by identifying the leaves of two rooted binary trees. The embedding of the trees in the plane is irrelevant for this application. We give an explicit formula to count the number of distinct binary rooted tanglegrams with n matched leaves, along with a simple asymptotic formula and an algorithm for choosing a tanglegram uniformly at random. The enumeration formula is then extended to count the number of tangled chains of binary trees of any length. This work gives a new formula for the number of binary trees with n leaves. Several open problems and conjectures are included along with pointers to several followup articles that have already appeared.]]> Wed, 22 Apr 2020 21:32:43 +0000 https://doi.org/10.46298/dmtcs.6348 https://doi.org/10.46298/dmtcs.6348 Billey, Sara, Konvalinka, Matjaz Matsen IV, Frderick, Billey, Sara, Konvalinka, Matjaz Matsen IV, Frderick, <![CDATA[Tanglegrams are a class of graphs arising in computer science and in biological research on cospeciation and coevolution. They are formed by identifying the leaves of two rooted binary trees. The embedding of the trees in the plane is irrelevant for this application. We give an explicit formula to count the number of distinct binary rooted tanglegrams with n matched leaves, along with a simple asymptotic formula and an algorithm for choosing a tanglegram uniformly at random. The enumeration formula is then extended to count the number of tangled chains of binary trees of any length. This work gives a new formula for the number of binary trees with n leaves. Several open problems and conjectures are included along with pointers to several followup articles that have already appeared.]]> 0 <![CDATA[Parabolic subgroups WI of Coxeter systems (W,S) and their ordinary and double cosets W/WI and WI\W/WJ appear in many contexts in combinatorics and Lie theory, including the geometry and topology of generalized flag varieties and the symmetry groups of regular polytopes. The set of ordinary cosets wWI , for I ⊆ S, forms the Coxeter complex of W , and is well-studied. In this extended abstract, we look at a less studied object: the set of all double cosets WIwWJ for I,J ⊆ S. Each double coset can be presented by many different triples (I, w, J). We describe what we call the lex-minimal presentation and prove that there exists a unique such choice for each double coset. Lex-minimal presentations can be enumerated via a finite automaton depending on the Coxeter graph for (W, S). In particular, we present a formula for the number of parabolic double cosets with a fixed minimal element when W is the symmetric group Sn. In that case, parabolic subgroups are also known as Young subgroups. Our formula is almost always linear time computable in n, and the formula can be generalized to any Coxeter group.]]> Wed, 22 Apr 2020 21:32:42 +0000 https://doi.org/10.46298/dmtcs.6347 https://doi.org/10.46298/dmtcs.6347 Billey, Sara, Konvalinka, Matjaz Petersen, T. Kyle Slofstra, William Tenner, Bridget, Billey, Sara, Konvalinka, Matjaz Petersen, T. Kyle Slofstra, William Tenner, Bridget, <![CDATA[Parabolic subgroups WI of Coxeter systems (W,S) and their ordinary and double cosets W/WI and WI\W/WJ appear in many contexts in combinatorics and Lie theory, including the geometry and topology of generalized flag varieties and the symmetry groups of regular polytopes. The set of ordinary cosets wWI , for I ⊆ S, forms the Coxeter complex of W , and is well-studied. In this extended abstract, we look at a less studied object: the set of all double cosets WIwWJ for I,J ⊆ S. Each double coset can be presented by many different triples (I, w, J). We describe what we call the lex-minimal presentation and prove that there exists a unique such choice for each double coset. Lex-minimal presentations can be enumerated via a finite automaton depending on the Coxeter graph for (W, S). In particular, we present a formula for the number of parabolic double cosets with a fixed minimal element when W is the symmetric group Sn. In that case, parabolic subgroups are also known as Young subgroups. Our formula is almost always linear time computable in n, and the formula can be generalized to any Coxeter group.]]> 0 <![CDATA[We study the enumeration of diagonally and antidiagonally symmetric alternating sign matrices (DAS- ASMs) of fixed odd order by introducing a case of the six-vertex model whose configurations are in bijection with such matrices. The model involves a grid graph on a triangle, with bulk and boundary weights which satisfy the Yang– Baxter and reflection equations. We obtain a general expression for the partition function of this model as a sum of two determinantal terms, and show that at a certain point each of these terms reduces to a Schur function. We are then able to prove a conjecture of Robbins from the mid 1980's that the total number of (2n + 1) × (2n + 1) DASASMs is∏n (3i)! ,andaconjectureofStroganovfrom2008thattheratiobetweenthenumbersof(2n+1)×(2n+1) i=0 (n+i)! DASASMs with central entry −1 and 1 is n/(n + 1). Among the several product formulae for the enumeration of symmetric alternating sign matrices which were conjectured in the 1980's, that for odd-order DASASMs is the last to have been proved.]]> Wed, 22 Apr 2020 21:32:42 +0000 https://doi.org/10.46298/dmtcs.6346 https://doi.org/10.46298/dmtcs.6346 Behrend, Roger, Fischer, Ilse Konvalinka, Matjaz Behrend, Roger, Fischer, Ilse Konvalinka, Matjaz <![CDATA[We study the enumeration of diagonally and antidiagonally symmetric alternating sign matrices (DAS- ASMs) of fixed odd order by introducing a case of the six-vertex model whose configurations are in bijection with such matrices. The model involves a grid graph on a triangle, with bulk and boundary weights which satisfy the Yang– Baxter and reflection equations. We obtain a general expression for the partition function of this model as a sum of two determinantal terms, and show that at a certain point each of these terms reduces to a Schur function. We are then able to prove a conjecture of Robbins from the mid 1980's that the total number of (2n + 1) × (2n + 1) DASASMs is∏n (3i)! ,andaconjectureofStroganovfrom2008thattheratiobetweenthenumbersof(2n+1)×(2n+1) i=0 (n+i)! DASASMs with central entry −1 and 1 is n/(n + 1). Among the several product formulae for the enumeration of symmetric alternating sign matrices which were conjectured in the 1980's, that for odd-order DASASMs is the last to have been proved.]]> 0 <![CDATA[We compute the generating function of some families of reduced Kronecker coefficients. We give a combi- natorial interpretation for these coefficients in terms of plane partitions. This unexpected relation allows us to check that the saturation hypothesis holds for the reduced Kronecker coefficients of our families. We also compute the quasipolynomial that govern these families, specifying the degree and period. Moving to the setting of Kronecker co- efficients, these results imply some observations related to the rate of growth experienced by the families of Kronecker coefficients associated to the reduced Kronecker coefficients already studied.]]> Wed, 22 Apr 2020 21:32:41 +0000 https://doi.org/10.46298/dmtcs.6345 https://doi.org/10.46298/dmtcs.6345 Colmenarejo, Laura Colmenarejo, Laura <![CDATA[We compute the generating function of some families of reduced Kronecker coefficients. We give a combi- natorial interpretation for these coefficients in terms of plane partitions. This unexpected relation allows us to check that the saturation hypothesis holds for the reduced Kronecker coefficients of our families. We also compute the quasipolynomial that govern these families, specifying the degree and period. Moving to the setting of Kronecker co- efficients, these results imply some observations related to the rate of growth experienced by the families of Kronecker coefficients associated to the reduced Kronecker coefficients already studied.]]> 0 <![CDATA[Following the lead of Stanley and Gessel, we consider a linear map which associates to an acyclic directed graph (or a poset) a quasi-symmetric function. The latter is naturally defined as multivariate generating series of non-decreasing functions on the graph (or of P -partitions of the poset).We describe the kernel of this linear map, using a simple combinatorial operation that we call cyclic inclusion- exclusion. Our result also holds for the natural non-commutative analog and for the commutative and non-commutative restrictions to bipartite graphs.]]> Wed, 22 Apr 2020 21:32:41 +0000 https://doi.org/10.46298/dmtcs.6344 https://doi.org/10.46298/dmtcs.6344 Féray, Valentin Féray, Valentin <![CDATA[Following the lead of Stanley and Gessel, we consider a linear map which associates to an acyclic directed graph (or a poset) a quasi-symmetric function. The latter is naturally defined as multivariate generating series of non-decreasing functions on the graph (or of P -partitions of the poset).We describe the kernel of this linear map, using a simple combinatorial operation that we call cyclic inclusion- exclusion. Our result also holds for the natural non-commutative analog and for the commutative and non-commutative restrictions to bipartite graphs.]]> 0 <![CDATA[Following the proof of the purity conjecture for weakly separated sets, recent years have revealed a variety of wider classes of pure domains in different settings. In this paper we show the purity for domains consisting of sets that are weakly separated from a pair of “generic” sets I and J. Our proof also gives a simple formula for the rank of these domains in terms of I and J. This is a new instance of the purity phenomenon which essentially differs from all previously known pure domains. We apply our result to calculate the cluster distance and to give lower bounds on the mutation distance between cluster variables in the cluster algebra structure on the coordinate ring of the Grassmannian. Using a linear projection that relates weak separation to the octahedron recurrence, we also find the exact mutation distances and cluster distances for a family of cluster variables.]]> Wed, 22 Apr 2020 21:32:40 +0000 https://doi.org/10.46298/dmtcs.6343 https://doi.org/10.46298/dmtcs.6343 Farber, Miriam Galashin, Pavel Farber, Miriam Galashin, Pavel <![CDATA[Following the proof of the purity conjecture for weakly separated sets, recent years have revealed a variety of wider classes of pure domains in different settings. In this paper we show the purity for domains consisting of sets that are weakly separated from a pair of “generic” sets I and J. Our proof also gives a simple formula for the rank of these domains in terms of I and J. This is a new instance of the purity phenomenon which essentially differs from all previously known pure domains. We apply our result to calculate the cluster distance and to give lower bounds on the mutation distance between cluster variables in the cluster algebra structure on the coordinate ring of the Grassmannian. Using a linear projection that relates weak separation to the octahedron recurrence, we also find the exact mutation distances and cluster distances for a family of cluster variables.]]> 0 <![CDATA[In this extended abstract we present colored generalizations of the symmetric algebra and its Koszul dual, the exterior algebra. The symmetric group Sn acts on the multilinear components of these algebras. While Sn acts trivially on the multilinear components of the colored symmetric algebra, we use poset topology techniques to describe the representation on its Koszul dual. We introduce an Sn-poset of weighted subsets that we call the weighted boolean algebra and we prove that the multilinear components of the colored exterior algebra are Sn- isomorphic to the top cohomology modules of its maximal intervals. We show that the two colored Koszul dual algebras are Koszul in the sense of Priddy et al.]]> Wed, 22 Apr 2020 21:32:39 +0000 https://doi.org/10.46298/dmtcs.6342 https://doi.org/10.46298/dmtcs.6342 Gonzalez D'Leon, Rafael S. Gonzalez D'Leon, Rafael S. <![CDATA[In this extended abstract we present colored generalizations of the symmetric algebra and its Koszul dual, the exterior algebra. The symmetric group Sn acts on the multilinear components of these algebras. While Sn acts trivially on the multilinear components of the colored symmetric algebra, we use poset topology techniques to describe the representation on its Koszul dual. We introduce an Sn-poset of weighted subsets that we call the weighted boolean algebra and we prove that the multilinear components of the colored exterior algebra are Sn- isomorphic to the top cohomology modules of its maximal intervals. We show that the two colored Koszul dual algebras are Koszul in the sense of Priddy et al.]]> 0 <![CDATA[This extended abstract proves that the number of fully packed loop configurations whose link pattern consists of two noncrossing matchings separated by m nested arches is a polynomial in m. This was conjectured by Zuber (2004) and for large values of m proved by Caselli et al. (2004)]]> Wed, 22 Apr 2020 21:32:38 +0000 https://doi.org/10.46298/dmtcs.6341 https://doi.org/10.46298/dmtcs.6341 Aigner, Florian Aigner, Florian <![CDATA[This extended abstract proves that the number of fully packed loop configurations whose link pattern consists of two noncrossing matchings separated by m nested arches is a polynomial in m. This was conjectured by Zuber (2004) and for large values of m proved by Caselli et al. (2004)]]> 0 <![CDATA[Consider the set Fn of factorizations of the full cycle (0 1 2 · · · n) ∈ S{0,1,...,n} into n transpositions. Write any such factorization (a1 b1) · · · (an bn) with all ai < bi to define its lower and upper sequences (a1, . . . , an) and (b1,...,bn), respectively. Remarkably, any factorization can be uniquely recovered from its lower (or upper) sequence. In fact, Biane (2002) showed that the simple map sending a factorization to its lower sequence is a bijection from Fn to the set Pn of parking functions of length n. Reversing this map to recover the factorization (and, hence, upper sequence) corresponding to a given lower sequence is nontrivial.]]> Wed, 22 Apr 2020 21:32:38 +0000 https://doi.org/10.46298/dmtcs.6340 https://doi.org/10.46298/dmtcs.6340 Irving, John Rattan, Amarpreet Irving, John Rattan, Amarpreet <![CDATA[Consider the set Fn of factorizations of the full cycle (0 1 2 · · · n) ∈ S{0,1,...,n} into n transpositions. Write any such factorization (a1 b1) · · · (an bn) with all ai < bi to define its lower and upper sequences (a1, . . . , an) and (b1,...,bn), respectively. Remarkably, any factorization can be uniquely recovered from its lower (or upper) sequence. In fact, Biane (2002) showed that the simple map sending a factorization to its lower sequence is a bijection from Fn to the set Pn of parking functions of length n. Reversing this map to recover the factorization (and, hence, upper sequence) corresponding to a given lower sequence is nontrivial.]]> 0 <![CDATA[In this paper, we use the multivariate analytic techniques of Pemantle and Wilson to find asymptotic for- mulae for the coefficients of a broad class of multivariate generating functions with algebraic singularities. Flajolet and Odlyzko (1990) analyzed the coefficients of a class of univariate generating functions with algebraic singularities. These results have been extended to classes of multivariate generating functions by Gao and Richmond (1992) and Hwang (1996, 1998), in both cases by immediately reducing the multivariate case to the univariate case. Pemantle and Wilson (2013) outlined new multivariate analytic techniques and used them to analyze the coefficients of rational generating functions.]]> Wed, 22 Apr 2020 21:32:37 +0000 https://doi.org/10.46298/dmtcs.6339 https://doi.org/10.46298/dmtcs.6339 Greenwood, Torin Greenwood, Torin <![CDATA[In this paper, we use the multivariate analytic techniques of Pemantle and Wilson to find asymptotic for- mulae for the coefficients of a broad class of multivariate generating functions with algebraic singularities. Flajolet and Odlyzko (1990) analyzed the coefficients of a class of univariate generating functions with algebraic singularities. These results have been extended to classes of multivariate generating functions by Gao and Richmond (1992) and Hwang (1996, 1998), in both cases by immediately reducing the multivariate case to the univariate case. Pemantle and Wilson (2013) outlined new multivariate analytic techniques and used them to analyze the coefficients of rational generating functions.]]> 0 <![CDATA[In this paper we introduce factorial characters for the classical groups and derive a number of central results. Classically, the factorial Schur function plays a fundamental role in traditional symmetric function theory and also in Schubert polynomial theory. Here we develop a parallel theory for the classical groups, offering combinatorial definitions of the factorial characters for the symplectic and orthogonal groups, and further establish flagged factorial Jacobi-Trudi identities and factorial Tokuyama identities, providing proofs in the symplectic case. These identities are established by manipulating determinants through the use of certain recurrence relations and by using lattice paths.]]> Wed, 22 Apr 2020 21:32:36 +0000 https://doi.org/10.46298/dmtcs.6338 https://doi.org/10.46298/dmtcs.6338 Hamel, Angèle M. King, Ronald C. Hamel, Angèle M. King, Ronald C. <![CDATA[In this paper we introduce factorial characters for the classical groups and derive a number of central results. Classically, the factorial Schur function plays a fundamental role in traditional symmetric function theory and also in Schubert polynomial theory. Here we develop a parallel theory for the classical groups, offering combinatorial definitions of the factorial characters for the symplectic and orthogonal groups, and further establish flagged factorial Jacobi-Trudi identities and factorial Tokuyama identities, providing proofs in the symplectic case. These identities are established by manipulating determinants through the use of certain recurrence relations and by using lattice paths.]]> 0 <![CDATA[Kenyon and Pemantle (2014) gave a formula for the entries of a square matrix in terms of connected principal and almost-principal minors. Each entry is an explicit Laurent polynomial whose terms are the weights of domino tilings of a half Aztec diamond. They conjectured an analogue of this parametrization for symmetric matrices, where the Laurent monomials are indexed by Catalan paths. In this paper we prove the Kenyon-Pemantle conjecture, and apply this to a statistics problem pioneered by Joe (2006). Correlation matrices are represented by an explicit bijection from the cube to the elliptope.]]> Wed, 22 Apr 2020 21:32:35 +0000 https://doi.org/10.46298/dmtcs.6337 https://doi.org/10.46298/dmtcs.6337 Tsukerman, Emmanuel Williams, Lauren Sturmfels, Bernd Tsukerman, Emmanuel Williams, Lauren Sturmfels, Bernd <![CDATA[Kenyon and Pemantle (2014) gave a formula for the entries of a square matrix in terms of connected principal and almost-principal minors. Each entry is an explicit Laurent polynomial whose terms are the weights of domino tilings of a half Aztec diamond. They conjectured an analogue of this parametrization for symmetric matrices, where the Laurent monomials are indexed by Catalan paths. In this paper we prove the Kenyon-Pemantle conjecture, and apply this to a statistics problem pioneered by Joe (2006). Correlation matrices are represented by an explicit bijection from the cube to the elliptope.]]> 0 <![CDATA[We define a new type of vertex coloring which generalizes vertex coloring in graphs, hypergraphs, andsimplicial complexes. To this coloring there is an associated symmetric function in noncommuting variables for whichwe give a deletion-contraction formula. In the case of graphs our symmetric function in noncommuting variablesagrees with the chromatic symmetric function in noncommuting variables of Gebhard and Sagan. Our vertex coloringis a special case of the scheduling problems defined by Breuer and Klivans. We show how the deletion-contractionlaw can be applied to scheduling problems.]]> Wed, 22 Apr 2020 21:32:35 +0000 https://doi.org/10.46298/dmtcs.6336 https://doi.org/10.46298/dmtcs.6336 Machacek, John Machacek, John <![CDATA[We define a new type of vertex coloring which generalizes vertex coloring in graphs, hypergraphs, andsimplicial complexes. To this coloring there is an associated symmetric function in noncommuting variables for whichwe give a deletion-contraction formula. In the case of graphs our symmetric function in noncommuting variablesagrees with the chromatic symmetric function in noncommuting variables of Gebhard and Sagan. Our vertex coloringis a special case of the scheduling problems defined by Breuer and Klivans. We show how the deletion-contractionlaw can be applied to scheduling problems.]]> 0 <![CDATA[We prove that among all flag 3-manifolds on n vertices, the join of two circles with [n 2] and [n 2] vertices respectively is the unique maximizer of the face numbers. This solves the first case of a conjecture due to Lutz and Nevo. Further, we establish a sharp upper bound on the number of edges of flag 5-manifolds and characterize the cases of equality. We also show that the inequality part of the flag upper bound conjecture continues to hold for all flag 3-dimensional Eulerian complexes and characterize the cases of equality in this class.]]> Wed, 22 Apr 2020 21:32:34 +0000 https://doi.org/10.46298/dmtcs.6335 https://doi.org/10.46298/dmtcs.6335 Zheng, Hailun Zheng, Hailun <![CDATA[We prove that among all flag 3-manifolds on n vertices, the join of two circles with [n 2] and [n 2] vertices respectively is the unique maximizer of the face numbers. This solves the first case of a conjecture due to Lutz and Nevo. Further, we establish a sharp upper bound on the number of edges of flag 5-manifolds and characterize the cases of equality. We also show that the inequality part of the flag upper bound conjecture continues to hold for all flag 3-dimensional Eulerian complexes and characterize the cases of equality in this class.]]> 0 <![CDATA[We investigate quasisymmetric functions coming from combinatorial Hopf monoids. We show that these invariants arise naturally in Ehrhart theory, and that some of their specializations are Hilbert functions for relative simplicial complexes. This class of complexes, called forbidden composition complexes, also forms a Hopf monoid, thus demonstrating a link between Hopf algebras, Ehrhart theory, and commutative algebra. We also study various specializations of quasisymmetric functions.]]> Wed, 22 Apr 2020 21:32:33 +0000 https://doi.org/10.46298/dmtcs.6334 https://doi.org/10.46298/dmtcs.6334 White, Jacob, White, Jacob, <![CDATA[We investigate quasisymmetric functions coming from combinatorial Hopf monoids. We show that these invariants arise naturally in Ehrhart theory, and that some of their specializations are Hilbert functions for relative simplicial complexes. This class of complexes, called forbidden composition complexes, also forms a Hopf monoid, thus demonstrating a link between Hopf algebras, Ehrhart theory, and commutative algebra. We also study various specializations of quasisymmetric functions.]]> 0 <![CDATA[A new triple product formulae for plane partitions with bounded size of parts is derived from a combinato- rial interpretation of biorthogonal polynomials in terms of lattice paths. Biorthogonal polynomials which generalize the little q-Laguerre polynomials are introduced to derive a new triple product formula which recovers the classical generating function in a triple product by MacMahon and generalizes the trace-type generating functions in double products by Stanley and Gansner.]]> Wed, 22 Apr 2020 21:32:32 +0000 https://doi.org/10.46298/dmtcs.6333 https://doi.org/10.46298/dmtcs.6333 Kamioka, Shuhei Kamioka, Shuhei <![CDATA[A new triple product formulae for plane partitions with bounded size of parts is derived from a combinato- rial interpretation of biorthogonal polynomials in terms of lattice paths. Biorthogonal polynomials which generalize the little q-Laguerre polynomials are introduced to derive a new triple product formula which recovers the classical generating function in a triple product by MacMahon and generalizes the trace-type generating functions in double products by Stanley and Gansner.]]> 0 <![CDATA[In this work triangular puzzles that are composed of unit triangles with labelled edges are considered. To be more precise, the labelled unit triangles that we allow are on the one hand the puzzle pieces that compute Schubert calculus and on the other hand the flipped K-theory puzzle piece. The motivation for studying such puzzles comes from the fact that they correspond to a class of oriented triangular fully packed loop configurations. The main result that is presented is an expression for the number of these puzzles with a fixed boundary in terms of Littlewood- Richardson coefficients.]]> Wed, 22 Apr 2020 21:32:32 +0000 https://doi.org/10.46298/dmtcs.6332 https://doi.org/10.46298/dmtcs.6332 Beil, Sabine Beil, Sabine <![CDATA[In this work triangular puzzles that are composed of unit triangles with labelled edges are considered. To be more precise, the labelled unit triangles that we allow are on the one hand the puzzle pieces that compute Schubert calculus and on the other hand the flipped K-theory puzzle piece. The motivation for studying such puzzles comes from the fact that they correspond to a class of oriented triangular fully packed loop configurations. The main result that is presented is an expression for the number of these puzzles with a fixed boundary in terms of Littlewood- Richardson coefficients.]]> 0 <![CDATA[The Tutte polynomial for matroids is not directly applicable to polymatroids. For instance, deletion- contraction properties do not hold. We construct a polynomial for polymatroids which behaves similarly to the Tutte polynomial of a matroid, and in fact contains the same information as the Tutte polynomial when we restrict to matroids. This polynomial is constructed using lattice point counts in the Minkowski sum of the base polytope of a polymatroid and scaled copies of the standard simplex. We also show that, in the matroid case, our polynomial has coefficients of alternating sign, with a combinatorial interpretation closely tied to the Dawson partition.]]> Wed, 22 Apr 2020 21:32:31 +0000 https://doi.org/10.46298/dmtcs.6331 https://doi.org/10.46298/dmtcs.6331 Cameron, Amanda Fink, Alex Cameron, Amanda Fink, Alex <![CDATA[The Tutte polynomial for matroids is not directly applicable to polymatroids. For instance, deletion- contraction properties do not hold. We construct a polynomial for polymatroids which behaves similarly to the Tutte polynomial of a matroid, and in fact contains the same information as the Tutte polynomial when we restrict to matroids. This polynomial is constructed using lattice point counts in the Minkowski sum of the base polytope of a polymatroid and scaled copies of the standard simplex. We also show that, in the matroid case, our polynomial has coefficients of alternating sign, with a combinatorial interpretation closely tied to the Dawson partition.]]> 0 <![CDATA[We consider self-avoiding polygons in a restricted geometry, namely an infinite L × M tube in Z3. These polygons are subjected to a force f, parallel to the infinite axis of the tube. When f > 0 the force stretches the polygons, while when f < 0 the force is compressive. In this extended abstract we obtain and prove the asymptotic form of the free energy in the limit f → −∞. We conjecture that the f → −∞ asymptote is the same as the free energy of Hamiltonian polygons, which visit every vertex in a L × M × N box.]]> Wed, 22 Apr 2020 21:32:30 +0000 https://doi.org/10.46298/dmtcs.6330 https://doi.org/10.46298/dmtcs.6330 Beaton, Nicholas, Eng, Jeremy Soteros, Christine Beaton, Nicholas, Eng, Jeremy Soteros, Christine <![CDATA[We consider self-avoiding polygons in a restricted geometry, namely an infinite L × M tube in Z3. These polygons are subjected to a force f, parallel to the infinite axis of the tube. When f > 0 the force stretches the polygons, while when f < 0 the force is compressive. In this extended abstract we obtain and prove the asymptotic form of the free energy in the limit f → −∞. We conjecture that the f → −∞ asymptote is the same as the free energy of Hamiltonian polygons, which visit every vertex in a L × M × N box.]]> 0 <![CDATA[We study random knotting by considering knot and link diagrams as decorated, (rooted) topological maps on spheres and sampling them with the counting measure on from sets of a fixed number of vertices n. We prove that random rooted knot diagrams are highly composite and hence almost surely knotted (this is the analogue of the Frisch-Wasserman-Delbruck conjecture) and extend this to unrooted knot diagrams by showing that almost all knot diagrams are asymmetric. The model is similar to one of Dunfield, et al.]]> Wed, 22 Apr 2020 21:32:30 +0000 https://doi.org/10.46298/dmtcs.6329 https://doi.org/10.46298/dmtcs.6329 Chapman, Harrison Chapman, Harrison <![CDATA[We study random knotting by considering knot and link diagrams as decorated, (rooted) topological maps on spheres and sampling them with the counting measure on from sets of a fixed number of vertices n. We prove that random rooted knot diagrams are highly composite and hence almost surely knotted (this is the analogue of the Frisch-Wasserman-Delbruck conjecture) and extend this to unrooted knot diagrams by showing that almost all knot diagrams are asymmetric. The model is similar to one of Dunfield, et al.]]> 0 <![CDATA[The generalized Lax conjecture asserts that each hyperbolicity cone is a linear slice of the cone of positive semidefinite matrices. Hyperbolic polynomials give rise to a class of (hyperbolic) matroids which properly contains the class of matroids representable over the complex numbers. This connection was used by the first author to construct counterexamples to algebraic (stronger) versions of the generalized Lax conjecture by considering a non- representable hyperbolic matroid. The Va ́mos matroid and a generalization of it are to this day the only known instances of non-representable hyperbolic matroids. We prove that the Non-Pappus and Non-Desargues matroids are non-representable hyperbolic matroids by exploiting a connection, due to Jordan, between Euclidean Jordan algebras and projective geometries. We further identify a large class of hyperbolic matroids that are parametrized by uniform hypergraphs and prove that many of them are non-representable. Finally we explore consequences to algebraic versions of the generalized Lax conjecture.]]> Wed, 22 Apr 2020 21:32:29 +0000 https://doi.org/10.46298/dmtcs.6328 https://doi.org/10.46298/dmtcs.6328 Amini, Nima Branden, Petter Amini, Nima Branden, Petter <![CDATA[The generalized Lax conjecture asserts that each hyperbolicity cone is a linear slice of the cone of positive semidefinite matrices. Hyperbolic polynomials give rise to a class of (hyperbolic) matroids which properly contains the class of matroids representable over the complex numbers. This connection was used by the first author to construct counterexamples to algebraic (stronger) versions of the generalized Lax conjecture by considering a non- representable hyperbolic matroid. The Va ́mos matroid and a generalization of it are to this day the only known instances of non-representable hyperbolic matroids. We prove that the Non-Pappus and Non-Desargues matroids are non-representable hyperbolic matroids by exploiting a connection, due to Jordan, between Euclidean Jordan algebras and projective geometries. We further identify a large class of hyperbolic matroids that are parametrized by uniform hypergraphs and prove that many of them are non-representable. Finally we explore consequences to algebraic versions of the generalized Lax conjecture.]]> 0 <![CDATA[We describe a type B analog of the much studied Lie representation of the symmetric group. The nth Lie representation of Sn restricts to the regular representation of Sn−1, and our generalization mimics this property. Specifically, we construct a representation of the type B Weyl group Bn that restricts to the regular representation of Bn−1. We view both of these representations as coming from the internal zonotopal algebra of the Gale dual of the corresponding reflection arrangements.]]> Wed, 22 Apr 2020 21:32:28 +0000 https://doi.org/10.46298/dmtcs.6327 https://doi.org/10.46298/dmtcs.6327 Berget, Andrew Berget, Andrew <![CDATA[We describe a type B analog of the much studied Lie representation of the symmetric group. The nth Lie representation of Sn restricts to the regular representation of Sn−1, and our generalization mimics this property. Specifically, we construct a representation of the type B Weyl group Bn that restricts to the regular representation of Bn−1. We view both of these representations as coming from the internal zonotopal algebra of the Gale dual of the corresponding reflection arrangements.]]> 0 <![CDATA[We construct the affine version of the Fomin-Kirillov algebra, called the affine FK algebra, to investigatethe combinatorics of affine Schubert calculus for typeA. We introduce Murnaghan-Nakayama elements and Dunklelements in the affine FK algebra. We show that they are commutative as Bruhat operators, and the commutativealgebra generated by these operators is isomorphic to the cohomology of the affine flag variety. As a byproduct, weobtain Murnaghan-Nakayama rules both for the affine Schubert polynomials and affine Stanley symmetric functions. This enable us to expressk-Schur functions in terms of power sum symmetric functions. We also provide the defi-nition of the affine Schubert polynomials, polynomial representatives of the Schubert basis in the cohomology of theaffine flag variety.]]> Wed, 22 Apr 2020 21:32:27 +0000 https://doi.org/10.46298/dmtcs.6326 https://doi.org/10.46298/dmtcs.6326 Lee, Seung Jin Lee, Seung Jin <![CDATA[We construct the affine version of the Fomin-Kirillov algebra, called the affine FK algebra, to investigatethe combinatorics of affine Schubert calculus for typeA. We introduce Murnaghan-Nakayama elements and Dunklelements in the affine FK algebra. We show that they are commutative as Bruhat operators, and the commutativealgebra generated by these operators is isomorphic to the cohomology of the affine flag variety. As a byproduct, weobtain Murnaghan-Nakayama rules both for the affine Schubert polynomials and affine Stanley symmetric functions. This enable us to expressk-Schur functions in terms of power sum symmetric functions. We also provide the defi-nition of the affine Schubert polynomials, polynomial representatives of the Schubert basis in the cohomology of theaffine flag variety.]]> 0 <![CDATA[A long-standing conjecture of Stanley states that every Cohen–Macaulay simplicial complex is partition- able. We disprove the conjecture by constructing an explicit counterexample. Due to a result of Herzog, Jahan and Yassemi, our construction also disproves the conjecture that the Stanley depth of a monomial ideal is always at least its depth.]]> Wed, 22 Apr 2020 21:32:26 +0000 https://doi.org/10.46298/dmtcs.6325 https://doi.org/10.46298/dmtcs.6325 Duval, Art M. Goeckner, Bennet Klivans, Caroline J. Martin, Jeremy Duval, Art M. Goeckner, Bennet Klivans, Caroline J. Martin, Jeremy <![CDATA[A long-standing conjecture of Stanley states that every Cohen–Macaulay simplicial complex is partition- able. We disprove the conjecture by constructing an explicit counterexample. Due to a result of Herzog, Jahan and Yassemi, our construction also disproves the conjecture that the Stanley depth of a monomial ideal is always at least its depth.]]> 0 <![CDATA[A fourientation of a graph G is a choice for each edge of the graph whether to orient that edge in either direction, leave it unoriented, or biorient it. We may naturally view fourientations as a mixture of subgraphs and graph orientations where unoriented and bioriented edges play the role of absent and present subgraph edges, respectively. Building on work of Backman and Hopkins (2015), we show that given a linear order and a reference orientation of the edge set, one can define activities for fourientations of G which allow for a new 12 variable expansion of the Tutte polynomial TG. Our formula specializes to both an orientation activities expansion of TG due to Las Vergnas (1984) and a generalized activities expansion of TG due to Gordon and Traldi (1990).]]> Wed, 22 Apr 2020 21:32:25 +0000 https://doi.org/10.46298/dmtcs.6324 https://doi.org/10.46298/dmtcs.6324 Backman, Spencer Hopkins, Sam Traldi, Lorenzo Backman, Spencer Hopkins, Sam Traldi, Lorenzo <![CDATA[A fourientation of a graph G is a choice for each edge of the graph whether to orient that edge in either direction, leave it unoriented, or biorient it. We may naturally view fourientations as a mixture of subgraphs and graph orientations where unoriented and bioriented edges play the role of absent and present subgraph edges, respectively. Building on work of Backman and Hopkins (2015), we show that given a linear order and a reference orientation of the edge set, one can define activities for fourientations of G which allow for a new 12 variable expansion of the Tutte polynomial TG. Our formula specializes to both an orientation activities expansion of TG due to Las Vergnas (1984) and a generalized activities expansion of TG due to Gordon and Traldi (1990).]]> 0 <![CDATA[We prove that the product of a monomial and a Demazure atom is a positive sum of Demazure atoms combinatorially. This result proves one particular case in a conjecture which provides an approach to a combinatorial proof of Schubert positivity property.]]> Wed, 22 Apr 2020 21:32:25 +0000 https://doi.org/10.46298/dmtcs.6323 https://doi.org/10.46298/dmtcs.6323 Ying Pun, Anna Ying Pun, Anna <![CDATA[We prove that the product of a monomial and a Demazure atom is a positive sum of Demazure atoms combinatorially. This result proves one particular case in a conjecture which provides an approach to a combinatorial proof of Schubert positivity property.]]> 0 <![CDATA[Goulden and Jackson (1996) introduced, using Jack symmetric functions, some multivariate generating series ψ(x, y, z; t, 1 + β) that might be interpreted as a continuous deformation of the rooted hypermap generating series. They made the following conjecture: coefficients of ψ(x, y, z; t, 1+β) are polynomials in β with nonnegative integer coefficients. We prove partially this conjecture, nowadays called b-conjecture, by showing that coefficients of ψ(x, y, z; t, 1 + β) are polynomials in β with rational coefficients. Until now, it was only known that they are rational functions of β. A key step of the proof is a strong factorization property of Jack polynomials when α → 0 that may be of independent interest.]]> Wed, 22 Apr 2020 21:32:24 +0000 https://doi.org/10.46298/dmtcs.6322 https://doi.org/10.46298/dmtcs.6322 Dolega, Maciej Féray, Valentin Dolega, Maciej Féray, Valentin <![CDATA[Goulden and Jackson (1996) introduced, using Jack symmetric functions, some multivariate generating series ψ(x, y, z; t, 1 + β) that might be interpreted as a continuous deformation of the rooted hypermap generating series. They made the following conjecture: coefficients of ψ(x, y, z; t, 1+β) are polynomials in β with nonnegative integer coefficients. We prove partially this conjecture, nowadays called b-conjecture, by showing that coefficients of ψ(x, y, z; t, 1 + β) are polynomials in β with rational coefficients. Until now, it was only known that they are rational functions of β. A key step of the proof is a strong factorization property of Jack polynomials when α → 0 that may be of independent interest.]]> 0 <![CDATA[In their 1987 paper Kraskiewicz and Pragacz defined certain modules, which we call KP modules, over the upper triangular Lie algebra whose characters are Schubert polynomials. In a previous work the author showed that the tensor product of Kraskiewicz-Pragacz modules always has KP filtration, i.e. a filtration whose each successive quotients are isomorphic to KP modules. In this paper we explicitly construct such filtrations for certain special cases of these tensor product modules, namely Sw Sd(Ki) and Sw Vd(Ki), corresponding to Pieri and dual Pieri rules for Schubert polynomials.]]> Wed, 22 Apr 2020 21:32:24 +0000 https://doi.org/10.46298/dmtcs.6321 https://doi.org/10.46298/dmtcs.6321 Watanabe, Masaki Watanabe, Masaki <![CDATA[In their 1987 paper Kraskiewicz and Pragacz defined certain modules, which we call KP modules, over the upper triangular Lie algebra whose characters are Schubert polynomials. In a previous work the author showed that the tensor product of Kraskiewicz-Pragacz modules always has KP filtration, i.e. a filtration whose each successive quotients are isomorphic to KP modules. In this paper we explicitly construct such filtrations for certain special cases of these tensor product modules, namely Sw Sd(Ki) and Sw Vd(Ki), corresponding to Pieri and dual Pieri rules for Schubert polynomials.]]> 0 <![CDATA[In this paper, we introduce therhombic alternative tableaux, whose weight generating functions providecombinatorial formulae to compute the steady state probabilities of the two-species ASEP. In the ASEP, there aretwo species of particles, oneheavyand onelight, on a one-dimensional finite lattice with open boundaries, and theparametersα,β, andqdescribe the hopping probabilities. The rhombic alternative tableaux are enumerated by theLah numbers, which also enumerate certainassembl ́ees of permutations. We describe a bijection between the rhombicalternative tableaux and these assembl ́ees. We also provide an insertion algorithm that gives a weight generatingfunction for the assemb ́ees. Combined, these results give a bijective proof for the weight generating function for therhombic alternative tableaux.]]> Wed, 22 Apr 2020 21:32:23 +0000 https://doi.org/10.46298/dmtcs.6320 https://doi.org/10.46298/dmtcs.6320 Mandelshtam, Olya Viennot, Xavier Mandelshtam, Olya Viennot, Xavier <![CDATA[In this paper, we introduce therhombic alternative tableaux, whose weight generating functions providecombinatorial formulae to compute the steady state probabilities of the two-species ASEP. In the ASEP, there aretwo species of particles, oneheavyand onelight, on a one-dimensional finite lattice with open boundaries, and theparametersα,β, andqdescribe the hopping probabilities. The rhombic alternative tableaux are enumerated by theLah numbers, which also enumerate certainassembl ́ees of permutations. We describe a bijection between the rhombicalternative tableaux and these assembl ́ees. We also provide an insertion algorithm that gives a weight generatingfunction for the assemb ́ees. Combined, these results give a bijective proof for the weight generating function for therhombic alternative tableaux.]]> 0 <![CDATA[We investigate the combinatorics of the symmetry relation H μ(x; q, t) = H μ∗ (x; t, q) on the transformed Macdonald polynomials, from the point of view of the combinatorial formula of Haglund, Haiman, and Loehr in terms of the inv and maj statistics on Young diagram fillings. By generalizing the Carlitz bijection on permutations, we provide a purely combinatorial proof of the relation in the case of Hall-Littlewood polynomials (q = 0) for the coefficients of the square-free monomials in the variables x. Our work in this case relates the Macdonald inv and maj statistics to the monomial basis of the modules Rμ studied by Garsia and Procesi. We also provide a new proof for the full Macdonald relation in the case when μ is a hook shape.]]> Wed, 22 Apr 2020 21:32:22 +0000 https://doi.org/10.46298/dmtcs.6319 https://doi.org/10.46298/dmtcs.6319 Gillespie, Maria Monks Gillespie, Maria Monks <![CDATA[We investigate the combinatorics of the symmetry relation H μ(x; q, t) = H μ∗ (x; t, q) on the transformed Macdonald polynomials, from the point of view of the combinatorial formula of Haglund, Haiman, and Loehr in terms of the inv and maj statistics on Young diagram fillings. By generalizing the Carlitz bijection on permutations, we provide a purely combinatorial proof of the relation in the case of Hall-Littlewood polynomials (q = 0) for the coefficients of the square-free monomials in the variables x. Our work in this case relates the Macdonald inv and maj statistics to the monomial basis of the modules Rμ studied by Garsia and Procesi. We also provide a new proof for the full Macdonald relation in the case when μ is a hook shape.]]> 0 <![CDATA[It is known that the number of minimal factorizations of the long cycle in the symmetric group into a product of k cycles of given lengths has a very simple formula: it is nk−1 where n is the rank of the underlying symmetric group and k is the number of factors. In particular, this is nn−2 for transposition factorizations. The goal of this work is to prove a multivariate generalization of this result. As a byproduct, we get a multivariate analog of Postnikov's hook length formula for trees, and a refined enumeration of final chains of noncrossing partitions.]]> Wed, 22 Apr 2020 21:32:22 +0000 https://doi.org/10.46298/dmtcs.6318 https://doi.org/10.46298/dmtcs.6318 Biane, Philippe Josuat-Vergès, Matthieu Biane, Philippe Josuat-Vergès, Matthieu <![CDATA[It is known that the number of minimal factorizations of the long cycle in the symmetric group into a product of k cycles of given lengths has a very simple formula: it is nk−1 where n is the rank of the underlying symmetric group and k is the number of factors. In particular, this is nn−2 for transposition factorizations. The goal of this work is to prove a multivariate generalization of this result. As a byproduct, we get a multivariate analog of Postnikov's hook length formula for trees, and a refined enumeration of final chains of noncrossing partitions.]]> 0 <![CDATA[We present simple graph-theoretic characterizations for the Cayley graphs of monoids, right-cancellative monoids, left-cancellative monoids, and groups.]]> Mon, 20 Apr 2020 07:16:02 +0000 https://doi.org/10.23638/DMTCS-21-1-16 https://doi.org/10.23638/DMTCS-21-1-16 Caucal, Didier Caucal, Didier <![CDATA[We present simple graph-theoretic characterizations for the Cayley graphs of monoids, right-cancellative monoids, left-cancellative monoids, and groups.]]> 0 <![CDATA[For positive integers $n,k$ and $t$, the uniform subset graph $G(n, k, t)$ has all $k$-subsets of $\{1,2,\ldots, n\}$ as vertices and two $k$-subsets are joined by an edge if they intersect at exactly $t$ elements. The Johnson graph $J(n,k)$ corresponds to $G(n,k,k-1)$, that is, two vertices of $J(n,k)$ are adjacent if the intersection of the corresponding $k$-subsets has size $k-1$. A super vertex-cut of a connected graph is a set of vertices whose removal disconnects the graph without isolating a vertex and the super-connectivity is the size of a minimum super vertex-cut. In this work, we fully determine the super-connectivity of the family of Johnson graphs $J(n,k)$ for $n\geq k\geq 1$.]]> Tue, 14 Apr 2020 09:13:25 +0000 https://doi.org/10.23638/DMTCS-22-1-12 https://doi.org/10.23638/DMTCS-22-1-12 Ekinci, Gülnaz Boruzanlı Gauci, John Baptist Ekinci, Gülnaz Boruzanlı Gauci, John Baptist <![CDATA[For positive integers $n,k$ and $t$, the uniform subset graph $G(n, k, t)$ has all $k$-subsets of $\{1,2,\ldots, n\}$ as vertices and two $k$-subsets are joined by an edge if they intersect at exactly $t$ elements. The Johnson graph $J(n,k)$ corresponds to $G(n,k,k-1)$, that is, two vertices of $J(n,k)$ are adjacent if the intersection of the corresponding $k$-subsets has size $k-1$. A super vertex-cut of a connected graph is a set of vertices whose removal disconnects the graph without isolating a vertex and the super-connectivity is the size of a minimum super vertex-cut. In this work, we fully determine the super-connectivity of the family of Johnson graphs $J(n,k)$ for $n\geq k\geq 1$.]]> 0 <![CDATA[Let $P$ be a set of $n\geq 4$ points in general position in the plane. Consider all the closed straight line segments with both endpoints in $P$. Suppose that these segments are colored with the rule that disjoint segments receive different colors. In this paper we show that if $P$ is the point configuration known as the double chain, with $k$ points in the upper convex chain and $l \ge k$ points in the lower convex chain, then $k+l- \left\lfloor \sqrt{2l+\frac{1}{4}} - \frac{1}{2}\right\rfloor$ colors are needed and that this number is sufficient.]]> Sun, 22 Mar 2020 08:58:41 +0000 https://doi.org/10.23638/DMTCS-22-1-11 https://doi.org/10.23638/DMTCS-22-1-11 Fabila-Monroy, Ruy Hidalgo-Toscano, Carlos Leaños, Jesús Lomelí-Haro, Mario Fabila-Monroy, Ruy Hidalgo-Toscano, Carlos Leaños, Jesús Lomelí-Haro, Mario <![CDATA[Let $P$ be a set of $n\geq 4$ points in general position in the plane. Consider all the closed straight line segments with both endpoints in $P$. Suppose that these segments are colored with the rule that disjoint segments receive different colors. In this paper we show that if $P$ is the point configuration known as the double chain, with $k$ points in the upper convex chain and $l \ge k$ points in the lower convex chain, then $k+l- \left\lfloor \sqrt{2l+\frac{1}{4}} - \frac{1}{2}\right\rfloor$ colors are needed and that this number is sufficient.]]> 0 <![CDATA[A monster is an automaton in which every function from states to states is represented by at least one letter. A modifier is a set of functions allowing one to transform a set of automata into one automaton. We revisit some language transformation algorithms in terms of modifier and monster. These new theoretical concepts allow one to find easily some state complexities. We illustrate this by retrieving the state complexity of the Star of Intersection and the one of the Square root operation.]]> Mon, 16 Mar 2020 09:28:28 +0000 https://doi.org/10.23638/DMTCS-22-1-9 https://doi.org/10.23638/DMTCS-22-1-9 Caron, Pascal court, Edwin Hamel-De le Luque, Jean-Gabriel Patrou, Bruno Caron, Pascal court, Edwin Hamel-De le Luque, Jean-Gabriel Patrou, Bruno <![CDATA[A monster is an automaton in which every function from states to states is represented by at least one letter. A modifier is a set of functions allowing one to transform a set of automata into one automaton. We revisit some language transformation algorithms in terms of modifier and monster. These new theoretical concepts allow one to find easily some state complexities. We illustrate this by retrieving the state complexity of the Star of Intersection and the one of the Square root operation.]]> 0 <![CDATA[In the eternal domination game, an attacker attacks a vertex at each turn and a team of guards must move a guard to the attacked vertex to defend it. The guards may only move to adjacent vertices and no more than one guard may occupy a vertex. The goal is to determine the eternal domination number of a graph which is the minimum number of guards required to defend the graph against an infinite sequence of attacks. In this paper, we continue the study of the eternal domination game on strong grids. Cartesian grids have been vastly studied with tight bounds for small grids such as 2×n, 3×n, 4×n, and 5×n grids, and recently it was proven in [Lamprou et al., CIAC 2017, 393-404] that the eternal domination number of these grids in general is within O(m + n) of their domination number which lower bounds the eternal domination number. Recently, Finbow et al. proved that the eternal domination number of strong grids is upper bounded by mn 6 + O(m + n). We adapt the techniques of [Lamprou et al., CIAC 2017, 393-404] to prove that the eternal domination number of strong grids is upper bounded by mn 7 + O(m + n). While this does not improve upon a recently announced bound of ⎡m/3⎤ x⎡n/3⎤ + O(m √ n) [Mc Inerney, Nisse, Pérennes, HAL archives, 2018; Mc Inerney, Nisse, Pérennes, CIAC 2019] in the general case, we show that our bound is an improvement in the case where the smaller of the two dimensions is at most 6179.]]> Mon, 16 Mar 2020 09:21:39 +0000 https://doi.org/10.23638/DMTCS-22-1-8 https://doi.org/10.23638/DMTCS-22-1-8 Gagnon, Alizée Hassler, Alexander Huang, Jerry Krim-Yee, Aaron Mc Inerney, Fionn Zacarías, Andrés, Seamone, Ben Virgile, Virgélot Gagnon, Alizée Hassler, Alexander Huang, Jerry Krim-Yee, Aaron Mc Inerney, Fionn Zacarías, Andrés, Seamone, Ben Virgile, Virgélot <![CDATA[In the eternal domination game, an attacker attacks a vertex at each turn and a team of guards must move a guard to the attacked vertex to defend it. The guards may only move to adjacent vertices and no more than one guard may occupy a vertex. The goal is to determine the eternal domination number of a graph which is the minimum number of guards required to defend the graph against an infinite sequence of attacks. In this paper, we continue the study of the eternal domination game on strong grids. Cartesian grids have been vastly studied with tight bounds for small grids such as 2×n, 3×n, 4×n, and 5×n grids, and recently it was proven in [Lamprou et al., CIAC 2017, 393-404] that the eternal domination number of these grids in general is within O(m + n) of their domination number which lower bounds the eternal domination number. Recently, Finbow et al. proved that the eternal domination number of strong grids is upper bounded by mn 6 + O(m + n). We adapt the techniques of [Lamprou et al., CIAC 2017, 393-404] to prove that the eternal domination number of strong grids is upper bounded by mn 7 + O(m + n). While this does not improve upon a recently announced bound of ⎡m/3⎤ x⎡n/3⎤ + O(m √ n) [Mc Inerney, Nisse, Pérennes, HAL archives, 2018; Mc Inerney, Nisse, Pérennes, CIAC 2019] in the general case, we show that our bound is an improvement in the case where the smaller of the two dimensions is at most 6179.]]> 0 <![CDATA[The flower at a point x in a Steiner triple system (X; B) is the set of all triples containing x. Denote by J3F(r) the set of all integers k such that there exists a collection of three STS(2r+1) mutually intersecting in the same set of k + r triples, r of them being the triples of a common flower. In this article we determine the set J3F(r) for any positive integer r = 0, 1 (mod 3) (only some cases are left undecided for r = 6, 7, 9, 24), and establish that J3F(r) = I3F(r) for r = 0, 1 (mod 3) where I3F(r) = {0, 1,..., 2r(r-1)/3-8, 2r(r-1)/3-6, 2r(r-1)/3}.]]> Mon, 16 Mar 2020 09:15:00 +0000 https://doi.org/10.23638/DMTCS-22-1-5 https://doi.org/10.23638/DMTCS-22-1-5 Amjadi, H. Soltankhah, N. Amjadi, H. Soltankhah, N. <![CDATA[The flower at a point x in a Steiner triple system (X; B) is the set of all triples containing x. Denote by J3F(r) the set of all integers k such that there exists a collection of three STS(2r+1) mutually intersecting in the same set of k + r triples, r of them being the triples of a common flower. In this article we determine the set J3F(r) for any positive integer r = 0, 1 (mod 3) (only some cases are left undecided for r = 6, 7, 9, 24), and establish that J3F(r) = I3F(r) for r = 0, 1 (mod 3) where I3F(r) = {0, 1,..., 2r(r-1)/3-8, 2r(r-1)/3-6, 2r(r-1)/3}.]]> 0 <![CDATA[A word of length $n$ is rich if it contains $n$ nonempty palindromic factors. An infinite word is rich if all of its finite factors are rich. Baranwal and Shallit produced an infinite binary rich word with critical exponent $2+\sqrt{2}/2$ ($\approx 2.707$) and conjectured that this was the least possible critical exponent for infinite binary rich words (i.e., that the repetition threshold for binary rich words is $2+\sqrt{2}/2$). In this article, we give a structure theorem for infinite binary rich words that avoid $14/5$-powers (i.e., repetitions with exponent at least 2.8). As a consequence, we deduce that the repetition threshold for binary rich words is $2+\sqrt{2}/2$, as conjectured by Baranwal and Shallit. This resolves an open problem of Vesti for the binary alphabet; the problem remains open for larger alphabets.]]> Mon, 24 Feb 2020 14:05:19 +0000 https://doi.org/10.23638/DMTCS-22-1-6 https://doi.org/10.23638/DMTCS-22-1-6 Currie, James D. Mol, Lucas Rampersad, Narad Currie, James D. Mol, Lucas Rampersad, Narad <![CDATA[A word of length $n$ is rich if it contains $n$ nonempty palindromic factors. An infinite word is rich if all of its finite factors are rich. Baranwal and Shallit produced an infinite binary rich word with critical exponent $2+\sqrt{2}/2$ ($\approx 2.707$) and conjectured that this was the least possible critical exponent for infinite binary rich words (i.e., that the repetition threshold for binary rich words is $2+\sqrt{2}/2$). In this article, we give a structure theorem for infinite binary rich words that avoid $14/5$-powers (i.e., repetitions with exponent at least 2.8). As a consequence, we deduce that the repetition threshold for binary rich words is $2+\sqrt{2}/2$, as conjectured by Baranwal and Shallit. This resolves an open problem of Vesti for the binary alphabet; the problem remains open for larger alphabets.]]> 0 <![CDATA[It is commonly believed that real networks are scale-free and fraction of nodes $P(k)$ with degree $k$ satisfies the power law $P(k) \propto k^{-\gamma} \text{ for } k > k_{min} > 0$. Preferential attachment is the mechanism that has been considered responsible for such organization of these networks. In many real networks, degree distribution before the $k_{min}$ varies very slowly to the extent of being uniform as compared with the degree distribution for $k > k_{min}$ . In this paper, we proposed a model that describe this particular degree distribution for the whole range of $k>0$. We adopt a two step approach. In the first step, at every time stamp we add a new node to the network and attach it with an existing node using preferential attachment method. In the second step, we add edges between existing pairs of nodes with the node selection based on the uniform probability distribution. Our approach generates weakly scale-free networks that closely follow the degree distribution of real-world networks. We perform comprehensive mathematical analysis of the model in the discrete domain and compare the degree distribution generated by these models with that of real-world networks.]]> Mon, 24 Feb 2020 14:02:25 +0000 https://doi.org/10.23638/DMTCS-22-1-7 https://doi.org/10.23638/DMTCS-22-1-7 Anwar, Raheel Yousuf, Muhammad Irfan Abid, Muhammad Anwar, Raheel Yousuf, Muhammad Irfan Abid, Muhammad <![CDATA[It is commonly believed that real networks are scale-free and fraction of nodes $P(k)$ with degree $k$ satisfies the power law $P(k) \propto k^{-\gamma} \text{ for } k > k_{min} > 0$. Preferential attachment is the mechanism that has been considered responsible for such organization of these networks. In many real networks, degree distribution before the $k_{min}$ varies very slowly to the extent of being uniform as compared with the degree distribution for $k > k_{min}$ . In this paper, we proposed a model that describe this particular degree distribution for the whole range of $k>0$. We adopt a two step approach. In the first step, at every time stamp we add a new node to the network and attach it with an existing node using preferential attachment method. In the second step, we add edges between existing pairs of nodes with the node selection based on the uniform probability distribution. Our approach generates weakly scale-free networks that closely follow the degree distribution of real-world networks. We perform comprehensive mathematical analysis of the model in the discrete domain and compare the degree distribution generated by these models with that of real-world networks.]]> 0 <![CDATA[A morphic word is obtained by iterating a morphism to generate an infinite word, and then applying a coding. We characterize morphic words with polynomial growth in terms of a new type of infinite word called a $\textit{zigzag word}$. A zigzag word is represented by an initial string, followed by a finite list of terms, each of which repeats for each $n \geq 1$ in one of three ways: it grows forward [$t(1)\ t(2)\ \dotsm\ t(n)]$, backward [$t(n)\ \dotsm\ t(2)\ t(1)$], or just occurs once [$t$]. Each term can recursively contain subterms with their own forward and backward repetitions. We show that an infinite word is morphic with growth $\Theta(n^k)$ iff it is a zigzag word of depth $k$. As corollaries, we obtain that the morphic words with growth $O(n)$ are exactly the ultimately periodic words, and the morphic words with growth $O(n^2)$ are exactly the multilinear words.]]> Thu, 06 Feb 2020 16:00:08 +0000 https://doi.org/10.23638/DMTCS-22-1-3 https://doi.org/10.23638/DMTCS-22-1-3 Smith, Tim Smith, Tim <![CDATA[A morphic word is obtained by iterating a morphism to generate an infinite word, and then applying a coding. We characterize morphic words with polynomial growth in terms of a new type of infinite word called a $\textit{zigzag word}$. A zigzag word is represented by an initial string, followed by a finite list of terms, each of which repeats for each $n \geq 1$ in one of three ways: it grows forward [$t(1)\ t(2)\ \dotsm\ t(n)]$, backward [$t(n)\ \dotsm\ t(2)\ t(1)$], or just occurs once [$t$]. Each term can recursively contain subterms with their own forward and backward repetitions. We show that an infinite word is morphic with growth $\Theta(n^k)$ iff it is a zigzag word of depth $k$. As corollaries, we obtain that the morphic words with growth $O(n)$ are exactly the ultimately periodic words, and the morphic words with growth $O(n^2)$ are exactly the multilinear words.]]> 0 <![CDATA[Linear chord diagrams are partitions of $\left[2n\right]$ into $n$ blocks of size two called chords. We refer to a block of the form $\{i,i+1\}$ as a short chord. In this paper, we study the distribution of the number of short chords on the set of linear chord diagrams, as a generalization of the Narayana distribution obtained when restricted to the set of noncrossing linear chord diagrams. We provide a combinatorial proof that this distribution is unimodal and has an expected value of one. We also study the number of pairs $(i,i+1)$ where $i$ is the minimal element of a chord and $i+1$ is the maximal element of a chord. We show that the distribution of this statistic on linear chord diagrams corresponds to the second-order Eulerian triangle and is log-concave.]]> Thu, 23 Jan 2020 16:56:22 +0000 https://doi.org/10.23638/DMTCS-21-2-11 https://doi.org/10.23638/DMTCS-21-2-11 Cameron, Naiomi T. Killpatrick, Kendra Cameron, Naiomi T. Killpatrick, Kendra <![CDATA[Linear chord diagrams are partitions of $\left[2n\right]$ into $n$ blocks of size two called chords. We refer to a block of the form $\{i,i+1\}$ as a short chord. In this paper, we study the distribution of the number of short chords on the set of linear chord diagrams, as a generalization of the Narayana distribution obtained when restricted to the set of noncrossing linear chord diagrams. We provide a combinatorial proof that this distribution is unimodal and has an expected value of one. We also study the number of pairs $(i,i+1)$ where $i$ is the minimal element of a chord and $i+1$ is the maximal element of a chord. We show that the distribution of this statistic on linear chord diagrams corresponds to the second-order Eulerian triangle and is log-concave.]]> 0 <![CDATA[It has been shown by Bokal et al. that deciding 2-colourability of digraphs is an NP-complete problem. This result was later on extended by Feder et al. to prove that deciding whether a digraph has a circular $p$-colouring is NP-complete for all rational $p>1$. In this paper, we consider the complexity of corresponding decision problems for related notions of fractional colourings for digraphs and graphs, including the star dichromatic number, the fractional dichromatic number and the circular vertex arboricity. We prove the following results: Deciding if the star dichromatic number of a digraph is at most $p$ is NP-complete for every rational $p>1$. Deciding if the fractional dichromatic number of a digraph is at most $p$ is NP-complete for every $p>1, p \neq 2$. Deciding if the circular vertex arboricity of a graph is at most $p$ is NP-complete for every rational $p>1$. To show these results, different techniques are required in each case. In order to prove the first result, we relate the star dichromatic number to a new notion of homomorphisms between digraphs, called circular homomorphisms, which might be of independent interest. We provide a classification of the computational complexities of the corresponding homomorphism colouring problems similar to the one derived by Feder et al. for acyclic homomorphisms.]]> Tue, 21 Jan 2020 14:55:24 +0000 https://doi.org/10.23638/DMTCS-22-1-4 https://doi.org/10.23638/DMTCS-22-1-4 Hochstättler, Winfried Schröder, Felix Steiner, Raphael Hochstättler, Winfried Schröder, Felix Steiner, Raphael <![CDATA[It has been shown by Bokal et al. that deciding 2-colourability of digraphs is an NP-complete problem. This result was later on extended by Feder et al. to prove that deciding whether a digraph has a circular $p$-colouring is NP-complete for all rational $p>1$. In this paper, we consider the complexity of corresponding decision problems for related notions of fractional colourings for digraphs and graphs, including the star dichromatic number, the fractional dichromatic number and the circular vertex arboricity. We prove the following results: Deciding if the star dichromatic number of a digraph is at most $p$ is NP-complete for every rational $p>1$. Deciding if the fractional dichromatic number of a digraph is at most $p$ is NP-complete for every $p>1, p \neq 2$. Deciding if the circular vertex arboricity of a graph is at most $p$ is NP-complete for every rational $p>1$. To show these results, different techniques are required in each case. In order to prove the first result, we relate the star dichromatic number to a new notion of homomorphisms between digraphs, called circular homomorphisms, which might be of independent interest. We provide a classification of the computational complexities of the corresponding homomorphism colouring problems similar to the one derived by Feder et al. for acyclic homomorphisms.]]> 0 <![CDATA[A strong edge-colouring of an undirected graph $G$ is an edge-colouring where every two edges at distance at most~$2$ receive distinct colours. The strong chromatic index of $G$ is the least number of colours in a strong edge-colouring of $G$. A conjecture of Erd\H{o}s and Ne\v{s}et\v{r}il, stated back in the $80$'s, asserts that every graph with maximum degree $\Delta$ should have strong chromatic index at most roughly $1.25 \Delta^2$. Several works in the last decades have confirmed this conjecture for various graph classes. In particular, lots of attention have been dedicated to planar graphs, for which the strong chromatic index decreases to roughly $4\Delta$, and even to smaller values under additional structural requirements.In this work, we initiate the study of the strong chromatic index of $1$-planar graphs, which are those graphs that can be drawn on the plane in such a way that every edge is crossed at most once. We provide constructions of $1$-planar graphs with maximum degree~$\Delta$ and strong chromatic index roughly $6\Delta$. As an upper bound, we prove that the strong chromatic index of a $1$-planar graph with maximum degree $\Delta$ is at most roughly $24\Delta$ (thus linear in $\Delta$). The proof of this result is based on the existence of light edges in $1$-planar graphs with minimum degree at least~$3$.]]> Thu, 09 Jan 2020 09:24:43 +0000 https://doi.org/10.23638/DMTCS-22-1-2 https://doi.org/10.23638/DMTCS-22-1-2 Bensmail, Julien Dross, François Hocquard, Hervé Sopena, Eric Bensmail, Julien Dross, François Hocquard, Hervé Sopena, Eric <![CDATA[A strong edge-colouring of an undirected graph $G$ is an edge-colouring where every two edges at distance at most~$2$ receive distinct colours. The strong chromatic index of $G$ is the least number of colours in a strong edge-colouring of $G$. A conjecture of Erd\H{o}s and Ne\v{s}et\v{r}il, stated back in the $80$'s, asserts that every graph with maximum degree $\Delta$ should have strong chromatic index at most roughly $1.25 \Delta^2$. Several works in the last decades have confirmed this conjecture for various graph classes. In particular, lots of attention have been dedicated to planar graphs, for which the strong chromatic index decreases to roughly $4\Delta$, and even to smaller values under additional structural requirements.In this work, we initiate the study of the strong chromatic index of $1$-planar graphs, which are those graphs that can be drawn on the plane in such a way that every edge is crossed at most once. We provide constructions of $1$-planar graphs with maximum degree~$\Delta$ and strong chromatic index roughly $6\Delta$. As an upper bound, we prove that the strong chromatic index of a $1$-planar graph with maximum degree $\Delta$ is at most roughly $24\Delta$ (thus linear in $\Delta$). The proof of this result is based on the existence of light edges in $1$-planar graphs with minimum degree at least~$3$.]]> 0 <![CDATA[In this paper, we study the complexity of the selection of a graph discretization order with a stepwise linear cost function. Finding such vertex ordering has been proved to be an essential step to solve discretizable distance geometry problems (DDGPs). DDGPs constitute a class of graph realization problems where the vertices can be ordered in such a way that the search space of possible positions becomes discrete, usually represented by a binary tree. In particular, it is useful to find discretization orders that minimize an indicator of the size of the search tree. Our stepwise linear cost function generalizes this situation and allows to discriminate the vertices into three categories depending on the number of adjacent predecessors of each vertex in the order and on two parameters K and U. We provide a complete study of NP-completeness for fixed values of K and U. Our main result is that the problem is NP-complete in general for all values of K and U such that U ≥ K + 1 and U ≥ 2. A consequence of this result is that the minimization of vertices with exactly K adjacent predecessors in a discretization order is also NP-complete.]]> Thu, 09 Jan 2020 09:13:29 +0000 https://doi.org/10.23638/DMTCS-22-1-1 https://doi.org/10.23638/DMTCS-22-1-1 Omer, Jérémy Migot, Tangi Omer, Jérémy Migot, Tangi <![CDATA[In this paper, we study the complexity of the selection of a graph discretization order with a stepwise linear cost function. Finding such vertex ordering has been proved to be an essential step to solve discretizable distance geometry problems (DDGPs). DDGPs constitute a class of graph realization problems where the vertices can be ordered in such a way that the search space of possible positions becomes discrete, usually represented by a binary tree. In particular, it is useful to find discretization orders that minimize an indicator of the size of the search tree. Our stepwise linear cost function generalizes this situation and allows to discriminate the vertices into three categories depending on the number of adjacent predecessors of each vertex in the order and on two parameters K and U. We provide a complete study of NP-completeness for fixed values of K and U. Our main result is that the problem is NP-complete in general for all values of K and U such that U ≥ K + 1 and U ≥ 2. A consequence of this result is that the minimization of vertices with exactly K adjacent predecessors in a discretization order is also NP-complete.]]> 0 <![CDATA[In this article we consider square permutations, a natural subclass of permutations defined in terms of geometric conditions, that can also be described in terms of pattern avoiding permutations, and convex permutoninoes, a related subclass of polyominoes. While these two classes of objects arised independently in various contexts, they play a natural role in the description of certain random horizontally and vertically convex grid configurations. We propose a common approach to the enumeration of these two classes of objets that allows us to explain the known common form of their generating functions, and to derive new refined formulas and linear time random generation algorithms for these objects and the associated grid configurations.]]> Mon, 30 Dec 2019 12:31:35 +0000 https://doi.org/10.23638/DMTCS-21-2-2 https://doi.org/10.23638/DMTCS-21-2-2 Duchi, Enrica Duchi, Enrica <![CDATA[In this article we consider square permutations, a natural subclass of permutations defined in terms of geometric conditions, that can also be described in terms of pattern avoiding permutations, and convex permutoninoes, a related subclass of polyominoes. While these two classes of objects arised independently in various contexts, they play a natural role in the description of certain random horizontally and vertically convex grid configurations. We propose a common approach to the enumeration of these two classes of objets that allows us to explain the known common form of their generating functions, and to derive new refined formulas and linear time random generation algorithms for these objects and the associated grid configurations.]]> 0 <![CDATA[We prove that the joint embedding property is undecidable for hereditary graph classes, via a reduction from the tiling problem. The proof is then adapted to show the undecidability of the joint homomorphism property as well.]]> Fri, 13 Dec 2019 17:05:17 +0000 https://doi.org/10.23638/DMTCS-21-2-9 https://doi.org/10.23638/DMTCS-21-2-9 Braunfeld, Samuel Braunfeld, Samuel <![CDATA[We prove that the joint embedding property is undecidable for hereditary graph classes, via a reduction from the tiling problem. The proof is then adapted to show the undecidability of the joint homomorphism property as well.]]> 0 <![CDATA[Power domination in graphs emerged from the problem of monitoring an electrical system by placing as few measurement devices in the system as possible. It corresponds to a variant of domination that includes the possibility of propagation. For measurement devices placed on a set S of vertices of a graph G, the set of monitored vertices is initially the set S together with all its neighbors. Then iteratively, whenever some monitored vertex v has a single neighbor u not yet monitored, u gets monitored. A set S is said to be a power dominating set of the graph G if all vertices of G eventually are monitored. The power domination number of a graph is the minimum size of a power dominating set. In this paper, we prove that any maximal planar graph of order n ≥ 6 admits a power dominating set of size at most (n−2)/4 .]]> Fri, 13 Dec 2019 15:01:55 +0000 https://doi.org/10.23638/DMTCS-21-4-18 https://doi.org/10.23638/DMTCS-21-4-18 Dorbec, Paul González, Antonio Pennarun, Claire Dorbec, Paul González, Antonio Pennarun, Claire <![CDATA[Power domination in graphs emerged from the problem of monitoring an electrical system by placing as few measurement devices in the system as possible. It corresponds to a variant of domination that includes the possibility of propagation. For measurement devices placed on a set S of vertices of a graph G, the set of monitored vertices is initially the set S together with all its neighbors. Then iteratively, whenever some monitored vertex v has a single neighbor u not yet monitored, u gets monitored. A set S is said to be a power dominating set of the graph G if all vertices of G eventually are monitored. The power domination number of a graph is the minimum size of a power dominating set. In this paper, we prove that any maximal planar graph of order n ≥ 6 admits a power dominating set of size at most (n−2)/4 .]]> 0 <![CDATA[Under what circumstances might every extension of a combinatorial structure contain more copies of another one than the original did? This property, which we call prolificity, holds universally in some cases (e.g., finite linear orders) and only trivially in others (e.g., permutations). Integer compositions, or equivalently layered permutations, provide a middle ground. In that setting, there are prolific compositions for a given pattern if and only if that pattern begins and ends with 1. For each pattern, there is an easily constructed automaton that recognises prolific compositions for that pattern. Some instances where there is a unique minimal prolific composition for a pattern are classified.]]> Fri, 13 Dec 2019 14:57:51 +0000 https://doi.org/10.23638/DMTCS-21-2-10 https://doi.org/10.23638/DMTCS-21-2-10 Tannock, Murray Albert, Michael Tannock, Murray Albert, Michael <![CDATA[Under what circumstances might every extension of a combinatorial structure contain more copies of another one than the original did? This property, which we call prolificity, holds universally in some cases (e.g., finite linear orders) and only trivially in others (e.g., permutations). Integer compositions, or equivalently layered permutations, provide a middle ground. In that setting, there are prolific compositions for a given pattern if and only if that pattern begins and ends with 1. For each pattern, there is an easily constructed automaton that recognises prolific compositions for that pattern. Some instances where there is a unique minimal prolific composition for a pattern are classified.]]> 0 <![CDATA[We complete the enumeration of cyclic permutations avoiding two patterns of length three each by providing explicit formulas for all but one of the pairs for which no such formulas were known. The pair $(123,231)$ proves to be the most difficult of these pairs. We also prove a lower bound for the growth rate of the number of cyclic permutations that avoid a single pattern $q$, where $q$ is an element of a certain infinite family of patterns.]]> Tue, 26 Nov 2019 13:45:09 +0000 https://doi.org/10.23638/DMTCS-21-2-8 https://doi.org/10.23638/DMTCS-21-2-8 Bona, Miklos Cory, Michael Bona, Miklos Cory, Michael <![CDATA[We complete the enumeration of cyclic permutations avoiding two patterns of length three each by providing explicit formulas for all but one of the pairs for which no such formulas were known. The pair $(123,231)$ proves to be the most difficult of these pairs. We also prove a lower bound for the growth rate of the number of cyclic permutations that avoid a single pattern $q$, where $q$ is an element of a certain infinite family of patterns.]]> 0 <![CDATA[Many researchers have studied symmetry properties of various Boolean functions. A class of Boolean functions, called nested canalyzing functions (NCFs), has been used to model certain biological phenomena. We identify some interesting relationships between NCFs, symmetric Boolean functions and a generalization of symmetric Boolean functions, which we call $r$-symmetric functions (where $r$ is the symmetry level). Using a normalized representation for NCFs, we develop a characterization of when two variables of an NCF are symmetric. Using this characterization, we show that the symmetry level of an NCF $f$ can be easily computed given a standard representation of $f$. We also present an algorithm for testing whether a given $r$-symmetric function is an NCF. Further, we show that for any NCF $f$ with $n$ variables, the notion of strong asymmetry considered in the literature is equivalent to the property that $f$ is $n$-symmetric. We use this result to derive a closed form expression for the number of $n$-variable Boolean functions that are NCFs and strongly asymmetric. We also identify all the Boolean functions that are NCFs and symmetric.]]> Tue, 26 Nov 2019 13:39:22 +0000 https://doi.org/10.23638/DMTCS-21-4-19 https://doi.org/10.23638/DMTCS-21-4-19 Rosenkrantz, Daniel J. Marathe, Madhav V. Ravi, S. S. Stearns, Richard E. Rosenkrantz, Daniel J. Marathe, Madhav V. Ravi, S. S. Stearns, Richard E. <![CDATA[Many researchers have studied symmetry properties of various Boolean functions. A class of Boolean functions, called nested canalyzing functions (NCFs), has been used to model certain biological phenomena. We identify some interesting relationships between NCFs, symmetric Boolean functions and a generalization of symmetric Boolean functions, which we call $r$-symmetric functions (where $r$ is the symmetry level). Using a normalized representation for NCFs, we develop a characterization of when two variables of an NCF are symmetric. Using this characterization, we show that the symmetry level of an NCF $f$ can be easily computed given a standard representation of $f$. We also present an algorithm for testing whether a given $r$-symmetric function is an NCF. Further, we show that for any NCF $f$ with $n$ variables, the notion of strong asymmetry considered in the literature is equivalent to the property that $f$ is $n$-symmetric. We use this result to derive a closed form expression for the number of $n$-variable Boolean functions that are NCFs and strongly asymmetric. We also identify all the Boolean functions that are NCFs and symmetric.]]> 0 <![CDATA[Super-strong Wilf equivalence classes of the symmetric group ${\mathcal S}_n$ on $n$ letters, with respect to the generalized factor order, were shown by Hadjiloucas, Michos and Savvidou (2018) to be in bijection with pyramidal sequences of consecutive differences. In this article we enumerate the latter by giving recursive formulae in terms of a two-dimensional analogue of non-interval permutations. As a by-product, we obtain a recursively defined set of representatives of super-strong Wilf equivalence classes in ${\mathcal S}_n$. We also provide a connection between super-strong Wilf equivalence and the geometric notion of shift equivalence---originally defined by Fidler, Glasscock, Miceli, Pantone, and Xu (2018) for words---by showing that an alternate way to characterize super-strong Wilf equivalence for permutations is by keeping only rigid shifts in the definition of shift equivalence. This allows us to fully describe shift equivalence classes for permutations of size $n$ and enumerate them, answering the corresponding problem posed by Fidler, Glasscock, Miceli, Pantone, and Xu (2018).]]> Mon, 11 Nov 2019 11:12:23 +0000 https://doi.org/10.23638/DMTCS-21-2-3 https://doi.org/10.23638/DMTCS-21-2-3 Michos, Ioannis Savvidou, Christina Michos, Ioannis Savvidou, Christina <![CDATA[Super-strong Wilf equivalence classes of the symmetric group ${\mathcal S}_n$ on $n$ letters, with respect to the generalized factor order, were shown by Hadjiloucas, Michos and Savvidou (2018) to be in bijection with pyramidal sequences of consecutive differences. In this article we enumerate the latter by giving recursive formulae in terms of a two-dimensional analogue of non-interval permutations. As a by-product, we obtain a recursively defined set of representatives of super-strong Wilf equivalence classes in ${\mathcal S}_n$. We also provide a connection between super-strong Wilf equivalence and the geometric notion of shift equivalence---originally defined by Fidler, Glasscock, Miceli, Pantone, and Xu (2018) for words---by showing that an alternate way to characterize super-strong Wilf equivalence for permutations is by keeping only rigid shifts in the definition of shift equivalence. This allows us to fully describe shift equivalence classes for permutations of size $n$ and enumerate them, answering the corresponding problem posed by Fidler, Glasscock, Miceli, Pantone, and Xu (2018).]]> 0 <![CDATA[The family of generalized Petersen graphs $G(n,k)$, introduced by Coxeter et al. [4] and named by Mark Watkins (1969), is a family of cubic graphs formed by connecting the vertices of a regular polygon to the corresponding vertices of a star polygon. The Kronecker cover $KC(G)$ of a simple undirected graph $G$ is a a special type of bipartite covering graph of $G$, isomorphic to the direct (tensor) product of $G$ and $K_2$. We characterize all the members of generalized Petersen graphs that are Kronecker covers, and describe the structure of their respective quotients. We observe that some of such quotients are again generalized Petersen graphs, and describe all such pairs.The results of this paper have been presented at EUROCOMB 2019 and an extended abstract has been published elsewhere.]]> Mon, 11 Nov 2019 10:41:35 +0000 https://doi.org/10.23638/DMTCS-21-4-15 https://doi.org/10.23638/DMTCS-21-4-15 Krnc, Matjaž Pisanski, Tomaž Krnc, Matjaž Pisanski, Tomaž <![CDATA[The family of generalized Petersen graphs $G(n,k)$, introduced by Coxeter et al. [4] and named by Mark Watkins (1969), is a family of cubic graphs formed by connecting the vertices of a regular polygon to the corresponding vertices of a star polygon. The Kronecker cover $KC(G)$ of a simple undirected graph $G$ is a a special type of bipartite covering graph of $G$, isomorphic to the direct (tensor) product of $G$ and $K_2$. We characterize all the members of generalized Petersen graphs that are Kronecker covers, and describe the structure of their respective quotients. We observe that some of such quotients are again generalized Petersen graphs, and describe all such pairs.The results of this paper have been presented at EUROCOMB 2019 and an extended abstract has been published elsewhere.]]> 0 <![CDATA[A graph $G$ is equitably $k$-choosable if, for any given $k$-uniform list assignment $L$, $G$ is $L$-colorable and each color appears on at most $\lceil\frac{|V(G)|}{k}\rceil$ vertices. A graph is equitably $k$-colorable if the vertex set $V(G)$ can be partitioned into $k$ independent subsets $V_1$, $V_2$, $\cdots$, $V_k$ such that $||V_i|-|V_j||\leq 1$ for $1\leq i, j\leq k$. In this paper, we prove that if $G$ is a planar graph without chordal $4$- and $6$-cycles, then $G$ is equitably $k$-colorable and equitably $k$-choosable where $k\geq\max\{\Delta(G), 7\}$.]]> Mon, 11 Nov 2019 10:31:41 +0000 https://doi.org/10.23638/DMTCS-21-3-16 https://doi.org/10.23638/DMTCS-21-3-16 Dong, Aijun Wu, Jianliang Dong, Aijun Wu, Jianliang <![CDATA[A graph $G$ is equitably $k$-choosable if, for any given $k$-uniform list assignment $L$, $G$ is $L$-colorable and each color appears on at most $\lceil\frac{|V(G)|}{k}\rceil$ vertices. A graph is equitably $k$-colorable if the vertex set $V(G)$ can be partitioned into $k$ independent subsets $V_1$, $V_2$, $\cdots$, $V_k$ such that $||V_i|-|V_j||\leq 1$ for $1\leq i, j\leq k$. In this paper, we prove that if $G$ is a planar graph without chordal $4$- and $6$-cycles, then $G$ is equitably $k$-colorable and equitably $k$-choosable where $k\geq\max\{\Delta(G), 7\}$.]]> 0 <![CDATA[Two permutations in a class are Wilf-equivalent if, for every size, $n$, the number of permutations in the class of size $n$ containing each of them is the same. Those infinite classes that have only one equivalence class in each size for this relation are characterised provided either that they avoid at least one permutation of size 3, or at least three permutations of size 4.]]> Mon, 04 Nov 2019 12:58:27 +0000 https://doi.org/10.23638/DMTCS-21-2-7 https://doi.org/10.23638/DMTCS-21-2-7 Albert, Michael Li, Jinge Albert, Michael Li, Jinge <![CDATA[Two permutations in a class are Wilf-equivalent if, for every size, $n$, the number of permutations in the class of size $n$ containing each of them is the same. Those infinite classes that have only one equivalence class in each size for this relation are characterised provided either that they avoid at least one permutation of size 3, or at least three permutations of size 4.]]> 0 <![CDATA[An inversion sequence of length $n$ is an integer sequence $e=e_{1}e_{2}\dots e_{n}$ such that $0\leq e_{i} Mon, 04 Nov 2019 12:54:33 +0000 https://doi.org/10.23638/DMTCS-21-2-6 https://doi.org/10.23638/DMTCS-21-2-6 Auli, Juan S. Elizalde, Sergi Auli, Juan S. Elizalde, Sergi <![CDATA[An inversion sequence of length $n$ is an integer sequence $e=e_{1}e_{2}\dots e_{n}$ such that $0\leq e_{i} 0 <![CDATA[Using existing classification results for the 7- and 8-cycles in the pancake graph, we determine the number of permutations that require 4 pancake flips (prefix reversals) to be sorted. A similar characterization of the 8-cycles in the burnt pancake graph, due to the authors, is used to derive a formula for the number of signed permutations requiring 4 (burnt) pancake flips to be sorted. We furthermore provide an analogous characterization of the 9-cycles in the burnt pancake graph. Finally we present numerical evidence that polynomial formulae exist giving the number of signed permutations that require $k$ flips to be sorted, with $5\leq k\leq9$.]]> Mon, 04 Nov 2019 12:50:08 +0000 https://doi.org/10.23638/DMTCS-21-2-5 https://doi.org/10.23638/DMTCS-21-2-5 Blanco, Saúl A. Buehrle, Charles Patidar, Akshay Blanco, Saúl A. Buehrle, Charles Patidar, Akshay <![CDATA[Using existing classification results for the 7- and 8-cycles in the pancake graph, we determine the number of permutations that require 4 pancake flips (prefix reversals) to be sorted. A similar characterization of the 8-cycles in the burnt pancake graph, due to the authors, is used to derive a formula for the number of signed permutations requiring 4 (burnt) pancake flips to be sorted. We furthermore provide an analogous characterization of the 9-cycles in the burnt pancake graph. Finally we present numerical evidence that polynomial formulae exist giving the number of signed permutations that require $k$ flips to be sorted, with $5\leq k\leq9$.]]> 0 <![CDATA[Classical pattern avoidance and occurrence are well studied in the symmetric group $\mathcal{S}_{n}$. In this paper, we provide explicit recurrence relations to the generating functions counting the number of classical pattern occurrence in the set of 132-avoiding permutations and the set of 123-avoiding permutations.]]> Mon, 04 Nov 2019 12:44:15 +0000 https://doi.org/10.23638/DMTCS-21-2-4 https://doi.org/10.23638/DMTCS-21-2-4 Qiu, Dun Remmel, Jeffrey Qiu, Dun Remmel, Jeffrey <![CDATA[Classical pattern avoidance and occurrence are well studied in the symmetric group $\mathcal{S}_{n}$. In this paper, we provide explicit recurrence relations to the generating functions counting the number of classical pattern occurrence in the set of 132-avoiding permutations and the set of 123-avoiding permutations.]]> 0 <![CDATA[Given a graph $G=(V,E)$ and a positive integer $t\geq2$, the task in the vertex cover $P_t$ ($VCP_t$) problem is to find a minimum subset of vertices $F\subseteq V$ such that every path of order $t$ in $G$ contains at least one vertex from $F$. The $VCP_t$ problem is NP-complete for any integer $t\geq2$ and has many applications in real world. Recently, the authors presented a dynamic programming algorithm running in time $4^p\cdot n^{O(1)}$ for the $VCP_3$ problem on $n$-vertex graphs with treewidth $p$. In this paper, we propose an improvement of it and improved the time-complexity to $3^p\cdot n^{O(1)}$. The connected vertex cover $P_3$ ($CVCP_3$) problem is the connected variation of the $VCP_3$ problem where $G[F]$ is required to be connected. Using the Cut\&Count technique, we give a randomized algorithm with runtime $4^p\cdot n^{O(1)}$ for the $CVCP_3$ problem on $n$-vertex graphs with treewidth $p$.]]> Mon, 04 Nov 2019 12:32:49 +0000 https://doi.org/10.23638/DMTCS-21-4-17 https://doi.org/10.23638/DMTCS-21-4-17 Bai, Zongwen Tu, Jianhua Shi, Yongtang Bai, Zongwen Tu, Jianhua Shi, Yongtang <![CDATA[Given a graph $G=(V,E)$ and a positive integer $t\geq2$, the task in the vertex cover $P_t$ ($VCP_t$) problem is to find a minimum subset of vertices $F\subseteq V$ such that every path of order $t$ in $G$ contains at least one vertex from $F$. The $VCP_t$ problem is NP-complete for any integer $t\geq2$ and has many applications in real world. Recently, the authors presented a dynamic programming algorithm running in time $4^p\cdot n^{O(1)}$ for the $VCP_3$ problem on $n$-vertex graphs with treewidth $p$. In this paper, we propose an improvement of it and improved the time-complexity to $3^p\cdot n^{O(1)}$. The connected vertex cover $P_3$ ($CVCP_3$) problem is the connected variation of the $VCP_3$ problem where $G[F]$ is required to be connected. Using the Cut\&Count technique, we give a randomized algorithm with runtime $4^p\cdot n^{O(1)}$ for the $CVCP_3$ problem on $n$-vertex graphs with treewidth $p$.]]> 0 <![CDATA[The quantity that captures the asymptotic value of the maximum number of appearances of a given topological tree (a rooted tree with no vertices of outdegree $1$) $S$ with $k$ leaves in an arbitrary tree with sufficiently large number of leaves is called the inducibility of $S$. Its precise value is known only for some specific families of trees, most of them exhibiting a symmetrical configuration. In an attempt to answer a recent question posed by Czabarka, Sz\'ekely, and the second author of this article, we provide bounds for the inducibility $J(A_5)$ of the $5$-leaf binary tree $A_5$ whose branches are a single leaf and the complete binary tree of height $2$. It was indicated before that $J(A_5)$ appears to be `close' to $1/4$. We can make this precise by showing that $0.24707\ldots \leq J(A_5) \leq 0.24745\ldots$. Furthermore, we also consider the problem of determining the inducibility of the tree $Q_4$, which is the only tree among $4$-leaf topological trees for which the inducibility is unknown.]]> Thu, 17 Oct 2019 13:50:52 +0000 https://doi.org/10.23638/DMTCS-21-4-13 https://doi.org/10.23638/DMTCS-21-4-13 Dossou-Olory, Audace A. V. Wagner, Stephan Dossou-Olory, Audace A. V. Wagner, Stephan <![CDATA[The quantity that captures the asymptotic value of the maximum number of appearances of a given topological tree (a rooted tree with no vertices of outdegree $1$) $S$ with $k$ leaves in an arbitrary tree with sufficiently large number of leaves is called the inducibility of $S$. Its precise value is known only for some specific families of trees, most of them exhibiting a symmetrical configuration. In an attempt to answer a recent question posed by Czabarka, Sz\'ekely, and the second author of this article, we provide bounds for the inducibility $J(A_5)$ of the $5$-leaf binary tree $A_5$ whose branches are a single leaf and the complete binary tree of height $2$. It was indicated before that $J(A_5)$ appears to be `close' to $1/4$. We can make this precise by showing that $0.24707\ldots \leq J(A_5) \leq 0.24745\ldots$. Furthermore, we also consider the problem of determining the inducibility of the tree $Q_4$, which is the only tree among $4$-leaf topological trees for which the inducibility is unknown.]]> 0 <![CDATA[After fixing a canonical ordering (or labeling) of the elements of a finite poset, one can associate each linear extension of the poset with a permutation. Some recent papers consider specific families of posets and ask how many linear extensions give rise to permutations that avoid certain patterns. We build off of two of these papers. We first consider pattern avoidance in $k$-ary heaps, where we obtain a general result that proves a conjecture of Levin, Pudwell, Riehl, and Sandberg in a special case. We then prove some conjectures that Anderson, Egge, Riehl, Ryan, Steinke, and Vaughan made about pattern-avoiding linear extensions of rectangular posets.]]> Wed, 02 Oct 2019 07:52:20 +0000 https://doi.org/10.23638/DMTCS-21-4-16 https://doi.org/10.23638/DMTCS-21-4-16 Defant, Colin Defant, Colin <![CDATA[After fixing a canonical ordering (or labeling) of the elements of a finite poset, one can associate each linear extension of the poset with a permutation. Some recent papers consider specific families of posets and ask how many linear extensions give rise to permutations that avoid certain patterns. We build off of two of these papers. We first consider pattern avoidance in $k$-ary heaps, where we obtain a general result that proves a conjecture of Levin, Pudwell, Riehl, and Sandberg in a special case. We then prove some conjectures that Anderson, Egge, Riehl, Ryan, Steinke, and Vaughan made about pattern-avoiding linear extensions of rectangular posets.]]> 0 <![CDATA[For integers $k\ge 2$ and $\ell\ge 0$, a $k$-uniform hypergraph is called a loose path of length $\ell$, and denoted by $P_\ell^{(k)}$, if it consists of $\ell $ edges $e_1,\dots,e_\ell$ such that $|e_i\cap e_j|=1$ if $|i-j|=1$ and $e_i\cap e_j=\emptyset$ if $|i-j|\ge2$. In other words, each pair of consecutive edges intersects on a single vertex, while all other pairs are disjoint. Let $R(P_\ell^{(k)};r)$ be the minimum integer $n$ such that every $r$-edge-coloring of the complete $k$-uniform hypergraph $K_n^{(k)}$ yields a monochromatic copy of $P_\ell^{(k)}$. In this paper we are mostly interested in constructive upper bounds on $R(P_\ell^{(k)};r)$, meaning that on the cost of possibly enlarging the order of the complete hypergraph, we would like to efficiently find a monochromatic copy of $P_\ell^{(k)}$ in every coloring. In particular, we show that there is a constant $c>0$ such that for all $k\ge 2$, $\ell\ge3$, $2\le r\le k-1$, and $n\ge k(\ell+1)r(1+\ln(r))$, there is an algorithm such that for every $r$-edge-coloring of the edges of $K_n^{(k)}$, it finds a monochromatic copy of $P_\ell^{(k)}$ in time at most $cn^k$. We also prove a non-constructive upper bound $R(P_\ell^{(k)};r)\le(k-1)\ell r$.]]> Wed, 02 Oct 2019 07:45:10 +0000 https://doi.org/10.23638/DMTCS-21-4-7 https://doi.org/10.23638/DMTCS-21-4-7 Dudek, Andrzej Ruciński, Andrzej Dudek, Andrzej Ruciński, Andrzej <![CDATA[For integers $k\ge 2$ and $\ell\ge 0$, a $k$-uniform hypergraph is called a loose path of length $\ell$, and denoted by $P_\ell^{(k)}$, if it consists of $\ell $ edges $e_1,\dots,e_\ell$ such that $|e_i\cap e_j|=1$ if $|i-j|=1$ and $e_i\cap e_j=\emptyset$ if $|i-j|\ge2$. In other words, each pair of consecutive edges intersects on a single vertex, while all other pairs are disjoint. Let $R(P_\ell^{(k)};r)$ be the minimum integer $n$ such that every $r$-edge-coloring of the complete $k$-uniform hypergraph $K_n^{(k)}$ yields a monochromatic copy of $P_\ell^{(k)}$. In this paper we are mostly interested in constructive upper bounds on $R(P_\ell^{(k)};r)$, meaning that on the cost of possibly enlarging the order of the complete hypergraph, we would like to efficiently find a monochromatic copy of $P_\ell^{(k)}$ in every coloring. In particular, we show that there is a constant $c>0$ such that for all $k\ge 2$, $\ell\ge3$, $2\le r\le k-1$, and $n\ge k(\ell+1)r(1+\ln(r))$, there is an algorithm such that for every $r$-edge-coloring of the edges of $K_n^{(k)}$, it finds a monochromatic copy of $P_\ell^{(k)}$ in time at most $cn^k$. We also prove a non-constructive upper bound $R(P_\ell^{(k)};r)\le(k-1)\ell r$.]]> 0 <![CDATA[Whitney's theorem states that every 3-connected planar graph is uniquely embeddable on the sphere. On the other hand, it has many inequivalent embeddings on another surface. We shall characterize structures of a $3$-connected $3$-regular planar graph $G$ embedded on the projective-plane, the torus and the Klein bottle, and give a one-to-one correspondence between inequivalent embeddings of $G$ on each surface and some subgraphs of the dual of $G$ embedded on the sphere. These results enable us to give explicit bounds for the number of inequivalent embeddings of $G$ on each surface, and propose effective algorithms for enumerating and counting these embeddings.]]> Fri, 27 Sep 2019 12:13:40 +0000 https://doi.org/10.23638/DMTCS-21-4-14 https://doi.org/10.23638/DMTCS-21-4-14 Enami, Kengo Enami, Kengo <![CDATA[Whitney's theorem states that every 3-connected planar graph is uniquely embeddable on the sphere. On the other hand, it has many inequivalent embeddings on another surface. We shall characterize structures of a $3$-connected $3$-regular planar graph $G$ embedded on the projective-plane, the torus and the Klein bottle, and give a one-to-one correspondence between inequivalent embeddings of $G$ on each surface and some subgraphs of the dual of $G$ embedded on the sphere. These results enable us to give explicit bounds for the number of inequivalent embeddings of $G$ on each surface, and propose effective algorithms for enumerating and counting these embeddings.]]> 0 <![CDATA[We show that one-way quantum one-counter automaton with zero-error is more powerful than its probabilistic counterpart on promise problems. Then, we obtain a similar separation result between Las Vegas one-way probabilistic one-counter automaton and one-way deterministic one-counter automaton. We also obtain new results on classical counter automata regarding language recognition. It was conjectured that one-way probabilistic one blind-counter automata cannot recognize Kleene closure of equality language [A. Yakaryilmaz: Superiority of one-way and realtime quantum machines. RAIRO - Theor. Inf. and Applic. 46(4): 615-641 (2012)]. We show that this conjecture is false, and also show several separation results for blind/non-blind counter automata.]]> Fri, 27 Sep 2019 11:34:14 +0000 https://doi.org/10.23638/DMTCS-21-4-12 https://doi.org/10.23638/DMTCS-21-4-12 Nakanishi, Masaki Yakaryılmaz, Abuzer Gainutdinova, Aida Nakanishi, Masaki Yakaryılmaz, Abuzer Gainutdinova, Aida <![CDATA[We show that one-way quantum one-counter automaton with zero-error is more powerful than its probabilistic counterpart on promise problems. Then, we obtain a similar separation result between Las Vegas one-way probabilistic one-counter automaton and one-way deterministic one-counter automaton. We also obtain new results on classical counter automata regarding language recognition. It was conjectured that one-way probabilistic one blind-counter automata cannot recognize Kleene closure of equality language [A. Yakaryilmaz: Superiority of one-way and realtime quantum machines. RAIRO - Theor. Inf. and Applic. 46(4): 615-641 (2012)]. We show that this conjecture is false, and also show several separation results for blind/non-blind counter automata.]]> 0 <![CDATA[We study the local limit of the fixed-point forest, a tree structure associated to a simple sorting algorithm on permutations. This local limit can be viewed as an infinite random tree that can be constructed from a Poisson point process configuration on $[0,1]^\mathbb{N}$. We generalize this random tree, and compute the expected size and expected number of leaves of a random rooted subtree in the generalized version. We also obtain bounds on the variance of the size.]]> Fri, 27 Sep 2019 09:53:22 +0000 https://doi.org/10.23638/DMTCS-21-2-1 https://doi.org/10.23638/DMTCS-21-2-1 Regan, Samuel Slivken, Erik Regan, Samuel Slivken, Erik <![CDATA[We study the local limit of the fixed-point forest, a tree structure associated to a simple sorting algorithm on permutations. This local limit can be viewed as an infinite random tree that can be constructed from a Poisson point process configuration on $[0,1]^\mathbb{N}$. We generalize this random tree, and compute the expected size and expected number of leaves of a random rooted subtree in the generalized version. We also obtain bounds on the variance of the size.]]> 0 <![CDATA[We consider the constrained graph alignment problem which has applications in biological network analysis. Given two input graphs $G_1=(V_1,E_1), G_2=(V_2,E_2)$, a pair of vertex mappings induces an {\it edge conservation} if the vertex pairs are adjacent in their respective graphs. %In general terms The goal is to provide a one-to-one mapping between the vertices of the input graphs in order to maximize edge conservation. However the allowed mappings are restricted since each vertex from $V_1$ (resp. $V_2$) is allowed to be mapped to at most $m_1$ (resp. $m_2$) specified vertices in $V_2$ (resp. $V_1$). Most of results in this paper deal with the case $m_2=1$ which attracted most attention in the related literature. We formulate the problem as a maximum independent set problem in a related {\em conflict graph} and investigate structural properties of this graph in terms of forbidden subgraphs. We are interested, in particular, in excluding certain wheals, fans, cliques or claws (all terms are defined in the paper), which corresponds in excluding certain cycles, paths, cliques or independent sets in the neighborhood of each vertex. Then, we investigate algorithmic consequences of some of these properties, which illustrates the potential of this approach and raises new horizons for further works. In particular this approach allows us to reinterpret a known polynomial case in terms of conflict graph and to improve known approximation and fixed-parameter tractability results through efficiently solving the maximum independent set problem in conflict graphs. Some of our new approximation results involve approximation ratios that are function of the optimal value, in particular its square root; this kind of results cannot be achieved for maximum independent set in general graphs.]]> Wed, 11 Sep 2019 07:18:44 +0000 https://doi.org/10.23638/DMTCS-21-4-10 https://doi.org/10.23638/DMTCS-21-4-10 Alkan, Ferhat Bıyıkoğlu, Türker Demange, Marc Erten, Cesim Alkan, Ferhat Bıyıkoğlu, Türker Demange, Marc Erten, Cesim <![CDATA[We consider the constrained graph alignment problem which has applications in biological network analysis. Given two input graphs $G_1=(V_1,E_1), G_2=(V_2,E_2)$, a pair of vertex mappings induces an {\it edge conservation} if the vertex pairs are adjacent in their respective graphs. %In general terms The goal is to provide a one-to-one mapping between the vertices of the input graphs in order to maximize edge conservation. However the allowed mappings are restricted since each vertex from $V_1$ (resp. $V_2$) is allowed to be mapped to at most $m_1$ (resp. $m_2$) specified vertices in $V_2$ (resp. $V_1$). Most of results in this paper deal with the case $m_2=1$ which attracted most attention in the related literature. We formulate the problem as a maximum independent set problem in a related {\em conflict graph} and investigate structural properties of this graph in terms of forbidden subgraphs. We are interested, in particular, in excluding certain wheals, fans, cliques or claws (all terms are defined in the paper), which corresponds in excluding certain cycles, paths, cliques or independent sets in the neighborhood of each vertex. Then, we investigate algorithmic consequences of some of these properties, which illustrates the potential of this approach and raises new horizons for further works. In particular this approach allows us to reinterpret a known polynomial case in terms of conflict graph and to improve known approximation and fixed-parameter tractability results through efficiently solving the maximum independent set problem in conflict graphs. Some of our new approximation results involve approximation ratios that are function of the optimal value, in particular its square root; this kind of results cannot be achieved for maximum independent set in general graphs.]]> 0 <![CDATA[Ear decompositions of graphs are a standard concept related to several major problems in graph theory like the Traveling Salesman Problem. For example, the Hamiltonian Cycle Problem, which is notoriously N P-complete, is equivalent to deciding whether a given graph admits an ear decomposition in which all ears except one are trivial (i.e. of length 1). On the other hand, a famous result of Lovász states that deciding whether a graph admits an ear decomposition with all ears of odd length can be done in polynomial time. In this paper, we study the complexity of deciding whether a graph admits an ear decomposition with prescribed ear lengths. We prove that deciding whether a graph admits an ear decomposition with all ears of length at most is polynomial-time solvable for all fixed positive integer. On the other hand, deciding whether a graph admits an ear decomposition without ears of length in F is N P-complete for any finite set F of positive integers. We also prove that, for any k ≥ 2, deciding whether a graph admits an ear decomposition with all ears of length 0 mod k is N P-complete. We also consider the directed analogue to ear decomposition, which we call handle decomposition, and prove analogous results : deciding whether a digraph admits a handle decomposition with all handles of length at most is polynomial-time solvable for all positive integer ; deciding whether a digraph admits a handle decomposition without handles of length in F is N P-complete for any finite set F of positive integers (and minimizing the number of handles of length in F is not approximable up to n(1 −)); for any k ≥ 2, deciding whether a digraph admits a handle decomposition with all handles of length 0 mod k is N P-complete. Also, in contrast with the result of Lovász, we prove that deciding whether a digraph admits a handle decomposition with all handles of odd length is N P-complete. Finally, we conjecture that, for every set A of integers, deciding whether a digraph has a handle decomposition with all handles of length in A is N P-complete, unless there exists h ∈ N such that A = {1, · · · , h}.]]> Mon, 02 Sep 2019 09:47:42 +0000 https://doi.org/10.23638/DMTCS-21-4-3 https://doi.org/10.23638/DMTCS-21-4-3 Havet, Frédéric Nisse, Nicolas Havet, Frédéric Nisse, Nicolas <![CDATA[Ear decompositions of graphs are a standard concept related to several major problems in graph theory like the Traveling Salesman Problem. For example, the Hamiltonian Cycle Problem, which is notoriously N P-complete, is equivalent to deciding whether a given graph admits an ear decomposition in which all ears except one are trivial (i.e. of length 1). On the other hand, a famous result of Lovász states that deciding whether a graph admits an ear decomposition with all ears of odd length can be done in polynomial time. In this paper, we study the complexity of deciding whether a graph admits an ear decomposition with prescribed ear lengths. We prove that deciding whether a graph admits an ear decomposition with all ears of length at most is polynomial-time solvable for all fixed positive integer. On the other hand, deciding whether a graph admits an ear decomposition without ears of length in F is N P-complete for any finite set F of positive integers. We also prove that, for any k ≥ 2, deciding whether a graph admits an ear decomposition with all ears of length 0 mod k is N P-complete. We also consider the directed analogue to ear decomposition, which we call handle decomposition, and prove analogous results : deciding whether a digraph admits a handle decomposition with all handles of length at most is polynomial-time solvable for all positive integer ; deciding whether a digraph admits a handle decomposition without handles of length in F is N P-complete for any finite set F of positive integers (and minimizing the number of handles of length in F is not approximable up to n(1 −)); for any k ≥ 2, deciding whether a digraph admits a handle decomposition with all handles of length 0 mod k is N P-complete. Also, in contrast with the result of Lovász, we prove that deciding whether a digraph admits a handle decomposition with all handles of odd length is N P-complete. Finally, we conjecture that, for every set A of integers, deciding whether a digraph has a handle decomposition with all handles of length in A is N P-complete, unless there exists h ∈ N such that A = {1, · · · , h}.]]> 0 <![CDATA[Let H =< V, S > be a hypergraph, where V is a set of vertices and S is a set of not necessarily disjoint clusters Si ⊆ V. The Clustered Spanning Tree problem is to find a spanning tree of G which satisfies that each cluster induces a subtree, when it exists. We provide an efficient and unique algorithm which finds a feasible solution tree for H when it exists, or states that no feasible solution exists. The paper also uses special structures of the intersection graph of H to construct a feasible solution more efficiently. For cases when the hypergraph does not have a feasible solution tree, we consider adding vertices to exactly one cluster in order to gain feasibility. We characterize when such addition can gain feasibility, find the appropriate cluster and a possible set of vertices to be added.]]> Tue, 27 Aug 2019 13:55:09 +0000 https://doi.org/10.23638/DMTCS-21-1-15 https://doi.org/10.23638/DMTCS-21-1-15 Guttmann-Beck, Nili Sorek, Zeev Stern, Michal Guttmann-Beck, Nili Sorek, Zeev Stern, Michal <![CDATA[Let H =< V, S > be a hypergraph, where V is a set of vertices and S is a set of not necessarily disjoint clusters Si ⊆ V. The Clustered Spanning Tree problem is to find a spanning tree of G which satisfies that each cluster induces a subtree, when it exists. We provide an efficient and unique algorithm which finds a feasible solution tree for H when it exists, or states that no feasible solution exists. The paper also uses special structures of the intersection graph of H to construct a feasible solution more efficiently. For cases when the hypergraph does not have a feasible solution tree, we consider adding vertices to exactly one cluster in order to gain feasibility. We characterize when such addition can gain feasibility, find the appropriate cluster and a possible set of vertices to be added.]]> 0 <![CDATA[A centroid node in a tree is a node for which the sum of the distances to all other nodes attains its minimum, or equivalently a node with the property that none of its branches contains more than half of the other nodes. We generalise some known results regarding the behaviour of centroid nodes in random recursive trees (due to Moon) to the class of very simple increasing trees, which also includes the families of plane-oriented and $d$-ary increasing trees. In particular, we derive limits of distributions and moments for the depth and label of the centroid node nearest to the root, as well as for the size of the subtree rooted at this node.]]> Tue, 27 Aug 2019 13:49:59 +0000 https://doi.org/10.23638/DMTCS-21-4-8 https://doi.org/10.23638/DMTCS-21-4-8 Durant, Kevin Wagner, Stephan Durant, Kevin Wagner, Stephan <![CDATA[A centroid node in a tree is a node for which the sum of the distances to all other nodes attains its minimum, or equivalently a node with the property that none of its branches contains more than half of the other nodes. We generalise some known results regarding the behaviour of centroid nodes in random recursive trees (due to Moon) to the class of very simple increasing trees, which also includes the families of plane-oriented and $d$-ary increasing trees. In particular, we derive limits of distributions and moments for the depth and label of the centroid node nearest to the root, as well as for the size of the subtree rooted at this node.]]> 0 <![CDATA[The \emph{matching preclusion number} of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings. As a generalization, Liu and Liu recently introduced the concept of fractional matching preclusion number. The \emph{fractional matching preclusion number} of $G$ is the minimum number of edges whose deletion leaves the resulting graph without a fractional perfect matching. The \emph{fractional strong matching preclusion number} of $G$ is the minimum number of vertices and edges whose deletion leaves the resulting graph without a fractional perfect matching. In this paper, we obtain the fractional matching preclusion number and the fractional strong matching preclusion number for generalized augmented cubes. In addition, all the optimal fractional strong matching preclusion sets of these graphs are categorized.]]> Tue, 13 Aug 2019 06:56:59 +0000 https://doi.org/10.23638/DMTCS-21-4-6 https://doi.org/10.23638/DMTCS-21-4-6 Ma, Tianlong Mao, Yaping Cheng, Eddie Melekian, Christopher Ma, Tianlong Mao, Yaping Cheng, Eddie Melekian, Christopher <![CDATA[The \emph{matching preclusion number} of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings. As a generalization, Liu and Liu recently introduced the concept of fractional matching preclusion number. The \emph{fractional matching preclusion number} of $G$ is the minimum number of edges whose deletion leaves the resulting graph without a fractional perfect matching. The \emph{fractional strong matching preclusion number} of $G$ is the minimum number of vertices and edges whose deletion leaves the resulting graph without a fractional perfect matching. In this paper, we obtain the fractional matching preclusion number and the fractional strong matching preclusion number for generalized augmented cubes. In addition, all the optimal fractional strong matching preclusion sets of these graphs are categorized.]]> 0 <![CDATA[The satisfiability problem is known to be $\mathbf{NP}$-complete in general and for many restricted cases. One way to restrict instances of $k$-SAT is to limit the number of times a variable can be occurred. It was shown that for an instance of 4-SAT with the property that every variable appears in exactly 4 clauses (2 times negated and 2 times not negated), determining whether there is an assignment for variables such that every clause contains exactly two true variables and two false variables is $\mathbf{NP}$-complete. In this work, we show that deciding the satisfiability of 3-SAT with the property that every variable appears in exactly four clauses (two times negated and two times not negated), and each clause contains at least two distinct variables is $ \mathbf{NP} $-complete. We call this problem $(2/2/3)$-SAT. For an $r$-regular graph $G = (V,E)$ with $r\geq 3$, it was asked in [Discrete Appl. Math., 160(15):2142--2146, 2012] to determine whether for a given independent set $T $ there is an independent dominating set $D$ that dominates $T$ such that $ T \cap D =\varnothing $? As an application of $(2/2/3)$-SAT problem we show that for every $r\geq 3$, this problem is $ \mathbf{NP} $-complete. Among other results, we study the relationship between 1-perfect codes and the incidence coloring of graphs and as another application of our complexity results, we prove that for a given cubic graph $G$ deciding whether $G$ is 4-incidence colorable is $ \mathbf{NP} $-complete.]]> Mon, 12 Aug 2019 09:38:12 +0000 https://doi.org/10.23638/DMTCS-21-4-9 https://doi.org/10.23638/DMTCS-21-4-9 Ahadi, Arash Dehghan, Ali Ahadi, Arash Dehghan, Ali <![CDATA[The satisfiability problem is known to be $\mathbf{NP}$-complete in general and for many restricted cases. One way to restrict instances of $k$-SAT is to limit the number of times a variable can be occurred. It was shown that for an instance of 4-SAT with the property that every variable appears in exactly 4 clauses (2 times negated and 2 times not negated), determining whether there is an assignment for variables such that every clause contains exactly two true variables and two false variables is $\mathbf{NP}$-complete. In this work, we show that deciding the satisfiability of 3-SAT with the property that every variable appears in exactly four clauses (two times negated and two times not negated), and each clause contains at least two distinct variables is $ \mathbf{NP} $-complete. We call this problem $(2/2/3)$-SAT. For an $r$-regular graph $G = (V,E)$ with $r\geq 3$, it was asked in [Discrete Appl. Math., 160(15):2142--2146, 2012] to determine whether for a given independent set $T $ there is an independent dominating set $D$ that dominates $T$ such that $ T \cap D =\varnothing $? As an application of $(2/2/3)$-SAT problem we show that for every $r\geq 3$, this problem is $ \mathbf{NP} $-complete. Among other results, we study the relationship between 1-perfect codes and the incidence coloring of graphs and as another application of our complexity results, we prove that for a given cubic graph $G$ deciding whether $G$ is 4-incidence colorable is $ \mathbf{NP} $-complete.]]> 0 <![CDATA[Let $f:V\rightarrow\mathbb{Z}_k$ be a vertex labeling of a hypergraph $H=(V,E)$. This labeling induces an~edge labeling of $H$ defined by $f(e)=\sum_{v\in e}f(v)$, where the sum is taken modulo $k$. We say that $f$ is $k$-cordial if for all $a, b \in \mathbb{Z}_k$ the number of vertices with label $a$ differs by at most $1$ from the number of vertices with label $b$ and the analogous condition holds also for labels of edges. If $H$ admits a $k$-cordial labeling then $H$ is called $k$-cordial. The existence of $k$-cordial labelings has been investigated for graphs for decades. Hovey~(1991) conjectured that every tree $T$ is $k$-cordial for every $k\ge 2$. Cichacz, G\"orlich and Tuza~(2013) were first to investigate the analogous problem for hypertrees, that is, connected hypergraphs without cycles. The main results of their work are that every $k$-uniform hypertree is $k$-cordial for every $k\ge 2$ and that every hypertree with $n$ or $m$ odd is $2$-cordial. Moreover, they conjectured that in fact all hypertrees are $2$-cordial. In this article, we confirm the conjecture of Cichacz et al. and make a step further by proving that for $k\in\{2,3\}$ every hypertree is $k$-cordial.]]> Wed, 07 Aug 2019 07:18:23 +0000 https://doi.org/10.23638/DMTCS-21-4-1 https://doi.org/10.23638/DMTCS-21-4-1 Tuczyński, Michał Wenus, Przemysław Węsek, Krzysztof Tuczyński, Michał Wenus, Przemysław Węsek, Krzysztof <![CDATA[Let $f:V\rightarrow\mathbb{Z}_k$ be a vertex labeling of a hypergraph $H=(V,E)$. This labeling induces an~edge labeling of $H$ defined by $f(e)=\sum_{v\in e}f(v)$, where the sum is taken modulo $k$. We say that $f$ is $k$-cordial if for all $a, b \in \mathbb{Z}_k$ the number of vertices with label $a$ differs by at most $1$ from the number of vertices with label $b$ and the analogous condition holds also for labels of edges. If $H$ admits a $k$-cordial labeling then $H$ is called $k$-cordial. The existence of $k$-cordial labelings has been investigated for graphs for decades. Hovey~(1991) conjectured that every tree $T$ is $k$-cordial for every $k\ge 2$. Cichacz, G\"orlich and Tuza~(2013) were first to investigate the analogous problem for hypertrees, that is, connected hypergraphs without cycles. The main results of their work are that every $k$-uniform hypertree is $k$-cordial for every $k\ge 2$ and that every hypertree with $n$ or $m$ odd is $2$-cordial. Moreover, they conjectured that in fact all hypertrees are $2$-cordial. In this article, we confirm the conjecture of Cichacz et al. and make a step further by proving that for $k\in\{2,3\}$ every hypertree is $k$-cordial.]]> 0 <![CDATA[Edge-connectivity is a classic measure for reliability of a network in the presence of edge failures. $k$-restricted edge-connectivity is one of the refined indicators for fault tolerance of large networks. Matching preclusion and conditional matching preclusion are two important measures for the robustness of networks in edge fault scenario. In this paper, we show that the DCell network $D_{k,n}$ is super-$\lambda$ for $k\geq2$ and $n\geq2$, super-$\lambda_2$ for $k\geq3$ and $n\geq2$, or $k=2$ and $n=2$, and super-$\lambda_3$ for $k\geq4$ and $n\geq3$. Moreover, as an application of $k$-restricted edge-connectivity, we study the matching preclusion number and conditional matching preclusion number, and characterize the corresponding optimal solutions of $D_{k,n}$. In particular, we have shown that $D_{1,n}$ is isomorphic to the $(n,k)$-star graph $S_{n+1,2}$ for $n\geq2$.]]> Tue, 30 Jul 2019 15:50:15 +0000 https://doi.org/10.23638/DMTCS-21-4-2 https://doi.org/10.23638/DMTCS-21-4-2 Lü, Huazhong Wu, Tingzeng Lü, Huazhong Wu, Tingzeng <![CDATA[Edge-connectivity is a classic measure for reliability of a network in the presence of edge failures. $k$-restricted edge-connectivity is one of the refined indicators for fault tolerance of large networks. Matching preclusion and conditional matching preclusion are two important measures for the robustness of networks in edge fault scenario. In this paper, we show that the DCell network $D_{k,n}$ is super-$\lambda$ for $k\geq2$ and $n\geq2$, super-$\lambda_2$ for $k\geq3$ and $n\geq2$, or $k=2$ and $n=2$, and super-$\lambda_3$ for $k\geq4$ and $n\geq3$. Moreover, as an application of $k$-restricted edge-connectivity, we study the matching preclusion number and conditional matching preclusion number, and characterize the corresponding optimal solutions of $D_{k,n}$. In particular, we have shown that $D_{1,n}$ is isomorphic to the $(n,k)$-star graph $S_{n+1,2}$ for $n\geq2$.]]> 0 <![CDATA[The problem of determining the number of "flooding operations" required to make a given coloured graph monochromatic in the one-player combinatorial game Flood-It has been studied extensively from an algorithmic point of view, but basic questions about the maximum number of moves that might be required in the worst case remain unanswered. We begin a systematic investigation of such questions, with the goal of determining, for a given graph, the maximum number of moves that may be required, taken over all possible colourings. We give several upper and lower bounds on this quantity for arbitrary graphs and show that all of the bounds are tight for trees; we also investigate how much the upper bounds can be improved if we restrict our attention to graphs with higher edge-density.]]> Tue, 30 Jul 2019 11:43:03 +0000 https://doi.org/10.23638/DMTCS-21-4-11 https://doi.org/10.23638/DMTCS-21-4-11 Meeks, Kitty Vu, Dominik K. Meeks, Kitty Vu, Dominik K. <![CDATA[The problem of determining the number of "flooding operations" required to make a given coloured graph monochromatic in the one-player combinatorial game Flood-It has been studied extensively from an algorithmic point of view, but basic questions about the maximum number of moves that might be required in the worst case remain unanswered. We begin a systematic investigation of such questions, with the goal of determining, for a given graph, the maximum number of moves that may be required, taken over all possible colourings. We give several upper and lower bounds on this quantity for arbitrary graphs and show that all of the bounds are tight for trees; we also investigate how much the upper bounds can be improved if we restrict our attention to graphs with higher edge-density.]]> 0 <![CDATA[A graph $G$ is almost hypohamiltonian (a.h.) if $G$ is non-hamiltonian, there exists a vertex $w$ in $G$ such that $G - w$ is non-hamiltonian, and $G - v$ is hamiltonian for every vertex $v \ne w$ in $G$. The second author asked in [J. Graph Theory 79 (2015) 63--81] for all orders for which a.h. graphs exist. Here we solve this problem. To this end, we present a specialised algorithm which generates complete sets of a.h. graphs for various orders. Furthermore, we show that the smallest cubic a.h. graphs have order 26. We provide a lower bound for the order of the smallest planar a.h. graph and improve the upper bound for the order of the smallest planar a.h. graph containing a cubic vertex. We also determine the smallest planar a.h. graphs of girth 5, both in the general and cubic case. Finally, we extend a result of Steffen on snarks and improve two bounds on longest paths and longest cycles in polyhedral graphs due to Jooyandeh, McKay, {\"O}sterg{\aa}rd, Pettersson, and the second author.]]> Tue, 30 Jul 2019 11:39:03 +0000 https://doi.org/10.23638/DMTCS-21-4-5 https://doi.org/10.23638/DMTCS-21-4-5 Goedgebeur, Jan Zamfirescu, Carol T. Goedgebeur, Jan Zamfirescu, Carol T. <![CDATA[A graph $G$ is almost hypohamiltonian (a.h.) if $G$ is non-hamiltonian, there exists a vertex $w$ in $G$ such that $G - w$ is non-hamiltonian, and $G - v$ is hamiltonian for every vertex $v \ne w$ in $G$. The second author asked in [J. Graph Theory 79 (2015) 63--81] for all orders for which a.h. graphs exist. Here we solve this problem. To this end, we present a specialised algorithm which generates complete sets of a.h. graphs for various orders. Furthermore, we show that the smallest cubic a.h. graphs have order 26. We provide a lower bound for the order of the smallest planar a.h. graph and improve the upper bound for the order of the smallest planar a.h. graph containing a cubic vertex. We also determine the smallest planar a.h. graphs of girth 5, both in the general and cubic case. Finally, we extend a result of Steffen on snarks and improve two bounds on longest paths and longest cycles in polyhedral graphs due to Jooyandeh, McKay, {\"O}sterg{\aa}rd, Pettersson, and the second author.]]> 0 <![CDATA[In the geodetic convexity, a set of vertices $S$ of a graph $G$ is $\textit{convex}$ if all vertices belonging to any shortest path between two vertices of $S$ lie in $S$. The cardinality $con(G)$ of a maximum proper convex set $S$ of $G$ is the $\textit{convexity number}$ of $G$. The $\textit{complementary prism}$ $G\overline{G}$ of a graph $G$ arises from the disjoint union of the graph $G$ and $\overline{G}$ by adding the edges of a perfect matching between the corresponding vertices of $G$ and $\overline{G}$. In this work, we we prove that the decision problem related to the convexity number is NP-complete even restricted to complementary prisms, we determine $con(G\overline{G})$ when $G$ is disconnected or $G$ is a cograph, and we present a lower bound when $diam(G) \neq 3$.]]> Tue, 30 Jul 2019 11:30:12 +0000 https://doi.org/10.23638/DMTCS-21-4-4 https://doi.org/10.23638/DMTCS-21-4-4 Castonguay, Diane Coelho, Erika M. M. Coelho, Hebert Nascimento, Julliano R. Castonguay, Diane Coelho, Erika M. M. Coelho, Hebert Nascimento, Julliano R. <![CDATA[In the geodetic convexity, a set of vertices $S$ of a graph $G$ is $\textit{convex}$ if all vertices belonging to any shortest path between two vertices of $S$ lie in $S$. The cardinality $con(G)$ of a maximum proper convex set $S$ of $G$ is the $\textit{convexity number}$ of $G$. The $\textit{complementary prism}$ $G\overline{G}$ of a graph $G$ arises from the disjoint union of the graph $G$ and $\overline{G}$ by adding the edges of a perfect matching between the corresponding vertices of $G$ and $\overline{G}$. In this work, we we prove that the decision problem related to the convexity number is NP-complete even restricted to complementary prisms, we determine $con(G\overline{G})$ when $G$ is disconnected or $G$ is a cograph, and we present a lower bound when $diam(G) \neq 3$.]]> 0 <![CDATA[We investigate graph colouring models for the purpose of optimizing TDMA link scheduling in Wireless Networks. Inspired by the BPRN-colouring model recently introduced by Rocha and Sasaki, we introduce a new colouring model, namely the BMRN-colouring model, which can be used to model link scheduling problems where particular types of collisions must be avoided during the node transmissions. In this paper, we initiate the study of the BMRN-colouring model by providing several bounds on the minimum number of colours needed to BMRN-colour digraphs, as well as several complexity results establishing the hardness of finding optimal colourings. We also give a special focus on these considerations for planar digraph topologies, for which we provide refined results.]]> Sat, 13 Jul 2019 12:04:12 +0000 https://doi.org/10.23638/DMTCS-21-3-24 https://doi.org/10.23638/DMTCS-21-3-24 Bensmail, Julien Blanc, Thibaut Cohen, Nathann Havet, Frédéric Rocha, Leonardo Bensmail, Julien Blanc, Thibaut Cohen, Nathann Havet, Frédéric Rocha, Leonardo <![CDATA[We investigate graph colouring models for the purpose of optimizing TDMA link scheduling in Wireless Networks. Inspired by the BPRN-colouring model recently introduced by Rocha and Sasaki, we introduce a new colouring model, namely the BMRN-colouring model, which can be used to model link scheduling problems where particular types of collisions must be avoided during the node transmissions. In this paper, we initiate the study of the BMRN-colouring model by providing several bounds on the minimum number of colours needed to BMRN-colour digraphs, as well as several complexity results establishing the hardness of finding optimal colourings. We also give a special focus on these considerations for planar digraph topologies, for which we provide refined results.]]> 0 <![CDATA[Generalizing Tur\'an's classical extremal problem, Alon and Shikhelman investigated the problem of maximizing the number of $T$ copies in an $H$-free graph, for a pair of graphs $T$ and $H$. Whereas Alon and Shikhelman were primarily interested in determining the order of magnitude for large classes of graphs $H$, we focus on the case when $T$ and $H$ are paths, where we find asymptotic and in some cases exact results. We also consider other structures like stars and the set of cycles of length at least $k$, where we derive asymptotically sharp estimates. Our results generalize well-known extremal theorems of Erd\H{o}s and Gallai.]]> Sat, 13 Jul 2019 08:02:11 +0000 https://doi.org/10.23638/DMTCS-21-1-14 https://doi.org/10.23638/DMTCS-21-1-14 Győri, Ervin Salia, Nika Tompkins, Casey Zamora, Oscar Győri, Ervin Salia, Nika Tompkins, Casey Zamora, Oscar <![CDATA[Generalizing Tur\'an's classical extremal problem, Alon and Shikhelman investigated the problem of maximizing the number of $T$ copies in an $H$-free graph, for a pair of graphs $T$ and $H$. Whereas Alon and Shikhelman were primarily interested in determining the order of magnitude for large classes of graphs $H$, we focus on the case when $T$ and $H$ are paths, where we find asymptotic and in some cases exact results. We also consider other structures like stars and the set of cycles of length at least $k$, where we derive asymptotically sharp estimates. Our results generalize well-known extremal theorems of Erd\H{o}s and Gallai.]]> 0 <![CDATA[The problem of estimating the number n of distinct keys of a large collection of N data is well known in computer science. A classical algorithm is the adaptive sampling (AS). n can be estimated by R2 J , where R is the final bucket size and J is the final depth at the end of the process. Several new interesting questions can be asked about AS (some of them were suggested by P.Flajolet and popularized by J.Lumbroso). The distribution of W = log(R2 J /n) is known, we rederive this distribution in a simpler way. We provide new results on the moments of J and W. We also analyze the final cache size R distribution. We consider colored keys: assume also that among the n distinct keys, m do have color K We show how to estimate p = m n. We study keys with some multiplicity : we provide a way to estimate the total number M of some color K keys among the total number N of keys. We consider the case where we know a priori the multiplicities but not the colors. There we want to estimate the total number of keys N. An appendix is devoted to the case where the hashing function provides bits with probability different from 1/2.]]> Tue, 25 Jun 2019 07:35:08 +0000 https://doi.org/10.23638/DMTCS-21-3-13 https://doi.org/10.23638/DMTCS-21-3-13 Drescher, Matthew Louchard, Guy Swan, Yvik Drescher, Matthew Louchard, Guy Swan, Yvik <![CDATA[The problem of estimating the number n of distinct keys of a large collection of N data is well known in computer science. A classical algorithm is the adaptive sampling (AS). n can be estimated by R2 J , where R is the final bucket size and J is the final depth at the end of the process. Several new interesting questions can be asked about AS (some of them were suggested by P.Flajolet and popularized by J.Lumbroso). The distribution of W = log(R2 J /n) is known, we rederive this distribution in a simpler way. We provide new results on the moments of J and W. We also analyze the final cache size R distribution. We consider colored keys: assume also that among the n distinct keys, m do have color K We show how to estimate p = m n. We study keys with some multiplicity : we provide a way to estimate the total number M of some color K keys among the total number N of keys. We consider the case where we know a priori the multiplicities but not the colors. There we want to estimate the total number of keys N. An appendix is devoted to the case where the hashing function provides bits with probability different from 1/2.]]> 0 <![CDATA[In 2001, Erwin introduced broadcast domination in graphs. It is a variant of classical domination where selected vertices may have different domination powers. The minimum cost of a dominating broadcast in a graph $G$ is denoted $\gamma_b(G)$. The dual of this problem is called multipacking: a multipacking is a set $M$ of vertices such that for any vertex $v$ and any positive integer $r$, the ball of radius $r$ around $v$ contains at most $r$ vertices of $M$ . The maximum size of a multipacking in a graph $G$ is denoted mp(G). Naturally mp(G) $\leq \gamma_b(G)$. Earlier results by Farber and by Lubiw show that broadcast and multipacking numbers are equal for strongly chordal graphs. In this paper, we show that all large grids (height at least 4 and width at least 7), which are far from being chordal, have their broadcast and multipacking numbers equal.]]> Thu, 20 Jun 2019 08:33:55 +0000 https://doi.org/10.23638/DMTCS-21-3-23 https://doi.org/10.23638/DMTCS-21-3-23 Beaudou, Laurent Brewster, Richard C. Beaudou, Laurent Brewster, Richard C. <![CDATA[In 2001, Erwin introduced broadcast domination in graphs. It is a variant of classical domination where selected vertices may have different domination powers. The minimum cost of a dominating broadcast in a graph $G$ is denoted $\gamma_b(G)$. The dual of this problem is called multipacking: a multipacking is a set $M$ of vertices such that for any vertex $v$ and any positive integer $r$, the ball of radius $r$ around $v$ contains at most $r$ vertices of $M$ . The maximum size of a multipacking in a graph $G$ is denoted mp(G). Naturally mp(G) $\leq \gamma_b(G)$. Earlier results by Farber and by Lubiw show that broadcast and multipacking numbers are equal for strongly chordal graphs. In this paper, we show that all large grids (height at least 4 and width at least 7), which are far from being chordal, have their broadcast and multipacking numbers equal.]]> 0 <![CDATA[A positional numeration system is given by a base and by a set of digits. The base is a real or complex number $\beta$ such that $|\beta|>1$, and the digit set $A$ is a finite set of digits including $0$. Thus a number can be seen as a finite or infinite string of digits. An on-line algorithm processes the input piece-by-piece in a serial fashion. On-line arithmetic, introduced by Trivedi and Ercegovac, is a mode of computation where operands and results flow through arithmetic units in a digit serial manner, starting with the most significant digit. In this paper, we first formulate a generalized version of the on-line algorithms for multiplication and division of Trivedi and Ercegovac for the cases that $\beta$ is any real or complex number, and digits are real or complex. We then define the so-called OL Property, and show that if $(\beta, A)$ has the OL Property, then on-line multiplication and division are feasible by the Trivedi-Ercegovac algorithms. For a real base $\beta$ and a digit set $A$ of contiguous integers, the system $(\beta, A)$ has the OL Property if $\# A > |\beta|$. For a complex base $\beta$ and symmetric digit set $A$ of contiguous integers, the system $(\beta, A)$ has the OL Property if $\# A > \beta\overline{\beta} + |\beta + \overline{\beta}|$. Provided that addition and subtraction are realizable in parallel in the system $(\beta, A)$ and that preprocessing of the denominator is possible, our on-line algorithms for multiplication and division have linear time complexity. Three examples are presented in detail: base $\beta=\frac{3+\sqrt{5}}{2}$ with digits $A=\{-1,0,1\}$; base $\beta=2i$ with digits $A = \{-2,-1, 0,1,2\}$; and base $\beta = -\frac{3}{2} + i \frac{\sqrt{3}}{2} = -1 + \omega$, where $\omega = \exp{\frac{2i\pi}{3}}$, with digits $A = \{0, \pm 1, \pm \omega, \pm \omega^2 \}$.]]> Thu, 20 Jun 2019 08:29:32 +0000 https://doi.org/10.23638/DMTCS-21-3-14 https://doi.org/10.23638/DMTCS-21-3-14 Frougny, Christiane Pavelka, Marta Pelantova, Edita Svobodova, Milena Frougny, Christiane Pavelka, Marta Pelantova, Edita Svobodova, Milena <![CDATA[A positional numeration system is given by a base and by a set of digits. The base is a real or complex number $\beta$ such that $|\beta|>1$, and the digit set $A$ is a finite set of digits including $0$. Thus a number can be seen as a finite or infinite string of digits. An on-line algorithm processes the input piece-by-piece in a serial fashion. On-line arithmetic, introduced by Trivedi and Ercegovac, is a mode of computation where operands and results flow through arithmetic units in a digit serial manner, starting with the most significant digit. In this paper, we first formulate a generalized version of the on-line algorithms for multiplication and division of Trivedi and Ercegovac for the cases that $\beta$ is any real or complex number, and digits are real or complex. We then define the so-called OL Property, and show that if $(\beta, A)$ has the OL Property, then on-line multiplication and division are feasible by the Trivedi-Ercegovac algorithms. For a real base $\beta$ and a digit set $A$ of contiguous integers, the system $(\beta, A)$ has the OL Property if $\# A > |\beta|$. For a complex base $\beta$ and symmetric digit set $A$ of contiguous integers, the system $(\beta, A)$ has the OL Property if $\# A > \beta\overline{\beta} + |\beta + \overline{\beta}|$. Provided that addition and subtraction are realizable in parallel in the system $(\beta, A)$ and that preprocessing of the denominator is possible, our on-line algorithms for multiplication and division have linear time complexity. Three examples are presented in detail: base $\beta=\frac{3+\sqrt{5}}{2}$ with digits $A=\{-1,0,1\}$; base $\beta=2i$ with digits $A = \{-2,-1, 0,1,2\}$; and base $\beta = -\frac{3}{2} + i \frac{\sqrt{3}}{2} = -1 + \omega$, where $\omega = \exp{\frac{2i\pi}{3}}$, with digits $A = \{0, \pm 1, \pm \omega, \pm \omega^2 \}$.]]> 0 <![CDATA[We initiate the study of a new problem on searching and fetching in a distributed environment concerning treasure-evacuation from a unit disk. A treasure and an exit are located at unknown positions on the perimeter of a disk and at known arc distance. A team of two robots start from the center of the disk, and their goal is to fetch the treasure to the exit. At any time the robots can move anywhere they choose on the disk, independently of each other, with the same speed. A robot detects an interesting point (treasure or exit) only if it passes over the exact location of that point. We are interested in designing distributed algorithms that minimize the worst-case treasure-evacuation time, i.e. the time it takes for the treasure to be discovered and brought (fetched) to the exit by any of the robots. The communication protocol between the robots is either wireless, where information is shared at any time, or face-to-face (i.e. non-wireless), where information can be shared only if the robots meet. For both models we obtain upper bounds for fetching the treasure to the exit. Our main technical contribution pertains to the face-to-face model. More specifically, we demonstrate how robots can exchange information without meeting, effectively achieving a highly efficient treasure-evacuation protocol which is minimally affected by the lack of distant communication. Finally, we complement our positive results above by providing a lower bound in the face-to-face model.]]> Thu, 13 Jun 2019 13:58:53 +0000 https://doi.org/10.23638/DMTCS-21-3-20 https://doi.org/10.23638/DMTCS-21-3-20 Georgiou, Konstantinos Karakostas, George Kranakis, Evangelos Georgiou, Konstantinos Karakostas, George Kranakis, Evangelos <![CDATA[We initiate the study of a new problem on searching and fetching in a distributed environment concerning treasure-evacuation from a unit disk. A treasure and an exit are located at unknown positions on the perimeter of a disk and at known arc distance. A team of two robots start from the center of the disk, and their goal is to fetch the treasure to the exit. At any time the robots can move anywhere they choose on the disk, independently of each other, with the same speed. A robot detects an interesting point (treasure or exit) only if it passes over the exact location of that point. We are interested in designing distributed algorithms that minimize the worst-case treasure-evacuation time, i.e. the time it takes for the treasure to be discovered and brought (fetched) to the exit by any of the robots. The communication protocol between the robots is either wireless, where information is shared at any time, or face-to-face (i.e. non-wireless), where information can be shared only if the robots meet. For both models we obtain upper bounds for fetching the treasure to the exit. Our main technical contribution pertains to the face-to-face model. More specifically, we demonstrate how robots can exchange information without meeting, effectively achieving a highly efficient treasure-evacuation protocol which is minimally affected by the lack of distant communication. Finally, we complement our positive results above by providing a lower bound in the face-to-face model.]]> 0 <![CDATA[End vertices of graph searches can exhibit strong structural properties and are crucial for many graph algorithms. The problem of deciding whether a given vertex of a graph is an end-vertex of a particular search was first introduced by Corneil, K\"ohler and Lanlignel in 2010. There they showed that this problem is in fact NP-complete for LBFS on weakly chordal graphs. A similar result for BFS was obtained by Charbit, Habib and Mamcarz in 2014. Here, we prove that the end-vertex problem is NP-complete for MNS on weakly chordal graphs and for MCS on general graphs. Moreover, building on previous results, we show that this problem is linear for various searches on split and unit interval graphs.]]> Thu, 13 Jun 2019 13:40:15 +0000 https://doi.org/10.23638/DMTCS-21-1-13 https://doi.org/10.23638/DMTCS-21-1-13 Beisegel, Jesse Denkert, Carolin Köhler, Ekkehard Krnc, Matjaž Pivač, Nevena Scheffler, Robert Strehler, Martin Beisegel, Jesse Denkert, Carolin Köhler, Ekkehard Krnc, Matjaž Pivač, Nevena Scheffler, Robert Strehler, Martin <![CDATA[End vertices of graph searches can exhibit strong structural properties and are crucial for many graph algorithms. The problem of deciding whether a given vertex of a graph is an end-vertex of a particular search was first introduced by Corneil, K\"ohler and Lanlignel in 2010. There they showed that this problem is in fact NP-complete for LBFS on weakly chordal graphs. A similar result for BFS was obtained by Charbit, Habib and Mamcarz in 2014. Here, we prove that the end-vertex problem is NP-complete for MNS on weakly chordal graphs and for MCS on general graphs. Moreover, building on previous results, we show that this problem is linear for various searches on split and unit interval graphs.]]> 0 <![CDATA[A path (cycle) in a $2$-edge-colored multigraph is alternating if no two consecutive edges have the same color. The problem of determining the existence of alternating Hamiltonian paths and cycles in $2$-edge-colored multigraphs is an $\mathcal{NP}$-complete problem and it has been studied by several authors. In Bang-Jensen and Gutin's book "Digraphs: Theory, Algorithms and Applications", it is devoted one chapter to survey the last results on this topic. Most results on the existence of alternating Hamiltonian paths and cycles concern on complete and bipartite complete multigraphs and a few ones on multigraphs with high monochromatic degrees or regular monochromatic subgraphs. In this work, we use a different approach imposing local conditions on the multigraphs and it is worthwhile to notice that the class of multigraphs we deal with is much larger than, and includes, complete multigraphs, and we provide a full characterization of this class. Given a $2$-edge-colored multigraph $G$, we say that $G$ is $2$-$\mathcal{M}$-closed (resp. $2$-$\mathcal{NM}$-closed)} if for every monochromatic (resp. non-monochromatic) $2$-path $P=(x_1, x_2, x_3)$, there exists an edge between $x_1$ and $x_3$. In this work we provide the following characterization: A $2$-$\mathcal{M}$-closed multigraph has an alternating Hamiltonian cycle if and only if it is color-connected and it has an alternating cycle factor. Furthermore, we construct an infinite family of $2$-$\mathcal{NM}$-closed graphs, color-connected, with an alternating cycle factor, and with no alternating Hamiltonian cycle.]]> Thu, 13 Jun 2019 13:33:15 +0000 https://doi.org/10.23638/DMTCS-21-1-12 https://doi.org/10.23638/DMTCS-21-1-12 Contreras-Balbuena, Alejandro Galeana-Sánchez, Hortensia Goldfeder, Ilan A. Contreras-Balbuena, Alejandro Galeana-Sánchez, Hortensia Goldfeder, Ilan A. <![CDATA[A path (cycle) in a $2$-edge-colored multigraph is alternating if no two consecutive edges have the same color. The problem of determining the existence of alternating Hamiltonian paths and cycles in $2$-edge-colored multigraphs is an $\mathcal{NP}$-complete problem and it has been studied by several authors. In Bang-Jensen and Gutin's book "Digraphs: Theory, Algorithms and Applications", it is devoted one chapter to survey the last results on this topic. Most results on the existence of alternating Hamiltonian paths and cycles concern on complete and bipartite complete multigraphs and a few ones on multigraphs with high monochromatic degrees or regular monochromatic subgraphs. In this work, we use a different approach imposing local conditions on the multigraphs and it is worthwhile to notice that the class of multigraphs we deal with is much larger than, and includes, complete multigraphs, and we provide a full characterization of this class. Given a $2$-edge-colored multigraph $G$, we say that $G$ is $2$-$\mathcal{M}$-closed (resp. $2$-$\mathcal{NM}$-closed)} if for every monochromatic (resp. non-monochromatic) $2$-path $P=(x_1, x_2, x_3)$, there exists an edge between $x_1$ and $x_3$. In this work we provide the following characterization: A $2$-$\mathcal{M}$-closed multigraph has an alternating Hamiltonian cycle if and only if it is color-connected and it has an alternating cycle factor. Furthermore, we construct an infinite family of $2$-$\mathcal{NM}$-closed graphs, color-connected, with an alternating cycle factor, and with no alternating Hamiltonian cycle.]]> 0 <![CDATA[Stable gonality is a multigraph parameter that measures the complexity of a graph. It is defined using maps to trees. Those maps, in some sense, divide the edges equally over the edges of the tree; stable gonality asks for the map with the minimum number of edges mapped to each edge of the tree. This parameter is related to treewidth, but unlike treewidth, it distinguishes multigraphs from their underlying simple graphs. Stable gonality is relevant for problems in number theory. In this paper, we show that deciding whether the stable gonality of a given graph is at most a given integer $k$ belongs to the class NP, and we give an algorithm that computes the stable gonality of a graph in $O((1.33n)^nm^m \text{poly}(n,m))$ time.]]> Thu, 13 Jun 2019 13:28:19 +0000 https://doi.org/10.23638/DMTCS-21-1-10 https://doi.org/10.23638/DMTCS-21-1-10 Koerkamp, Ragnar Groot van der Wegen, Marieke Koerkamp, Ragnar Groot van der Wegen, Marieke <![CDATA[Stable gonality is a multigraph parameter that measures the complexity of a graph. It is defined using maps to trees. Those maps, in some sense, divide the edges equally over the edges of the tree; stable gonality asks for the map with the minimum number of edges mapped to each edge of the tree. This parameter is related to treewidth, but unlike treewidth, it distinguishes multigraphs from their underlying simple graphs. Stable gonality is relevant for problems in number theory. In this paper, we show that deciding whether the stable gonality of a given graph is at most a given integer $k$ belongs to the class NP, and we give an algorithm that computes the stable gonality of a graph in $O((1.33n)^nm^m \text{poly}(n,m))$ time.]]> 0 <![CDATA[In this paper we address the problem of generating all elements obtained by the saturation of an initial set by some operations. More precisely, we prove that we can generate the closure of a boolean relation (a set of boolean vectors) by polymorphisms with a polynomial delay. Therefore we can compute with polynomial delay the closure of a family of sets by any set of "set operations": union, intersection, symmetric difference, subsets, supersets $\dots$). To do so, we study the $Membership_{\mathcal{F}}$ problem: for a set of operations $\mathcal{F}$, decide whether an element belongs to the closure by $\mathcal{F}$ of a family of elements. In the boolean case, we prove that $Membership_{\mathcal{F}}$ is in P for any set of boolean operations $\mathcal{F}$. When the input vectors are over a domain larger than two elements, we prove that the generic enumeration method fails, since $Membership_{\mathcal{F}}$ is NP-hard for some $\mathcal{F}$. We also study the problem of generating minimal or maximal elements of closures and prove that some of them are related to well known enumeration problems such as the enumeration of the circuits of a matroid or the enumeration of maximal independent sets of a hypergraph. This article improves on previous works of the same authors.]]> Thu, 13 Jun 2019 13:23:30 +0000 https://doi.org/10.23638/DMTCS-21-3-22 https://doi.org/10.23638/DMTCS-21-3-22 Mary, Arnaud Strozecki, Yann Mary, Arnaud Strozecki, Yann <![CDATA[In this paper we address the problem of generating all elements obtained by the saturation of an initial set by some operations. More precisely, we prove that we can generate the closure of a boolean relation (a set of boolean vectors) by polymorphisms with a polynomial delay. Therefore we can compute with polynomial delay the closure of a family of sets by any set of "set operations": union, intersection, symmetric difference, subsets, supersets $\dots$). To do so, we study the $Membership_{\mathcal{F}}$ problem: for a set of operations $\mathcal{F}$, decide whether an element belongs to the closure by $\mathcal{F}$ of a family of elements. In the boolean case, we prove that $Membership_{\mathcal{F}}$ is in P for any set of boolean operations $\mathcal{F}$. When the input vectors are over a domain larger than two elements, we prove that the generic enumeration method fails, since $Membership_{\mathcal{F}}$ is NP-hard for some $\mathcal{F}$. We also study the problem of generating minimal or maximal elements of closures and prove that some of them are related to well known enumeration problems such as the enumeration of the circuits of a matroid or the enumeration of maximal independent sets of a hypergraph. This article improves on previous works of the same authors.]]> 0 <![CDATA[It is well-known that the set $\mathbf I_n$ of involutions of the symmetric group $\mathbf S_n$ corresponds bijectively - by the Foata map $F$ - to the set of $n$-permutations that avoid the two vincular patterns $\underline{123},$ $\underline{132}.$ We consider a bijection $\Gamma$ from the set $\mathbf S_n$ to the set of histoires de Laguerre, namely, bicolored Motzkin paths with labelled steps, and study its properties when restricted to $\mathbf S_n(1\underline{23},1\underline{32}).$ In particular, we show that the set $\mathbf S_n(\underline{123},{132})$ of permutations that avoids the consecutive pattern $\underline{123}$ and the classical pattern $132$ corresponds via $\Gamma$ to the set of Motzkin paths, while its image under $F$ is the set of restricted involutions $\mathbf I_n(3412).$ We exploit these results to determine the joint distribution of the statistics des and inv over $\mathbf S_n(\underline{123},{132})$ and over $\mathbf I_n(3412).$ Moreover, we determine the distribution in these two sets of every consecutive pattern of length three. To this aim, we use a modified version of the well-known Goulden-Jacson cluster method.]]> Wed, 05 Jun 2019 09:23:05 +0000 https://doi.org/10.23638/DMTCS-21-3-21 https://doi.org/10.23638/DMTCS-21-3-21 Barnabei, M. Bonetti, F. Castronuovo, N. Silimbani, M. Barnabei, M. Bonetti, F. Castronuovo, N. Silimbani, M. <![CDATA[It is well-known that the set $\mathbf I_n$ of involutions of the symmetric group $\mathbf S_n$ corresponds bijectively - by the Foata map $F$ - to the set of $n$-permutations that avoid the two vincular patterns $\underline{123},$ $\underline{132}.$ We consider a bijection $\Gamma$ from the set $\mathbf S_n$ to the set of histoires de Laguerre, namely, bicolored Motzkin paths with labelled steps, and study its properties when restricted to $\mathbf S_n(1\underline{23},1\underline{32}).$ In particular, we show that the set $\mathbf S_n(\underline{123},{132})$ of permutations that avoids the consecutive pattern $\underline{123}$ and the classical pattern $132$ corresponds via $\Gamma$ to the set of Motzkin paths, while its image under $F$ is the set of restricted involutions $\mathbf I_n(3412).$ We exploit these results to determine the joint distribution of the statistics des and inv over $\mathbf S_n(\underline{123},{132})$ and over $\mathbf I_n(3412).$ Moreover, we determine the distribution in these two sets of every consecutive pattern of length three. To this aim, we use a modified version of the well-known Goulden-Jacson cluster method.]]> 0 <![CDATA[In the Planar 3-SAT problem, we are given a 3-SAT formula together with its incidence graph, which is planar, and are asked whether this formula is satisfiable. Since Lichtenstein's proof that this problem is NP-complete, it has been used as a starting point for a large number of reductions. In the course of this research, different restrictions on the incidence graph of the formula have been devised, for which the problem also remains hard. In this paper, we investigate the restriction in which we require that the incidence graph can be augmented by the edges of a Hamiltonian cycle that first passes through all variables and then through all clauses, in a way that the resulting graph is still planar. We show that the problem of deciding satisfiability of a 3-SAT formula remains NP-complete even if the incidence graph is restricted in that way and the Hamiltonian cycle is given. This complements previous results demanding cycles only through either the variables or clauses. The problem remains hard for monotone formulas, as well as for instances with exactly three distinct variables per clause. In the course of this investigation, we show that monotone instances of Planar 3-SAT with exactly three distinct variables per clause are always satisfiable, thus settling the question by Darmann, D\"ocker, and Dorn on the complexity of this problem variant in a surprising way.]]> Wed, 05 Jun 2019 09:16:09 +0000 https://doi.org/10.23638/DMTCS-21-3-18 https://doi.org/10.23638/DMTCS-21-3-18 Pilz, Alexander Pilz, Alexander <![CDATA[In the Planar 3-SAT problem, we are given a 3-SAT formula together with its incidence graph, which is planar, and are asked whether this formula is satisfiable. Since Lichtenstein's proof that this problem is NP-complete, it has been used as a starting point for a large number of reductions. In the course of this research, different restrictions on the incidence graph of the formula have been devised, for which the problem also remains hard. In this paper, we investigate the restriction in which we require that the incidence graph can be augmented by the edges of a Hamiltonian cycle that first passes through all variables and then through all clauses, in a way that the resulting graph is still planar. We show that the problem of deciding satisfiability of a 3-SAT formula remains NP-complete even if the incidence graph is restricted in that way and the Hamiltonian cycle is given. This complements previous results demanding cycles only through either the variables or clauses. The problem remains hard for monotone formulas, as well as for instances with exactly three distinct variables per clause. In the course of this investigation, we show that monotone instances of Planar 3-SAT with exactly three distinct variables per clause are always satisfiable, thus settling the question by Darmann, D\"ocker, and Dorn on the complexity of this problem variant in a surprising way.]]> 0 <![CDATA[In this paper, we give a lengthy proof of a small result! A graph is bisplit if its vertex set can be partitioned into three stable sets with two of them inducing a complete bipartite graph. We prove that these graphs satisfy the Chen-Chv\'atal conjecture: their metric space (in the usual sense) has a universal line (in an unusual sense) or at least as many lines as the number of vertices.]]> Wed, 29 May 2019 07:26:16 +0000 https://doi.org/10.23638/DMTCS-21-1-5 https://doi.org/10.23638/DMTCS-21-1-5 Beaudou, Laurent Kahn, Giacomo Rosenfeld, Matthieu Beaudou, Laurent Kahn, Giacomo Rosenfeld, Matthieu <![CDATA[In this paper, we give a lengthy proof of a small result! A graph is bisplit if its vertex set can be partitioned into three stable sets with two of them inducing a complete bipartite graph. We prove that these graphs satisfy the Chen-Chv\'atal conjecture: their metric space (in the usual sense) has a universal line (in an unusual sense) or at least as many lines as the number of vertices.]]> 0 <![CDATA[The chromatic number of a graph $G$, denoted by $\chi(G)$, is the minimum $k$ such that $G$ admits a $k$-coloring of its vertex set in such a way that each color class is an independent set (a set of pairwise non-adjacent vertices). The dichromatic number of a digraph $D$, denoted by $\chi_A(D)$, is the minimum $k$ such that $D$ admits a $k$-coloring of its vertex set in such a way that each color class is acyclic. In 1976, Bondy proved that the chromatic number of a digraph $D$ is at most its circumference, the length of a longest cycle. Given a digraph $D$, we will construct three different graphs whose chromatic numbers bound $\chi_A(D)$. Moreover, we prove: i) for integers $k\geq 2$, $s\geq 1$ and $r_1, \ldots, r_s$ with $k\geq r_i\geq 0$ and $r_i\neq 1$ for each $i\in[s]$, that if all cycles in $D$ have length $r$ modulo $k$ for some $r\in\{r_1,\ldots,r_s\}$, then $\chi_A(D)\leq 2s+1$; ii) if $D$ has girth $g$ and there are integers $k$ and $p$, with $k\geq g-1\geq p\geq 1$ such that $D$ contains no cycle of length $r$ modulo $\lceil \frac{k}{p} \rceil p$ for each $r\in \{-p+2,\ldots,0,\ldots,p\}$, then $\chi_A (D)\leq \lceil \frac{k}{p} \rceil$; iii) if $D$ has girth $g$, the length of a shortest cycle, and circumference $c$, then $\chi_A(D)\leq \lceil \frac{c-1}{g-1} \rceil +1$, which improves, substantially, the bound proposed by Bondy. Our results show that if we have more information about the lengths of cycles in a digraph, then we can improve the bounds for the dichromatic number known until now.]]> Thu, 23 May 2019 15:34:39 +0000 https://doi.org/10.23638/DMTCS-21-1-7 https://doi.org/10.23638/DMTCS-21-1-7 Cordero-Michel, Narda Galeana-Sánchez, Hortensia Cordero-Michel, Narda Galeana-Sánchez, Hortensia <![CDATA[The chromatic number of a graph $G$, denoted by $\chi(G)$, is the minimum $k$ such that $G$ admits a $k$-coloring of its vertex set in such a way that each color class is an independent set (a set of pairwise non-adjacent vertices). The dichromatic number of a digraph $D$, denoted by $\chi_A(D)$, is the minimum $k$ such that $D$ admits a $k$-coloring of its vertex set in such a way that each color class is acyclic. In 1976, Bondy proved that the chromatic number of a digraph $D$ is at most its circumference, the length of a longest cycle. Given a digraph $D$, we will construct three different graphs whose chromatic numbers bound $\chi_A(D)$. Moreover, we prove: i) for integers $k\geq 2$, $s\geq 1$ and $r_1, \ldots, r_s$ with $k\geq r_i\geq 0$ and $r_i\neq 1$ for each $i\in[s]$, that if all cycles in $D$ have length $r$ modulo $k$ for some $r\in\{r_1,\ldots,r_s\}$, then $\chi_A(D)\leq 2s+1$; ii) if $D$ has girth $g$ and there are integers $k$ and $p$, with $k\geq g-1\geq p\geq 1$ such that $D$ contains no cycle of length $r$ modulo $\lceil \frac{k}{p} \rceil p$ for each $r\in \{-p+2,\ldots,0,\ldots,p\}$, then $\chi_A (D)\leq \lceil \frac{k}{p} \rceil$; iii) if $D$ has girth $g$, the length of a shortest cycle, and circumference $c$, then $\chi_A(D)\leq \lceil \frac{c-1}{g-1} \rceil +1$, which improves, substantially, the bound proposed by Bondy. Our results show that if we have more information about the lengths of cycles in a digraph, then we can improve the bounds for the dichromatic number known until now.]]> 0 <![CDATA[We prove that, given a closure function the smallest preimage of a closed set can be calculated in polynomial time in the number of closed sets. This implies that there is a polynomial time algorithm to compute the convex hull number of a graph, when all its convex subgraphs are given as input. We then show that deciding if the smallest preimage of a closed set is logarithmic in the size of the ground set is LOGSNP-hard if only the ground set is given. A special instance of this problem is to compute the dimension of a poset given its linear extension graph, that is conjectured to be in P.The intent to show that the latter problem is LOGSNP-complete leads to several interesting questions and to the definition of the isometric hull, i.e., a smallest isometric subgraph containing a given set of vertices $S$. While for $|S|=2$ an isometric hull is just a shortest path, we show that computing the isometric hull of a set of vertices is NP-complete even if $|S|=3$. Finally, we consider the problem of computing the isometric hull number of a graph and show that computing it is $\Sigma^P_2$ complete.]]> Thu, 23 May 2019 13:14:19 +0000 https://doi.org/10.23638/DMTCS-21-1-11 https://doi.org/10.23638/DMTCS-21-1-11 Knauer, Kolja Nisse, Nicolas Knauer, Kolja Nisse, Nicolas <![CDATA[We prove that, given a closure function the smallest preimage of a closed set can be calculated in polynomial time in the number of closed sets. This implies that there is a polynomial time algorithm to compute the convex hull number of a graph, when all its convex subgraphs are given as input. We then show that deciding if the smallest preimage of a closed set is logarithmic in the size of the ground set is LOGSNP-hard if only the ground set is given. A special instance of this problem is to compute the dimension of a poset given its linear extension graph, that is conjectured to be in P.The intent to show that the latter problem is LOGSNP-complete leads to several interesting questions and to the definition of the isometric hull, i.e., a smallest isometric subgraph containing a given set of vertices $S$. While for $|S|=2$ an isometric hull is just a shortest path, we show that computing the isometric hull of a set of vertices is NP-complete even if $|S|=3$. Finally, we consider the problem of computing the isometric hull number of a graph and show that computing it is $\Sigma^P_2$ complete.]]> 0 <![CDATA[Consider a vertex colouring game played on a simple graph with $k$ permissible colours. Two players, a maker and a breaker, take turns to colour an uncoloured vertex such that adjacent vertices receive different colours. The game ends once the graph is fully coloured, in which case the maker wins, or the graph can no longer be fully coloured, in which case the breaker wins. In the game $g_B$, the breaker makes the first move. Our main focus is on the class of $g_B$-perfect graphs: graphs such that for every induced subgraph $H$, the game $g_B$ played on $H$ admits a winning strategy for the maker with only $\omega(H)$ colours, where $\omega(H)$ denotes the clique number of $H$. Complementing analogous results for other variations of the game, we characterise $g_B$-perfect graphs in two ways, by forbidden induced subgraphs and by explicit structural descriptions. We also present a clique module decomposition, which may be of independent interest, that allows us to efficiently recognise $g_B$-perfect graphs.]]> Thu, 23 May 2019 09:54:49 +0000 https://doi.org/10.23638/DMTCS-21-1-6 https://doi.org/10.23638/DMTCS-21-1-6 Andres, Dominique Lock, Edwin Andres, Dominique Lock, Edwin <![CDATA[Consider a vertex colouring game played on a simple graph with $k$ permissible colours. Two players, a maker and a breaker, take turns to colour an uncoloured vertex such that adjacent vertices receive different colours. The game ends once the graph is fully coloured, in which case the maker wins, or the graph can no longer be fully coloured, in which case the breaker wins. In the game $g_B$, the breaker makes the first move. Our main focus is on the class of $g_B$-perfect graphs: graphs such that for every induced subgraph $H$, the game $g_B$ played on $H$ admits a winning strategy for the maker with only $\omega(H)$ colours, where $\omega(H)$ denotes the clique number of $H$. Complementing analogous results for other variations of the game, we characterise $g_B$-perfect graphs in two ways, by forbidden induced subgraphs and by explicit structural descriptions. We also present a clique module decomposition, which may be of independent interest, that allows us to efficiently recognise $g_B$-perfect graphs.]]> 0 <![CDATA[The minimal number of rooted subtree prune and regraft (rSPR) operations needed to transform one phylogenetic tree into another one induces a metric on phylogenetic trees - the rSPR-distance. The rSPR-distance between two phylogenetic trees $T$ and $T'$ can be characterised by a maximum agreement forest; a forest with a minimum number of components that covers both $T$ and $T'$. The rSPR operation has recently been generalised to phylogenetic networks with, among others, the subnetwork prune and regraft (SNPR) operation. Here, we introduce maximum agreement graphs as an explicit representations of differences of two phylogenetic networks, thus generalising maximum agreement forests. We show that maximum agreement graphs induce a metric on phylogenetic networks - the agreement distance. While this metric does not characterise the distances induced by SNPR and other generalisations of rSPR, we prove that it still bounds these distances with constant factors.]]> Thu, 23 May 2019 09:16:07 +0000 https://doi.org/10.23638/DMTCS-21-3-19 https://doi.org/10.23638/DMTCS-21-3-19 Klawitter, Jonathan Klawitter, Jonathan <![CDATA[The minimal number of rooted subtree prune and regraft (rSPR) operations needed to transform one phylogenetic tree into another one induces a metric on phylogenetic trees - the rSPR-distance. The rSPR-distance between two phylogenetic trees $T$ and $T'$ can be characterised by a maximum agreement forest; a forest with a minimum number of components that covers both $T$ and $T'$. The rSPR operation has recently been generalised to phylogenetic networks with, among others, the subnetwork prune and regraft (SNPR) operation. Here, we introduce maximum agreement graphs as an explicit representations of differences of two phylogenetic networks, thus generalising maximum agreement forests. We show that maximum agreement graphs induce a metric on phylogenetic networks - the agreement distance. While this metric does not characterise the distances induced by SNPR and other generalisations of rSPR, we prove that it still bounds these distances with constant factors.]]> 0 <![CDATA[A geographic network is a graph whose vertices are restricted to lie in a prescribed region in the plane. In this paper we begin to study the following fundamental problem for geographic networks: can a given geographic network be drawn without crossings? We focus on the seemingly simple setting where each region is a vertical segment, and one wants to connect pairs of segments with a path that lies inside the convex hull of the two segments. We prove that when paths must be drawn as straight line segments, it is NP-complete to determine if a crossing-free solution exists, even if all vertical segments have unit length. In contrast, we show that when paths must be monotone curves, the question can be answered in polynomial time. In the more general case of paths that can have any shape, we show that the problem is polynomial under certain assumptions.]]> Thu, 23 May 2019 09:12:53 +0000 https://doi.org/10.23638/DMTCS-21-3-15 https://doi.org/10.23638/DMTCS-21-3-15 Silveira, Rodrigo I. Speckmann, Bettina Verbeek, Kevin Silveira, Rodrigo I. Speckmann, Bettina Verbeek, Kevin <![CDATA[A geographic network is a graph whose vertices are restricted to lie in a prescribed region in the plane. In this paper we begin to study the following fundamental problem for geographic networks: can a given geographic network be drawn without crossings? We focus on the seemingly simple setting where each region is a vertical segment, and one wants to connect pairs of segments with a path that lies inside the convex hull of the two segments. We prove that when paths must be drawn as straight line segments, it is NP-complete to determine if a crossing-free solution exists, even if all vertical segments have unit length. In contrast, we show that when paths must be monotone curves, the question can be answered in polynomial time. In the more general case of paths that can have any shape, we show that the problem is polynomial under certain assumptions.]]> 0 <![CDATA[We investigate the 2-domination number for grid graphs, that is the size of a smallest set $D$ of vertices of the grid such that each vertex of the grid belongs to $D$ or has at least two neighbours in $D$. We give a closed formula giving the 2-domination number of any $n \!\times\! m$ grid, hereby confirming the results found by Lu and Xu, and Shaheen et al. for $n \leq 4$ and slightly correct the value of Shaheen et al. for $n = 5$. The proof relies on some dynamic programming algorithms, using transfer matrices in (min,+)-algebra. We also apply the method to solve the Roman domination problem on grid graphs.]]> Thu, 23 May 2019 09:08:17 +0000 https://doi.org/10.23638/DMTCS-21-1-9 https://doi.org/10.23638/DMTCS-21-1-9 Rao, Michaël Talon, Alexandre Rao, Michaël Talon, Alexandre <![CDATA[We investigate the 2-domination number for grid graphs, that is the size of a smallest set $D$ of vertices of the grid such that each vertex of the grid belongs to $D$ or has at least two neighbours in $D$. We give a closed formula giving the 2-domination number of any $n \!\times\! m$ grid, hereby confirming the results found by Lu and Xu, and Shaheen et al. for $n \leq 4$ and slightly correct the value of Shaheen et al. for $n = 5$. The proof relies on some dynamic programming algorithms, using transfer matrices in (min,+)-algebra. We also apply the method to solve the Roman domination problem on grid graphs.]]> 0 <![CDATA[A graph on $n$ vertices is equitably $k$-colorable if it is $k$-colorable and every color is used either $\left\lfloor n/k \right\rfloor$ or $\left\lceil n/k \right\rceil$ times. Such a problem appears to be considerably harder than vertex coloring, being $\mathsf{NP\text{-}Complete}$ even for cographs and interval graphs. In this work, we prove that it is $\mathsf{W[1]\text{-}Hard}$ for block graphs and for disjoint union of split graphs when parameterized by the number of colors; and $\mathsf{W[1]\text{-}Hard}$ for $K_{1,4}$-free interval graphs when parameterized by treewidth, number of colors and maximum degree, generalizing a result by Fellows et al. (2014) through a much simpler reduction. Using a previous result due to Dominique de Werra (1985), we establish a dichotomy for the complexity of equitable coloring of chordal graphs based on the size of the largest induced star. Finally, we show that \textsc{equitable coloring} is $\mathsf{FPT}$ when parameterized by the treewidth of the complement graph.]]> Thu, 16 May 2019 11:24:04 +0000 https://doi.org/10.23638/DMTCS-21-1-8 https://doi.org/10.23638/DMTCS-21-1-8 Gomes, Guilherme de C. M. Lima, Carlos V. G. C. Santos, Vinícius F. dos Gomes, Guilherme de C. M. Lima, Carlos V. G. C. Santos, Vinícius F. dos <![CDATA[A graph on $n$ vertices is equitably $k$-colorable if it is $k$-colorable and every color is used either $\left\lfloor n/k \right\rfloor$ or $\left\lceil n/k \right\rceil$ times. Such a problem appears to be considerably harder than vertex coloring, being $\mathsf{NP\text{-}Complete}$ even for cographs and interval graphs. In this work, we prove that it is $\mathsf{W[1]\text{-}Hard}$ for block graphs and for disjoint union of split graphs when parameterized by the number of colors; and $\mathsf{W[1]\text{-}Hard}$ for $K_{1,4}$-free interval graphs when parameterized by treewidth, number of colors and maximum degree, generalizing a result by Fellows et al. (2014) through a much simpler reduction. Using a previous result due to Dominique de Werra (1985), we establish a dichotomy for the complexity of equitable coloring of chordal graphs based on the size of the largest induced star. Finally, we show that \textsc{equitable coloring} is $\mathsf{FPT}$ when parameterized by the treewidth of the complement graph.]]> 0 <![CDATA[A new recursive function on discrete interval exchange transformation associated to a composition of length $r$, and the permutation $\sigma(i) = r -i +1$ is defined. Acting on composition $c$, this recursive function counts the number of orbits of the discrete interval exchange transformation associated to the composition $c$. Moreover, minimal discrete interval exchanges transformation i.e. the ones having only one orbit, are reduced to the composition which label the root of the Raney tree. Therefore, we describe a generalization of the Raney tree using our recursive function.]]> Thu, 16 May 2019 11:19:54 +0000 https://doi.org/10.23638/DMTCS-21-3-17 https://doi.org/10.23638/DMTCS-21-3-17 Lapointe, Mélodie Lapointe, Mélodie <![CDATA[A new recursive function on discrete interval exchange transformation associated to a composition of length $r$, and the permutation $\sigma(i) = r -i +1$ is defined. Acting on composition $c$, this recursive function counts the number of orbits of the discrete interval exchange transformation associated to the composition $c$. Moreover, minimal discrete interval exchanges transformation i.e. the ones having only one orbit, are reduced to the composition which label the root of the Raney tree. Therefore, we describe a generalization of the Raney tree using our recursive function.]]> 0 <![CDATA[In this paper we study three domination-like problems, namely identifying codes, locating-dominating codes, and locating-total-dominating codes. We are interested in finding the minimum cardinality of such codes in circular and infinite grid graphs of given height. We provide an alternate proof for already known results, as well as new results. These were obtained by a computer search based on a generic framework, that we developed earlier, for the search of a minimum labeling satisfying a pseudo-d-local property in rotagraphs.]]> Thu, 16 May 2019 11:15:52 +0000 https://doi.org/10.23638/DMTCS-21-3-12 https://doi.org/10.23638/DMTCS-21-3-12 Bouznif, Marwane Darlay, Julien Moncel, Julien Preissmann, Myriam Bouznif, Marwane Darlay, Julien Moncel, Julien Preissmann, Myriam <![CDATA[In this paper we study three domination-like problems, namely identifying codes, locating-dominating codes, and locating-total-dominating codes. We are interested in finding the minimum cardinality of such codes in circular and infinite grid graphs of given height. We provide an alternate proof for already known results, as well as new results. These were obtained by a computer search based on a generic framework, that we developed earlier, for the search of a minimum labeling satisfying a pseudo-d-local property in rotagraphs.]]> 0 <![CDATA[Locating-dominating codes in a graph find their application in sensor networks and have been studied extensively over the years. A locating-dominating code can locate one object in a sensor network, but if there is more than one object, it may lead to false conclusions. In this paper, we consider stronger types of locating-dominating codes which can locate one object and detect if there are multiple objects. We study the properties of these codes and provide bounds on the smallest possible size of these codes, for example, with the aid of the Dilworth number and Sperner families. Moreover, these codes are studied in trees and Cartesian products of graphs. We also give the complete realization theorems for the coexistence of the smallest possible size of these codes and the optimal locating-dominating codes in a graph.]]> Sat, 11 May 2019 10:10:31 +0000 https://doi.org/10.23638/DMTCS-21-1-1 https://doi.org/10.23638/DMTCS-21-1-1 Junnila, Ville Laihonen, Tero Lehtilä, Tuomo Puertas, María Luz Junnila, Ville Laihonen, Tero Lehtilä, Tuomo Puertas, María Luz <![CDATA[Locating-dominating codes in a graph find their application in sensor networks and have been studied extensively over the years. A locating-dominating code can locate one object in a sensor network, but if there is more than one object, it may lead to false conclusions. In this paper, we consider stronger types of locating-dominating codes which can locate one object and detect if there are multiple objects. We study the properties of these codes and provide bounds on the smallest possible size of these codes, for example, with the aid of the Dilworth number and Sperner families. Moreover, these codes are studied in trees and Cartesian products of graphs. We also give the complete realization theorems for the coexistence of the smallest possible size of these codes and the optimal locating-dominating codes in a graph.]]> 0 <![CDATA[Given a proper edge coloring $\varphi$ of a graph $G$, we define the palette $S_{G}(v,\varphi)$ of a vertex $v \in V(G)$ as the set of all colors appearing on edges incident with $v$. The palette index $\check s(G)$ of $G$ is the minimum number of distinct palettes occurring in a proper edge coloring of $G$. In this paper we give various upper and lower bounds on the palette index of $G$ in terms of the vertex degrees of $G$, particularly for the case when $G$ is a bipartite graph with small vertex degrees. Some of our results concern $(a,b)$-biregular graphs; that is, bipartite graphs where all vertices in one part have degree $a$ and all vertices in the other part have degree $b$. We conjecture that if $G$ is $(a,b)$-biregular, then $\check{s}(G)\leq 1+\max\{a,b\}$, and we prove that this conjecture holds for several families of $(a,b)$-biregular graphs. Additionally, we characterize the graphs whose palette index equals the number of vertices.]]> Sat, 11 May 2019 10:03:15 +0000 https://doi.org/10.23638/DMTCS-21-3-11 https://doi.org/10.23638/DMTCS-21-3-11 Casselgren, C. J. Petrosyan, Petros A. Casselgren, C. J. Petrosyan, Petros A. <![CDATA[Given a proper edge coloring $\varphi$ of a graph $G$, we define the palette $S_{G}(v,\varphi)$ of a vertex $v \in V(G)$ as the set of all colors appearing on edges incident with $v$. The palette index $\check s(G)$ of $G$ is the minimum number of distinct palettes occurring in a proper edge coloring of $G$. In this paper we give various upper and lower bounds on the palette index of $G$ in terms of the vertex degrees of $G$, particularly for the case when $G$ is a bipartite graph with small vertex degrees. Some of our results concern $(a,b)$-biregular graphs; that is, bipartite graphs where all vertices in one part have degree $a$ and all vertices in the other part have degree $b$. We conjecture that if $G$ is $(a,b)$-biregular, then $\check{s}(G)\leq 1+\max\{a,b\}$, and we prove that this conjecture holds for several families of $(a,b)$-biregular graphs. Additionally, we characterize the graphs whose palette index equals the number of vertices.]]> 0 <![CDATA[Given a graph $G$, let $vc(G)$ and $vc^+(G)$ be the sizes of a minimum and a maximum minimal vertex covers of $G$, respectively. We say that $G$ is well covered if $vc(G)=vc^+(G)$ (that is, all minimal vertex covers have the same size). Determining if a graph is well covered is a coNP-complete problem. In this paper, we obtain $O^*(2^{vc})$-time and $O^*(1.4656^{vc^+})$-time algorithms to decide well coveredness, improving results of Boria et. al. (2015). Moreover, using crown decomposition, we show that such problems admit kernels having linear number of vertices. In 2018, Alves et. al. (2018) proved that recognizing well covered graphs is coW[2]-hard when the independence number $\alpha(G)=n-vc(G)$ is the parameter. Contrasting with such coW[2]-hardness, we present an FPT algorithm to decide well coveredness when $\alpha(G)$ and the degeneracy of the input graph $G$ are aggregate parameters. Finally, we use the primeval decomposition technique to obtain a linear time algorithm for extended $P_4$-laden graphs and $(q,q-4)$-graphs, which is FPT parameterized by $q$, improving results of Klein et al (2013).]]> Tue, 02 Apr 2019 12:20:11 +0000 https://doi.org/10.23638/DMTCS-21-1-3 https://doi.org/10.23638/DMTCS-21-1-3 Araujo, Rafael Costa, Eurinardo Klein, Sulamita Sampaio, Rudini Souza, Ueverton S. Araujo, Rafael Costa, Eurinardo Klein, Sulamita Sampaio, Rudini Souza, Ueverton S. <![CDATA[Given a graph $G$, let $vc(G)$ and $vc^+(G)$ be the sizes of a minimum and a maximum minimal vertex covers of $G$, respectively. We say that $G$ is well covered if $vc(G)=vc^+(G)$ (that is, all minimal vertex covers have the same size). Determining if a graph is well covered is a coNP-complete problem. In this paper, we obtain $O^*(2^{vc})$-time and $O^*(1.4656^{vc^+})$-time algorithms to decide well coveredness, improving results of Boria et. al. (2015). Moreover, using crown decomposition, we show that such problems admit kernels having linear number of vertices. In 2018, Alves et. al. (2018) proved that recognizing well covered graphs is coW[2]-hard when the independence number $\alpha(G)=n-vc(G)$ is the parameter. Contrasting with such coW[2]-hardness, we present an FPT algorithm to decide well coveredness when $\alpha(G)$ and the degeneracy of the input graph $G$ are aggregate parameters. Finally, we use the primeval decomposition technique to obtain a linear time algorithm for extended $P_4$-laden graphs and $(q,q-4)$-graphs, which is FPT parameterized by $q$, improving results of Klein et al (2013).]]> 0 <![CDATA[A univariate graph polynomial P(G;X) is weakly distinguishing if for almost all finite graphs G there is a finite graph H with P(G;X)=P(H;X). We show that the clique polynomial and the independence polynomial are weakly distinguishing. Furthermore, we show that generating functions of induced subgraphs with property C are weakly distinguishing provided that C is of bounded degeneracy or tree-width. The same holds for the harmonious chromatic polynomial.]]> Tue, 02 Apr 2019 09:34:53 +0000 https://doi.org/10.23638/DMTCS-21-1-4 https://doi.org/10.23638/DMTCS-21-1-4 Makowsky, Johann A. Rakita, Vsevolod Makowsky, Johann A. Rakita, Vsevolod <![CDATA[A univariate graph polynomial P(G;X) is weakly distinguishing if for almost all finite graphs G there is a finite graph H with P(G;X)=P(H;X). We show that the clique polynomial and the independence polynomial are weakly distinguishing. Furthermore, we show that generating functions of induced subgraphs with property C are weakly distinguishing provided that C is of bounded degeneracy or tree-width. The same holds for the harmonious chromatic polynomial.]]> 0 <![CDATA[How can one distinguish the adjacent vertices of a graph through an edge-weighting? In the last decades, this question has been attracting increasing attention, which resulted in the active field of distinguishing labellings. One of its most popular problems is the one where neighbours must be distinguishable via their incident sums of weights. An edge-weighting verifying this is said neighbour-sum-distinguishing. The popularity of this notion arises from two reasons. A first one is that designing a neighbour-sum-distinguishing edge-weighting showed up to be equivalent to turning a simple graph into a locally irregular (i.e., without neighbours with the same degree) multigraph by adding parallel edges, which is motivated by the concept of irregularity in graphs. Another source of popularity is probably the influence of the famous 1-2-3 Conjecture, which claims that such weightings with weights in {1,2,3} exist for graphs with no isolated edge. The 1-2-3 Conjecture has recently been investigated from a decompositional angle, via so-called locally irregular decompositions, which are edge-partitions into locally irregular subgraphs. Through several recent studies, it was shown that this concept is quite related to the 1-2-3 Conjecture. However, the full connexion between all those concepts was not clear. In this work, we propose an approach that generalizes all concepts above, involving coloured weights and sums. As a consequence, we get another interpretation of several existing results related to the 1-2-3 Conjecture. We also come up with new related conjectures, to which we give some support.]]> Tue, 02 Apr 2019 09:27:40 +0000 https://doi.org/10.23638/DMTCS-21-1-2 https://doi.org/10.23638/DMTCS-21-1-2 Baudon, Olivier Bensmail, Julien Davot, Tom Hocquard, Hervé Przybyło, Jakub Senhaji, Mohammed Sopena, Eric Woźniak, Mariusz Baudon, Olivier Bensmail, Julien Davot, Tom Hocquard, Hervé Przybyło, Jakub Senhaji, Mohammed Sopena, Eric Woźniak, Mariusz <![CDATA[How can one distinguish the adjacent vertices of a graph through an edge-weighting? In the last decades, this question has been attracting increasing attention, which resulted in the active field of distinguishing labellings. One of its most popular problems is the one where neighbours must be distinguishable via their incident sums of weights. An edge-weighting verifying this is said neighbour-sum-distinguishing. The popularity of this notion arises from two reasons. A first one is that designing a neighbour-sum-distinguishing edge-weighting showed up to be equivalent to turning a simple graph into a locally irregular (i.e., without neighbours with the same degree) multigraph by adding parallel edges, which is motivated by the concept of irregularity in graphs. Another source of popularity is probably the influence of the famous 1-2-3 Conjecture, which claims that such weightings with weights in {1,2,3} exist for graphs with no isolated edge. The 1-2-3 Conjecture has recently been investigated from a decompositional angle, via so-called locally irregular decompositions, which are edge-partitions into locally irregular subgraphs. Through several recent studies, it was shown that this concept is quite related to the 1-2-3 Conjecture. However, the full connexion between all those concepts was not clear. In this work, we propose an approach that generalizes all concepts above, involving coloured weights and sums. As a consequence, we get another interpretation of several existing results related to the 1-2-3 Conjecture. We also come up with new related conjectures, to which we give some support.]]> 0 <![CDATA[Let $n_g(k)$ denote the smallest order of a $k$-chromatic graph of girth at least $g$. We consider the problem of determining $n_g(k)$ for small values of $k$ and $g$. After giving an overview of what is known about $n_g(k)$, we provide some new lower bounds based on exhaustive searches, and then obtain several new upper bounds using computer algorithms for the construction of witnesses, and for the verification of their correctness. We also present the first examples of reasonably small order for $k = 4$ and $g > 5$. In particular, the new bounds include: $n_4(7) \leq 77$, $26 \leq n_6(4) \leq 66$, $30 \leq n_7(4) \leq 171$.]]> Mon, 11 Mar 2019 15:08:17 +0000 https://doi.org/10.23638/DMTCS-21-3-9 https://doi.org/10.23638/DMTCS-21-3-9 Exoo, Geoffrey Goedgebeur, Jan Exoo, Geoffrey Goedgebeur, Jan <![CDATA[Let $n_g(k)$ denote the smallest order of a $k$-chromatic graph of girth at least $g$. We consider the problem of determining $n_g(k)$ for small values of $k$ and $g$. After giving an overview of what is known about $n_g(k)$, we provide some new lower bounds based on exhaustive searches, and then obtain several new upper bounds using computer algorithms for the construction of witnesses, and for the verification of their correctness. We also present the first examples of reasonably small order for $k = 4$ and $g > 5$. In particular, the new bounds include: $n_4(7) \leq 77$, $26 \leq n_6(4) \leq 66$, $30 \leq n_7(4) \leq 171$.]]> 0 <![CDATA[Slimness of a graph measures the local deviation of its metric from a tree metric. In a graph $G=(V,E)$, a geodesic triangle $\bigtriangleup(x,y,z)$ with $x, y, z\in V$ is the union $P(x,y) \cup P(x,z) \cup P(y,z)$ of three shortest paths connecting these vertices. A geodesic triangle $\bigtriangleup(x,y,z)$ is called $\delta$-slim if for any vertex $u\in V$ on any side $P(x,y)$ the distance from $u$ to $P(x,z) \cup P(y,z)$ is at most $\delta$, i.e. each path is contained in the union of the $\delta$-neighborhoods of two others. A graph $G$ is called $\delta$-slim, if all geodesic triangles in $G$ are $\delta$-slim. The smallest value $\delta$ for which $G$ is $\delta$-slim is called the slimness of $G$. In this paper, using the layering partition technique, we obtain sharp bounds on slimness of such families of graphs as (1) graphs with cluster-diameter $\Delta(G)$ of a layering partition of $G$, (2) graphs with tree-length $\lambda$, (3) graphs with tree-breadth $\rho$, (4) $k$-chordal graphs, AT-free graphs and HHD-free graphs. Additionally, we show that the slimness of every 4-chordal graph is at most 2 and characterize those 4-chordal graphs for which the slimness of every of its induced subgraph is at most 1.]]> Mon, 04 Mar 2019 15:04:50 +0000 https://doi.org/10.23638/DMTCS-21-3-10 https://doi.org/10.23638/DMTCS-21-3-10 Dragan, Feodor F. Mohammed, Abdulhakeem Dragan, Feodor F. Mohammed, Abdulhakeem <![CDATA[Slimness of a graph measures the local deviation of its metric from a tree metric. In a graph $G=(V,E)$, a geodesic triangle $\bigtriangleup(x,y,z)$ with $x, y, z\in V$ is the union $P(x,y) \cup P(x,z) \cup P(y,z)$ of three shortest paths connecting these vertices. A geodesic triangle $\bigtriangleup(x,y,z)$ is called $\delta$-slim if for any vertex $u\in V$ on any side $P(x,y)$ the distance from $u$ to $P(x,z) \cup P(y,z)$ is at most $\delta$, i.e. each path is contained in the union of the $\delta$-neighborhoods of two others. A graph $G$ is called $\delta$-slim, if all geodesic triangles in $G$ are $\delta$-slim. The smallest value $\delta$ for which $G$ is $\delta$-slim is called the slimness of $G$. In this paper, using the layering partition technique, we obtain sharp bounds on slimness of such families of graphs as (1) graphs with cluster-diameter $\Delta(G)$ of a layering partition of $G$, (2) graphs with tree-length $\lambda$, (3) graphs with tree-breadth $\rho$, (4) $k$-chordal graphs, AT-free graphs and HHD-free graphs. Additionally, we show that the slimness of every 4-chordal graph is at most 2 and characterize those 4-chordal graphs for which the slimness of every of its induced subgraph is at most 1.]]> 0 <![CDATA[The packing chromatic number $\chi_{\rho}(G)$ of a graph $G$ is the smallest integer $k$ such that the vertex set of $G$ can be partitioned into sets $V_i$, $i\in [k]$, where vertices in $V_i$ are pairwise at distance at least $i+1$. Packing chromatic vertex-critical graphs, $\chi_{\rho}$-critical for short, are introduced as the graphs $G$ for which $\chi_{\rho}(G-x) < \chi_{\rho}(G)$ holds for every vertex $x$ of $G$. If $\chi_{\rho}(G) = k$, then $G$ is $k$-$\chi_{\rho}$-critical. It is shown that if $G$ is $\chi_{\rho}$-critical, then the set $\{\chi_{\rho}(G) - \chi_{\rho}(G-x):\ x\in V(G)\}$ can be almost arbitrary. The $3$-$\chi_{\rho}$-critical graphs are characterized, and $4$-$\chi_{\rho}$-critical graphs are characterized in the case when they contain a cycle of length at least $5$ which is not congruent to $0$ modulo $4$. It is shown that for every integer $k\ge 2$ there exists a $k$-$\chi_{\rho}$-critical tree and that a $k$-$\chi_{\rho}$-critical caterpillar exists if and only if $k\le 7$. Cartesian products are also considered and in particular it is proved that if $G$ and $H$ are vertex-transitive graphs and ${\rm diam(G)} + {\rm diam}(H) \le \chi_{\rho}(G)$, then $G\,\square\, H$ is $\chi_{\rho}$-critical.]]> Mon, 18 Feb 2019 14:45:17 +0000 https://doi.org/10.23638/DMTCS-21-3-8 https://doi.org/10.23638/DMTCS-21-3-8 Klavžar, Sandi Rall, Douglas F. Klavžar, Sandi Rall, Douglas F. <![CDATA[The packing chromatic number $\chi_{\rho}(G)$ of a graph $G$ is the smallest integer $k$ such that the vertex set of $G$ can be partitioned into sets $V_i$, $i\in [k]$, where vertices in $V_i$ are pairwise at distance at least $i+1$. Packing chromatic vertex-critical graphs, $\chi_{\rho}$-critical for short, are introduced as the graphs $G$ for which $\chi_{\rho}(G-x) < \chi_{\rho}(G)$ holds for every vertex $x$ of $G$. If $\chi_{\rho}(G) = k$, then $G$ is $k$-$\chi_{\rho}$-critical. It is shown that if $G$ is $\chi_{\rho}$-critical, then the set $\{\chi_{\rho}(G) - \chi_{\rho}(G-x):\ x\in V(G)\}$ can be almost arbitrary. The $3$-$\chi_{\rho}$-critical graphs are characterized, and $4$-$\chi_{\rho}$-critical graphs are characterized in the case when they contain a cycle of length at least $5$ which is not congruent to $0$ modulo $4$. It is shown that for every integer $k\ge 2$ there exists a $k$-$\chi_{\rho}$-critical tree and that a $k$-$\chi_{\rho}$-critical caterpillar exists if and only if $k\le 7$. Cartesian products are also considered and in particular it is proved that if $G$ and $H$ are vertex-transitive graphs and ${\rm diam(G)} + {\rm diam}(H) \le \chi_{\rho}(G)$, then $G\,\square\, H$ is $\chi_{\rho}$-critical.]]> 0 <![CDATA[The packing chromatic number $\chi_{\rho}(G)$ of a graph $G$ is the smallest integer $c$ such that the vertex set $V(G)$ can be partitioned into sets $X_1, . . . , X_c$, with the condition that vertices in $X_i$ have pairwise distance greater than $i$. In this paper, we consider the packing chromatic number of several families of Sierpinski-type graphs. We establish the packing chromatic numbers of generalized Sierpinski graphs $S^n_G$ where $G$ is a path or a cycle (with exception of a cycle of length five) as well as a connected graph of order four. Furthermore, we prove that the packing chromatic number in the family of Sierpinski-triangle graphs $ST_4^n$ is bounded from above by 20.]]> Fri, 08 Feb 2019 10:36:37 +0000 https://doi.org/10.23638/DMTCS-21-3-7 https://doi.org/10.23638/DMTCS-21-3-7 Korze, Danilo Vesel, Aleksander Korze, Danilo Vesel, Aleksander <![CDATA[The packing chromatic number $\chi_{\rho}(G)$ of a graph $G$ is the smallest integer $c$ such that the vertex set $V(G)$ can be partitioned into sets $X_1, . . . , X_c$, with the condition that vertices in $X_i$ have pairwise distance greater than $i$. In this paper, we consider the packing chromatic number of several families of Sierpinski-type graphs. We establish the packing chromatic numbers of generalized Sierpinski graphs $S^n_G$ where $G$ is a path or a cycle (with exception of a cycle of length five) as well as a connected graph of order four. Furthermore, we prove that the packing chromatic number in the family of Sierpinski-triangle graphs $ST_4^n$ is bounded from above by 20.]]> 0 <![CDATA[In algebraic terms, the insertion of $n$-powers in words may be modelled at the language level by considering the pseudovariety of ordered monoids defined by the inequality $1\le x^n$. We compare this pseudovariety with several other natural pseudovarieties of ordered monoids and of monoids associated with the Burnside pseudovariety of groups defined by the identity $x^n=1$. In particular, we are interested in determining the pseudovariety of monoids that it generates, which can be viewed as the problem of determining the Boolean closure of the class of regular languages closed under $n$-power insertions. We exhibit a simple upper bound and show that it satisfies all pseudoidentities which are provable from $1\le x^n$ in which both sides are regular elements with respect to the upper bound.]]> Tue, 05 Feb 2019 15:08:07 +0000 https://doi.org/10.23638/DMTCS-21-3-5 https://doi.org/10.23638/DMTCS-21-3-5 Almeida, J. Klíma, O. Almeida, J. Klíma, O. <![CDATA[In algebraic terms, the insertion of $n$-powers in words may be modelled at the language level by considering the pseudovariety of ordered monoids defined by the inequality $1\le x^n$. We compare this pseudovariety with several other natural pseudovarieties of ordered monoids and of monoids associated with the Burnside pseudovariety of groups defined by the identity $x^n=1$. In particular, we are interested in determining the pseudovariety of monoids that it generates, which can be viewed as the problem of determining the Boolean closure of the class of regular languages closed under $n$-power insertions. We exhibit a simple upper bound and show that it satisfies all pseudoidentities which are provable from $1\le x^n$ in which both sides are regular elements with respect to the upper bound.]]> 0 <![CDATA[We say that a finite set of red and blue points in the plane in general position can be $K_{1,3}$-covered if the set can be partitioned into subsets of size $4$, with $3$ points of one color and $1$ point of the other color, in such a way that, if at each subset the fourth point is connected by straight-line segments to the same-colored points, then the resulting set of all segments has no crossings. We consider the following problem: Given a set $R$ of $r$ red points and a set $B$ of $b$ blue points in the plane in general position, how many points of $R\cup B$ can be $K_{1,3}$-covered? and we prove the following results: (1) If $r=3g+h$ and $b=3h+g$, for some non-negative integers $g$ and $h$, then there are point sets $R\cup B$, like $\{1,3\}$-equitable sets (i.e., $r=3b$ or $b=3r$) and linearly separable sets, that can be $K_{1,3}$-covered. (2) If $r=3g+h$, $b=3h+g$ and the points in $R\cup B$ are in convex position, then at least $r+b-4$ points can be $K_{1,3}$-covered, and this bound is tight. (3) There are arbitrarily large point sets $R\cup B$ in general position, with $r=b+1$, such that at most $r+b-5$ points can be $K_{1,3}$-covered. (4) If $b\le r\le 3b$, then at least $\frac{8}{9}(r+b-8)$ points of $R\cup B$ can be $K_{1,3}$-covered. For $r>3b$, there are too many red points and at least $r-3b$ of them will remain uncovered in any $K_{1,3}$-covering. Furthermore, in all the cases we provide efficient algorithms to compute the corresponding coverings.]]> Thu, 31 Jan 2019 16:01:00 +0000 https://doi.org/10.23638/DMTCS-21-3-6 https://doi.org/10.23638/DMTCS-21-3-6 Ábrego, Bernardo M. Fernández-Merchant, Silvia Kano, Mikio Orden, David Pérez-Lantero, Pablo Seara, Carlos Tejel, Javier Ábrego, Bernardo M. Fernández-Merchant, Silvia Kano, Mikio Orden, David Pérez-Lantero, Pablo Seara, Carlos Tejel, Javier <![CDATA[We say that a finite set of red and blue points in the plane in general position can be $K_{1,3}$-covered if the set can be partitioned into subsets of size $4$, with $3$ points of one color and $1$ point of the other color, in such a way that, if at each subset the fourth point is connected by straight-line segments to the same-colored points, then the resulting set of all segments has no crossings. We consider the following problem: Given a set $R$ of $r$ red points and a set $B$ of $b$ blue points in the plane in general position, how many points of $R\cup B$ can be $K_{1,3}$-covered? and we prove the following results: (1) If $r=3g+h$ and $b=3h+g$, for some non-negative integers $g$ and $h$, then there are point sets $R\cup B$, like $\{1,3\}$-equitable sets (i.e., $r=3b$ or $b=3r$) and linearly separable sets, that can be $K_{1,3}$-covered. (2) If $r=3g+h$, $b=3h+g$ and the points in $R\cup B$ are in convex position, then at least $r+b-4$ points can be $K_{1,3}$-covered, and this bound is tight. (3) There are arbitrarily large point sets $R\cup B$ in general position, with $r=b+1$, such that at most $r+b-5$ points can be $K_{1,3}$-covered. (4) If $b\le r\le 3b$, then at least $\frac{8}{9}(r+b-8)$ points of $R\cup B$ can be $K_{1,3}$-covered. For $r>3b$, there are too many red points and at least $r-3b$ of them will remain uncovered in any $K_{1,3}$-covering. Furthermore, in all the cases we provide efficient algorithms to compute the corresponding coverings.]]> 0 <![CDATA[We consider decision problems for relations over finite and infinite words defined by finite automata. We prove that the equivalence problem for binary deterministic rational relations over infinite words is undecidable in contrast to the case of finite words, where the problem is decidable. Furthermore, we show that it is decidable in doubly exponential time for an automatic relation over infinite words whether it is a recognizable relation. We also revisit this problem in the context of finite words and improve the complexity of the decision procedure to single exponential time. The procedure is based on a polynomial time regularity test for deterministic visibly pushdown automata, which is a result of independent interest.]]> Thu, 31 Jan 2019 15:49:43 +0000 https://doi.org/10.23638/DMTCS-21-3-4 https://doi.org/10.23638/DMTCS-21-3-4 Löding, Christof Spinrath, Christopher Löding, Christof Spinrath, Christopher <![CDATA[We consider decision problems for relations over finite and infinite words defined by finite automata. We prove that the equivalence problem for binary deterministic rational relations over infinite words is undecidable in contrast to the case of finite words, where the problem is decidable. Furthermore, we show that it is decidable in doubly exponential time for an automatic relation over infinite words whether it is a recognizable relation. We also revisit this problem in the context of finite words and improve the complexity of the decision procedure to single exponential time. The procedure is based on a polynomial time regularity test for deterministic visibly pushdown automata, which is a result of independent interest.]]> 0 <![CDATA[We present a polynomial time algorithm, which solves a nonstandard Variation of the well-known PARTITION-problem: Given positive integers $n, k$ and $t$ such that $t \geq n$ and $k \cdot t = {n+1 \choose 2}$, the algorithm partitions the elements of the set $I_n = \{1, \ldots, n\}$ into $k$ mutually disjoint subsets $T_j$ such that $\cup_{j=1}^k T_j = I_n$ and $\sum_{x \in T_{j}} x = t$ for each $j \in \{1,2, \ldots, k\}$. The algorithm needs $\mathcal{O}(n \cdot ( \frac{n}{2k} + \log \frac{n(n+1)}{2k} ))$ steps to insert the $n$ elements of $I_n$ into the $k$ sets $T_j$.]]> Thu, 24 Jan 2019 15:36:20 +0000 https://doi.org/10.23638/DMTCS-20-2-18 https://doi.org/10.23638/DMTCS-20-2-18 Büchel, Alexander Gilleßen, Ulrich Witt, Kurt-Ulrich Büchel, Alexander Gilleßen, Ulrich Witt, Kurt-Ulrich <![CDATA[We present a polynomial time algorithm, which solves a nonstandard Variation of the well-known PARTITION-problem: Given positive integers $n, k$ and $t$ such that $t \geq n$ and $k \cdot t = {n+1 \choose 2}$, the algorithm partitions the elements of the set $I_n = \{1, \ldots, n\}$ into $k$ mutually disjoint subsets $T_j$ such that $\cup_{j=1}^k T_j = I_n$ and $\sum_{x \in T_{j}} x = t$ for each $j \in \{1,2, \ldots, k\}$. The algorithm needs $\mathcal{O}(n \cdot ( \frac{n}{2k} + \log \frac{n(n+1)}{2k} ))$ steps to insert the $n$ elements of $I_n$ into the $k$ sets $T_j$.]]> 0 <![CDATA[We propose the conjecture that every tree with order $n$ at least $2$ and total domination number $\gamma_t$ has at most $\left(\frac{n-\frac{\gamma_t}{2}}{\frac{\gamma_t}{2}}\right)^{\frac{\gamma_t}{2}}$ minimum total dominating sets. As a relaxation of this conjecture, we show that every forest $F$ with order $n$, no isolated vertex, and total domination number $\gamma_t$ has at most $\min\left\{\left(8\sqrt{e}\, \right)^{\gamma_t}\left(\frac{n-\frac{\gamma_t}{2}}{\frac{\gamma_t}{2}}\right)^{\frac{\gamma_t}{2}}, (1+\sqrt{2})^{n-\gamma_t},1.4865^n\right\}$ minimum total dominating sets.]]> Wed, 23 Jan 2019 10:30:07 +0000 https://doi.org/10.23638/DMTCS-21-3-3 https://doi.org/10.23638/DMTCS-21-3-3 Henning, Michael A. Mohr, Elena Rautenbach, Dieter Henning, Michael A. Mohr, Elena Rautenbach, Dieter <![CDATA[We propose the conjecture that every tree with order $n$ at least $2$ and total domination number $\gamma_t$ has at most $\left(\frac{n-\frac{\gamma_t}{2}}{\frac{\gamma_t}{2}}\right)^{\frac{\gamma_t}{2}}$ minimum total dominating sets. As a relaxation of this conjecture, we show that every forest $F$ with order $n$, no isolated vertex, and total domination number $\gamma_t$ has at most $\min\left\{\left(8\sqrt{e}\, \right)^{\gamma_t}\left(\frac{n-\frac{\gamma_t}{2}}{\frac{\gamma_t}{2}}\right)^{\frac{\gamma_t}{2}}, (1+\sqrt{2})^{n-\gamma_t},1.4865^n\right\}$ minimum total dominating sets.]]> 0 <![CDATA[A connected graph $G$ with at least $2m + 2n + 2$ vertices which contains a perfect matching is $E(m, n)$-{\it extendable}, if for any two sets of disjoint independent edges $M$ and $N$ with $|M| = m$ and $|N|= n$, there is a perfect matching $F$ in $G$ such that $M\subseteq F$ and $N\cap F=\emptyset$. Similarly, a connected graph with at least $n+2k+2$ vertices is called $(n,k)$-{\it extendable} if for any vertex set $S$ of size $n$ and any matching $M$ of size $k$ of $G-S$, $G-S-V(M)$ contains a perfect matching. Let $\varepsilon$ be a small positive constant, $b(G)$ and $t(G)$ be the binding number and toughness of a graph $G$. The two main theorems of this paper are: for every graph $G$ with sufficiently large order, 1) if $b(G)\geq 4/3+\varepsilon$, then $G$ is $E(m,n)$-extendable and also $(n,k)$-extendable; 2) if $t(G)\geq 1+\varepsilon$ and $G$ has a high connectivity, then $G$ is $E(m,n)$-extendable and also $(n,k)$-extendable. It is worth to point out that the binding number and toughness conditions for the existence of the general matching extension properties are almost same as that for the existence of perfect matchings.]]> Thu, 17 Jan 2019 12:37:26 +0000 https://doi.org/10.23638/DMTCS-21-3-1 https://doi.org/10.23638/DMTCS-21-3-1 Lu, Hongliang Yu, Qinglin Lu, Hongliang Yu, Qinglin <![CDATA[A connected graph $G$ with at least $2m + 2n + 2$ vertices which contains a perfect matching is $E(m, n)$-{\it extendable}, if for any two sets of disjoint independent edges $M$ and $N$ with $|M| = m$ and $|N|= n$, there is a perfect matching $F$ in $G$ such that $M\subseteq F$ and $N\cap F=\emptyset$. Similarly, a connected graph with at least $n+2k+2$ vertices is called $(n,k)$-{\it extendable} if for any vertex set $S$ of size $n$ and any matching $M$ of size $k$ of $G-S$, $G-S-V(M)$ contains a perfect matching. Let $\varepsilon$ be a small positive constant, $b(G)$ and $t(G)$ be the binding number and toughness of a graph $G$. The two main theorems of this paper are: for every graph $G$ with sufficiently large order, 1) if $b(G)\geq 4/3+\varepsilon$, then $G$ is $E(m,n)$-extendable and also $(n,k)$-extendable; 2) if $t(G)\geq 1+\varepsilon$ and $G$ has a high connectivity, then $G$ is $E(m,n)$-extendable and also $(n,k)$-extendable. It is worth to point out that the binding number and toughness conditions for the existence of the general matching extension properties are almost same as that for the existence of perfect matchings.]]> 0 <![CDATA[Identifying and locating-dominating codes have been widely studied in circulant graphs of type $C_n(1,2, \ldots, r)$, which can also be viewed as power graphs of cycles. Recently, Ghebleh and Niepel (2013) considered identification and location-domination in the circulant graphs $C_n(1,3)$. They showed that the smallest cardinality of a locating-dominating code in $C_n(1,3)$ is at least $\lceil n/3 \rceil$ and at most $\lceil n/3 \rceil + 1$ for all $n \geq 9$. Moreover, they proved that the lower bound is strict when $n \equiv 0, 1, 4 \pmod{6}$ and conjectured that the lower bound can be increased by one for other $n$. In this paper, we prove their conjecture. Similarly, they showed that the smallest cardinality of an identifying code in $C_n(1,3)$ is at least $\lceil 4n/11 \rceil$ and at most $\lceil 4n/11 \rceil + 1$ for all $n \geq 11$. Furthermore, they proved that the lower bound is attained for most of the lengths $n$ and conjectured that in the rest of the cases the lower bound can improved by one. This conjecture is also proved in the paper. The proofs of the conjectures are based on a novel approach which, instead of making use of the local properties of the graphs as is usual to identification and location-domination, also manages to take advantage of the global properties of the codes and the underlying graphs.]]> Tue, 08 Jan 2019 15:00:19 +0000 https://doi.org/10.23638/DMTCS-21-3-2 https://doi.org/10.23638/DMTCS-21-3-2 Junnila, Ville Laihonen, Tero Paris, Gabrielle Junnila, Ville Laihonen, Tero Paris, Gabrielle <![CDATA[Identifying and locating-dominating codes have been widely studied in circulant graphs of type $C_n(1,2, \ldots, r)$, which can also be viewed as power graphs of cycles. Recently, Ghebleh and Niepel (2013) considered identification and location-domination in the circulant graphs $C_n(1,3)$. They showed that the smallest cardinality of a locating-dominating code in $C_n(1,3)$ is at least $\lceil n/3 \rceil$ and at most $\lceil n/3 \rceil + 1$ for all $n \geq 9$. Moreover, they proved that the lower bound is strict when $n \equiv 0, 1, 4 \pmod{6}$ and conjectured that the lower bound can be increased by one for other $n$. In this paper, we prove their conjecture. Similarly, they showed that the smallest cardinality of an identifying code in $C_n(1,3)$ is at least $\lceil 4n/11 \rceil$ and at most $\lceil 4n/11 \rceil + 1$ for all $n \geq 11$. Furthermore, they proved that the lower bound is attained for most of the lengths $n$ and conjectured that in the rest of the cases the lower bound can improved by one. This conjecture is also proved in the paper. The proofs of the conjectures are based on a novel approach which, instead of making use of the local properties of the graphs as is usual to identification and location-domination, also manages to take advantage of the global properties of the codes and the underlying graphs.]]> 0 <![CDATA[A $\textit{sigma partitioning}$ of a graph $G$ is a partition of the vertices into sets $P_1, \ldots, P_k$ such that for every two adjacent vertices $u$ and $v$ there is an index $i$ such that $u$ and $v$ have different numbers of neighbors in $P_i$. The $\textit{ sigma number}$ of a graph $G$, denoted by $\sigma(G)$, is the minimum number $k$ such that $ G $ has a sigma partitioning $P_1, \ldots, P_k$. Also, a $\textit{ lucky labeling}$ of a graph $G$ is a function $ \ell :V(G) \rightarrow \mathbb{N}$, such that for every two adjacent vertices $ v $ and $ u$ of $ G $, $ \sum_{w \sim v}\ell(w)\neq \sum_{w \sim u}\ell(w) $ ($ x \sim y $ means that $ x $ and $y$ are adjacent). The $\textit{ lucky number}$ of $ G $, denoted by $\eta(G)$, is the minimum number $k $ such that $ G $ has a lucky labeling $ \ell :V(G) \rightarrow \mathbb{N}_k$. It was conjectured in [Inform. Process. Lett., 112(4):109--112, 2012] that it is $ \mathbf{NP} $-complete to decide whether $ \eta(G)=2$ for a given 3-regular graph $G$. In this work, we prove this conjecture. Among other results, we give an upper bound of five for the sigma number of a uniformly random graph.]]> Mon, 17 Dec 2018 14:39:19 +0000 https://doi.org/10.23638/DMTCS-20-2-19 https://doi.org/10.23638/DMTCS-20-2-19 Dehghan, Ali Sadeghi, Mohammad-Reza Ahadi, Arash Dehghan, Ali Sadeghi, Mohammad-Reza Ahadi, Arash <![CDATA[A $\textit{sigma partitioning}$ of a graph $G$ is a partition of the vertices into sets $P_1, \ldots, P_k$ such that for every two adjacent vertices $u$ and $v$ there is an index $i$ such that $u$ and $v$ have different numbers of neighbors in $P_i$. The $\textit{ sigma number}$ of a graph $G$, denoted by $\sigma(G)$, is the minimum number $k$ such that $ G $ has a sigma partitioning $P_1, \ldots, P_k$. Also, a $\textit{ lucky labeling}$ of a graph $G$ is a function $ \ell :V(G) \rightarrow \mathbb{N}$, such that for every two adjacent vertices $ v $ and $ u$ of $ G $, $ \sum_{w \sim v}\ell(w)\neq \sum_{w \sim u}\ell(w) $ ($ x \sim y $ means that $ x $ and $y$ are adjacent). The $\textit{ lucky number}$ of $ G $, denoted by $\eta(G)$, is the minimum number $k $ such that $ G $ has a lucky labeling $ \ell :V(G) \rightarrow \mathbb{N}_k$. It was conjectured in [Inform. Process. Lett., 112(4):109--112, 2012] that it is $ \mathbf{NP} $-complete to decide whether $ \eta(G)=2$ for a given 3-regular graph $G$. In this work, we prove this conjecture. Among other results, we give an upper bound of five for the sigma number of a uniformly random graph.]]> 0 <![CDATA[A digraph such that every proper induced subdigraph has a kernel is said to be \emph{kernel perfect} (KP for short) (\emph{critical kernel imperfect} (CKI for short) resp.) if the digraph has a kernel (does not have a kernel resp.). The unique CKI-tournament is $\overrightarrow{C}_3$ and the unique KP-tournaments are the transitive tournaments, however bipartite tournaments are KP. In this paper we characterize the CKI- and KP-digraphs for the following families of digraphs: locally in-/out-semicomplete, asymmetric arc-locally in-/out-semicomplete, asymmetric $3$-quasi-transitive and asymmetric $3$-anti-quasi-transitive $TT_3$-free and we state that the problem of determining whether a digraph of one of these families is CKI is polynomial, giving a solution to a problem closely related to the following conjecture posted by Bang-Jensen in 1998: the kernel problem is polynomially solvable for locally in-semicomplete digraphs.]]> Tue, 11 Dec 2018 09:24:21 +0000 https://doi.org/10.23638/DMTCS-20-2-16 https://doi.org/10.23638/DMTCS-20-2-16 Galeana-Sánchez, H. Olsen, M. Galeana-Sánchez, H. Olsen, M. <![CDATA[A digraph such that every proper induced subdigraph has a kernel is said to be \emph{kernel perfect} (KP for short) (\emph{critical kernel imperfect} (CKI for short) resp.) if the digraph has a kernel (does not have a kernel resp.). The unique CKI-tournament is $\overrightarrow{C}_3$ and the unique KP-tournaments are the transitive tournaments, however bipartite tournaments are KP. In this paper we characterize the CKI- and KP-digraphs for the following families of digraphs: locally in-/out-semicomplete, asymmetric arc-locally in-/out-semicomplete, asymmetric $3$-quasi-transitive and asymmetric $3$-anti-quasi-transitive $TT_3$-free and we state that the problem of determining whether a digraph of one of these families is CKI is polynomial, giving a solution to a problem closely related to the following conjecture posted by Bang-Jensen in 1998: the kernel problem is polynomially solvable for locally in-semicomplete digraphs.]]> 0 <![CDATA[Using techniques from Poisson approximation, we prove explicit error bounds on the number of permutations that avoid any pattern. Most generally, we bound the total variation distance between the joint distribution of pattern occurrences and a corresponding joint distribution of independent Bernoulli random variables, which as a corollary yields a Poisson approximation for the distribution of the number of occurrences of any pattern. We also investigate occurrences of consecutive patterns in random Mallows permutations, of which uniform random permutations are a special case. These bounds allow us to estimate the probability that a pattern occurs any number of times and, in particular, the probability that a random permutation avoids a given pattern.]]> Tue, 04 Dec 2018 14:42:38 +0000 https://doi.org/10.23638/DMTCS-19-2-13 https://doi.org/10.23638/DMTCS-19-2-13 Crane, Harry DeSalvo, Stephen Crane, Harry DeSalvo, Stephen <![CDATA[Using techniques from Poisson approximation, we prove explicit error bounds on the number of permutations that avoid any pattern. Most generally, we bound the total variation distance between the joint distribution of pattern occurrences and a corresponding joint distribution of independent Bernoulli random variables, which as a corollary yields a Poisson approximation for the distribution of the number of occurrences of any pattern. We also investigate occurrences of consecutive patterns in random Mallows permutations, of which uniform random permutations are a special case. These bounds allow us to estimate the probability that a pattern occurs any number of times and, in particular, the probability that a random permutation avoids a given pattern.]]> 0 <![CDATA[For oriented graphs $G$ and $H$, a homomorphism $f: G \rightarrow H$ is locally-injective if, for every $v \in V(G)$, it is injective when restricted to some combination of the in-neighbourhood and out-neighbourhood of $v$. Two of the possible definitions of local-injectivity are examined. In each case it is shown that the associated homomorphism problem is NP-complete when $H$ is a reflexive tournament on three or more vertices with a loop at every vertex, and solvable in polynomial time when $H$ is a reflexive tournament on two or fewer vertices.]]> Fri, 30 Nov 2018 10:33:11 +0000 https://doi.org/10.23638/DMTCS-20-2-4 https://doi.org/10.23638/DMTCS-20-2-4 Bard, Stefan Bellitto, Thomas Duffy, Christopher MacGillivray, Gary Yang, Feiran Bard, Stefan Bellitto, Thomas Duffy, Christopher MacGillivray, Gary Yang, Feiran <![CDATA[For oriented graphs $G$ and $H$, a homomorphism $f: G \rightarrow H$ is locally-injective if, for every $v \in V(G)$, it is injective when restricted to some combination of the in-neighbourhood and out-neighbourhood of $v$. Two of the possible definitions of local-injectivity are examined. In each case it is shown that the associated homomorphism problem is NP-complete when $H$ is a reflexive tournament on three or more vertices with a loop at every vertex, and solvable in polynomial time when $H$ is a reflexive tournament on two or fewer vertices.]]> 0 <![CDATA[A graph $G$ is {\em matching-decyclable} if it has a matching $M$ such that $G-M$ is acyclic. Deciding whether $G$ is matching-decyclable is an NP-complete problem even if $G$ is 2-connected, planar, and subcubic. In this work we present results on matching-decyclability in the following classes: Hamiltonian subcubic graphs, chordal graphs, and distance-hereditary graphs. In Hamiltonian subcubic graphs we show that deciding matching-decyclability is NP-complete even if there are exactly two vertices of degree two. For chordal and distance-hereditary graphs, we present characterizations of matching-decyclability that lead to $O(n)$-time recognition algorithms.]]> Tue, 20 Nov 2018 10:33:15 +0000 https://doi.org/10.23638/DMTCS-20-2-15 https://doi.org/10.23638/DMTCS-20-2-15 Protti, Fábio Souza, Uéverton S. Protti, Fábio Souza, Uéverton S. <![CDATA[A graph $G$ is {\em matching-decyclable} if it has a matching $M$ such that $G-M$ is acyclic. Deciding whether $G$ is matching-decyclable is an NP-complete problem even if $G$ is 2-connected, planar, and subcubic. In this work we present results on matching-decyclability in the following classes: Hamiltonian subcubic graphs, chordal graphs, and distance-hereditary graphs. In Hamiltonian subcubic graphs we show that deciding matching-decyclability is NP-complete even if there are exactly two vertices of degree two. For chordal and distance-hereditary graphs, we present characterizations of matching-decyclability that lead to $O(n)$-time recognition algorithms.]]> 0 <![CDATA[We consider a relaxation of the concept of well-covered graphs, which are graphs with all maximal independent sets of the same size. The extent to which a graph fails to be well-covered can be measured by its independence gap, defined as the difference between the maximum and minimum sizes of a maximal independent set in $G$. While the well-covered graphs are exactly the graphs of independence gap zero, we investigate in this paper graphs of independence gap one, which we also call almost well-covered graphs. Previous works due to Finbow et al. (1994) and Barbosa et al. (2013) have implications for the structure of almost well-covered graphs of girth at least $k$ for $k\in \{7,8\}$. We focus on almost well-covered graphs of girth at least $6$. We show that every graph in this class has at most two vertices each of which is adjacent to exactly $2$ leaves. We give efficiently testable characterizations of almost well-covered graphs of girth at least $6$ having exactly one or exactly two such vertices. Building on these results, we develop a polynomial-time recognition algorithm of almost well-covered $\{C_3,C_4,C_5,C_7\}$-free graphs.]]> Tue, 20 Nov 2018 10:28:34 +0000 https://doi.org/10.23638/DMTCS-20-2-17 https://doi.org/10.23638/DMTCS-20-2-17 Ekim, Tınaz Gözüpek, Didem Hujdurović, Ademir Milanič, Martin Ekim, Tınaz Gözüpek, Didem Hujdurović, Ademir Milanič, Martin <![CDATA[We consider a relaxation of the concept of well-covered graphs, which are graphs with all maximal independent sets of the same size. The extent to which a graph fails to be well-covered can be measured by its independence gap, defined as the difference between the maximum and minimum sizes of a maximal independent set in $G$. While the well-covered graphs are exactly the graphs of independence gap zero, we investigate in this paper graphs of independence gap one, which we also call almost well-covered graphs. Previous works due to Finbow et al. (1994) and Barbosa et al. (2013) have implications for the structure of almost well-covered graphs of girth at least $k$ for $k\in \{7,8\}$. We focus on almost well-covered graphs of girth at least $6$. We show that every graph in this class has at most two vertices each of which is adjacent to exactly $2$ leaves. We give efficiently testable characterizations of almost well-covered graphs of girth at least $6$ having exactly one or exactly two such vertices. Building on these results, we develop a polynomial-time recognition algorithm of almost well-covered $\{C_3,C_4,C_5,C_7\}$-free graphs.]]> 0 <![CDATA[Rectangulations are partitions of a square into axis-aligned rectangles. A number of results provide bijections between combinatorial equivalence classes of rectangulations and families of pattern-avoiding permutations. Other results deal with local changes involving a single edge of a rectangulation, referred to as flips, edge rotations, or edge pivoting. Such operations induce a graph on equivalence classes of rectangulations, related to so-called flip graphs on triangulations and other families of geometric partitions. In this note, we consider a family of flip operations on the equivalence classes of diagonal rectangulations, and their interpretation as transpositions in the associated Baxter permutations, avoiding the vincular patterns { 3{14}2, 2{41}3 }. This complements results from Law and Reading (JCTA, 2012) and provides a complete characterization of flip operations on diagonal rectangulations, in both geometric and combinatorial terms.]]> Fri, 09 Nov 2018 14:34:53 +0000 https://doi.org/10.23638/DMTCS-20-2-14 https://doi.org/10.23638/DMTCS-20-2-14 Cardinal, Jean Sacristán, Vera Silveira, Rodrigo I. Cardinal, Jean Sacristán, Vera Silveira, Rodrigo I. <![CDATA[Rectangulations are partitions of a square into axis-aligned rectangles. A number of results provide bijections between combinatorial equivalence classes of rectangulations and families of pattern-avoiding permutations. Other results deal with local changes involving a single edge of a rectangulation, referred to as flips, edge rotations, or edge pivoting. Such operations induce a graph on equivalence classes of rectangulations, related to so-called flip graphs on triangulations and other families of geometric partitions. In this note, we consider a family of flip operations on the equivalence classes of diagonal rectangulations, and their interpretation as transpositions in the associated Baxter permutations, avoiding the vincular patterns { 3{14}2, 2{41}3 }. This complements results from Law and Reading (JCTA, 2012) and provides a complete characterization of flip operations on diagonal rectangulations, in both geometric and combinatorial terms.]]> 0 <![CDATA[Dominating broadcasting is a domination-type structure that models a transmission antenna network. In this paper, we study a limited version of this structure, that was proposed as a common framework for both broadcast and classical domination. In this limited version, the broadcast function is upper bounded by an integer $k$ and the minimum cost of such function is the dominating $k$-broadcast number. Our main result is a unified upper bound on this parameter for any value of $k$ in general graphs, in terms of both $k$ and the order of the graph. We also study the computational complexity of the associated decision problem.]]> Mon, 29 Oct 2018 16:37:28 +0000 https://doi.org/10.23638/DMTCS-20-2-13 https://doi.org/10.23638/DMTCS-20-2-13 Cáceres, José Hernando, Carmen Mora, Mercè Pelayo, Ignacio M. Puertas, María Luz Cáceres, José Hernando, Carmen Mora, Mercè Pelayo, Ignacio M. Puertas, María Luz <![CDATA[Dominating broadcasting is a domination-type structure that models a transmission antenna network. In this paper, we study a limited version of this structure, that was proposed as a common framework for both broadcast and classical domination. In this limited version, the broadcast function is upper bounded by an integer $k$ and the minimum cost of such function is the dominating $k$-broadcast number. Our main result is a unified upper bound on this parameter for any value of $k$ in general graphs, in terms of both $k$ and the order of the graph. We also study the computational complexity of the associated decision problem.]]> 0 <![CDATA[In this paper, we facilitate the reasoning about impure programming languages, by annotating terms with “decorations”that describe what computational (side) effect evaluation of a term may involve. In a point-free categorical language,called the “decorated logic”, we formalize the mutable state and the exception effects first separately, exploiting anice duality between them, and then combined. The combined decorated logic is used as the target language forthe denotational semantics of the IMP+Exc imperative programming language, and allows us to prove equivalencesbetween programs written in IMP+Exc. The combined logic is encoded in Coq, and this encoding is used to certifysome program equivalence proofs.]]> Mon, 29 Oct 2018 16:18:58 +0000 https://doi.org/10.23638/DMTCS-20-2-11 https://doi.org/10.23638/DMTCS-20-2-11 Ekici , Burak Ekici , Burak <![CDATA[In this paper, we facilitate the reasoning about impure programming languages, by annotating terms with “decorations”that describe what computational (side) effect evaluation of a term may involve. In a point-free categorical language,called the “decorated logic”, we formalize the mutable state and the exception effects first separately, exploiting anice duality between them, and then combined. The combined decorated logic is used as the target language forthe denotational semantics of the IMP+Exc imperative programming language, and allows us to prove equivalencesbetween programs written in IMP+Exc. The combined logic is encoded in Coq, and this encoding is used to certifysome program equivalence proofs.]]> 0 <![CDATA[In this paper, we consider pattern avoidance in a subset of words on $\{1,1,2,2,\dots,n,n\}$ called reverse double lists. In particular a reverse double list is a word formed by concatenating a permutation with its reversal. We enumerate reverse double lists avoiding any permutation pattern of length at most 4 and completely determine the corresponding Wilf classes. For permutation patterns $\rho$ of length 5 or more, we characterize when the number of $\rho$-avoiding reverse double lists on $n$ letters has polynomial growth. We also determine the number of $1\cdots k$-avoiders of maximum length for any positive integer $k$.]]> Mon, 29 Oct 2018 16:10:26 +0000 https://doi.org/10.23638/DMTCS-19-2-14 https://doi.org/10.23638/DMTCS-19-2-14 Anderson, Monica Diepenbroek, Marika Pudwell, Lara Stoll, Alex Anderson, Monica Diepenbroek, Marika Pudwell, Lara Stoll, Alex <![CDATA[In this paper, we consider pattern avoidance in a subset of words on $\{1,1,2,2,\dots,n,n\}$ called reverse double lists. In particular a reverse double list is a word formed by concatenating a permutation with its reversal. We enumerate reverse double lists avoiding any permutation pattern of length at most 4 and completely determine the corresponding Wilf classes. For permutation patterns $\rho$ of length 5 or more, we characterize when the number of $\rho$-avoiding reverse double lists on $n$ letters has polynomial growth. We also determine the number of $1\cdots k$-avoiders of maximum length for any positive integer $k$.]]> 0 <![CDATA[We present two families of Wilf-equivalences for consecutive and quasi-consecutive vincular patterns. These give new proofs of the classification of consecutive patterns of length $4$ and $5$. We then prove additional equivalences to explicitly classify all quasi-consecutive patterns of length $5$ into 26 Wilf-equivalence classes.]]> Wed, 24 Oct 2018 09:21:40 +0000 https://doi.org/10.23638/DMTCS-20-2-12 https://doi.org/10.23638/DMTCS-20-2-12 Chen, Evan Narayanan, Shyam Chen, Evan Narayanan, Shyam <![CDATA[We present two families of Wilf-equivalences for consecutive and quasi-consecutive vincular patterns. These give new proofs of the classification of consecutive patterns of length $4$ and $5$. We then prove additional equivalences to explicitly classify all quasi-consecutive patterns of length $5$ into 26 Wilf-equivalence classes.]]> 0 <![CDATA[For a connected graph $G$ of order at least $2$ and $S\subseteq V(G)$, the \emph{Steiner distance} $d_G(S)$ among the vertices of $S$ is the minimum size among all connected subgraphs whose vertex sets contain $S$. Let $n$ and $k$ be two integers with $2\leq k\leq n$. Then the \emph{Steiner $k$-eccentricity $e_k(v)$} of a vertex $v$ of $G$ is defined by $e_k(v)=\max \{d_G(S)\,|\,S\subseteq V(G), \ |S|=k, \ and \ v\in S\}$. Furthermore, the \emph{Steiner $k$-diameter} of $G$ is $sdiam_k(G)=\max \{e_k(v)\,|\, v\in V(G)\}$. In this paper, we investigate the Steiner distance and Steiner $k$-diameter of Cartesian and lexicographical product graphs. Also, we study the Steiner $k$-diameter of some networks.]]> Tue, 23 Oct 2018 09:14:34 +0000 https://doi.org/10.23638/DMTCS-20-2-8 https://doi.org/10.23638/DMTCS-20-2-8 Mao, Yaping Cheng, Eddie Wang, Zhao Mao, Yaping Cheng, Eddie Wang, Zhao <![CDATA[For a connected graph $G$ of order at least $2$ and $S\subseteq V(G)$, the \emph{Steiner distance} $d_G(S)$ among the vertices of $S$ is the minimum size among all connected subgraphs whose vertex sets contain $S$. Let $n$ and $k$ be two integers with $2\leq k\leq n$. Then the \emph{Steiner $k$-eccentricity $e_k(v)$} of a vertex $v$ of $G$ is defined by $e_k(v)=\max \{d_G(S)\,|\,S\subseteq V(G), \ |S|=k, \ and \ v\in S\}$. Furthermore, the \emph{Steiner $k$-diameter} of $G$ is $sdiam_k(G)=\max \{e_k(v)\,|\, v\in V(G)\}$. In this paper, we investigate the Steiner distance and Steiner $k$-diameter of Cartesian and lexicographical product graphs. Also, we study the Steiner $k$-diameter of some networks.]]> 0 <![CDATA[By means of inversion techniques and several known hypergeometric series identities, summation formulas for Fox-Wright function are explored. They give some new hypergeometric series identities when the parameters are specified.]]> Mon, 22 Oct 2018 12:29:30 +0000 https://doi.org/10.23638/DMTCS-20-2-9 https://doi.org/10.23638/DMTCS-20-2-9 Wei, Chuanan Liu, Lily Li Gong, Dianxuan Wei, Chuanan Liu, Lily Li Gong, Dianxuan <![CDATA[By means of inversion techniques and several known hypergeometric series identities, summation formulas for Fox-Wright function are explored. They give some new hypergeometric series identities when the parameters are specified.]]> 0 <![CDATA[We study a recently introduced generalization of the Vertex Cover (VC) problem, called Power Vertex Cover (PVC). In this problem, each edge of the input graph is supplied with a positive integer demand. A solution is an assignment of (power) values to the vertices, so that for each edge one of its endpoints has value as high as the demand, and the total sum of power values assigned is minimized. We investigate how this generalization affects the parameterized complexity of Vertex Cover. On the positive side, when parameterized by the value of the optimal P, we give an O*(1.274^P)-time branching algorithm (O* is used to hide factors polynomial in the input size), and also an O*(1.325^P)-time algorithm for the more general asymmetric case of the problem, where the demand of each edge may differ for its two endpoints. When the parameter is the number of vertices k that receive positive value, we give O*(1.619^k) and O*(k^k)-time algorithms for the symmetric and asymmetric cases respectively, as well as a simple quadratic kernel for the asymmetric case. We also show that PVC becomes significantly harder than classical VC when parameterized by the graph's treewidth t. More specifically, we prove that unless the ETH is false, there is no n^o(t)-time algorithm for PVC. We give a method to overcome this hardness by designing an FPT approximation scheme which gives a (1+epsilon)-approximation to the optimal solution in time FPT in parameters t and 1/epsilon.]]> Mon, 08 Oct 2018 13:44:44 +0000 https://doi.org/10.23638/DMTCS-20-2-10 https://doi.org/10.23638/DMTCS-20-2-10 Angel, Eric Bampis, Evripidis Escoffier, Bruno Lampis, Michael Angel, Eric Bampis, Evripidis Escoffier, Bruno Lampis, Michael <![CDATA[We study a recently introduced generalization of the Vertex Cover (VC) problem, called Power Vertex Cover (PVC). In this problem, each edge of the input graph is supplied with a positive integer demand. A solution is an assignment of (power) values to the vertices, so that for each edge one of its endpoints has value as high as the demand, and the total sum of power values assigned is minimized. We investigate how this generalization affects the parameterized complexity of Vertex Cover. On the positive side, when parameterized by the value of the optimal P, we give an O*(1.274^P)-time branching algorithm (O* is used to hide factors polynomial in the input size), and also an O*(1.325^P)-time algorithm for the more general asymmetric case of the problem, where the demand of each edge may differ for its two endpoints. When the parameter is the number of vertices k that receive positive value, we give O*(1.619^k) and O*(k^k)-time algorithms for the symmetric and asymmetric cases respectively, as well as a simple quadratic kernel for the asymmetric case. We also show that PVC becomes significantly harder than classical VC when parameterized by the graph's treewidth t. More specifically, we prove that unless the ETH is false, there is no n^o(t)-time algorithm for PVC. We give a method to overcome this hardness by designing an FPT approximation scheme which gives a (1+epsilon)-approximation to the optimal solution in time FPT in parameters t and 1/epsilon.]]> 0 <![CDATA[We study the biased $(1:b)$ Maker--Breaker positional games, played on the edge set of the complete graph on $n$ vertices, $K_n$. Given Breaker's bias $b$, possibly depending on $n$, we determine the bounds for the minimal number of moves, depending on $b$, in which Maker can win in each of the two standard graph games, the Perfect Matching game and the Hamilton Cycle game.]]> Mon, 08 Oct 2018 13:37:12 +0000 https://doi.org/10.23638/DMTCS-20-2-6 https://doi.org/10.23638/DMTCS-20-2-6 Mikalački, Mirjana Stojaković, Miloš Mikalački, Mirjana Stojaković, Miloš <![CDATA[We study the biased $(1:b)$ Maker--Breaker positional games, played on the edge set of the complete graph on $n$ vertices, $K_n$. Given Breaker's bias $b$, possibly depending on $n$, we determine the bounds for the minimal number of moves, depending on $b$, in which Maker can win in each of the two standard graph games, the Perfect Matching game and the Hamilton Cycle game.]]> 0 <![CDATA[The 1-2 Conjecture raised by Przybylo and Wozniak in 2010 asserts that every undirected graph admits a 2-total-weighting such that the sums of weights "incident" to the vertices yield a proper vertex-colouring. Following several recent works bringing related problems and notions (such as the well-known 1-2-3 Conjecture, and the notion of locally irregular decompositions) to digraphs, we here introduce and study several variants of the 1-2 Conjecture for digraphs. For every such variant, we raise conjectures concerning the number of weights necessary to obtain a desired total-weighting in any digraph. We verify some of these conjectures, while we obtain close results towards the ones that are still open.]]> Mon, 01 Oct 2018 12:11:10 +0000 https://doi.org/10.23638/DMTCS-20-2-7 https://doi.org/10.23638/DMTCS-20-2-7 Baudon , Olivier Bensmail , Julien Przybyło , Jakub Woźniak , Mariusz Baudon , Olivier Bensmail , Julien Przybyło , Jakub Woźniak , Mariusz <![CDATA[The 1-2 Conjecture raised by Przybylo and Wozniak in 2010 asserts that every undirected graph admits a 2-total-weighting such that the sums of weights "incident" to the vertices yield a proper vertex-colouring. Following several recent works bringing related problems and notions (such as the well-known 1-2-3 Conjecture, and the notion of locally irregular decompositions) to digraphs, we here introduce and study several variants of the 1-2 Conjecture for digraphs. For every such variant, we raise conjectures concerning the number of weights necessary to obtain a desired total-weighting in any digraph. We verify some of these conjectures, while we obtain close results towards the ones that are still open.]]> 0 <![CDATA[In this paper, we study a parameter that is squeezed between arguably the two important domination parameters, namely the domination number, $\gamma(G)$, and the total domination number, $\gamma_t(G)$. A set $S$ of vertices in $G$ is a semitotal dominating set of $G$ if it is a dominating set of $G$ and every vertex in S is within distance $2$ of another vertex of $S$. The semitotal domination number, $\gamma_{t2}(G)$, is the minimum cardinality of a semitotal dominating set of $G$. We observe that $\gamma(G)\leq \gamma_{t2}(G)\leq \gamma_t(G)$. In this paper, we give a lower bound for the semitotal domination number of trees and we characterize the extremal trees. In addition, we characterize trees with equal domination and semitotal domination numbers.]]> Fri, 28 Sep 2018 15:11:41 +0000 https://doi.org/10.23638/DMTCS-20-2-5 https://doi.org/10.23638/DMTCS-20-2-5 Wei, Zhuang Guoliang, Hao Wei, Zhuang Guoliang, Hao <![CDATA[In this paper, we study a parameter that is squeezed between arguably the two important domination parameters, namely the domination number, $\gamma(G)$, and the total domination number, $\gamma_t(G)$. A set $S$ of vertices in $G$ is a semitotal dominating set of $G$ if it is a dominating set of $G$ and every vertex in S is within distance $2$ of another vertex of $S$. The semitotal domination number, $\gamma_{t2}(G)$, is the minimum cardinality of a semitotal dominating set of $G$. We observe that $\gamma(G)\leq \gamma_{t2}(G)\leq \gamma_t(G)$. In this paper, we give a lower bound for the semitotal domination number of trees and we characterize the extremal trees. In addition, we characterize trees with equal domination and semitotal domination numbers.]]> 0 <![CDATA[This is the first of three papers that develop structures which are counted by a "parabolic" generalization of Catalan numbers. Fix a subset R of {1,..,n-1}. Consider the ordered partitions of {1,..,n} whose block sizes are determined by R. These are the "inverses" of (parabolic) multipermutations whose multiplicities are determined by R. The standard forms of the ordered partitions are refered to as "R-permutations". The notion of 312-avoidance is extended from permutations to R-permutations. Let lambda be a partition of N such that the set of column lengths in its shape is R or R union {n}. Fix an R-permutation pi. The type A Demazure character (key polynomial) in x_1, .., x_n that is indexed by lambda and pi can be described as the sum of the weight monomials for some of the semistandard Young tableau of shape lambda that are used to describe the Schur function indexed by lambda. Descriptions of these "Demazure" tableaux developed by the authors in earlier papers are used to prove that the set of these tableaux is convex in Z^N if and only if pi is R-312-avoiding if and only if the tableau set is the entire principal ideal generated by the key of pi. These papers were inspired by results of Reiner and Shimozono and by Postnikov and Stanley concerning coincidences between Demazure characters and flagged Schur functions. This convexity result is used in the next paper to deepen those results from the level of polynomials to the level of tableau sets. The R-parabolic Catalan number is defined to be the number of R-312-avoiding permutations. These special R-permutations are reformulated as "R-rightmost clump deleting" chains of subsets of {1,..,n} and as "gapless R-tuples"; the latter n-tuples arise in multiple contexts in these papers.]]> Thu, 16 Aug 2018 08:27:14 +0000 https://doi.org/10.23638/DMTCS-20-2-3 https://doi.org/10.23638/DMTCS-20-2-3 Proctor, Robert A. Willis, Matthew J. Proctor, Robert A. Willis, Matthew J. <![CDATA[This is the first of three papers that develop structures which are counted by a "parabolic" generalization of Catalan numbers. Fix a subset R of {1,..,n-1}. Consider the ordered partitions of {1,..,n} whose block sizes are determined by R. These are the "inverses" of (parabolic) multipermutations whose multiplicities are determined by R. The standard forms of the ordered partitions are refered to as "R-permutations". The notion of 312-avoidance is extended from permutations to R-permutations. Let lambda be a partition of N such that the set of column lengths in its shape is R or R union {n}. Fix an R-permutation pi. The type A Demazure character (key polynomial) in x_1, .., x_n that is indexed by lambda and pi can be described as the sum of the weight monomials for some of the semistandard Young tableau of shape lambda that are used to describe the Schur function indexed by lambda. Descriptions of these "Demazure" tableaux developed by the authors in earlier papers are used to prove that the set of these tableaux is convex in Z^N if and only if pi is R-312-avoiding if and only if the tableau set is the entire principal ideal generated by the key of pi. These papers were inspired by results of Reiner and Shimozono and by Postnikov and Stanley concerning coincidences between Demazure characters and flagged Schur functions. This convexity result is used in the next paper to deepen those results from the level of polynomials to the level of tableau sets. The R-parabolic Catalan number is defined to be the number of R-312-avoiding permutations. These special R-permutations are reformulated as "R-rightmost clump deleting" chains of subsets of {1,..,n} and as "gapless R-tuples"; the latter n-tuples arise in multiple contexts in these papers.]]> 0 <![CDATA[Given a permutation $\sigma = \sigma_1 \ldots \sigma_n$ in the symmetric group $\mathcal{S}_{n}$, we say that $\sigma_i$ matches the quadrant marked mesh pattern $\mathrm{MMP}(a,b,c,d)$ in $\sigma$ if there are at least $a$ points to the right of $\sigma_i$ in $\sigma$ which are greater than $\sigma_i$, at least $b$ points to the left of $\sigma_i$ in $\sigma$ which are greater than $\sigma_i$, at least $c$ points to the left of $\sigma_i$ in $\sigma$ which are smaller than $\sigma_i$, and at least $d$ points to the right of $\sigma_i$ in $\sigma$ which are smaller than $\sigma_i$. Kitaev, Remmel, and Tiefenbruck systematically studied the distribution of the number of matches of $\mathrm{MMP}(a,b,c,d)$ in 132-avoiding permutations. The operation of reverse and complement on permutations allow one to translate their results to find the distribution of the number of $\mathrm{MMP}(a,b,c,d)$ matches in 231-avoiding, 213-avoiding, and 312-avoiding permutations. In this paper, we study the distribution of the number of matches of $\mathrm{MMP}(a,b,c,d)$ in 123-avoiding permutations. We provide explicit recurrence relations to enumerate our objects which can be used to give closed forms for the generating functions associated with such distributions. In many cases, we provide combinatorial explanations of the coefficients that appear in our generating functions.]]> Fri, 03 Aug 2018 09:38:59 +0000 https://doi.org/10.23638/DMTCS-19-2-12 https://doi.org/10.23638/DMTCS-19-2-12 Qiu, Dun Remmel, Jeffrey B. Qiu, Dun Remmel, Jeffrey B. <![CDATA[Given a permutation $\sigma = \sigma_1 \ldots \sigma_n$ in the symmetric group $\mathcal{S}_{n}$, we say that $\sigma_i$ matches the quadrant marked mesh pattern $\mathrm{MMP}(a,b,c,d)$ in $\sigma$ if there are at least $a$ points to the right of $\sigma_i$ in $\sigma$ which are greater than $\sigma_i$, at least $b$ points to the left of $\sigma_i$ in $\sigma$ which are greater than $\sigma_i$, at least $c$ points to the left of $\sigma_i$ in $\sigma$ which are smaller than $\sigma_i$, and at least $d$ points to the right of $\sigma_i$ in $\sigma$ which are smaller than $\sigma_i$. Kitaev, Remmel, and Tiefenbruck systematically studied the distribution of the number of matches of $\mathrm{MMP}(a,b,c,d)$ in 132-avoiding permutations. The operation of reverse and complement on permutations allow one to translate their results to find the distribution of the number of $\mathrm{MMP}(a,b,c,d)$ matches in 231-avoiding, 213-avoiding, and 312-avoiding permutations. In this paper, we study the distribution of the number of matches of $\mathrm{MMP}(a,b,c,d)$ in 123-avoiding permutations. We provide explicit recurrence relations to enumerate our objects which can be used to give closed forms for the generating functions associated with such distributions. In many cases, we provide combinatorial explanations of the coefficients that appear in our generating functions.]]> 0 <![CDATA[In a barter exchange market, agents bring items and seek to exchange their items with one another. Agents may agree to a k-way exchange involving a cycle of k agents. A barter exchange market can be represented by a digraph where the vertices represent items and the edges out of a vertex indicate the items that an agent is willing to accept in exchange for that item. It is known that the problem of finding a set of vertex-disjoint cycles with the maximum total number of vertices (MAX-SIZE-EXCHANGE) can be solved in polynomial time. We consider a barter exchange where each agent may bring multiple items, and items of the same agent are represented by vertices with the same color. A set of cycles is said to be tropical if for every color there is a cycle that contains a vertex of that color. We show that the problem of determining whether there exists a tropical set of vertex-disjoint cycles in a digraph (TROPICAL-EXCHANGE) is NP-complete and APX-hard. This is equivalent to determining whether it is possible to arrange an exchange of items among agents such that every agent trades away at least one item. TROPICAL-MAX-SIZE-EXCHANGE is a similar problem, where the goal is to find a set of vertex-disjoint cycles that contains the maximum number of vertices and also contains all of the colors in the graph. We show that this problem is likewise NP-complete and APX-hard. For the restricted case where there are at most two vertices of each color (corresponding to a restriction that each agent may bring at most two items), both problems remain NP-hard but are in APX. Finally, we consider MAX-SIZE-TROPICAL-EXCHANGE, where the set of cycles must primarily include as many colors as possible and secondarily include as many vertices as possible. We show that this problem is NP-hard.]]> Tue, 31 Jul 2018 10:03:57 +0000 https://doi.org/10.23638/DMTCS-20-2-1 https://doi.org/10.23638/DMTCS-20-2-1 Highley, Timothy Le, Hoang Highley, Timothy Le, Hoang <![CDATA[In a barter exchange market, agents bring items and seek to exchange their items with one another. Agents may agree to a k-way exchange involving a cycle of k agents. A barter exchange market can be represented by a digraph where the vertices represent items and the edges out of a vertex indicate the items that an agent is willing to accept in exchange for that item. It is known that the problem of finding a set of vertex-disjoint cycles with the maximum total number of vertices (MAX-SIZE-EXCHANGE) can be solved in polynomial time. We consider a barter exchange where each agent may bring multiple items, and items of the same agent are represented by vertices with the same color. A set of cycles is said to be tropical if for every color there is a cycle that contains a vertex of that color. We show that the problem of determining whether there exists a tropical set of vertex-disjoint cycles in a digraph (TROPICAL-EXCHANGE) is NP-complete and APX-hard. This is equivalent to determining whether it is possible to arrange an exchange of items among agents such that every agent trades away at least one item. TROPICAL-MAX-SIZE-EXCHANGE is a similar problem, where the goal is to find a set of vertex-disjoint cycles that contains the maximum number of vertices and also contains all of the colors in the graph. We show that this problem is likewise NP-complete and APX-hard. For the restricted case where there are at most two vertices of each color (corresponding to a restriction that each agent may bring at most two items), both problems remain NP-hard but are in APX. Finally, we consider MAX-SIZE-TROPICAL-EXCHANGE, where the set of cycles must primarily include as many colors as possible and secondarily include as many vertices as possible. We show that this problem is NP-hard.]]> 0 <![CDATA[A mixed dominating set for a graph $G = (V,E)$ is a set $S\subseteq V \cup E$ such that every element $x \in (V \cup E) \backslash S$ is either adjacent or incident to an element of $S$. The mixed domination number of a graph $G$, denoted by $\gamma_m(G)$, is the minimum cardinality of mixed dominating sets of $G$. Any mixed dominating set with the cardinality of $\gamma_m(G)$ is called a minimum mixed dominating set. The mixed domination set (MDS) problem is to find a minimum mixed dominating set for a graph $G$ and is known to be an NP-complete problem. In this paper, we present a novel approach to find all of the mixed dominating sets, called the AMDS problem, of a graph with bounded tree-width $tw$. Our new technique of assigning power values to edges and vertices, and combining with dynamic programming, leads to a fixed-parameter algorithm of time $O(3^{tw^{2}}\times tw^2 \times |V|)$. This shows that MDS is fixed-parameter tractable with respect to tree-width. In addition, we theoretically improve the proposed algorithm to solve the MDS problem in $O(6^{tw} \times |V|)$ time.]]> Tue, 31 Jul 2018 09:58:49 +0000 https://doi.org/10.23638/DMTCS-20-2-2 https://doi.org/10.23638/DMTCS-20-2-2 Rajaati, M. Hooshmandasl, M. R. Dinneen, M. J. Shakiba, A. Rajaati, M. Hooshmandasl, M. R. Dinneen, M. J. Shakiba, A. <![CDATA[A mixed dominating set for a graph $G = (V,E)$ is a set $S\subseteq V \cup E$ such that every element $x \in (V \cup E) \backslash S$ is either adjacent or incident to an element of $S$. The mixed domination number of a graph $G$, denoted by $\gamma_m(G)$, is the minimum cardinality of mixed dominating sets of $G$. Any mixed dominating set with the cardinality of $\gamma_m(G)$ is called a minimum mixed dominating set. The mixed domination set (MDS) problem is to find a minimum mixed dominating set for a graph $G$ and is known to be an NP-complete problem. In this paper, we present a novel approach to find all of the mixed dominating sets, called the AMDS problem, of a graph with bounded tree-width $tw$. Our new technique of assigning power values to edges and vertices, and combining with dynamic programming, leads to a fixed-parameter algorithm of time $O(3^{tw^{2}}\times tw^2 \times |V|)$. This shows that MDS is fixed-parameter tractable with respect to tree-width. In addition, we theoretically improve the proposed algorithm to solve the MDS problem in $O(6^{tw} \times |V|)$ time.]]> 0 <![CDATA[We introduce weighted regular tree grammars with storage as combination of (a) regular tree grammars with storage and (b) weighted tree automata over multioperator monoids. Each weighted regular tree grammar with storage generates a weighted tree language, which is a mapping from the set of trees to the multioperator monoid. We prove that, for multioperator monoids canonically associated to particular strong bi-monoids, the support of the generated weighted tree languages can be generated by (unweighted) regular tree grammars with storage. We characterize the class of all generated weighted tree languages by the composition of three basic concepts. Moreover, we prove results on the elimination of chain rules and of finite storage types, and we characterize weighted regular tree grammars with storage by a new weighted MSO-logic.]]> Tue, 03 Jul 2018 14:54:58 +0000 https://doi.org/10.23638/DMTCS-20-1-26 https://doi.org/10.23638/DMTCS-20-1-26 Fülöp, Zoltán Herrmann, Luisa Vogler, Heiko Fülöp, Zoltán Herrmann, Luisa Vogler, Heiko <![CDATA[We introduce weighted regular tree grammars with storage as combination of (a) regular tree grammars with storage and (b) weighted tree automata over multioperator monoids. Each weighted regular tree grammar with storage generates a weighted tree language, which is a mapping from the set of trees to the multioperator monoid. We prove that, for multioperator monoids canonically associated to particular strong bi-monoids, the support of the generated weighted tree languages can be generated by (unweighted) regular tree grammars with storage. We characterize the class of all generated weighted tree languages by the composition of three basic concepts. Moreover, we prove results on the elimination of chain rules and of finite storage types, and we characterize weighted regular tree grammars with storage by a new weighted MSO-logic.]]> 0 <![CDATA[For a given graph $G$, the least integer $k\geq 2$ such that for every Abelian group $\mathcal{G}$ of order $k$ there exists a proper edge labeling $f:E(G)\rightarrow \mathcal{G}$ so that $\sum_{x\in N(u)}f(xu)\neq \sum_{x\in N(v)}f(xv)$ for each edge $uv\in E(G)$ is called the \textit{group twin chromatic index} of $G$ and denoted by $\chi'_g(G)$. This graph invariant is related to a few well-known problems in the field of neighbor distinguishing graph colorings. We conjecture that $\chi'_g(G)\leq \Delta(G)+3$ for all graphs without isolated edges, where $\Delta(G)$ is the maximum degree of $G$, and provide an infinite family of connected graph (trees) for which the equality holds. We prove that this conjecture is valid for all trees, and then apply this result as the base case for proving a general upper bound for all graphs $G$ without isolated edges: $\chi'_g(G)\leq 2(\Delta(G)+{\rm col}(G))-5$, where ${\rm col}(G)$ denotes the coloring number of $G$. This improves the best known upper bound known previously only for the case of cyclic groups $\mathbb{Z}_k$.]]> Tue, 26 Jun 2018 15:14:31 +0000 https://doi.org/10.23638/DMTCS-20-1-24 https://doi.org/10.23638/DMTCS-20-1-24 Cichacz, Sylwia Przybyło, Jakub Cichacz, Sylwia Przybyło, Jakub <![CDATA[For a given graph $G$, the least integer $k\geq 2$ such that for every Abelian group $\mathcal{G}$ of order $k$ there exists a proper edge labeling $f:E(G)\rightarrow \mathcal{G}$ so that $\sum_{x\in N(u)}f(xu)\neq \sum_{x\in N(v)}f(xv)$ for each edge $uv\in E(G)$ is called the \textit{group twin chromatic index} of $G$ and denoted by $\chi'_g(G)$. This graph invariant is related to a few well-known problems in the field of neighbor distinguishing graph colorings. We conjecture that $\chi'_g(G)\leq \Delta(G)+3$ for all graphs without isolated edges, where $\Delta(G)$ is the maximum degree of $G$, and provide an infinite family of connected graph (trees) for which the equality holds. We prove that this conjecture is valid for all trees, and then apply this result as the base case for proving a general upper bound for all graphs $G$ without isolated edges: $\chi'_g(G)\leq 2(\Delta(G)+{\rm col}(G))-5$, where ${\rm col}(G)$ denotes the coloring number of $G$. This improves the best known upper bound known previously only for the case of cyclic groups $\mathbb{Z}_k$.]]> 0 <![CDATA[When considering binary strings, it's natural to wonder how many distinct subsequences might exist in a given string. Given that there is an existing algorithm which provides a straightforward way to compute the number of distinct subsequences in a fixed string, we might next be interested in the expected number of distinct subsequences in random strings. This expected value is already known for random binary strings where each letter in the string is, independently, equally likely to be a 1 or a 0. We generalize this result to random strings where the letter 1 appears independently with probability $\alpha \in [0,1]$. Also, we make some progress in the case of random strings from an arbitrary alphabet as well as when the string is generated by a two-state Markov chain.]]> Tue, 26 Jun 2018 09:19:47 +0000 https://doi.org/10.23638/DMTCS-19-2-10 https://doi.org/10.23638/DMTCS-19-2-10 Biers-Ariel, Yonah Godbole, Anant Kelley, Elizabeth Biers-Ariel, Yonah Godbole, Anant Kelley, Elizabeth <![CDATA[When considering binary strings, it's natural to wonder how many distinct subsequences might exist in a given string. Given that there is an existing algorithm which provides a straightforward way to compute the number of distinct subsequences in a fixed string, we might next be interested in the expected number of distinct subsequences in random strings. This expected value is already known for random binary strings where each letter in the string is, independently, equally likely to be a 1 or a 0. We generalize this result to random strings where the letter 1 appears independently with probability $\alpha \in [0,1]$. Also, we make some progress in the case of random strings from an arbitrary alphabet as well as when the string is generated by a two-state Markov chain.]]> 0 <![CDATA[We explore a bijection between permutations and colored Motzkin paths that has been used in different forms by Foata and Zeilberger, Biane, and Corteel. By giving a visual representation of this bijection in terms of so-called cycle diagrams, we find simple translations of some statistics on permutations (and subsets of permutations) into statistics on colored Motzkin paths, which are amenable to the use of continued fractions. We obtain new enumeration formulas for subsets of permutations with respect to fixed points, excedances, double excedances, cycles, and inversions. In particular, we prove that cyclic permutations whose excedances are increasing are counted by the Bell numbers.]]> Mon, 25 Jun 2018 09:08:15 +0000 https://doi.org/10.23638/DMTCS-19-2-11 https://doi.org/10.23638/DMTCS-19-2-11 Elizalde, Sergi Elizalde, Sergi <![CDATA[We explore a bijection between permutations and colored Motzkin paths that has been used in different forms by Foata and Zeilberger, Biane, and Corteel. By giving a visual representation of this bijection in terms of so-called cycle diagrams, we find simple translations of some statistics on permutations (and subsets of permutations) into statistics on colored Motzkin paths, which are amenable to the use of continued fractions. We obtain new enumeration formulas for subsets of permutations with respect to fixed points, excedances, double excedances, cycles, and inversions. In particular, we prove that cyclic permutations whose excedances are increasing are counted by the Bell numbers.]]> 0 <![CDATA[We give a sufficient condition for a degree sequence to be graphic based on its largest and smallest elements, length, and sum. This bound generalizes a result of Zverovich and Zverovich.]]> Mon, 25 Jun 2018 09:02:23 +0000 https://doi.org/10.23638/DMTCS-20-1-25 https://doi.org/10.23638/DMTCS-20-1-25 Cloteaux, Brian Cloteaux, Brian <![CDATA[We give a sufficient condition for a degree sequence to be graphic based on its largest and smallest elements, length, and sum. This bound generalizes a result of Zverovich and Zverovich.]]> 0 <![CDATA[Let $\gamma(G)$ and $\gamma_t(G)$ denote the domination number and the total domination number, respectively, of a graph $G$ with no isolated vertices. It is well-known that $\gamma_t(G) \leq 2\gamma(G)$. We provide a characterization of a large family of graphs (including chordal graphs) satisfying $\gamma_t(G)= 2\gamma(G)$, strictly generalizing the results of Henning (2001) and Hou et al. (2010), and partially answering an open question of Henning (2009).]]> Mon, 04 Jun 2018 08:37:11 +0000 https://doi.org/10.23638/DMTCS-20-1-23 https://doi.org/10.23638/DMTCS-20-1-23 Bahadır, Selim Gözüpek, Didem Bahadır, Selim Gözüpek, Didem <![CDATA[Let $\gamma(G)$ and $\gamma_t(G)$ denote the domination number and the total domination number, respectively, of a graph $G$ with no isolated vertices. It is well-known that $\gamma_t(G) \leq 2\gamma(G)$. We provide a characterization of a large family of graphs (including chordal graphs) satisfying $\gamma_t(G)= 2\gamma(G)$, strictly generalizing the results of Henning (2001) and Hou et al. (2010), and partially answering an open question of Henning (2009).]]> 0 <![CDATA[A path in an edge-colored graph $G$ is rainbow if no two edges of it are colored the same. The graph $G$ is rainbow-connected if there is a rainbow path between every pair of vertices. If there is a rainbow shortest path between every pair of vertices, the graph $G$ is strongly rainbow-connected. The minimum number of colors needed to make $G$ rainbow-connected is known as the rainbow connection number of $G$, and is denoted by $\text{rc}(G)$. Similarly, the minimum number of colors needed to make $G$ strongly rainbow-connected is known as the strong rainbow connection number of $G$, and is denoted by $\text{src}(G)$. We prove that for every $k \geq 3$, deciding whether $\text{src}(G) \leq k$ is NP-complete for split graphs, which form a subclass of chordal graphs. Furthermore, there exists no polynomial-time algorithm for approximating the strong rainbow connection number of an $n$-vertex split graph with a factor of $n^{1/2-\epsilon}$ for any $\epsilon > 0$ unless P = NP. We then turn our attention to block graphs, which also form a subclass of chordal graphs. We determine the strong rainbow connection number of block graphs, and show it can be computed in linear time. Finally, we provide a polynomial-time characterization of bridgeless block graphs with rainbow connection number at most 4.]]> Mon, 04 Jun 2018 08:30:35 +0000 https://doi.org/10.23638/DMTCS-20-1-22 https://doi.org/10.23638/DMTCS-20-1-22 Keranen, Melissa Lauri, Juho Keranen, Melissa Lauri, Juho <![CDATA[A path in an edge-colored graph $G$ is rainbow if no two edges of it are colored the same. The graph $G$ is rainbow-connected if there is a rainbow path between every pair of vertices. If there is a rainbow shortest path between every pair of vertices, the graph $G$ is strongly rainbow-connected. The minimum number of colors needed to make $G$ rainbow-connected is known as the rainbow connection number of $G$, and is denoted by $\text{rc}(G)$. Similarly, the minimum number of colors needed to make $G$ strongly rainbow-connected is known as the strong rainbow connection number of $G$, and is denoted by $\text{src}(G)$. We prove that for every $k \geq 3$, deciding whether $\text{src}(G) \leq k$ is NP-complete for split graphs, which form a subclass of chordal graphs. Furthermore, there exists no polynomial-time algorithm for approximating the strong rainbow connection number of an $n$-vertex split graph with a factor of $n^{1/2-\epsilon}$ for any $\epsilon > 0$ unless P = NP. We then turn our attention to block graphs, which also form a subclass of chordal graphs. We determine the strong rainbow connection number of block graphs, and show it can be computed in linear time. Finally, we provide a polynomial-time characterization of bridgeless block graphs with rainbow connection number at most 4.]]> 0 <![CDATA[Let $S$ be a set of integers. A graph G is said to have the S-property if there exists an S-edge-weighting $w : E(G) \rightarrow S$ such that any two adjacent vertices have different sums of incident edge-weights. In this paper we characterise all bridgeless bipartite graphs and all trees without the $\{0,1\}$-property. In particular this problem belongs to P for these graphs while it is NP-complete for all graphs.]]> Mon, 04 Jun 2018 08:27:12 +0000 https://doi.org/10.23638/DMTCS-20-1-21 https://doi.org/10.23638/DMTCS-20-1-21 Lyngsie, Kasper Szabo Lyngsie, Kasper Szabo <![CDATA[Let $S$ be a set of integers. A graph G is said to have the S-property if there exists an S-edge-weighting $w : E(G) \rightarrow S$ such that any two adjacent vertices have different sums of incident edge-weights. In this paper we characterise all bridgeless bipartite graphs and all trees without the $\{0,1\}$-property. In particular this problem belongs to P for these graphs while it is NP-complete for all graphs.]]> 0 <![CDATA[We show that the permutation complexity of the image of a Sturmian word by a binary marked morphism is $n+k$ for some constant $k$ and all lengths $n$ sufficiently large.]]> Mon, 04 Jun 2018 08:22:33 +0000 https://doi.org/10.23638/DMTCS-20-1-20 https://doi.org/10.23638/DMTCS-20-1-20 Borchert, Adam Rampersad, Narad Borchert, Adam Rampersad, Narad <![CDATA[We show that the permutation complexity of the image of a Sturmian word by a binary marked morphism is $n+k$ for some constant $k$ and all lengths $n$ sufficiently large.]]> 0 <![CDATA[In this paper, we characterize the sets $\mathcal{H}$ of connected graphs such that there exists a constant $c=c(\mathcal{H})$ satisfying $\gamma (G)\leq c$ for every connected $\mathcal{H}$-free graph $G$, where $\gamma (G)$ is the domination number of $G$.]]> Mon, 04 Jun 2018 08:19:22 +0000 https://doi.org/10.23638/DMTCS-20-1-19 https://doi.org/10.23638/DMTCS-20-1-19 Furuya, Michitaka Furuya, Michitaka <![CDATA[In this paper, we characterize the sets $\mathcal{H}$ of connected graphs such that there exists a constant $c=c(\mathcal{H})$ satisfying $\gamma (G)\leq c$ for every connected $\mathcal{H}$-free graph $G$, where $\gamma (G)$ is the domination number of $G$.]]> 0 <![CDATA[An antimagic labelling of a graph $G$ is a bijection $f:E(G)\to\{1,\ldots,E(G)\}$ such that the sums $S_v=\sum_{e\ni v}f(e)$ distinguish all vertices. A well-known conjecture of Hartsfield and Ringel (1994) is that every connected graph other than $K_2$ admits an antimagic labelling. Recently, two sets of authors (Arumugam, Premalatha, Ba\v{c}a \& Semani\v{c}ov\'a-Fe\v{n}ov\v{c}\'ikov\'a (2017), and Bensmail, Senhaji \& Lyngsie (2017)) independently introduced the weaker notion of a local antimagic labelling, where only adjacent vertices must be distinguished. Both sets of authors conjectured that any connected graph other than $K_2$ admits a local antimagic labelling. We prove this latter conjecture using the probabilistic method. Thus the parameter of local antimagic chromatic number, introduced by Arumugam et al., is well-defined for every connected graph other than $K_2$ .]]> Mon, 04 Jun 2018 08:15:49 +0000 https://doi.org/10.23638/DMTCS-20-1-18 https://doi.org/10.23638/DMTCS-20-1-18 Haslegrave, John Haslegrave, John <![CDATA[An antimagic labelling of a graph $G$ is a bijection $f:E(G)\to\{1,\ldots,E(G)\}$ such that the sums $S_v=\sum_{e\ni v}f(e)$ distinguish all vertices. A well-known conjecture of Hartsfield and Ringel (1994) is that every connected graph other than $K_2$ admits an antimagic labelling. Recently, two sets of authors (Arumugam, Premalatha, Ba\v{c}a \& Semani\v{c}ov\'a-Fe\v{n}ov\v{c}\'ikov\'a (2017), and Bensmail, Senhaji \& Lyngsie (2017)) independently introduced the weaker notion of a local antimagic labelling, where only adjacent vertices must be distinguished. Both sets of authors conjectured that any connected graph other than $K_2$ admits a local antimagic labelling. We prove this latter conjecture using the probabilistic method. Thus the parameter of local antimagic chromatic number, introduced by Arumugam et al., is well-defined for every connected graph other than $K_2$ .]]> 0 <![CDATA[We define a weakly threshold sequence to be a degree sequence $d=(d_1,\dots,d_n)$ of a graph having the property that $\sum_{i \leq k} d_i \geq k(k-1)+\sum_{i > k} \min\{k,d_i\} - 1$ for all positive $k \leq \max\{i:d_i \geq i-1\}$. The weakly threshold graphs are the realizations of the weakly threshold sequences. The weakly threshold graphs properly include the threshold graphs and satisfy pleasing extensions of many properties of threshold graphs. We demonstrate a majorization property of weakly threshold sequences and an iterative construction algorithm for weakly threshold graphs, as well as a forbidden induced subgraph characterization. We conclude by exactly enumerating weakly threshold sequences and graphs.]]> Mon, 04 Jun 2018 08:08:17 +0000 https://doi.org/10.23638/DMTCS-20-1-15 https://doi.org/10.23638/DMTCS-20-1-15 Barrus, Michael D. Barrus, Michael D. <![CDATA[We define a weakly threshold sequence to be a degree sequence $d=(d_1,\dots,d_n)$ of a graph having the property that $\sum_{i \leq k} d_i \geq k(k-1)+\sum_{i > k} \min\{k,d_i\} - 1$ for all positive $k \leq \max\{i:d_i \geq i-1\}$. The weakly threshold graphs are the realizations of the weakly threshold sequences. The weakly threshold graphs properly include the threshold graphs and satisfy pleasing extensions of many properties of threshold graphs. We demonstrate a majorization property of weakly threshold sequences and an iterative construction algorithm for weakly threshold graphs, as well as a forbidden induced subgraph characterization. We conclude by exactly enumerating weakly threshold sequences and graphs.]]> 0 <![CDATA[We generalize the notion of the toggle group, as defined in [P. Cameron-D. Fon-der-Flaass '95] and further explored in [J. Striker-N. Williams '12], from the set of order ideals of a poset to any family of subsets of a finite set. We prove structure theorems for certain finite generalized toggle groups, similar to the theorem of Cameron and Fon-der-Flaass in the case of order ideals. We apply these theorems and find other results on generalized toggle groups in the following settings: chains, antichains, and interval-closed sets of a poset; independent sets, vertex covers, acyclic subgraphs, and spanning subgraphs of a graph; matroids and convex geometries. We generalize rowmotion, an action studied on order ideals in [P. Cameron-D. Fon-der-Flaass '95] and [J. Striker-N. Williams '12], to a map we call cover-closure on closed sets of a closure operator. We show that cover-closure is bijective if and only if the set of closed sets is isomorphic to the set of order ideals of a poset, which implies rowmotion is the only bijective cover-closure map.]]> Fri, 25 May 2018 15:04:47 +0000 https://doi.org/10.23638/DMTCS-20-1-17 https://doi.org/10.23638/DMTCS-20-1-17 Striker, Jessica Striker, Jessica <![CDATA[We generalize the notion of the toggle group, as defined in [P. Cameron-D. Fon-der-Flaass '95] and further explored in [J. Striker-N. Williams '12], from the set of order ideals of a poset to any family of subsets of a finite set. We prove structure theorems for certain finite generalized toggle groups, similar to the theorem of Cameron and Fon-der-Flaass in the case of order ideals. We apply these theorems and find other results on generalized toggle groups in the following settings: chains, antichains, and interval-closed sets of a poset; independent sets, vertex covers, acyclic subgraphs, and spanning subgraphs of a graph; matroids and convex geometries. We generalize rowmotion, an action studied on order ideals in [P. Cameron-D. Fon-der-Flaass '95] and [J. Striker-N. Williams '12], to a map we call cover-closure on closed sets of a closure operator. We show that cover-closure is bijective if and only if the set of closed sets is isomorphic to the set of order ideals of a poset, which implies rowmotion is the only bijective cover-closure map.]]> 0 <![CDATA[A thrackle is a drawing of a graph in which each pair of edges meets precisely once. Conway's Thrackle Conjecture asserts that a thrackle drawing of a graph on the plane cannot have more edges than vertices. We prove the Conjecture for thrackle drawings all of whose vertices lie on the boundaries of $d \le 3$ connected domains in the complement of the drawing. We also give a detailed description of thrackle drawings corresponding to the cases when $d=2$ (annular thrackles) and $d=3$ (pants thrackles).]]> Fri, 25 May 2018 14:59:37 +0000 https://doi.org/10.23638/DMTCS-20-1-16 https://doi.org/10.23638/DMTCS-20-1-16 Misereh, Grace Nikolayevsky, Yuri Misereh, Grace Nikolayevsky, Yuri <![CDATA[A thrackle is a drawing of a graph in which each pair of edges meets precisely once. Conway's Thrackle Conjecture asserts that a thrackle drawing of a graph on the plane cannot have more edges than vertices. We prove the Conjecture for thrackle drawings all of whose vertices lie on the boundaries of $d \le 3$ connected domains in the complement of the drawing. We also give a detailed description of thrackle drawings corresponding to the cases when $d=2$ (annular thrackles) and $d=3$ (pants thrackles).]]> 0 <![CDATA[A total dominating set of a graph $G=(V,E)$ is a subset $D \subseteq V$ such that every vertex in $V$ is adjacent to some vertex in $D$. Finding a total dominating set of minimum size is NP-hard on planar graphs and W[2]-complete on general graphs when parameterized by the solution size. By the meta-theorem of Bodlaender et al. [J. ACM, 2016], there exists a linear kernel for Total Dominating Set on graphs of bounded genus. Nevertheless, it is not clear how such a kernel can be effectively constructed, and how to obtain explicit reduction rules with reasonably small constants. Following the approach of Alber et al. [J. ACM, 2004], we provide an explicit kernel for Total Dominating Set on planar graphs with at most $410k$ vertices, where $k$ is the size of the solution. This result complements several known constructive linear kernels on planar graphs for other domination problems such as Dominating Set, Edge Dominating Set, Efficient Dominating Set, Connected Dominating Set, or Red-Blue Dominating Set.]]> Wed, 16 May 2018 14:19:58 +0000 https://doi.org/10.23638/DMTCS-20-1-14 https://doi.org/10.23638/DMTCS-20-1-14 Garnero, Valentin Sau, Ignasi Garnero, Valentin Sau, Ignasi <![CDATA[A total dominating set of a graph $G=(V,E)$ is a subset $D \subseteq V$ such that every vertex in $V$ is adjacent to some vertex in $D$. Finding a total dominating set of minimum size is NP-hard on planar graphs and W[2]-complete on general graphs when parameterized by the solution size. By the meta-theorem of Bodlaender et al. [J. ACM, 2016], there exists a linear kernel for Total Dominating Set on graphs of bounded genus. Nevertheless, it is not clear how such a kernel can be effectively constructed, and how to obtain explicit reduction rules with reasonably small constants. Following the approach of Alber et al. [J. ACM, 2004], we provide an explicit kernel for Total Dominating Set on planar graphs with at most $410k$ vertices, where $k$ is the size of the solution. This result complements several known constructive linear kernels on planar graphs for other domination problems such as Dominating Set, Edge Dominating Set, Efficient Dominating Set, Connected Dominating Set, or Red-Blue Dominating Set.]]> 0 <![CDATA[Recently, Araujo et al. [Manuscript in preparation, 2017] introduced the notion of Cycle Convexity of graphs. In their seminal work, they studied the graph convexity parameter called hull number for this new graph convexity they proposed, and they presented some of its applications in Knot theory. Roughly, the tunnel number of a knot embedded in a plane is upper bounded by the hull number of a corresponding planar 4-regular graph in cycle convexity. In this paper, we go further in the study of this new graph convexity and we study the interval number of a graph in cycle convexity. This parameter is, alongside the hull number, one of the most studied parameters in the literature about graph convexities. Precisely, given a graph G, its interval number in cycle convexity, denoted by $in_{cc} (G)$, is the minimum cardinality of a set S ⊆ V (G) such that every vertex w ∈ V (G) \ S has two distinct neighbors u, v ∈ S such that u and v lie in same connected component of G[S], i.e. the subgraph of G induced by the vertices in S.In this work, first we provide bounds on $in_{cc} (G)$ and its relations to other graph convexity parameters, and explore its behavior on grids. Then, we present some hardness results by showing that deciding whether $in_{cc} (G)$ ≤ k is NP-complete, even if G is a split graph or a bounded-degree planar graph, and that the problem is W[2]-hard in bipartite graphs when k is the parameter. As a consequence, we obtainthat $in_{cc} (G)$ cannot be approximated up to a constant factor in the classes of split graphs and bipartite graphs (unless P = N P ).On the positive side, we present polynomial-time algorithms to compute $in_{cc} (G)$ for outerplanar graphs, cobipartite graphs and interval graphs. We also present fixed-parameter tractable (FPT) algorithms to compute it for (q, q − 4)-graphs when q is the parameter and for general graphs G when parameterized either by the treewidth or the neighborhood diversity of G.Some of our hardness results and positive results are not known to hold for related graph convexities and domination problems. We hope that the design of our new reductions and polynomial-time algorithms can be helpful in order to advance in the study of related graph problems.]]> Mon, 07 May 2018 08:26:53 +0000 https://doi.org/10.23638/DMTCS-20-1-13 https://doi.org/10.23638/DMTCS-20-1-13 Araujo , Julio Ducoffe , Guillaume Nisse , Nicolas Suchan , Karol Araujo , Julio Ducoffe , Guillaume Nisse , Nicolas Suchan , Karol <![CDATA[Recently, Araujo et al. [Manuscript in preparation, 2017] introduced the notion of Cycle Convexity of graphs. In their seminal work, they studied the graph convexity parameter called hull number for this new graph convexity they proposed, and they presented some of its applications in Knot theory. Roughly, the tunnel number of a knot embedded in a plane is upper bounded by the hull number of a corresponding planar 4-regular graph in cycle convexity. In this paper, we go further in the study of this new graph convexity and we study the interval number of a graph in cycle convexity. This parameter is, alongside the hull number, one of the most studied parameters in the literature about graph convexities. Precisely, given a graph G, its interval number in cycle convexity, denoted by $in_{cc} (G)$, is the minimum cardinality of a set S ⊆ V (G) such that every vertex w ∈ V (G) \ S has two distinct neighbors u, v ∈ S such that u and v lie in same connected component of G[S], i.e. the subgraph of G induced by the vertices in S.In this work, first we provide bounds on $in_{cc} (G)$ and its relations to other graph convexity parameters, and explore its behavior on grids. Then, we present some hardness results by showing that deciding whether $in_{cc} (G)$ ≤ k is NP-complete, even if G is a split graph or a bounded-degree planar graph, and that the problem is W[2]-hard in bipartite graphs when k is the parameter. As a consequence, we obtainthat $in_{cc} (G)$ cannot be approximated up to a constant factor in the classes of split graphs and bipartite graphs (unless P = N P ).On the positive side, we present polynomial-time algorithms to compute $in_{cc} (G)$ for outerplanar graphs, cobipartite graphs and interval graphs. We also present fixed-parameter tractable (FPT) algorithms to compute it for (q, q − 4)-graphs when q is the parameter and for general graphs G when parameterized either by the treewidth or the neighborhood diversity of G.Some of our hardness results and positive results are not known to hold for related graph convexities and domination problems. We hope that the design of our new reductions and polynomial-time algorithms can be helpful in order to advance in the study of related graph problems.]]> 0 <![CDATA[We study the number of occurrences of any fixed vincular permutation pattern. We show that this statistics on uniform random permutations is asymptotically normal and describe the speed of convergence. To prove this central limit theorem, we use the method of dependency graphs. The main difficulty is then to estimate the variance of our statistics. We need a lower bound on the variance, for which we introduce a recursive technique based on the law of total variance.]]> Mon, 26 Mar 2018 09:21:09 +0000 https://doi.org/10.23638/DMTCS-19-2-9 https://doi.org/10.23638/DMTCS-19-2-9 Hofer, Lisa Hofer, Lisa <![CDATA[We study the number of occurrences of any fixed vincular permutation pattern. We show that this statistics on uniform random permutations is asymptotically normal and describe the speed of convergence. To prove this central limit theorem, we use the method of dependency graphs. The main difficulty is then to estimate the variance of our statistics. We need a lower bound on the variance, for which we introduce a recursive technique based on the law of total variance.]]> 0 <![CDATA[Grebinski and Kucherov (1998) and Alon et al. (2004-2005) study the problem of learning a hidden graph for some especial cases, such as hamiltonian cycle, cliques, stars, and matchings. This problem is motivated by problems in chemical reactions, molecular biology and genome sequencing. In this paper, we present a generalization of this problem. Precisely, we consider a graph G and a subgraph H of G and we assume that G contains exactly one defective subgraph isomorphic to H. The goal is to find the defective subgraph by testing whether an induced subgraph contains an edge of the defective subgraph, with the minimum number of tests. We present an upper bound for the number of tests to find the defective subgraph by using the symmetric and high probability variation of Lov\'asz Local Lemma.]]> Mon, 26 Mar 2018 09:13:44 +0000 https://doi.org/10.23638/DMTCS-20-1-9 https://doi.org/10.23638/DMTCS-20-1-9 Kameli, Hamid Kameli, Hamid <![CDATA[Grebinski and Kucherov (1998) and Alon et al. (2004-2005) study the problem of learning a hidden graph for some especial cases, such as hamiltonian cycle, cliques, stars, and matchings. This problem is motivated by problems in chemical reactions, molecular biology and genome sequencing. In this paper, we present a generalization of this problem. Precisely, we consider a graph G and a subgraph H of G and we assume that G contains exactly one defective subgraph isomorphic to H. The goal is to find the defective subgraph by testing whether an induced subgraph contains an edge of the defective subgraph, with the minimum number of tests. We present an upper bound for the number of tests to find the defective subgraph by using the symmetric and high probability variation of Lov\'asz Local Lemma.]]> 0 <![CDATA[In a rooted tree, protected nodes are neither leaves nor parents of any leaves. They have some practical motivations, e.g., in organizational schemes, security models and social-network models. Protected node profile measures the number of protected nodes with the same distance from the root in rooted trees. For no rooted tree, protected node profile has been investigated so far. Here, we present the asymptotic expectations, variances, covariance and limiting bivariate distribution of protected node profile and non-protected internal node profile in random tries, an important data structure on words in computer science. Also we investigate the fraction of these expectations asymptotically. These results are derived by the methods of analytic combinatorics such as generating functions, Mellin transform, Poissonization and depoissonization, saddle point method and singularity analysis.]]> Mon, 26 Mar 2018 09:04:02 +0000 https://doi.org/10.23638/DMTCS-20-1-12 https://doi.org/10.23638/DMTCS-20-1-12 Javanian, Mehri Javanian, Mehri <![CDATA[In a rooted tree, protected nodes are neither leaves nor parents of any leaves. They have some practical motivations, e.g., in organizational schemes, security models and social-network models. Protected node profile measures the number of protected nodes with the same distance from the root in rooted trees. For no rooted tree, protected node profile has been investigated so far. Here, we present the asymptotic expectations, variances, covariance and limiting bivariate distribution of protected node profile and non-protected internal node profile in random tries, an important data structure on words in computer science. Also we investigate the fraction of these expectations asymptotically. These results are derived by the methods of analytic combinatorics such as generating functions, Mellin transform, Poissonization and depoissonization, saddle point method and singularity analysis.]]> 0 <![CDATA[We establish asymptotic bounds for the number of partitions of $[n]$ avoiding a given partition in Klazar's sense, obtaining the correct answer to within an exponential for the block case. This technique also enables us to establish a general lower bound. Additionally, we consider a graph theoretic restatement of partition avoidance problems, and propose several conjectures.]]> Mon, 19 Mar 2018 10:02:07 +0000 https://doi.org/10.23638/DMTCS-19-2-7 https://doi.org/10.23638/DMTCS-19-2-7 Alweiss, Ryan Alweiss, Ryan <![CDATA[We establish asymptotic bounds for the number of partitions of $[n]$ avoiding a given partition in Klazar's sense, obtaining the correct answer to within an exponential for the block case. This technique also enables us to establish a general lower bound. Additionally, we consider a graph theoretic restatement of partition avoidance problems, and propose several conjectures.]]> 0 <![CDATA[Every rational number p/q defines a rational base numeration system in which every integer has a unique finite representation, up to leading zeroes. This work is a contribution to the study of the set of the representations of integers. This prefix-closed subset of the free monoid is naturally represented as a highly non-regular tree. Its nodes are the integers, its edges bear labels taken in {0,1,...,p-1}, and its subtrees are all distinct. We associate with each subtree (or with its root n) three infinite words. The bottom word of n is the lexicographically smallest word that is the label of a branch of the subtree. The top word of n is defined similarly. The span-word of n is the digitwise difference between the latter and the former. First, we show that the set of all the span-words is accepted by an infinite automaton whose underlying graph is essentially the same as the tree itself. Second, we study the function that computes for all n the bottom word associated with n+1 from the one associated with n, and show that it is realised by an infinite sequential transducer whose underlying graph is once again essentially the same as the tree itself. An infinite word may be interpreted as an expansion in base p/q after the radix point, hence evaluated to a real number. If T is a subtree whose root is n, then the evaluations of the labels of the branches of T form an interval of $\mathbb{R}$. The length of this interval is called the span of n and is equal to the evaluation of the span-word of n. The set of all spans is then a subset of R and we use the preceding construction to study its topological closure. We show that it is an interval when p is greater than or equal to 2q-1, and a Cantor set of measure zero otherwise.]]> Mon, 05 Mar 2018 08:53:22 +0000 https://doi.org/10.23638/DMTCS-20-1-10 https://doi.org/10.23638/DMTCS-20-1-10 Akiyama, Shigeki Marsault, Victor Sakarovitch, Jacques Akiyama, Shigeki Marsault, Victor Sakarovitch, Jacques <![CDATA[Every rational number p/q defines a rational base numeration system in which every integer has a unique finite representation, up to leading zeroes. This work is a contribution to the study of the set of the representations of integers. This prefix-closed subset of the free monoid is naturally represented as a highly non-regular tree. Its nodes are the integers, its edges bear labels taken in {0,1,...,p-1}, and its subtrees are all distinct. We associate with each subtree (or with its root n) three infinite words. The bottom word of n is the lexicographically smallest word that is the label of a branch of the subtree. The top word of n is defined similarly. The span-word of n is the digitwise difference between the latter and the former. First, we show that the set of all the span-words is accepted by an infinite automaton whose underlying graph is essentially the same as the tree itself. Second, we study the function that computes for all n the bottom word associated with n+1 from the one associated with n, and show that it is realised by an infinite sequential transducer whose underlying graph is once again essentially the same as the tree itself. An infinite word may be interpreted as an expansion in base p/q after the radix point, hence evaluated to a real number. If T is a subtree whose root is n, then the evaluations of the labels of the branches of T form an interval of $\mathbb{R}$. The length of this interval is called the span of n and is equal to the evaluation of the span-word of n. The set of all spans is then a subset of R and we use the preceding construction to study its topological closure. We show that it is an interval when p is greater than or equal to 2q-1, and a Cantor set of measure zero otherwise.]]> 0 <![CDATA[The construction of the random intersection graph model is based on a random family of sets. Such structures, which are derived from intersections of sets, appear in a natural manner in many applications. In this article we study the problem of finding a Hamilton cycle in a random intersection graph. To this end we analyse a classical algorithm for finding Hamilton cycles in random graphs (algorithm HAM) and study its efficiency on graphs from a family of random intersection graphs (denoted here by G(n,m,p)). We prove that the threshold function for the property of HAM constructing a Hamilton cycle in G(n,m,p) is the same as the threshold function for the minimum degree at least two. Until now, known algorithms for finding Hamilton cycles in G(n,m,p) were designed to work in very small ranges of parameters and, unlike HAM, used the structure of the family of random sets.]]> Mon, 05 Mar 2018 08:48:23 +0000 https://doi.org/10.23638/DMTCS-20-1-8 https://doi.org/10.23638/DMTCS-20-1-8 Rybarczyk, Katarzyna Rybarczyk, Katarzyna <![CDATA[The construction of the random intersection graph model is based on a random family of sets. Such structures, which are derived from intersections of sets, appear in a natural manner in many applications. In this article we study the problem of finding a Hamilton cycle in a random intersection graph. To this end we analyse a classical algorithm for finding Hamilton cycles in random graphs (algorithm HAM) and study its efficiency on graphs from a family of random intersection graphs (denoted here by G(n,m,p)). We prove that the threshold function for the property of HAM constructing a Hamilton cycle in G(n,m,p) is the same as the threshold function for the minimum degree at least two. Until now, known algorithms for finding Hamilton cycles in G(n,m,p) were designed to work in very small ranges of parameters and, unlike HAM, used the structure of the family of random sets.]]> 0 <![CDATA[Stanley lists the class of Dyck paths where all returns to the axis are of odd length as one of the many objects enumerated by (shifted) Catalan numbers. By the standard bijection in this context, these special Dyck paths correspond to a class of rooted plane trees, so-called Catalan-Stanley trees. This paper investigates a deterministic growth procedure for these trees by which any Catalan-Stanley tree can be grown from the tree of size one after some number of rounds; a parameter that will be referred to as the age of the tree. Asymptotic analyses are carried out for the age of a random Catalan-Stanley tree of given size as well as for the "speed" of the growth process by comparing the size of a given tree to the size of its ancestors.]]> Wed, 28 Feb 2018 15:37:21 +0000 https://doi.org/10.23638/DMTCS-20-1-11 https://doi.org/10.23638/DMTCS-20-1-11 Hackl, Benjamin Prodinger, Helmut Hackl, Benjamin Prodinger, Helmut <![CDATA[Stanley lists the class of Dyck paths where all returns to the axis are of odd length as one of the many objects enumerated by (shifted) Catalan numbers. By the standard bijection in this context, these special Dyck paths correspond to a class of rooted plane trees, so-called Catalan-Stanley trees. This paper investigates a deterministic growth procedure for these trees by which any Catalan-Stanley tree can be grown from the tree of size one after some number of rounds; a parameter that will be referred to as the age of the tree. Asymptotic analyses are carried out for the age of a random Catalan-Stanley tree of given size as well as for the "speed" of the growth process by comparing the size of a given tree to the size of its ancestors.]]> 0 <![CDATA[We review and extend what is known about the generating functions for consecutive pattern-avoiding permutations of length 4, 5 and beyond, and their asymptotic behaviour. There are respectively, seven length-4 and twenty-five length-5 consecutive-Wilf classes. D-finite differential equations are known for the reciprocal of the exponential generating functions for four of the length-4 and eight of the length-5 classes. We give the solutions of some of these ODEs. An unsolved functional equation is known for one more class of length-4, length-5 and beyond. We give the solution of this functional equation, and use it to show that the solution is not D-finite. For three further length-5 c-Wilf classes we give recurrences for two and a differential-functional equation for a third. For a fourth class we find a new algebraic solution. We give a polynomial-time algorithm to generate the coefficients of the generating functions which is faster than existing algorithms, and use this to (a) calculate the asymptotics for all classes of length 4 and length 5 to significantly greater precision than previously, and (b) use these extended series to search, unsuccessfully, for D-finite solutions for the unsolved classes, leading us to conjecture that the solutions are not D-finite. We have also searched, unsuccessfully, for differentially algebraic solutions.]]> Mon, 26 Feb 2018 09:53:53 +0000 https://doi.org/10.23638/DMTCS-19-2-8 https://doi.org/10.23638/DMTCS-19-2-8 Beaton, Nicholas R Conway, Andrew R Guttmann, Anthony J Beaton, Nicholas R Conway, Andrew R Guttmann, Anthony J <![CDATA[We review and extend what is known about the generating functions for consecutive pattern-avoiding permutations of length 4, 5 and beyond, and their asymptotic behaviour. There are respectively, seven length-4 and twenty-five length-5 consecutive-Wilf classes. D-finite differential equations are known for the reciprocal of the exponential generating functions for four of the length-4 and eight of the length-5 classes. We give the solutions of some of these ODEs. An unsolved functional equation is known for one more class of length-4, length-5 and beyond. We give the solution of this functional equation, and use it to show that the solution is not D-finite. For three further length-5 c-Wilf classes we give recurrences for two and a differential-functional equation for a third. For a fourth class we find a new algebraic solution. We give a polynomial-time algorithm to generate the coefficients of the generating functions which is faster than existing algorithms, and use this to (a) calculate the asymptotics for all classes of length 4 and length 5 to significantly greater precision than previously, and (b) use these extended series to search, unsuccessfully, for D-finite solutions for the unsolved classes, leading us to conjecture that the solutions are not D-finite. We have also searched, unsuccessfully, for differentially algebraic solutions.]]> 0 <![CDATA[Two mesh patterns are coincident if they are avoided by the same set of permutations, and are Wilf-equivalent if they have the same number of avoiders of each length. We provide sufficient conditions for coincidence of mesh patterns, when only permutations also avoiding a longer classical pattern are considered. Using these conditions we completely classify coincidences between families containing a mesh pattern of length 2 and a classical pattern of length 3. Furthermore, we completely Wilf-classify mesh patterns of length 2 inside the class of 231-avoiding permutations.]]> Fri, 09 Feb 2018 09:58:50 +0000 https://doi.org/10.23638/DMTCS-19-2-6 https://doi.org/10.23638/DMTCS-19-2-6 Tannock, Murray Ulfarsson, Henning Tannock, Murray Ulfarsson, Henning <![CDATA[Two mesh patterns are coincident if they are avoided by the same set of permutations, and are Wilf-equivalent if they have the same number of avoiders of each length. We provide sufficient conditions for coincidence of mesh patterns, when only permutations also avoiding a longer classical pattern are considered. Using these conditions we completely classify coincidences between families containing a mesh pattern of length 2 and a classical pattern of length 3. Furthermore, we completely Wilf-classify mesh patterns of length 2 inside the class of 231-avoiding permutations.]]> 0 <![CDATA[We consolidate what is currently known about packing densities of 4-point permutations and in the process improve the lower bounds for the packing densities of 1324 and 1342. We also provide rigorous upper bounds for the packing densities of 1324, 1342, and 2413. All our bounds are within $10^{-4}$ of the true packing densities. Together with the known bounds, this gives us a fairly complete picture of all 4-point packing densities. We also provide new upper bounds for several small permutations of length at least five. Our main tool for the upper bounds is the framework of flag algebras introduced by Razborov in 2007.]]> Tue, 06 Feb 2018 16:39:16 +0000 https://doi.org/10.23638/DMTCS-19-2-3 https://doi.org/10.23638/DMTCS-19-2-3 Sliacan, Jakub Stromquist, Walter Sliacan, Jakub Stromquist, Walter <![CDATA[We consolidate what is currently known about packing densities of 4-point permutations and in the process improve the lower bounds for the packing densities of 1324 and 1342. We also provide rigorous upper bounds for the packing densities of 1324, 1342, and 2413. All our bounds are within $10^{-4}$ of the true packing densities. Together with the known bounds, this gives us a fairly complete picture of all 4-point packing densities. We also provide new upper bounds for several small permutations of length at least five. Our main tool for the upper bounds is the framework of flag algebras introduced by Razborov in 2007.]]> 0 <![CDATA[We examine $t$-colourings of oriented graphs in which, for a fixed integer $k \geq 1$, vertices joined by a directed path of length at most $k$ must be assigned different colours. A homomorphism model that extends the ideas of Sherk for the case $k=2$ is described. Dichotomy theorems for the complexity of the problem of deciding, for fixed $k$ and $t$, whether there exists such a $t$-colouring are proved.]]> Thu, 01 Feb 2018 13:42:05 +0000 https://doi.org/10.23638/DMTCS-20-1-6 https://doi.org/10.23638/DMTCS-20-1-6 Duffy, Christopher MacGillivray, Gary Sopena, Éric Duffy, Christopher MacGillivray, Gary Sopena, Éric <![CDATA[We examine $t$-colourings of oriented graphs in which, for a fixed integer $k \geq 1$, vertices joined by a directed path of length at most $k$ must be assigned different colours. A homomorphism model that extends the ideas of Sherk for the case $k=2$ is described. Dichotomy theorems for the complexity of the problem of deciding, for fixed $k$ and $t$, whether there exists such a $t$-colouring are proved.]]> 0 <![CDATA[The goal of this paper is to prove that several variants of deciding whether a poset can be (weakly) embedded into a small Boolean lattice, or to a few consecutive levels of a Boolean lattice, are NP-complete, answering a question of Griggs and of Patkos. As an equivalent reformulation of one of these problems, we also derive that it is NP-complete to decide whether a given graph can be embedded to the two middle levels of some hypercube.]]> Wed, 24 Jan 2018 13:06:55 +0000 https://doi.org/10.23638/DMTCS-20-1-7 https://doi.org/10.23638/DMTCS-20-1-7 Pálvölgyi, Dömötör Pálvölgyi, Dömötör <![CDATA[The goal of this paper is to prove that several variants of deciding whether a poset can be (weakly) embedded into a small Boolean lattice, or to a few consecutive levels of a Boolean lattice, are NP-complete, answering a question of Griggs and of Patkos. As an equivalent reformulation of one of these problems, we also derive that it is NP-complete to decide whether a given graph can be embedded to the two middle levels of some hypercube.]]> 0 <![CDATA[Matchings are frequently used to model RNA secondary structures; however, not all matchings can be realized as RNA motifs. One class of matchings, called the L $\&$ P matchings, is the most restrictive model for RNA secondary structures in the Largest Hairpin Family (LHF). The L $\&$ P matchings were enumerated in $2015$ by Jefferson, and they are equinumerous with the set of nesting-similarity classes of matchings, enumerated by Klazar. We provide a bijection between these two sets. This bijection preserves noncrossing matchings, and preserves the sequence obtained reading left to right of whether an edge begins or ends at that vertex.]]> Mon, 22 Jan 2018 07:55:56 +0000 https://doi.org/10.23638/DMTCS-19-2-1 https://doi.org/10.23638/DMTCS-19-2-1 Martinez, Megan A. Riehl, Manda Martinez, Megan A. Riehl, Manda <![CDATA[Matchings are frequently used to model RNA secondary structures; however, not all matchings can be realized as RNA motifs. One class of matchings, called the L $\&$ P matchings, is the most restrictive model for RNA secondary structures in the Largest Hairpin Family (LHF). The L $\&$ P matchings were enumerated in $2015$ by Jefferson, and they are equinumerous with the set of nesting-similarity classes of matchings, enumerated by Klazar. We provide a bijection between these two sets. This bijection preserves noncrossing matchings, and preserves the sequence obtained reading left to right of whether an edge begins or ends at that vertex.]]> 0 <![CDATA[The Erd\H{o}s-P\'osa property relates parameters of covering and packing of combinatorial structures and has been mostly studied in the setting of undirected graphs. In this note, we use results of Chudnovsky, Fradkin, Kim, and Seymour to show that, for every directed graph $H$ (resp. strongly-connected directed graph $H$), the class of directed graphs that contain $H$ as a strong minor (resp. butterfly minor, topological minor) has the vertex-Erd\H{o}s-P\'osa property in the class of tournaments. We also prove that if $H$ is a strongly-connected directed graph, the class of directed graphs containing $H$ as an immersion has the edge-Erd\H{o}s-P\'osa property in the class of tournaments.]]> Wed, 17 Jan 2018 08:38:58 +0000 https://doi.org/10.23638/DMTCS-20-1-5 https://doi.org/10.23638/DMTCS-20-1-5 Raymond, Jean-Florent Raymond, Jean-Florent <![CDATA[The Erd\H{o}s-P\'osa property relates parameters of covering and packing of combinatorial structures and has been mostly studied in the setting of undirected graphs. In this note, we use results of Chudnovsky, Fradkin, Kim, and Seymour to show that, for every directed graph $H$ (resp. strongly-connected directed graph $H$), the class of directed graphs that contain $H$ as a strong minor (resp. butterfly minor, topological minor) has the vertex-Erd\H{o}s-P\'osa property in the class of tournaments. We also prove that if $H$ is a strongly-connected directed graph, the class of directed graphs containing $H$ as an immersion has the edge-Erd\H{o}s-P\'osa property in the class of tournaments.]]> 0 <![CDATA[We introduce a natural variant of the parallel chip-firing game, called the diffusion game. Chips are initially assigned to vertices of a graph. At every step, all vertices simultaneously send one chip to each neighbour with fewer chips. As the dynamics of the parallel chip-firing game occur on a finite set the process is inherently periodic. However the diffusion game is not obviously periodic: even if $2|E(G)|$ chips are assigned to vertices of graph G, there may exist time steps where some vertices have a negative number of chips. We investigate the process, prove periodicity for a number of graph classes, and pose some questions for future research.]]> Wed, 17 Jan 2018 08:30:10 +0000 https://doi.org/10.23638/DMTCS-20-1-4 https://doi.org/10.23638/DMTCS-20-1-4 Duffy, C. Lidbetter, T. F. Messinger, M. E. Nowakowski, R. J. Duffy, C. Lidbetter, T. F. Messinger, M. E. Nowakowski, R. J. <![CDATA[We introduce a natural variant of the parallel chip-firing game, called the diffusion game. Chips are initially assigned to vertices of a graph. At every step, all vertices simultaneously send one chip to each neighbour with fewer chips. As the dynamics of the parallel chip-firing game occur on a finite set the process is inherently periodic. However the diffusion game is not obviously periodic: even if $2|E(G)|$ chips are assigned to vertices of graph G, there may exist time steps where some vertices have a negative number of chips. We investigate the process, prove periodicity for a number of graph classes, and pose some questions for future research.]]> 0 <![CDATA[In connection with Fulkerson's conjecture on cycle covers, Fan and Raspaud proposed a weaker conjecture: For every bridgeless cubic graph $G$, there are three perfect matchings $M_1$, $M_2$, and $M_3$ such that $M_1\cap M_2 \cap M_3=\emptyset$. We call the property specified in this conjecture the three matching intersection property (and 3PM property for short). We study this property on matching covered graphs. The main results are a necessary and sufficient condition and its applications to characterization of special graphs, such as the Halin graphs and 4-regular graphs.]]> Mon, 15 Jan 2018 10:43:15 +0000 https://doi.org/10.23638/DMTCS-19-3-16 https://doi.org/10.23638/DMTCS-19-3-16 Lin, Hao Wang, Xiumei Lin, Hao Wang, Xiumei <![CDATA[In connection with Fulkerson's conjecture on cycle covers, Fan and Raspaud proposed a weaker conjecture: For every bridgeless cubic graph $G$, there are three perfect matchings $M_1$, $M_2$, and $M_3$ such that $M_1\cap M_2 \cap M_3=\emptyset$. We call the property specified in this conjecture the three matching intersection property (and 3PM property for short). We study this property on matching covered graphs. The main results are a necessary and sufficient condition and its applications to characterization of special graphs, such as the Halin graphs and 4-regular graphs.]]> 0 <![CDATA[We study the computational complexity of several problems connected with finding a maximal distance-$k$ matching of minimum cardinality or minimum weight in a given graph. We introduce the class of $k$-equimatchable graphs which is an edge analogue of $k$-equipackable graphs. We prove that the recognition of $k$-equimatchable graphs is co-NP-complete for any fixed $k \ge 2$. We provide a simple characterization for the class of strongly chordal graphs with equal $k$-packing and $k$-domination numbers. We also prove that for any fixed integer $\ell \ge 1$ the problem of finding a minimum weight maximal distance-$2\ell$ matching and the problem of finding a minimum weight $(2 \ell - 1)$-independent dominating set cannot be approximated in polynomial time in chordal graphs within a factor of $\delta \ln |V(G)|$ unless $\mathrm{P} = \mathrm{NP}$, where $\delta$ is a fixed constant (thereby improving the NP-hardness result of Chang for the independent domination case). Finally, we show the NP-hardness of the minimum maximal induced matching and independent dominating set problems in large-girth planar graphs.]]> Thu, 11 Jan 2018 13:10:17 +0000 https://doi.org/10.23638/DMTCS-20-1-3 https://doi.org/10.23638/DMTCS-20-1-3 Kartynnik, Yury Ryzhikov, Andrew Kartynnik, Yury Ryzhikov, Andrew <![CDATA[We study the computational complexity of several problems connected with finding a maximal distance-$k$ matching of minimum cardinality or minimum weight in a given graph. We introduce the class of $k$-equimatchable graphs which is an edge analogue of $k$-equipackable graphs. We prove that the recognition of $k$-equimatchable graphs is co-NP-complete for any fixed $k \ge 2$. We provide a simple characterization for the class of strongly chordal graphs with equal $k$-packing and $k$-domination numbers. We also prove that for any fixed integer $\ell \ge 1$ the problem of finding a minimum weight maximal distance-$2\ell$ matching and the problem of finding a minimum weight $(2 \ell - 1)$-independent dominating set cannot be approximated in polynomial time in chordal graphs within a factor of $\delta \ln |V(G)|$ unless $\mathrm{P} = \mathrm{NP}$, where $\delta$ is a fixed constant (thereby improving the NP-hardness result of Chang for the independent domination case). Finally, we show the NP-hardness of the minimum maximal induced matching and independent dominating set problems in large-girth planar graphs.]]> 0 <![CDATA[Given a tree and a set P of non-trivial simple paths on it, VPT(P) is the VPT graph (i.e. the vertex intersection graph) of the paths P, and EPT(P) is the EPT graph (i.e. the edge intersection graph) of P. These graphs have been extensively studied in the literature. Given two (edge) intersecting paths in a graph, their split vertices is the set of vertices having degree at least 3 in their union. A pair of (edge) intersecting paths is termed non-splitting if they do not have split vertices (namely if their union is a path). We define the graph ENPT(P) of edge intersecting non-splitting paths of a tree, termed the ENPT graph, as the graph having a vertex for each path in P, and an edge between every pair of vertices representing two paths that are both edge-intersecting and non-splitting. A graph G is an ENPT graph if there is a tree T and a set of paths P of T such that G=ENPT(P), and we say that is a representation of G. Our goal is to characterize the representation of chordless ENPT cycles (holes). To achieve this goal, we first assume that the EPT graph induced by the vertices of an ENPT hole is given. In [2] we introduce three assumptions (P1), (P2), (P3) defined on EPT, ENPT pairs of graphs. In the same study, we define two problems HamiltonianPairRec, P3-HamiltonianPairRec and characterize the representations of ENPT holes that satisfy (P1), (P2), (P3). In this work, we continue our work by relaxing these three assumptions one by one. We characterize the representations of ENPT holes satisfying (P3) by providing a polynomial-time algorithm to solve P3-HamiltonianPairRec. We also show that there does not exist a polynomial-time algorithm to solve HamiltonianPairRec, unless P=NP.]]> Thu, 11 Jan 2018 12:36:58 +0000 https://doi.org/10.23638/DMTCS-20-1-2 https://doi.org/10.23638/DMTCS-20-1-2 Boyacı, Arman Ekim, Tınaz Shalom, Mordechai Zaks, Shmuel Boyacı, Arman Ekim, Tınaz Shalom, Mordechai Zaks, Shmuel <![CDATA[Given a tree and a set P of non-trivial simple paths on it, VPT(P) is the VPT graph (i.e. the vertex intersection graph) of the paths P, and EPT(P) is the EPT graph (i.e. the edge intersection graph) of P. These graphs have been extensively studied in the literature. Given two (edge) intersecting paths in a graph, their split vertices is the set of vertices having degree at least 3 in their union. A pair of (edge) intersecting paths is termed non-splitting if they do not have split vertices (namely if their union is a path). We define the graph ENPT(P) of edge intersecting non-splitting paths of a tree, termed the ENPT graph, as the graph having a vertex for each path in P, and an edge between every pair of vertices representing two paths that are both edge-intersecting and non-splitting. A graph G is an ENPT graph if there is a tree T and a set of paths P of T such that G=ENPT(P), and we say that is a representation of G. Our goal is to characterize the representation of chordless ENPT cycles (holes). To achieve this goal, we first assume that the EPT graph induced by the vertices of an ENPT hole is given. In [2] we introduce three assumptions (P1), (P2), (P3) defined on EPT, ENPT pairs of graphs. In the same study, we define two problems HamiltonianPairRec, P3-HamiltonianPairRec and characterize the representations of ENPT holes that satisfy (P1), (P2), (P3). In this work, we continue our work by relaxing these three assumptions one by one. We characterize the representations of ENPT holes satisfying (P3) by providing a polynomial-time algorithm to solve P3-HamiltonianPairRec. We also show that there does not exist a polynomial-time algorithm to solve HamiltonianPairRec, unless P=NP.]]> 0 <![CDATA[We study the following problem: Given $k$ paths that share the same vertex set, is there a simultaneous geometric embedding of these paths such that each individual drawing is monotone in some direction? We prove that for any dimension $d\geq 2$, there is a set of $d + 1$ paths that does not admit a monotone simultaneous geometric embedding.]]> Fri, 05 Jan 2018 10:41:38 +0000 https://doi.org/10.23638/DMTCS-20-1-1 https://doi.org/10.23638/DMTCS-20-1-1 Bremner, David Devillers, Olivier Glisse, Marc Lazard, Sylvain Liotta, Giuseppe Mchedlidze, Tamara Moroz, Guillaume Whitesides, Sue Wismath, Stephen Bremner, David Devillers, Olivier Glisse, Marc Lazard, Sylvain Liotta, Giuseppe Mchedlidze, Tamara Moroz, Guillaume Whitesides, Sue Wismath, Stephen <![CDATA[We study the following problem: Given $k$ paths that share the same vertex set, is there a simultaneous geometric embedding of these paths such that each individual drawing is monotone in some direction? We prove that for any dimension $d\geq 2$, there is a set of $d + 1$ paths that does not admit a monotone simultaneous geometric embedding.]]> 0 <![CDATA[We study a randomized algorithm for graph domination, by which, according to a uniformly chosen permutation, vertices are revealed and added to the dominating set if not already dominated. We determine the expected size of the dominating set produced by the algorithm for the path graph $P_n$ and use this to derive the expected size for some related families of graphs. We then provide a much-refined analysis of the worst and best cases of this algorithm on $P_n$ and enumerate the permutations for which the algorithm has the worst-possible performance and best-possible performance. The case of dominating the path graph has connections to previous work of Bouwer and Star, and of Gessel on greedily coloring the path.]]> Wed, 20 Dec 2017 10:48:15 +0000 https://doi.org/10.23638/DMTCS-19-2-2 https://doi.org/10.23638/DMTCS-19-2-2 Coscia, Christopher DeWitt, Jonathan Yang, Fan Zhang, Yiguang Coscia, Christopher DeWitt, Jonathan Yang, Fan Zhang, Yiguang <![CDATA[We study a randomized algorithm for graph domination, by which, according to a uniformly chosen permutation, vertices are revealed and added to the dominating set if not already dominated. We determine the expected size of the dominating set produced by the algorithm for the path graph $P_n$ and use this to derive the expected size for some related families of graphs. We then provide a much-refined analysis of the worst and best cases of this algorithm on $P_n$ and enumerate the permutations for which the algorithm has the worst-possible performance and best-possible performance. The case of dominating the path graph has connections to previous work of Bouwer and Star, and of Gessel on greedily coloring the path.]]> 0 <![CDATA[Total dominating set, connected vertex cover and Steiner tree are well-known graph problems. Despite the fact that they are NP-complete to optimize, it is easy (even trivial) to find solutions, regardless of their size. In this paper, we study a variant of these problems by adding conflicts, that are pairs of vertices that cannot be both in a solution. This new constraint leads to situations where it is NP-complete to decide if there exists a solution avoiding conflicts. This paper proposes NP-completeness proofs of the existence of a solution for different restricted classes of graphs and conflicts, improving recent results. We also propose polynomial time constructions in several restricted cases and we introduce a new parameter, the stretch, to capture the locality of the conflicts.]]> Wed, 20 Dec 2017 10:43:45 +0000 https://doi.org/10.23638/DMTCS-19-3-17 https://doi.org/10.23638/DMTCS-19-3-17 Cornet, Alexis Laforest, Christian Cornet, Alexis Laforest, Christian <![CDATA[Total dominating set, connected vertex cover and Steiner tree are well-known graph problems. Despite the fact that they are NP-complete to optimize, it is easy (even trivial) to find solutions, regardless of their size. In this paper, we study a variant of these problems by adding conflicts, that are pairs of vertices that cannot be both in a solution. This new constraint leads to situations where it is NP-complete to decide if there exists a solution avoiding conflicts. This paper proposes NP-completeness proofs of the existence of a solution for different restricted classes of graphs and conflicts, improving recent results. We also propose polynomial time constructions in several restricted cases and we introduce a new parameter, the stretch, to capture the locality of the conflicts.]]> 0 <![CDATA[In this article we focus on the parameterized complexity of the Multidimensional Binary Vector Assignment problem (called \BVA). An input of this problem is defined by $m$ disjoint sets $V^1, V^2, \dots, V^m$, each composed of $n$ binary vectors of size $p$. An output is a set of $n$ disjoint $m$-tuples of vectors, where each $m$-tuple is obtained by picking one vector from each set $V^i$. To each $m$-tuple we associate a $p$ dimensional vector by applying the bit-wise AND operation on the $m$ vectors of the tuple. The objective is to minimize the total number of zeros in these $n$ vectors. mBVA can be seen as a variant of multidimensional matching where hyperedges are implicitly locally encoded via labels attached to vertices, but was originally introduced in the context of integrated circuit manufacturing. We provide for this problem FPT algorithms and negative results ($ETH$-based results, $W$[2]-hardness and a kernel lower bound) according to several parameters: the standard parameter $k$ i.e. the total number of zeros), as well as two parameters above some guaranteed values.]]> Wed, 20 Dec 2017 10:33:20 +0000 https://doi.org/10.23638/DMTCS-19-4-3 https://doi.org/10.23638/DMTCS-19-4-3 Bougeret, Marin Duvillié, Guillerme Giroudeau, Rodolphe Watrigant, Rémi Bougeret, Marin Duvillié, Guillerme Giroudeau, Rodolphe Watrigant, Rémi <![CDATA[In this article we focus on the parameterized complexity of the Multidimensional Binary Vector Assignment problem (called \BVA). An input of this problem is defined by $m$ disjoint sets $V^1, V^2, \dots, V^m$, each composed of $n$ binary vectors of size $p$. An output is a set of $n$ disjoint $m$-tuples of vectors, where each $m$-tuple is obtained by picking one vector from each set $V^i$. To each $m$-tuple we associate a $p$ dimensional vector by applying the bit-wise AND operation on the $m$ vectors of the tuple. The objective is to minimize the total number of zeros in these $n$ vectors. mBVA can be seen as a variant of multidimensional matching where hyperedges are implicitly locally encoded via labels attached to vertices, but was originally introduced in the context of integrated circuit manufacturing. We provide for this problem FPT algorithms and negative results ($ETH$-based results, $W$[2]-hardness and a kernel lower bound) according to several parameters: the standard parameter $k$ i.e. the total number of zeros), as well as two parameters above some guaranteed values.]]> 0 <![CDATA[For a variety of pattern-avoiding classes, we describe the limiting distribution for the number of fixed points for involutions chosen uniformly at random from that class. In particular we consider monotone patterns of arbitrary length as well as all patterns of length 3. For monotone patterns we utilize the connection with standard Young tableaux with at most $k$ rows and involutions avoiding a monotone pattern of length $k$. For every pattern of length 3 we give the bivariate generating function with respect to fixed points for the involutions that avoid that pattern, and where applicable apply tools from analytic combinatorics to extract information about the limiting distribution from the generating function. Many well-known distributions appear.]]> Mon, 11 Dec 2017 12:37:42 +0000 https://doi.org/10.23638/DMTCS-19-2-5 https://doi.org/10.23638/DMTCS-19-2-5 Miner, Samuel Rizzolo, Douglas Slivken, Erik Miner, Samuel Rizzolo, Douglas Slivken, Erik <![CDATA[For a variety of pattern-avoiding classes, we describe the limiting distribution for the number of fixed points for involutions chosen uniformly at random from that class. In particular we consider monotone patterns of arbitrary length as well as all patterns of length 3. For monotone patterns we utilize the connection with standard Young tableaux with at most $k$ rows and involutions avoiding a monotone pattern of length $k$. For every pattern of length 3 we give the bivariate generating function with respect to fixed points for the involutions that avoid that pattern, and where applicable apply tools from analytic combinatorics to extract information about the limiting distribution from the generating function. Many well-known distributions appear.]]> 0 <![CDATA[The separability problem for word languages of a class $\mathcal{C}$ by languages of a class $\mathcal{S}$ asks, for two given languages $I$ and $E$ from $\mathcal{C}$, whether there exists a language $S$ from $\mathcal{S}$ that includes $I$ and excludes $E$, that is, $I \subseteq S$ and $S\cap E = \emptyset$. In this work, we assume some mild closure properties for $\mathcal{C}$ and study for which such classes separability by a piecewise testable language (PTL) is decidable. We characterize these classes in terms of decidability of (two variants of) an unboundedness problem. From this, we deduce that separability by PTL is decidable for a number of language classes, such as the context-free languages and languages of labeled vector addition systems. Furthermore, it follows that separability by PTL is decidable if and only if one can compute for any language of the class its downward closure wrt. the scattered substring ordering (i.e., if the set of scattered substrings of any language of the class is effectively regular). The obtained decidability results contrast some undecidability results. In fact, for all (non-regular) language classes that we present as examples with decidable separability, it is undecidable whether a given language is a PTL itself. Our characterization involves a result of independent interest, which states that for any kind of languages $I$ and $E$, non-separability by PTL is equivalent to the existence of common patterns in $I$ and $E$.]]> Mon, 11 Dec 2017 10:46:30 +0000 https://doi.org/10.23638/DMTCS-19-4-1 https://doi.org/10.23638/DMTCS-19-4-1 Czerwiński, Wojciech Martens, Wim van Rooijen, Lorijn Zeitoun, Marc Zetzsche, Georg Czerwiński, Wojciech Martens, Wim van Rooijen, Lorijn Zeitoun, Marc Zetzsche, Georg <![CDATA[The separability problem for word languages of a class $\mathcal{C}$ by languages of a class $\mathcal{S}$ asks, for two given languages $I$ and $E$ from $\mathcal{C}$, whether there exists a language $S$ from $\mathcal{S}$ that includes $I$ and excludes $E$, that is, $I \subseteq S$ and $S\cap E = \emptyset$. In this work, we assume some mild closure properties for $\mathcal{C}$ and study for which such classes separability by a piecewise testable language (PTL) is decidable. We characterize these classes in terms of decidability of (two variants of) an unboundedness problem. From this, we deduce that separability by PTL is decidable for a number of language classes, such as the context-free languages and languages of labeled vector addition systems. Furthermore, it follows that separability by PTL is decidable if and only if one can compute for any language of the class its downward closure wrt. the scattered substring ordering (i.e., if the set of scattered substrings of any language of the class is effectively regular). The obtained decidability results contrast some undecidability results. In fact, for all (non-regular) language classes that we present as examples with decidable separability, it is undecidable whether a given language is a PTL itself. Our characterization involves a result of independent interest, which states that for any kind of languages $I$ and $E$, non-separability by PTL is equivalent to the existence of common patterns in $I$ and $E$.]]> 0 <![CDATA[A permutation class $C$ is splittable if it is contained in a merge of two of its proper subclasses, and it is 1-amalgamable if given two permutations $\sigma$ and $\tau$ in $C$, each with a marked element, we can find a permutation $\pi$ in $C$ containing both $\sigma$ and $\tau$ such that the two marked elements coincide. It was previously shown that unsplittability implies 1-amalgamability. We prove that unsplittability and 1-amalgamability are not equivalent properties of permutation classes by showing that the class $Av(1423, 1342)$ is both splittable and 1-amalgamable. Our construction is based on the concept of LR-inflations, which we introduce here and which may be of independent interest.]]> Tue, 05 Dec 2017 09:56:28 +0000 https://doi.org/10.23638/DMTCS-19-2-4 https://doi.org/10.23638/DMTCS-19-2-4 Jelínek, Vít Opler, Michal Jelínek, Vít Opler, Michal <![CDATA[A permutation class $C$ is splittable if it is contained in a merge of two of its proper subclasses, and it is 1-amalgamable if given two permutations $\sigma$ and $\tau$ in $C$, each with a marked element, we can find a permutation $\pi$ in $C$ containing both $\sigma$ and $\tau$ such that the two marked elements coincide. It was previously shown that unsplittability implies 1-amalgamability. We prove that unsplittability and 1-amalgamability are not equivalent properties of permutation classes by showing that the class $Av(1423, 1342)$ is both splittable and 1-amalgamable. Our construction is based on the concept of LR-inflations, which we introduce here and which may be of independent interest.]]> 0 <![CDATA[Fix an integer partition lambda that has no more than n parts. Let beta be a weakly increasing n-tuple with entries from {1,..,n}. The flagged Schur function indexed by lambda and beta is a polynomial generating function in x_1, .., x_n for certain semistandard tableaux of shape lambda. Let pi be an n-permutation. The type A Demazure character (key polynomial, Demazure polynomial) indexed by lambda and pi is another such polynomial generating function. Reiner and Shimozono and then Postnikov and Stanley studied coincidences between these two families of polynomials. Here their results are sharpened by the specification of unique representatives for the equivalence classes of indexes for both families of polynomials, extended by the consideration of more general beta, and deepened by proving that the polynomial coincidences also hold at the level of the underlying tableau sets. Let R be the set of lengths of columns in the shape of lambda that are less than n. Ordered set partitions of {1,..,n} with block sizes determined by R, called R-permutations, are used to describe the minimal length representatives for the parabolic quotient of the nth symmetric group specified by the set {1,..,n-1}\R. The notion of 312-avoidance is generalized from n-permutations to these set partitions. The R-parabolic Catalan number is defined to be the number of these. Every flagged Schur function arises as a Demazure polynomial. Those Demazure polynomials are precisely indexed by the R-312-avoiding R-permutations. Hence the number of flagged Schur functions that are distinct as polynomials is shown to be the R-parabolic Catalan number. The projecting and lifting processes that relate the notions of 312-avoidance and of R-312-avoidance are described with maps developed for other purposes.]]> Tue, 05 Dec 2017 09:50:02 +0000 https://doi.org/10.23638/DMTCS-19-3-15 https://doi.org/10.23638/DMTCS-19-3-15 Proctor, Robert A. Willis, Matthew J. Proctor, Robert A. Willis, Matthew J. <![CDATA[Fix an integer partition lambda that has no more than n parts. Let beta be a weakly increasing n-tuple with entries from {1,..,n}. The flagged Schur function indexed by lambda and beta is a polynomial generating function in x_1, .., x_n for certain semistandard tableaux of shape lambda. Let pi be an n-permutation. The type A Demazure character (key polynomial, Demazure polynomial) indexed by lambda and pi is another such polynomial generating function. Reiner and Shimozono and then Postnikov and Stanley studied coincidences between these two families of polynomials. Here their results are sharpened by the specification of unique representatives for the equivalence classes of indexes for both families of polynomials, extended by the consideration of more general beta, and deepened by proving that the polynomial coincidences also hold at the level of the underlying tableau sets. Let R be the set of lengths of columns in the shape of lambda that are less than n. Ordered set partitions of {1,..,n} with block sizes determined by R, called R-permutations, are used to describe the minimal length representatives for the parabolic quotient of the nth symmetric group specified by the set {1,..,n-1}\R. The notion of 312-avoidance is generalized from n-permutations to these set partitions. The R-parabolic Catalan number is defined to be the number of these. Every flagged Schur function arises as a Demazure polynomial. Those Demazure polynomials are precisely indexed by the R-312-avoiding R-permutations. Hence the number of flagged Schur functions that are distinct as polynomials is shown to be the R-parabolic Catalan number. The projecting and lifting processes that relate the notions of 312-avoidance and of R-312-avoidance are described with maps developed for other purposes.]]> 0 <![CDATA[We deal with the problem of maintaining a shortest-path tree rooted at some process r in a network that may be disconnected after topological changes. The goal is then to maintain a shortest-path tree rooted at r in its connected component, V_r, and make all processes of other components detecting that r is not part of their connected component. We propose, in the composite atomicity model, a silent self-stabilizing algorithm for this problem working in semi-anonymous networks, where edges have strictly positive weights. This algorithm does not require any a priori knowledge about global parameters of the network. We prove its correctness assuming the distributed unfair daemon, the most general daemon. Its stabilization time in rounds is at most 3nmax+D, where nmax is the maximum number of non-root processes in a connected component and D is the hop-diameter of V_r. Furthermore, if we additionally assume that edge weights are positive integers, then it stabilizes in a polynomial number of steps: namely, we exhibit a bound in O(maxi nmax^3 n), where maxi is the maximum weight of an edge and n is the number of processes.]]> Thu, 30 Nov 2017 16:05:03 +0000 https://doi.org/10.23638/DMTCS-19-3-14 https://doi.org/10.23638/DMTCS-19-3-14 Devismes, Stéphane Ilcinkas, David Johnen, Colette Devismes, Stéphane Ilcinkas, David Johnen, Colette <![CDATA[We deal with the problem of maintaining a shortest-path tree rooted at some process r in a network that may be disconnected after topological changes. The goal is then to maintain a shortest-path tree rooted at r in its connected component, V_r, and make all processes of other components detecting that r is not part of their connected component. We propose, in the composite atomicity model, a silent self-stabilizing algorithm for this problem working in semi-anonymous networks, where edges have strictly positive weights. This algorithm does not require any a priori knowledge about global parameters of the network. We prove its correctness assuming the distributed unfair daemon, the most general daemon. Its stabilization time in rounds is at most 3nmax+D, where nmax is the maximum number of non-root processes in a connected component and D is the hop-diameter of V_r. Furthermore, if we additionally assume that edge weights are positive integers, then it stabilizes in a polynomial number of steps: namely, we exhibit a bound in O(maxi nmax^3 n), where maxi is the maximum weight of an edge and n is the number of processes.]]> 0 <![CDATA[A binary triangle of size $n$ is a triangle of zeroes and ones, with $n$ rows, built with the same local rule as the standard Pascal triangle modulo $2$. A binary triangle is said to be balanced if the absolute difference between the numbers of zeroes and ones that constitute this triangle is at most $1$. In this paper, the existence of balanced binary triangles of size $n$, for all positive integers $n$, is shown. This is achieved by considering periodic balanced binary triangles, that are balanced binary triangles where each row, column or diagonal is a periodic sequence.]]> Tue, 28 Nov 2017 19:32:42 +0000 https://doi.org/10.23638/DMTCS-19-3-13 https://doi.org/10.23638/DMTCS-19-3-13 Chappelon, Jonathan Chappelon, Jonathan <![CDATA[A binary triangle of size $n$ is a triangle of zeroes and ones, with $n$ rows, built with the same local rule as the standard Pascal triangle modulo $2$. A binary triangle is said to be balanced if the absolute difference between the numbers of zeroes and ones that constitute this triangle is at most $1$. In this paper, the existence of balanced binary triangles of size $n$, for all positive integers $n$, is shown. This is achieved by considering periodic balanced binary triangles, that are balanced binary triangles where each row, column or diagonal is a periodic sequence.]]> 0 <![CDATA[In this paper we introduce and study a new family of combinatorial simplicial complexes, which we call immediate snapshot complexes. Our construction and terminology is strongly motivated by theoretical distributed computing, as these complexes are combinatorial models of the standard protocol complexes associated to immediate snapshot read/write shared memory communication model. In order to define the immediate snapshot complexes we need a new combinatorial object, which we call a witness structure. These objects are indexing the simplices in the immediate snapshot complexes, while a special operation on them, called ghosting, describes the combinatorics of taking simplicial boundary. In general, we develop the theory of witness structures and use it to prove several combinatorial as well as topological properties of the immediate snapshot complexes.]]> Tue, 28 Nov 2017 10:19:21 +0000 https://doi.org/10.23638/DMTCS-19-3-12 https://doi.org/10.23638/DMTCS-19-3-12 Kozlov, Dmitry N. Kozlov, Dmitry N. <![CDATA[In this paper we introduce and study a new family of combinatorial simplicial complexes, which we call immediate snapshot complexes. Our construction and terminology is strongly motivated by theoretical distributed computing, as these complexes are combinatorial models of the standard protocol complexes associated to immediate snapshot read/write shared memory communication model. In order to define the immediate snapshot complexes we need a new combinatorial object, which we call a witness structure. These objects are indexing the simplices in the immediate snapshot complexes, while a special operation on them, called ghosting, describes the combinatorics of taking simplicial boundary. In general, we develop the theory of witness structures and use it to prove several combinatorial as well as topological properties of the immediate snapshot complexes.]]> 0 <![CDATA[We describe a new type of sufficient condition for a balanced bipartite digraph to be hamiltonian. Let $D$ be a balanced bipartite digraph and $x,y$ be distinct vertices in $D$. $\{x, y\}$ dominates a vertex $z$ if $x\rightarrow z$ and $y\rightarrow z$; in this case, we call the pair $\{x, y\}$ dominating. In this paper, we prove that a strong balanced bipartite digraph $D$ on $2a$ vertices contains a hamiltonian cycle if, for every dominating pair of vertices $\{x, y\}$, either $d(x)\ge 2a-1$ and $d(y)\ge a+1$ or $d(x)\ge a+1$ and $d(y)\ge 2a-1$. The lower bound in the result is sharp.]]> Fri, 10 Nov 2017 12:31:54 +0000 https://doi.org/10.23638/DMTCS-19-3-11 https://doi.org/10.23638/DMTCS-19-3-11 Wang, Ruixia Wang, Ruixia <![CDATA[We describe a new type of sufficient condition for a balanced bipartite digraph to be hamiltonian. Let $D$ be a balanced bipartite digraph and $x,y$ be distinct vertices in $D$. $\{x, y\}$ dominates a vertex $z$ if $x\rightarrow z$ and $y\rightarrow z$; in this case, we call the pair $\{x, y\}$ dominating. In this paper, we prove that a strong balanced bipartite digraph $D$ on $2a$ vertices contains a hamiltonian cycle if, for every dominating pair of vertices $\{x, y\}$, either $d(x)\ge 2a-1$ and $d(y)\ge a+1$ or $d(x)\ge a+1$ and $d(y)\ge 2a-1$. The lower bound in the result is sharp.]]> 0 <![CDATA[A pair of non-adjacent edges is said to be separated in a circular ordering of vertices, if the endpoints of the two edges do not alternate in the ordering. The circular separation dimension of a graph $G$, denoted by $\pi^\circ(G)$, is the minimum number of circular orderings of the vertices of $G$ such that every pair of non-adjacent edges is separated in at least one of the circular orderings. This notion is introduced by Loeb and West in their recent paper. In this article, we consider two subclasses of planar graphs, namely $2$-outerplanar graphs and series-parallel graphs. A $2$-outerplanar graph has a planar embedding such that the subgraph obtained by removal of the vertices of the exterior face is outerplanar. We prove that if $G$ is $2$-outerplanar then $\pi^\circ(G) = 2$. We also prove that if $G$ is a series-parallel graph then $\pi^\circ(G) \leq 2$.]]> Fri, 03 Nov 2017 12:57:46 +0000 https://doi.org/10.23638/DMTCS-19-3-8 https://doi.org/10.23638/DMTCS-19-3-8 Bharathi, Arpitha P. De, Minati Lahiri, Abhiruk Bharathi, Arpitha P. De, Minati Lahiri, Abhiruk <![CDATA[A pair of non-adjacent edges is said to be separated in a circular ordering of vertices, if the endpoints of the two edges do not alternate in the ordering. The circular separation dimension of a graph $G$, denoted by $\pi^\circ(G)$, is the minimum number of circular orderings of the vertices of $G$ such that every pair of non-adjacent edges is separated in at least one of the circular orderings. This notion is introduced by Loeb and West in their recent paper. In this article, we consider two subclasses of planar graphs, namely $2$-outerplanar graphs and series-parallel graphs. A $2$-outerplanar graph has a planar embedding such that the subgraph obtained by removal of the vertices of the exterior face is outerplanar. We prove that if $G$ is $2$-outerplanar then $\pi^\circ(G) = 2$. We also prove that if $G$ is a series-parallel graph then $\pi^\circ(G) \leq 2$.]]> 0 <![CDATA[We study Bennett deep sequences in the context of recursion theory; in particular we investigate the notions of O(1)-deepK, O(1)-deepC , order-deep K and order-deep C sequences. Our main results are that Martin-Loef random sets are not order-deepC , that every many-one degree contains a set which is not O(1)-deepC , that O(1)-deepC sets and order-deepK sets have high or DNR Turing degree and that no K-trival set is O(1)-deepK.]]> Thu, 26 Oct 2017 08:22:26 +0000 https://doi.org/10.23638/DMTCS-19-4-2 https://doi.org/10.23638/DMTCS-19-4-2 Moser, Philippe Stephan, Frank Moser, Philippe Stephan, Frank <![CDATA[We study Bennett deep sequences in the context of recursion theory; in particular we investigate the notions of O(1)-deepK, O(1)-deepC , order-deep K and order-deep C sequences. Our main results are that Martin-Loef random sets are not order-deepC , that every many-one degree contains a set which is not O(1)-deepC , that O(1)-deepC sets and order-deepK sets have high or DNR Turing degree and that no K-trival set is O(1)-deepK.]]> 0 <![CDATA[In this work, we study conditions for the existence of length-constrained path-cycle decompositions, that is, partitions of the edge set of a graph into paths and cycles of a given minimum length. Our main contribution is the characterization of the class of all triangle-free graphs with odd distance at least $3$ that admit a path-cycle decomposition with elements of length at least $4$. As a consequence, it follows that Gallai's conjecture on path decomposition holds in a broad class of sparse graphs.]]> Thu, 26 Oct 2017 08:17:39 +0000 https://doi.org/10.23638/DMTCS-19-3-7 https://doi.org/10.23638/DMTCS-19-3-7 Jiménez, Andrea Wakabayashi, Yoshiko Jiménez, Andrea Wakabayashi, Yoshiko <![CDATA[In this work, we study conditions for the existence of length-constrained path-cycle decompositions, that is, partitions of the edge set of a graph into paths and cycles of a given minimum length. Our main contribution is the characterization of the class of all triangle-free graphs with odd distance at least $3$ that admit a path-cycle decomposition with elements of length at least $4$. As a consequence, it follows that Gallai's conjecture on path decomposition holds in a broad class of sparse graphs.]]> 0 <![CDATA[Let $G$ be a simple graph with a perfect matching. Deng and Zhang showed that the maximum anti-forcing number of $G$ is no more than the cyclomatic number. In this paper, we get a novel upper bound on the maximum anti-forcing number of $G$ and investigate the extremal graphs. If $G$ has a perfect matching $M$ whose anti-forcing number attains this upper bound, then we say $G$ is an extremal graph and $M$ is a nice perfect matching. We obtain an equivalent condition for the nice perfect matchings of $G$ and establish a one-to-one correspondence between the nice perfect matchings and the edge-involutions of $G$, which are the automorphisms $\alpha$ of order two such that $v$ and $\alpha(v)$ are adjacent for every vertex $v$. We demonstrate that all extremal graphs can be constructed from $K_2$ by implementing two expansion operations, and $G$ is extremal if and only if one factor in a Cartesian decomposition of $G$ is extremal. As examples, we have that all perfect matchings of the complete graph $K_{2n}$ and the complete bipartite graph $K_{n, n}$ are nice. Also we show that the hypercube $Q_n$, the folded hypercube $FQ_n$ ($n\geq4$) and the enhanced hypercube $Q_{n, k}$ ($0\leq k\leq n-4$) have exactly $n$, $n+1$ and $n+1$ nice perfect matchings respectively.]]> Tue, 17 Oct 2017 09:07:09 +0000 https://doi.org/10.23638/DMTCS-19-3-9 https://doi.org/10.23638/DMTCS-19-3-9 Shi, Lingjuan Zhang, Heping Shi, Lingjuan Zhang, Heping <![CDATA[Let $G$ be a simple graph with a perfect matching. Deng and Zhang showed that the maximum anti-forcing number of $G$ is no more than the cyclomatic number. In this paper, we get a novel upper bound on the maximum anti-forcing number of $G$ and investigate the extremal graphs. If $G$ has a perfect matching $M$ whose anti-forcing number attains this upper bound, then we say $G$ is an extremal graph and $M$ is a nice perfect matching. We obtain an equivalent condition for the nice perfect matchings of $G$ and establish a one-to-one correspondence between the nice perfect matchings and the edge-involutions of $G$, which are the automorphisms $\alpha$ of order two such that $v$ and $\alpha(v)$ are adjacent for every vertex $v$. We demonstrate that all extremal graphs can be constructed from $K_2$ by implementing two expansion operations, and $G$ is extremal if and only if one factor in a Cartesian decomposition of $G$ is extremal. As examples, we have that all perfect matchings of the complete graph $K_{2n}$ and the complete bipartite graph $K_{n, n}$ are nice. Also we show that the hypercube $Q_n$, the folded hypercube $FQ_n$ ($n\geq4$) and the enhanced hypercube $Q_{n, k}$ ($0\leq k\leq n-4$) have exactly $n$, $n+1$ and $n+1$ nice perfect matchings respectively.]]> 0 <![CDATA[A gapped repeat (respectively, palindrome) occurring in a word $w$ is a factor $uvu$ (respectively, $u^Rvu$) of $w$. In such a repeat (palindrome) $u$ is called the arm of the repeat (respectively, palindrome), while $v$ is called the gap. We show how to compute efficiently, for every position $i$ of the word $w$, the longest gapped repeat and palindrome occurring at that position, provided that the length of the gap is subject to various types of restrictions. That is, that for each position $i$ we compute the longest prefix $u$ of $w[i..n]$ such that $uv$ (respectively, $u^Rv$) is a suffix of $w[1..i-1]$ (defining thus a gapped repeat $uvu$ -- respectively, palindrome $u^Rvu$), and the length of $v$ is subject to the aforementioned restrictions.]]> Fri, 13 Oct 2017 12:32:21 +0000 https://doi.org/10.23638/DMTCS-19-4-4 https://doi.org/10.23638/DMTCS-19-4-4 Dumitran, Marius Gawrychowski, Paweł Manea, Florin Dumitran, Marius Gawrychowski, Paweł Manea, Florin <![CDATA[A gapped repeat (respectively, palindrome) occurring in a word $w$ is a factor $uvu$ (respectively, $u^Rvu$) of $w$. In such a repeat (palindrome) $u$ is called the arm of the repeat (respectively, palindrome), while $v$ is called the gap. We show how to compute efficiently, for every position $i$ of the word $w$, the longest gapped repeat and palindrome occurring at that position, provided that the length of the gap is subject to various types of restrictions. That is, that for each position $i$ we compute the longest prefix $u$ of $w[i..n]$ such that $uv$ (respectively, $u^Rv$) is a suffix of $w[1..i-1]$ (defining thus a gapped repeat $uvu$ -- respectively, palindrome $u^Rvu$), and the length of $v$ is subject to the aforementioned restrictions.]]> 0 <![CDATA[We present a class of (diamond, even hole)-free graphs with no clique cutset that has unbounded rank-width. In general, even-hole-free graphs have unbounded rank-width, because chordal graphs are even-hole-free. A.A. da Silva, A. Silva and C. Linhares-Sales (2010) showed that planar even-hole-free graphs have bounded rank-width, and N.K. Le (2016) showed that even-hole-free graphs with no star cutset have bounded rank-width. A natural question is to ask, whether even-hole-free graphs with no clique cutsets have bounded rank-width. Our result gives a negative answer. Hence we cannot apply Courcelle and Makowsky's meta-theorem which would provide efficient algorithms for a large number of problems, including the maximum independent set problem, whose complexity remains open for (diamond, even hole)-free graphs.]]> Thu, 05 Oct 2017 11:39:06 +0000 https://doi.org/10.23638/DMTCS-19-1-24 https://doi.org/10.23638/DMTCS-19-1-24 Adler, Isolde Le, Ngoc Khang Müller, Haiko Radovanović, Marko Trotignon, Nicolas Vušković, Kristina Adler, Isolde Le, Ngoc Khang Müller, Haiko Radovanović, Marko Trotignon, Nicolas Vušković, Kristina <![CDATA[We present a class of (diamond, even hole)-free graphs with no clique cutset that has unbounded rank-width. In general, even-hole-free graphs have unbounded rank-width, because chordal graphs are even-hole-free. A.A. da Silva, A. Silva and C. Linhares-Sales (2010) showed that planar even-hole-free graphs have bounded rank-width, and N.K. Le (2016) showed that even-hole-free graphs with no star cutset have bounded rank-width. A natural question is to ask, whether even-hole-free graphs with no clique cutsets have bounded rank-width. Our result gives a negative answer. Hence we cannot apply Courcelle and Makowsky's meta-theorem which would provide efficient algorithms for a large number of problems, including the maximum independent set problem, whose complexity remains open for (diamond, even hole)-free graphs.]]> 0 <![CDATA[Let $[K_n,f,\pi]$ be the (global) SDS map of a sequential dynamical system (SDS) defined over the complete graph $K_n$ using the update order $\pi\in S_n$ in which all vertex functions are equal to the same function $f\colon\mathbb F_2^n\to\mathbb F_2^n$. Let $\eta_n$ denote the maximum number of periodic orbits of period $2$ that an SDS map of the form $[K_n,f,\pi]$ can have. We show that $\eta_n$ is equal to the maximum number of codewords in a binary code of length $n-1$ with minimum distance at least $3$. This result is significant because it represents the first interpretation of this fascinating coding-theoretic sequence other than its original definition.]]> Tue, 03 Oct 2017 09:40:34 +0000 https://doi.org/10.23638/DMTCS-19-3-10 https://doi.org/10.23638/DMTCS-19-3-10 Defant, Colin Defant, Colin <![CDATA[Let $[K_n,f,\pi]$ be the (global) SDS map of a sequential dynamical system (SDS) defined over the complete graph $K_n$ using the update order $\pi\in S_n$ in which all vertex functions are equal to the same function $f\colon\mathbb F_2^n\to\mathbb F_2^n$. Let $\eta_n$ denote the maximum number of periodic orbits of period $2$ that an SDS map of the form $[K_n,f,\pi]$ can have. We show that $\eta_n$ is equal to the maximum number of codewords in a binary code of length $n-1$ with minimum distance at least $3$. This result is significant because it represents the first interpretation of this fascinating coding-theoretic sequence other than its original definition.]]> 0 <![CDATA[In queuing theory, it is usual to have some models with a "reset" of the queue. In terms of lattice paths, it is like having the possibility of jumping from any altitude to zero. These objects have the interesting feature that they do not have the same intuitive probabilistic behaviour as classical Dyck paths (the typical properties of which are strongly related to Brownian motion theory), and this article quantifies some relations between these two types of paths. We give a bijection with some other lattice paths and a link with a continued fraction expansion. Furthermore, we prove several formulae for related combinatorial structures conjectured in the On-Line Encyclopedia of Integer Sequences. Thanks to the kernel method and via analytic combinatorics, we provide the enumeration and limit laws of these "lattice paths with catastrophes" for any finite set of jumps. We end with an algorithm to generate such lattice paths uniformly at random.]]> Fri, 29 Sep 2017 14:58:21 +0000 https://doi.org/10.23638/DMTCS-19-1-23 https://doi.org/10.23638/DMTCS-19-1-23 Banderier, Cyril Wallner, Michael Banderier, Cyril Wallner, Michael <![CDATA[In queuing theory, it is usual to have some models with a "reset" of the queue. In terms of lattice paths, it is like having the possibility of jumping from any altitude to zero. These objects have the interesting feature that they do not have the same intuitive probabilistic behaviour as classical Dyck paths (the typical properties of which are strongly related to Brownian motion theory), and this article quantifies some relations between these two types of paths. We give a bijection with some other lattice paths and a link with a continued fraction expansion. Furthermore, we prove several formulae for related combinatorial structures conjectured in the On-Line Encyclopedia of Integer Sequences. Thanks to the kernel method and via analytic combinatorics, we provide the enumeration and limit laws of these "lattice paths with catastrophes" for any finite set of jumps. We end with an algorithm to generate such lattice paths uniformly at random.]]> 0 <![CDATA[An irreversible $k$-threshold process (also a $k$-neighbor bootstrap percolation) is a dynamic process on a graph where vertices change color from white to black if they have at least $k$ black neighbors. An irreversible $k$-conversion set of a graph $G$ is a subset $S$ of vertices of $G$ such that the irreversible $k$-threshold process starting with $S$ black eventually changes all vertices of $G$ to black. We show that deciding the existence of an irreversible 2-conversion set of a given size is NP-complete, even for graphs of maximum degree 4, which answers a question of Dreyer and Roberts. Conversely, we show that for graphs of maximum degree 3, the minimum size of an irreversible 2-conversion set can be computed in polynomial time. Moreover, we find an optimal irreversible 3-conversion set for the toroidal grid, simplifying constructions of Pike and Zou.]]> Tue, 26 Sep 2017 11:54:32 +0000 https://doi.org/10.23638/DMTCS-19-3-5 https://doi.org/10.23638/DMTCS-19-3-5 Kynčl, Jan Lidický, Bernard Vyskočil, Tomáš Kynčl, Jan Lidický, Bernard Vyskočil, Tomáš <![CDATA[An irreversible $k$-threshold process (also a $k$-neighbor bootstrap percolation) is a dynamic process on a graph where vertices change color from white to black if they have at least $k$ black neighbors. An irreversible $k$-conversion set of a graph $G$ is a subset $S$ of vertices of $G$ such that the irreversible $k$-threshold process starting with $S$ black eventually changes all vertices of $G$ to black. We show that deciding the existence of an irreversible 2-conversion set of a given size is NP-complete, even for graphs of maximum degree 4, which answers a question of Dreyer and Roberts. Conversely, we show that for graphs of maximum degree 3, the minimum size of an irreversible 2-conversion set can be computed in polynomial time. Moreover, we find an optimal irreversible 3-conversion set for the toroidal grid, simplifying constructions of Pike and Zou.]]> 0 <![CDATA[We examine inkdots placed on the input string as a way of providing advice to finite automata, and establish the relations between this model and the previously studied models of advised finite automata. The existence of an infinite hierarchy of classes of languages that can be recognized with the help of increasing numbers of inkdots as advice is shown. The effects of different forms of advice on the succinctness of the advised machines are examined. We also study randomly placed inkdots as advice to probabilistic finite automata, and demonstrate the superiority of this model over its deterministic version. Even very slowly growing amounts of space can become a resource of meaningful use if the underlying advised model is extended with access to secondary memory, while it is famously known that such small amounts of space are not useful for unadvised one-way Turing machines.]]> Tue, 26 Sep 2017 11:50:45 +0000 https://doi.org/10.23638/DMTCS-19-3-1 https://doi.org/10.23638/DMTCS-19-3-1 Küçük, Uğur Say, A. C. Cem Yakaryılmaz, Abuzer Küçük, Uğur Say, A. C. Cem Yakaryılmaz, Abuzer <![CDATA[We examine inkdots placed on the input string as a way of providing advice to finite automata, and establish the relations between this model and the previously studied models of advised finite automata. The existence of an infinite hierarchy of classes of languages that can be recognized with the help of increasing numbers of inkdots as advice is shown. The effects of different forms of advice on the succinctness of the advised machines are examined. We also study randomly placed inkdots as advice to probabilistic finite automata, and demonstrate the superiority of this model over its deterministic version. Even very slowly growing amounts of space can become a resource of meaningful use if the underlying advised model is extended with access to secondary memory, while it is famously known that such small amounts of space are not useful for unadvised one-way Turing machines.]]> 0 <![CDATA[Tree-like tableaux are certain fillings of Ferrers diagrams originally introduced by Aval et al., which are in simple bijections with permutation tableaux coming from Postnikov's study of totally nonnegative Grassmanian and alternative tableaux introduced by Viennot. In this paper, we confirm two conjectures of Gao et al. on the refined enumeration of non-occupied corners in tree-like tableaux and symmetric tree-like tableaux via intermediate structures of alternative tableaux, linked partitions, type $B$ alternative tableaux and type $B$ linked partitions.]]> Tue, 26 Sep 2017 11:47:44 +0000 https://doi.org/10.23638/DMTCS-19-3-6 https://doi.org/10.23638/DMTCS-19-3-6 Yan, Sherry H. F. Zhou, Robin D. P. Yan, Sherry H. F. Zhou, Robin D. P. <![CDATA[Tree-like tableaux are certain fillings of Ferrers diagrams originally introduced by Aval et al., which are in simple bijections with permutation tableaux coming from Postnikov's study of totally nonnegative Grassmanian and alternative tableaux introduced by Viennot. In this paper, we confirm two conjectures of Gao et al. on the refined enumeration of non-occupied corners in tree-like tableaux and symmetric tree-like tableaux via intermediate structures of alternative tableaux, linked partitions, type $B$ alternative tableaux and type $B$ linked partitions.]]> 0 <![CDATA[By a tight tour in a $k$-uniform hypergraph $H$ we mean any sequence of its vertices $(w_0,w_1,\ldots,w_{s-1})$ such that for all $i=0,\ldots,s-1$ the set $e_i=\{w_i,w_{i+1}\ldots,w_{i+k-1}\}$ is an edge of $H$ (where operations on indices are computed modulo $s$) and the sets $e_i$ for $i=0,\ldots,s-1$ are pairwise different. A tight tour in $H$ is a tight Euler tour if it contains all edges of $H$. We prove that the problem of deciding if a given $3$-uniform hypergraph has a tight Euler tour is NP-complete, and that it cannot be solved in time $2^{o(m)}$ (where $m$ is the number of edges in the input hypergraph), unless the ETH fails. We also present an exact exponential algorithm for the problem, whose time complexity matches this lower bound, and the space complexity is polynomial. In fact, this algorithm solves a more general problem of computing the number of tight Euler tours in a given uniform hypergraph.]]> Tue, 26 Sep 2017 11:43:07 +0000 https://doi.org/10.23638/DMTCS-19-3-2 https://doi.org/10.23638/DMTCS-19-3-2 Lonc, Zbigniew Naroski, Paweł Rzążewski, Paweł Lonc, Zbigniew Naroski, Paweł Rzążewski, Paweł <![CDATA[By a tight tour in a $k$-uniform hypergraph $H$ we mean any sequence of its vertices $(w_0,w_1,\ldots,w_{s-1})$ such that for all $i=0,\ldots,s-1$ the set $e_i=\{w_i,w_{i+1}\ldots,w_{i+k-1}\}$ is an edge of $H$ (where operations on indices are computed modulo $s$) and the sets $e_i$ for $i=0,\ldots,s-1$ are pairwise different. A tight tour in $H$ is a tight Euler tour if it contains all edges of $H$. We prove that the problem of deciding if a given $3$-uniform hypergraph has a tight Euler tour is NP-complete, and that it cannot be solved in time $2^{o(m)}$ (where $m$ is the number of edges in the input hypergraph), unless the ETH fails. We also present an exact exponential algorithm for the problem, whose time complexity matches this lower bound, and the space complexity is polynomial. In fact, this algorithm solves a more general problem of computing the number of tight Euler tours in a given uniform hypergraph.]]> 0 <![CDATA[The PASEP (Partially Asymmetric Simple Exclusion Process) is a probabilistic model of moving particles, which is of great interest in combinatorics, since it appeared that its partition function counts some tableaux. These tableaux have several variants such as permutations tableaux, alternative tableaux, tree- like tableaux, Dyck tableaux, etc. We introduce in this context certain excursions in Young's lattice, that we call stammering tableaux (by analogy with oscillating tableaux, vacillating tableaux, hesitating tableaux). Some natural bijections make a link with rook placements in a double staircase, chains of Dyck paths obtained by successive addition of ribbons, Laguerre histories, Dyck tableaux, etc.]]> Fri, 15 Sep 2017 11:36:00 +0000 https://doi.org/10.23638/DMTCS-19-3-3 https://doi.org/10.23638/DMTCS-19-3-3 Josuat-Vergès, Matthieu Josuat-Vergès, Matthieu <![CDATA[The PASEP (Partially Asymmetric Simple Exclusion Process) is a probabilistic model of moving particles, which is of great interest in combinatorics, since it appeared that its partition function counts some tableaux. These tableaux have several variants such as permutations tableaux, alternative tableaux, tree- like tableaux, Dyck tableaux, etc. We introduce in this context certain excursions in Young's lattice, that we call stammering tableaux (by analogy with oscillating tableaux, vacillating tableaux, hesitating tableaux). Some natural bijections make a link with rook placements in a double staircase, chains of Dyck paths obtained by successive addition of ribbons, Laguerre histories, Dyck tableaux, etc.]]> 0 <![CDATA[We discuss cellular automata over arbitrary finitely generated groups. We call a cellular automaton post-surjective if for any pair of asymptotic configurations, every pre-image of one is asymptotic to a pre-image of the other. The well known dual concept is pre-injectivity: a cellular automaton is pre-injective if distinct asymptotic configurations have distinct images. We prove that pre-injective, post-surjective cellular automata are reversible. Moreover, on sofic groups, post-surjectivity alone implies reversibility. We also prove that reversible cellular automata over arbitrary groups are balanced, that is, they preserve the uniform measure on the configuration space.]]> Fri, 15 Sep 2017 11:30:58 +0000 https://doi.org/10.23638/DMTCS-19-3-4 https://doi.org/10.23638/DMTCS-19-3-4 Capobianco, Silvio Kari, Jarkko Taati, Siamak Capobianco, Silvio Kari, Jarkko Taati, Siamak <![CDATA[We discuss cellular automata over arbitrary finitely generated groups. We call a cellular automaton post-surjective if for any pair of asymptotic configurations, every pre-image of one is asymptotic to a pre-image of the other. The well known dual concept is pre-injectivity: a cellular automaton is pre-injective if distinct asymptotic configurations have distinct images. We prove that pre-injective, post-surjective cellular automata are reversible. Moreover, on sofic groups, post-surjectivity alone implies reversibility. We also prove that reversible cellular automata over arbitrary groups are balanced, that is, they preserve the uniform measure on the configuration space.]]> 0 <![CDATA[A graph is said to be well-dominated if all its minimal dominating sets are of the same size. The class of well-dominated graphs forms a subclass of the well studied class of well-covered graphs. While the recognition problem for the class of well-covered graphs is known to be co-NP-complete, the recognition complexity of well-dominated graphs is open. In this paper we introduce the notion of an irreducible dominating set, a variant of dominating set generalizing both minimal dominating sets and minimal total dominating sets. Based on this notion, we characterize the family of minimal dominating sets in a lexicographic product of two graphs and derive a characterization of the well-dominated lexicographic product graphs. As a side result motivated by this study, we give a polynomially testable characterization of well-dominated graphs with domination number two, and show, more generally, that well-dominated graphs can be recognized in polynomial time in any class of graphs with bounded domination number. Our results include a characterization of dominating sets in lexicographic product graphs, which generalizes the expression for the domination number of such graphs following from works of Zhang et al. (2011) and of \v{S}umenjak et al. (2012).]]> Wed, 30 Aug 2017 13:49:35 +0000 https://doi.org/10.23638/DMTCS-19-1-25 https://doi.org/10.23638/DMTCS-19-1-25 Gözüpek, Didem Hujdurović, Ademir Milanič, Martin Gözüpek, Didem Hujdurović, Ademir Milanič, Martin <![CDATA[A graph is said to be well-dominated if all its minimal dominating sets are of the same size. The class of well-dominated graphs forms a subclass of the well studied class of well-covered graphs. While the recognition problem for the class of well-covered graphs is known to be co-NP-complete, the recognition complexity of well-dominated graphs is open. In this paper we introduce the notion of an irreducible dominating set, a variant of dominating set generalizing both minimal dominating sets and minimal total dominating sets. Based on this notion, we characterize the family of minimal dominating sets in a lexicographic product of two graphs and derive a characterization of the well-dominated lexicographic product graphs. As a side result motivated by this study, we give a polynomially testable characterization of well-dominated graphs with domination number two, and show, more generally, that well-dominated graphs can be recognized in polynomial time in any class of graphs with bounded domination number. Our results include a characterization of dominating sets in lexicographic product graphs, which generalizes the expression for the domination number of such graphs following from works of Zhang et al. (2011) and of \v{S}umenjak et al. (2012).]]> 0 <![CDATA[This paper is dedicated to studying the following question: Is it always possible to injectively assign the weights 1, ..., |E(G)| to the edges of any given graph G (with no component isomorphic to K2) so that every two adjacent vertices of G get distinguished by their sums of incident weights? One may see this question as a combination of the well-known 1-2-3 Conjecture and the Antimagic Labelling Conjecture. Throughout this paper, we exhibit evidence that this question might be true. Benefiting from the investigations on the Antimagic Labelling Conjecture, we first point out that several classes of graphs, such as regular graphs, indeed admit such assignments. We then show that trees also do, answering a recent conjecture of Arumugam, Premalatha, Bača and Semaničová-Feňovčíková. Towards a general answer to the question above, we then prove that claimed assignments can be constructed for any graph, provided we are allowed to use some number of additional edge weights. For some classes of sparse graphs, namely 2-degenerate graphs and graphs with maximum average degree 3, we show that only a small (constant) number of such additional weights suffices.]]> Tue, 08 Aug 2017 10:14:00 +0000 https://doi.org/10.23638/DMTCS-19-1-21 https://doi.org/10.23638/DMTCS-19-1-21 Bensmail, Julien Senhaji, Mohammed Szabo Lyngsie, Kasper Bensmail, Julien Senhaji, Mohammed Szabo Lyngsie, Kasper <![CDATA[This paper is dedicated to studying the following question: Is it always possible to injectively assign the weights 1, ..., |E(G)| to the edges of any given graph G (with no component isomorphic to K2) so that every two adjacent vertices of G get distinguished by their sums of incident weights? One may see this question as a combination of the well-known 1-2-3 Conjecture and the Antimagic Labelling Conjecture. Throughout this paper, we exhibit evidence that this question might be true. Benefiting from the investigations on the Antimagic Labelling Conjecture, we first point out that several classes of graphs, such as regular graphs, indeed admit such assignments. We then show that trees also do, answering a recent conjecture of Arumugam, Premalatha, Bača and Semaničová-Feňovčíková. Towards a general answer to the question above, we then prove that claimed assignments can be constructed for any graph, provided we are allowed to use some number of additional edge weights. For some classes of sparse graphs, namely 2-degenerate graphs and graphs with maximum average degree 3, we show that only a small (constant) number of such additional weights suffices.]]> 0 <![CDATA[Consider $ m $ copies of an irreducible, aperiodic Markov chain $ Y $ taking values in a finite state space. The asymptotics as $ m $ tends to infinity, of the first time from which on the trajectories of the $ m $ copies differ, have been studied by Szpankowski (1991) in the setting of tries. We use a different approach and model the $ m $ trajectories by a variant of the occupancy scheme, where we consider a nested sequence of boxes. This approach will enable us to extend the result to the case when the transition probabilities are random. We moreover use the same techniques to study the asymptotics as $ m $ tends to infinity of the time up to which we have observed all the possible trajectories of $ Y $ in random and nonrandom scenery.]]> Fri, 04 Aug 2017 18:31:07 +0000 https://doi.org/10.23638/DMTCS-19-1-22 https://doi.org/10.23638/DMTCS-19-1-22 Businger, Silvia Businger, Silvia <![CDATA[Consider $ m $ copies of an irreducible, aperiodic Markov chain $ Y $ taking values in a finite state space. The asymptotics as $ m $ tends to infinity, of the first time from which on the trajectories of the $ m $ copies differ, have been studied by Szpankowski (1991) in the setting of tries. We use a different approach and model the $ m $ trajectories by a variant of the occupancy scheme, where we consider a nested sequence of boxes. This approach will enable us to extend the result to the case when the transition probabilities are random. We moreover use the same techniques to study the asymptotics as $ m $ tends to infinity of the time up to which we have observed all the possible trajectories of $ Y $ in random and nonrandom scenery.]]> 0 <![CDATA[The study of patterns in permutations associated with forests of binary shrubs was initiated by D. Bevan et al.. In this paper, we study five different types of rise statistics that can be associated with such permutations and find the generating functions for the distribution of such rise statistics.]]> Wed, 19 Jul 2017 09:55:50 +0000 https://doi.org/10.23638/DMTCS-19-1-15 https://doi.org/10.23638/DMTCS-19-1-15 Remmel, Jeffrey Zheng, Sai-nan Remmel, Jeffrey Zheng, Sai-nan <![CDATA[The study of patterns in permutations associated with forests of binary shrubs was initiated by D. Bevan et al.. In this paper, we study five different types of rise statistics that can be associated with such permutations and find the generating functions for the distribution of such rise statistics.]]> 0 <![CDATA[We consider a sorting machine consisting of two stacks in series where the first stack has the added restriction that entries in the stack must be in decreasing order from top to bottom. The class of permutations sortable by this machine are known to be enumerated by the Schröder numbers. In this paper, we give a bijection between these sortable permutations of length $n$ and Schröder paths -- the lattice paths from $(0,0)$ to $(n-1,n-1)$ composed of East steps $(1,0)$, North steps $(0,1)$, and Diagonal steps $(1,1)$ that travel weakly below the line $y=x$.]]> Wed, 19 Jul 2017 09:46:44 +0000 https://doi.org/10.46298/dmtcs.1326 https://doi.org/10.46298/dmtcs.1326 Schroeder, Michael W. Smith, Rebecca Schroeder, Michael W. Smith, Rebecca <![CDATA[We consider a sorting machine consisting of two stacks in series where the first stack has the added restriction that entries in the stack must be in decreasing order from top to bottom. The class of permutations sortable by this machine are known to be enumerated by the Schröder numbers. In this paper, we give a bijection between these sortable permutations of length $n$ and Schröder paths -- the lattice paths from $(0,0)$ to $(n-1,n-1)$ composed of East steps $(1,0)$, North steps $(0,1)$, and Diagonal steps $(1,1)$ that travel weakly below the line $y=x$.]]> 0 <![CDATA[Using $q$-series identities and series rearrangement, we establish several extensions of $q$-Watson formulas with two extra integer parameters. Then they and Sears' transformation formula are utilized to derive some generalizations of $q$-Dixon formulas and $q$-Whipple formulas with two extra integer parameters. As special cases of these results, many interesting evaluations of series of $q$-Watson,$q$-Dixon, and $q$-Whipple type are displayed.]]> Tue, 27 Jun 2017 08:30:43 +0000 https://doi.org/10.23638/DMTCS-19-1-19 https://doi.org/10.23638/DMTCS-19-1-19 Wei, Chuanan Wang, Xiaoxia Wei, Chuanan Wang, Xiaoxia <![CDATA[Using $q$-series identities and series rearrangement, we establish several extensions of $q$-Watson formulas with two extra integer parameters. Then they and Sears' transformation formula are utilized to derive some generalizations of $q$-Dixon formulas and $q$-Whipple formulas with two extra integer parameters. As special cases of these results, many interesting evaluations of series of $q$-Watson,$q$-Dixon, and $q$-Whipple type are displayed.]]> 0 <![CDATA[A repetition is a sequence of symbols in which the first half is the same as the second half. An edge-coloring of a graph is repetition-free or nonrepetitive if there is no path with a color pattern that is a repetition. The minimum number of colors so that a graph has a nonrepetitive edge-coloring is called its Thue edge-chromatic number. We improve on the best known general upper bound of $4\Delta-4$ for the Thue edge-chromatic number of trees of maximum degree $\Delta$ due to Alon, Grytczuk, Ha{\l}uszczak and Riordan (2002) by providing a simple nonrepetitive edge-coloring with $3\Delta-2$ colors.]]> Tue, 27 Jun 2017 08:27:45 +0000 https://doi.org/10.23638/DMTCS-19-1-18 https://doi.org/10.23638/DMTCS-19-1-18 Kündgen, A. Talbot, T. Kündgen, A. Talbot, T. <![CDATA[A repetition is a sequence of symbols in which the first half is the same as the second half. An edge-coloring of a graph is repetition-free or nonrepetitive if there is no path with a color pattern that is a repetition. The minimum number of colors so that a graph has a nonrepetitive edge-coloring is called its Thue edge-chromatic number. We improve on the best known general upper bound of $4\Delta-4$ for the Thue edge-chromatic number of trees of maximum degree $\Delta$ due to Alon, Grytczuk, Ha{\l}uszczak and Riordan (2002) by providing a simple nonrepetitive edge-coloring with $3\Delta-2$ colors.]]> 0 <![CDATA[The overlap graphs of subtrees of a tree are equivalent to subtree filament graphs, the overlap graphs of subtrees of a star are cocomparability graphs, and the overlap graphs of subtrees of a caterpillar are interval filament graphs. In this paper, we show the equivalence of many more classes of subtree overlap and subtree filament graphs, and equate them to classes of complements of cochordal-mixed graphs. Our results generalize the previously known results mentioned above.]]> Tue, 20 Jun 2017 09:37:51 +0000 https://doi.org/10.23638/DMTCS-19-1-20 https://doi.org/10.23638/DMTCS-19-1-20 Enright, Jessica Stewart, Lorna Enright, Jessica Stewart, Lorna <![CDATA[The overlap graphs of subtrees of a tree are equivalent to subtree filament graphs, the overlap graphs of subtrees of a star are cocomparability graphs, and the overlap graphs of subtrees of a caterpillar are interval filament graphs. In this paper, we show the equivalence of many more classes of subtree overlap and subtree filament graphs, and equate them to classes of complements of cochordal-mixed graphs. Our results generalize the previously known results mentioned above.]]> 0 <![CDATA[A finite deterministic (semi)automaton A = (Q, Σ, δ) is k-compressible if there is some word w ∈ Σ + such that theimage of its state set Q under the natural action of w is reduced by at least k states. Such word w, if it exists, is calleda k-compressing word for A and A is said to be k-compressed by w. A word is k-collapsing if it is k-compressing foreach k-compressible automaton, and it is k-synchronizing if it is k-compressing for all k-compressible automata withk+1 states. We compute a set W of short words such that each 3-compressible automaton on a two-letter alphabetis 3-compressed at least by a word in W. Then we construct a shortest common superstring of the words in W and,with a further refinement, we obtain a 3-collapsing word of length 53. Moreover, as previously announced, we showthat the shortest 3-synchronizing word is not 3-collapsing, illustrating the new bounds 34 ≤ c(2, 3) ≤ 53 for the length c(2, 3) of the shortest 3-collapsing word on a two-letter alphabet.]]> Tue, 20 Jun 2017 09:34:52 +0000 https://doi.org/10.23638/DMTCS-19-1-17 https://doi.org/10.23638/DMTCS-19-1-17 Cherubini, Alessandra Frigeri, Achille Liu, Zuhua Cherubini, Alessandra Frigeri, Achille Liu, Zuhua <![CDATA[A finite deterministic (semi)automaton A = (Q, Σ, δ) is k-compressible if there is some word w ∈ Σ + such that theimage of its state set Q under the natural action of w is reduced by at least k states. Such word w, if it exists, is calleda k-compressing word for A and A is said to be k-compressed by w. A word is k-collapsing if it is k-compressing foreach k-compressible automaton, and it is k-synchronizing if it is k-compressing for all k-compressible automata withk+1 states. We compute a set W of short words such that each 3-compressible automaton on a two-letter alphabetis 3-compressed at least by a word in W. Then we construct a shortest common superstring of the words in W and,with a further refinement, we obtain a 3-collapsing word of length 53. Moreover, as previously announced, we showthat the shortest 3-synchronizing word is not 3-collapsing, illustrating the new bounds 34 ≤ c(2, 3) ≤ 53 for the length c(2, 3) of the shortest 3-collapsing word on a two-letter alphabet.]]> 0 <![CDATA[The families EPT (resp. EPG) Edge Intersection Graphs of Paths in a tree (resp. in a grid) are well studied graph classes. Recently we introduced the graph classes Edge-Intersecting and Non-Splitting Paths in a Tree ENPT, and in a Grid (ENPG). It was shown that ENPG contains an infinite hierarchy of subclasses that are obtained by restricting the number of bends in the paths. Motivated by this result, in this work we focus on one bend {ENPG} graphs. We show that one bend ENPG graphs are properly included in two bend ENPG graphs. We also show that trees and cycles are one bend ENPG graphs, and characterize the split graphs and co-bipartite graphs that are one bend ENPG. We prove that the recognition problem of one bend ENPG split graphs is NP-complete even in a very restricted subfamily of split graphs. Last we provide a linear time recognition algorithm for one bend ENPG co-bipartite graphs.]]> Mon, 12 Jun 2017 11:18:19 +0000 https://doi.org/10.23638/DMTCS-19-1-13 https://doi.org/10.23638/DMTCS-19-1-13 Boyacı, Arman Ekim, Tınaz Shalom, Mordechai Zaks, Shmuel Boyacı, Arman Ekim, Tınaz Shalom, Mordechai Zaks, Shmuel <![CDATA[The families EPT (resp. EPG) Edge Intersection Graphs of Paths in a tree (resp. in a grid) are well studied graph classes. Recently we introduced the graph classes Edge-Intersecting and Non-Splitting Paths in a Tree ENPT, and in a Grid (ENPG). It was shown that ENPG contains an infinite hierarchy of subclasses that are obtained by restricting the number of bends in the paths. Motivated by this result, in this work we focus on one bend {ENPG} graphs. We show that one bend ENPG graphs are properly included in two bend ENPG graphs. We also show that trees and cycles are one bend ENPG graphs, and characterize the split graphs and co-bipartite graphs that are one bend ENPG. We prove that the recognition problem of one bend ENPG split graphs is NP-complete even in a very restricted subfamily of split graphs. Last we provide a linear time recognition algorithm for one bend ENPG co-bipartite graphs.]]> 0 <![CDATA[A graph $G$ is signed if each edge is assigned $+$ or $-$. A signed graph is balanced if there is a bipartition of its vertex set such that an edge has sign $-$ if and only if its endpoints are in different parts. The Edwards-Erd\"os bound states that every graph with $n$ vertices and $m$ edges has a balanced subgraph with at least $\frac{m}{2}+\frac{n-1}{4}$ edges. In the Signed Max Cut Above Tight Lower Bound (Signed Max Cut ATLB) problem, given a signed graph $G$ and a parameter $k$, the question is whether $G$ has a balanced subgraph with at least $\frac{m}{2}+\frac{n-1}{4}+\frac{k}{4}$ edges. This problem generalizes Max Cut Above Tight Lower Bound, for which a kernel with $O(k^5)$ vertices was given by Crowston et al. [ICALP 2012, Algorithmica 2015]. Crowston et al. [TCS 2013] improved this result by providing a kernel with $O(k^3)$ vertices for the more general Signed Max Cut ATLB problem. In this article we are interested in improving the size of the kernels for Signed Max Cut ATLB on restricted graph classes for which the problem remains hard. For two integers $r,\ell \geq 0$, a graph $G$ is an $(r,\ell)$-graph if $V(G)$ can be partitioned into $r$ independent sets and $\ell$ cliques. Building on the techniques of Crowston et al. [TCS 2013], we provide a kernel with $O(k^2)$ vertices on $(r,\ell)$-graphs for any fixed $r,\ell \geq 0$, and a simple linear kernel on subclasses of split graphs for which we prove that the problem is still NP-hard.]]> Wed, 07 Jun 2017 09:25:02 +0000 https://doi.org/10.23638/DMTCS-19-1-14 https://doi.org/10.23638/DMTCS-19-1-14 Faria, Luerbio Klein, Sulamita Sau, Ignasi Sucupira, Rubens Faria, Luerbio Klein, Sulamita Sau, Ignasi Sucupira, Rubens <![CDATA[A graph $G$ is signed if each edge is assigned $+$ or $-$. A signed graph is balanced if there is a bipartition of its vertex set such that an edge has sign $-$ if and only if its endpoints are in different parts. The Edwards-Erd\"os bound states that every graph with $n$ vertices and $m$ edges has a balanced subgraph with at least $\frac{m}{2}+\frac{n-1}{4}$ edges. In the Signed Max Cut Above Tight Lower Bound (Signed Max Cut ATLB) problem, given a signed graph $G$ and a parameter $k$, the question is whether $G$ has a balanced subgraph with at least $\frac{m}{2}+\frac{n-1}{4}+\frac{k}{4}$ edges. This problem generalizes Max Cut Above Tight Lower Bound, for which a kernel with $O(k^5)$ vertices was given by Crowston et al. [ICALP 2012, Algorithmica 2015]. Crowston et al. [TCS 2013] improved this result by providing a kernel with $O(k^3)$ vertices for the more general Signed Max Cut ATLB problem. In this article we are interested in improving the size of the kernels for Signed Max Cut ATLB on restricted graph classes for which the problem remains hard. For two integers $r,\ell \geq 0$, a graph $G$ is an $(r,\ell)$-graph if $V(G)$ can be partitioned into $r$ independent sets and $\ell$ cliques. Building on the techniques of Crowston et al. [TCS 2013], we provide a kernel with $O(k^2)$ vertices on $(r,\ell)$-graphs for any fixed $r,\ell \geq 0$, and a simple linear kernel on subclasses of split graphs for which we prove that the problem is still NP-hard.]]> 0 <![CDATA[A universal word for a finite alphabet $A$ and some integer $n\geq 1$ is a word over $A$ such that every word in $A^n$ appears exactly once as a subword (cyclically or linearly). It is well-known and easy to prove that universal words exist for any $A$ and $n$. In this work we initiate the systematic study of universal partial words. These are words that in addition to the letters from $A$ may contain an arbitrary number of occurrences of a special `joker' symbol $\Diamond\notin A$, which can be substituted by any symbol from $A$. For example, $u=0\Diamond 011100$ is a linear partial word for the binary alphabet $A=\{0,1\}$ and for $n=3$ (e.g., the first three letters of $u$ yield the subwords $000$ and $010$). We present results on the existence and non-existence of linear and cyclic universal partial words in different situations (depending on the number of $\Diamond$s and their positions), including various explicit constructions. We also provide numerous examples of universal partial words that we found with the help of a computer.]]> Wed, 31 May 2017 09:00:18 +0000 https://doi.org/10.23638/DMTCS-19-1-16 https://doi.org/10.23638/DMTCS-19-1-16 Chen, Herman Z. Q. Kitaev, Sergey Mütze, Torsten Sun, Brian Y. Chen, Herman Z. Q. Kitaev, Sergey Mütze, Torsten Sun, Brian Y. <![CDATA[A universal word for a finite alphabet $A$ and some integer $n\geq 1$ is a word over $A$ such that every word in $A^n$ appears exactly once as a subword (cyclically or linearly). It is well-known and easy to prove that universal words exist for any $A$ and $n$. In this work we initiate the systematic study of universal partial words. These are words that in addition to the letters from $A$ may contain an arbitrary number of occurrences of a special `joker' symbol $\Diamond\notin A$, which can be substituted by any symbol from $A$. For example, $u=0\Diamond 011100$ is a linear partial word for the binary alphabet $A=\{0,1\}$ and for $n=3$ (e.g., the first three letters of $u$ yield the subwords $000$ and $010$). We present results on the existence and non-existence of linear and cyclic universal partial words in different situations (depending on the number of $\Diamond$s and their positions), including various explicit constructions. We also provide numerous examples of universal partial words that we found with the help of a computer.]]> 0 <![CDATA[Let $Sz(G),Sz^*(G)$ and $W(G)$ be the Szeged index, revised Szeged index and Wiener index of a graph $G.$ In this paper, the graphs with the fourth, fifth, sixth and seventh largest Wiener indices among all unicyclic graphs of order $n\geqslant 10$ are characterized; as well the graphs with the first, second, third, and fourth largest Wiener indices among all bicyclic graphs are identified. Based on these results, further relation on the quotients between the (revised) Szeged index and the Wiener index are studied. Sharp lower bound on $Sz(G)/W(G)$ is determined for all connected graphs each of which contains at least one non-complete block. As well the connected graph with the second smallest value on $Sz^*(G)/W(G)$ is identified for $G$ containing at least one cycle.]]> Tue, 09 May 2017 08:48:50 +0000 https://doi.org/10.23638/DMTCS-19-1-12 https://doi.org/10.23638/DMTCS-19-1-12 Zhang, Huihui Chen, Jing Li, Shuchao Zhang, Huihui Chen, Jing Li, Shuchao <![CDATA[Let $Sz(G),Sz^*(G)$ and $W(G)$ be the Szeged index, revised Szeged index and Wiener index of a graph $G.$ In this paper, the graphs with the fourth, fifth, sixth and seventh largest Wiener indices among all unicyclic graphs of order $n\geqslant 10$ are characterized; as well the graphs with the first, second, third, and fourth largest Wiener indices among all bicyclic graphs are identified. Based on these results, further relation on the quotients between the (revised) Szeged index and the Wiener index are studied. Sharp lower bound on $Sz(G)/W(G)$ is determined for all connected graphs each of which contains at least one non-complete block. As well the connected graph with the second smallest value on $Sz^*(G)/W(G)$ is identified for $G$ containing at least one cycle.]]> 0 <![CDATA[Codes with various kinds of decipherability, weaker than the usual unique decipherability, have been studied since multiset decipherability was introduced in mid-1980s. We consider decipherability of directed figure codes, where directed figures are defined as labelled polyominoes with designated start and end points, equipped with catenation operation that may use a merging function to resolve possible conflicts. This is one of possible extensions generalizing words and variable-length codes to planar structures. Here, verification whether a given set is a code is no longer decidable in general. We study the decidability status of figure codes depending on catenation type (with or without a merging function), decipherability kind (unique, multiset, set or numeric) and code geometry (several classes determined by relative positions of start and end points of figures). We give decidability or undecidability proofs in all but two cases that remain open.]]> Wed, 03 May 2017 08:17:40 +0000 https://doi.org/10.23638/DMTCS-19-1-11 https://doi.org/10.23638/DMTCS-19-1-11 Moczurad, Włodzimierz Moczurad, Włodzimierz <![CDATA[Codes with various kinds of decipherability, weaker than the usual unique decipherability, have been studied since multiset decipherability was introduced in mid-1980s. We consider decipherability of directed figure codes, where directed figures are defined as labelled polyominoes with designated start and end points, equipped with catenation operation that may use a merging function to resolve possible conflicts. This is one of possible extensions generalizing words and variable-length codes to planar structures. Here, verification whether a given set is a code is no longer decidable in general. We study the decidability status of figure codes depending on catenation type (with or without a merging function), decipherability kind (unique, multiset, set or numeric) and code geometry (several classes determined by relative positions of start and end points of figures). We give decidability or undecidability proofs in all but two cases that remain open.]]> 0 <![CDATA[This paper deals with two-sided matching market with two disjoint sets, i.e. the set of buyers and the set of sellers. Each seller can trade with at most with one buyer and vice versa. Money is transferred from sellers to buyers for an indivisible goods that buyers own. Valuation functions, for participants of both sides, are represented by strictly increasing functions with money considered as discrete variable. An algorithm is devised to prove the existence of stability for this model.]]> Wed, 12 Apr 2017 12:50:47 +0000 https://doi.org/10.23638/DMTCS-19-1-10 https://doi.org/10.23638/DMTCS-19-1-10 Ali, Yasir Javaid, Asma Ali, Yasir Javaid, Asma <![CDATA[This paper deals with two-sided matching market with two disjoint sets, i.e. the set of buyers and the set of sellers. Each seller can trade with at most with one buyer and vice versa. Money is transferred from sellers to buyers for an indivisible goods that buyers own. Valuation functions, for participants of both sides, are represented by strictly increasing functions with money considered as discrete variable. An algorithm is devised to prove the existence of stability for this model.]]> 0 <![CDATA[We determine the structure of permutations avoiding the patterns 4213 and 2143. Each such permutation consists of the skew sum of a sequence of plane trees, together with an increasing sequence of points above and an increasing sequence of points to its left. We use this characterisation to establish the generating function enumerating these permutations. We also investigate the properties of a typical large permutation in the class and prove that if a large permutation that avoids 4213 and 2143 is chosen uniformly at random, then it is more likely than not to avoid 2413 as well.]]> Tue, 04 Apr 2017 14:33:15 +0000 https://doi.org/10.46298/dmtcs.1309 https://doi.org/10.46298/dmtcs.1309 Bevan, David Bevan, David <![CDATA[We determine the structure of permutations avoiding the patterns 4213 and 2143. Each such permutation consists of the skew sum of a sequence of plane trees, together with an increasing sequence of points above and an increasing sequence of points to its left. We use this characterisation to establish the generating function enumerating these permutations. We also investigate the properties of a typical large permutation in the class and prove that if a large permutation that avoids 4213 and 2143 is chosen uniformly at random, then it is more likely than not to avoid 2413 as well.]]> 0 <![CDATA[An S-restricted composition of a positive integer n is an ordered partition of n where each summand is drawn from a given subset S of positive integers. There are various problems regarding such compositions which have received attention in recent years. This paper is an attempt at finding a closed- form formula for the number of S-restricted compositions of n. To do so, we reduce the problem to finding solutions to corresponding so-called interpreters which are linear homogeneous recurrence relations with constant coefficients. Then, we reduce interpreters to Diophantine equations. Such equations are not in general solvable. Thus, we restrict our attention to those S-restricted composition problems whose interpreters have a small number of coefficients, thereby leading to solvable Diophantine equations. The formalism developed is then used to study the integer sequences related to some well-known cases of the S-restricted composition problem.]]> Tue, 28 Mar 2017 08:24:04 +0000 https://doi.org/10.23638/DMTCS-19-1-9 https://doi.org/10.23638/DMTCS-19-1-9 Zolfaghari, Behrouz Fallah, Mehran S. Sedighi, Mehdi Zolfaghari, Behrouz Fallah, Mehran S. Sedighi, Mehdi <![CDATA[An S-restricted composition of a positive integer n is an ordered partition of n where each summand is drawn from a given subset S of positive integers. There are various problems regarding such compositions which have received attention in recent years. This paper is an attempt at finding a closed- form formula for the number of S-restricted compositions of n. To do so, we reduce the problem to finding solutions to corresponding so-called interpreters which are linear homogeneous recurrence relations with constant coefficients. Then, we reduce interpreters to Diophantine equations. Such equations are not in general solvable. Thus, we restrict our attention to those S-restricted composition problems whose interpreters have a small number of coefficients, thereby leading to solvable Diophantine equations. The formalism developed is then used to study the integer sequences related to some well-known cases of the S-restricted composition problem.]]> 0 <![CDATA[Chordal graph shelling antimatroids have received little attention with regard to their combinatorial properties and related optimization problems, as compared to the case of poset shelling antimatroids. Here we consider a special case of these antimatroids, namely the split graph shelling antimatroids. We show that the feasible sets of such an antimatroid relate to some poset shelling antimatroids constructed from the graph. We discuss a few applications, obtaining in particular a simple polynomial-time algorithm to find a maximum weight feasible set. We also provide a simple description of the circuits and the free sets.]]> Fri, 24 Mar 2017 16:07:54 +0000 https://doi.org/10.23638/DMTCS-19-1-7 https://doi.org/10.23638/DMTCS-19-1-7 Cardinal, Jean Doignon, Jean-Paul Merckx, Keno Cardinal, Jean Doignon, Jean-Paul Merckx, Keno <![CDATA[Chordal graph shelling antimatroids have received little attention with regard to their combinatorial properties and related optimization problems, as compared to the case of poset shelling antimatroids. Here we consider a special case of these antimatroids, namely the split graph shelling antimatroids. We show that the feasible sets of such an antimatroid relate to some poset shelling antimatroids constructed from the graph. We discuss a few applications, obtaining in particular a simple polynomial-time algorithm to find a maximum weight feasible set. We also provide a simple description of the circuits and the free sets.]]> 0 <![CDATA[This is the second of two papers in which we determine all 242 Wilf classes of triples of 4-letter patterns by showing that there are 32 non-singleton Wilf classes. There are 317 symmetry classes of triples of 4-letter patterns and after computer calculation of initial terms, the problem reduces to showing that counting sequences that appear to be the same (agree in the first 16 terms) are in fact identical. This amounts to counting avoiders for 107 representative triples. The insertion encoding algorithm (INSENC) applies to many of them and some others have been previously counted. There remain 36 triples and the first paper dealt with the first 18. In this paper, we find the generating function for the last 18 triples which turns out to be algebraic in each case. Our methods are both combinatorial and analytic, including decomposi-tions by left-right maxima and by initial letters. Sometimes this leads to an algebraic equation for the generating function, sometimes to a functional equation or a multi-index recurrence that succumbs to the kernel method. A particularly nice so-called cell decomposition is used in one of the cases (Case 238).]]> Fri, 24 Mar 2017 15:40:12 +0000 https://doi.org/10.23638/DMTCS-19-1-6 https://doi.org/10.23638/DMTCS-19-1-6 Callan, David Mansour, Toufik Shattuck, Mark Callan, David Mansour, Toufik Shattuck, Mark <![CDATA[This is the second of two papers in which we determine all 242 Wilf classes of triples of 4-letter patterns by showing that there are 32 non-singleton Wilf classes. There are 317 symmetry classes of triples of 4-letter patterns and after computer calculation of initial terms, the problem reduces to showing that counting sequences that appear to be the same (agree in the first 16 terms) are in fact identical. This amounts to counting avoiders for 107 representative triples. The insertion encoding algorithm (INSENC) applies to many of them and some others have been previously counted. There remain 36 triples and the first paper dealt with the first 18. In this paper, we find the generating function for the last 18 triples which turns out to be algebraic in each case. Our methods are both combinatorial and analytic, including decomposi-tions by left-right maxima and by initial letters. Sometimes this leads to an algebraic equation for the generating function, sometimes to a functional equation or a multi-index recurrence that succumbs to the kernel method. A particularly nice so-called cell decomposition is used in one of the cases (Case 238).]]> 0 <![CDATA[This is the first of two papers in which we determine all 242 Wilf classes of triples of 4-letter patterns by showing that there are 32 non-singleton Wilf classes. There are 317 symmetry classes of triples of 4-letter patterns and after computer calculation of initial terms, the problem reduces to showing that counting sequences that appear to be the same (agree in the first 16 terms) are in fact identical. This amounts to counting avoiders for 107 representative triples. The insertion encoding algorithm (INSENC) applies to many of them and some others have been previously counted. Thus there remain 36 triples. In this paper, we find the generating function for the first 18 of these triples and in a second paper, we treat the other 18. The generating function turns out to be algebraic in each case. Our methods are both combinatorial and analytic, including decompositions by left-right maxima and by initial letters. Sometimes this leads to an algebraic equation for the generating function, sometimes to a functional equation or a multi-index recurrence that succumbs to the kernel method. A bijection is used in one of the cases (Case 50).]]> Fri, 24 Mar 2017 15:37:57 +0000 https://doi.org/10.23638/DMTCS-19-1-5 https://doi.org/10.23638/DMTCS-19-1-5 Callan, David Mansour, Toufik Shattuck, Mark Callan, David Mansour, Toufik Shattuck, Mark <![CDATA[This is the first of two papers in which we determine all 242 Wilf classes of triples of 4-letter patterns by showing that there are 32 non-singleton Wilf classes. There are 317 symmetry classes of triples of 4-letter patterns and after computer calculation of initial terms, the problem reduces to showing that counting sequences that appear to be the same (agree in the first 16 terms) are in fact identical. This amounts to counting avoiders for 107 representative triples. The insertion encoding algorithm (INSENC) applies to many of them and some others have been previously counted. Thus there remain 36 triples. In this paper, we find the generating function for the first 18 of these triples and in a second paper, we treat the other 18. The generating function turns out to be algebraic in each case. Our methods are both combinatorial and analytic, including decompositions by left-right maxima and by initial letters. Sometimes this leads to an algebraic equation for the generating function, sometimes to a functional equation or a multi-index recurrence that succumbs to the kernel method. A bijection is used in one of the cases (Case 50).]]> 0 <![CDATA[Recent research on the combinatorics of finite sets has explored the structure of symmetric difference-closed sets, and recent research in combinatorial group theory has concerned the enumeration of commuting involutions in $S_{n}$ and $A_{n}$. In this article, we consider an interesting combination of these two subjects, by introducing classes of symmetric difference-closed sets of elements which correspond in a natural way to commuting involutions in $S_{n}$ and $A_{n}$. We consider the natural combinatorial problem of enumerating symmetric difference-closed sets consisting of subsets of sets consisting of pairwise disjoint $2$-subsets of $[n]$, and the problem of enumerating symmetric difference-closed sets consisting of elements which correspond to commuting involutions in $A_{n}$. We prove explicit combinatorial formulas for symmetric difference-closed sets of these forms, and we prove a number of conjectured properties related to such sets which had previously been discovered experimentally using the On-Line Encyclopedia of Integer Sequences.]]> Thu, 23 Mar 2017 10:33:33 +0000 https://doi.org/10.23638/DMTCS-19-1-8 https://doi.org/10.23638/DMTCS-19-1-8 Campbell, John Campbell, John <![CDATA[Recent research on the combinatorics of finite sets has explored the structure of symmetric difference-closed sets, and recent research in combinatorial group theory has concerned the enumeration of commuting involutions in $S_{n}$ and $A_{n}$. In this article, we consider an interesting combination of these two subjects, by introducing classes of symmetric difference-closed sets of elements which correspond in a natural way to commuting involutions in $S_{n}$ and $A_{n}$. We consider the natural combinatorial problem of enumerating symmetric difference-closed sets consisting of subsets of sets consisting of pairwise disjoint $2$-subsets of $[n]$, and the problem of enumerating symmetric difference-closed sets consisting of elements which correspond to commuting involutions in $A_{n}$. We prove explicit combinatorial formulas for symmetric difference-closed sets of these forms, and we prove a number of conjectured properties related to such sets which had previously been discovered experimentally using the On-Line Encyclopedia of Integer Sequences.]]> 0 <![CDATA[Given permutations σ of size k and π of size n with k < n, the permutation pattern matching problem is to decide whether σ occurs in π as an order-isomorphic subsequence. We give a linear-time algorithm in case both π and σ avoid the two size-3 permutations 213 and 231. For the special case where only σ avoids 213 and 231, we present a O(max(kn 2 , n 2 log log n)-time algorithm. We extend our research to bivincular patterns that avoid 213 and 231 and present a O(kn 4)-time algorithm. Finally we look at the related problem of the longest subsequence which avoids 213 and 231.]]> Wed, 22 Mar 2017 10:10:58 +0000 https://doi.org/10.46298/dmtcs.1329 https://doi.org/10.46298/dmtcs.1329 Neou, Both, Rizzi, Romeo Vialette, Stéphane Neou, Both, Rizzi, Romeo Vialette, Stéphane <![CDATA[Given permutations σ of size k and π of size n with k < n, the permutation pattern matching problem is to decide whether σ occurs in π as an order-isomorphic subsequence. We give a linear-time algorithm in case both π and σ avoid the two size-3 permutations 213 and 231. For the special case where only σ avoids 213 and 231, we present a O(max(kn 2 , n 2 log log n)-time algorithm. We extend our research to bivincular patterns that avoid 213 and 231 and present a O(kn 4)-time algorithm. Finally we look at the related problem of the longest subsequence which avoids 213 and 231.]]> 0 <![CDATA[A digraph is \emph{traceable} if it has a path that visits every vertex. A digraph $D$ is \emph{hypotraceable} if $D$ is not traceable but $D-v$ is traceable for every vertex $v\in V(D)$. It is known that there exists a planar hypotraceable digraph of order $n$ for every $n\geq 7$, but no examples of planar hypotraceable oriented graphs (digraphs without 2-cycles) have yet appeared in the literature. We show that there exists a planar hypotraceable oriented graph of order $n$ for every even $n \geq 10$, with the possible exception of $n = 14$.]]> Thu, 16 Mar 2017 13:27:09 +0000 https://doi.org/10.23638/DMTCS-19-1-4 https://doi.org/10.23638/DMTCS-19-1-4 van Aardt, Susan, Burger, Alewyn Petrus Frick, Marietjie van Aardt, Susan, Burger, Alewyn Petrus Frick, Marietjie <![CDATA[A digraph is \emph{traceable} if it has a path that visits every vertex. A digraph $D$ is \emph{hypotraceable} if $D$ is not traceable but $D-v$ is traceable for every vertex $v\in V(D)$. It is known that there exists a planar hypotraceable digraph of order $n$ for every $n\geq 7$, but no examples of planar hypotraceable oriented graphs (digraphs without 2-cycles) have yet appeared in the literature. We show that there exists a planar hypotraceable oriented graph of order $n$ for every even $n \geq 10$, with the possible exception of $n = 14$.]]> 0 <![CDATA[A set $S$ of vertices in a graph $G$ is a $2$-dominating set if every vertex of $G$ not in $S$ is adjacent to at least two vertices in $S$, and $S$ is a $2$-independent set if every vertex in $S$ is adjacent to at most one vertex of $S$. The $2$-domination number $\gamma_2(G)$ is the minimum cardinality of a $2$-dominating set in $G$, and the $2$-independence number $\alpha_2(G)$ is the maximum cardinality of a $2$-independent set in $G$. Chellali and Meddah [{\it Trees with equal $2$-domination and $2$-independence numbers,} Discussiones Mathematicae Graph Theory 32 (2012), 263--270] provided a constructive characterization of trees with equal $2$-domination and $2$-independence numbers. Their characterization is in terms of global properties of a tree, and involves properties of minimum $2$-dominating and maximum $2$-independent sets in the tree at each stage of the construction. We provide a constructive characterization that relies only on local properties of the tree at each stage of the construction.]]> Thu, 09 Mar 2017 07:53:33 +0000 https://doi.org/10.23638/DMTCS-19-1-1 https://doi.org/10.23638/DMTCS-19-1-1 Brause, Christoph Henning, Michael A. Krzywkowski, Marcin Brause, Christoph Henning, Michael A. Krzywkowski, Marcin <![CDATA[A set $S$ of vertices in a graph $G$ is a $2$-dominating set if every vertex of $G$ not in $S$ is adjacent to at least two vertices in $S$, and $S$ is a $2$-independent set if every vertex in $S$ is adjacent to at most one vertex of $S$. The $2$-domination number $\gamma_2(G)$ is the minimum cardinality of a $2$-dominating set in $G$, and the $2$-independence number $\alpha_2(G)$ is the maximum cardinality of a $2$-independent set in $G$. Chellali and Meddah [{\it Trees with equal $2$-domination and $2$-independence numbers,} Discussiones Mathematicae Graph Theory 32 (2012), 263--270] provided a constructive characterization of trees with equal $2$-domination and $2$-independence numbers. Their characterization is in terms of global properties of a tree, and involves properties of minimum $2$-dominating and maximum $2$-independent sets in the tree at each stage of the construction. We provide a constructive characterization that relies only on local properties of the tree at each stage of the construction.]]> 0 <![CDATA[The distinguishing number of a graph $G$ is a symmetry related graph invariant whose study started two decades ago. The distinguishing number $D(G)$ is the least integer $d$ such that $G$ has a $d$-distinguishing coloring. A distinguishing $d$-coloring is a coloring $c:V(G)\rightarrow\{1,...,d\}$ invariant only under the trivial automorphism. In this paper, we introduce a game variant of the distinguishing number. The distinguishing game is a game with two players, the Gentle and the Rascal, with antagonist goals. This game is played on a graph $G$ with a set of $d\in\mathbb N^*$ colors. Alternately, the two players choose a vertex of $G$ and color it with one of the $d$ colors. The game ends when all the vertices have been colored. Then the Gentle wins if the coloring is distinguishing and the Rascal wins otherwise. This game leads to define two new invariants for a graph $G$, which are the minimum numbers of colors needed to ensure that the Gentle has a winning strategy, depending on who starts. These invariants could be infinite, thus we start by giving sufficient conditions to have infinite game distinguishing numbers. We also show that for graphs with cyclic automorphisms group of prime odd order, both game invariants are finite. After that, we define a class of graphs, the involutive graphs, for which the game distinguishing number can be quadratically bounded above by the classical distinguishing number. The definition of this class is closely related to imprimitive actions whose blocks have size $2$. Then, we apply results on involutive graphs to compute the exact value of these invariants for hypercubes and even cycles. Finally, we study odd cycles, for which we are able to compute the exact value when their order is not prime. In the prime order case, we give an upper bound of $3$.]]> Thu, 02 Mar 2017 14:48:34 +0000 https://doi.org/10.23638/DMTCS-19-1-2 https://doi.org/10.23638/DMTCS-19-1-2 Gravier, Sylvain Meslem, Kahina Schmidt, Simon Slimani, Souad Gravier, Sylvain Meslem, Kahina Schmidt, Simon Slimani, Souad <![CDATA[The distinguishing number of a graph $G$ is a symmetry related graph invariant whose study started two decades ago. The distinguishing number $D(G)$ is the least integer $d$ such that $G$ has a $d$-distinguishing coloring. A distinguishing $d$-coloring is a coloring $c:V(G)\rightarrow\{1,...,d\}$ invariant only under the trivial automorphism. In this paper, we introduce a game variant of the distinguishing number. The distinguishing game is a game with two players, the Gentle and the Rascal, with antagonist goals. This game is played on a graph $G$ with a set of $d\in\mathbb N^*$ colors. Alternately, the two players choose a vertex of $G$ and color it with one of the $d$ colors. The game ends when all the vertices have been colored. Then the Gentle wins if the coloring is distinguishing and the Rascal wins otherwise. This game leads to define two new invariants for a graph $G$, which are the minimum numbers of colors needed to ensure that the Gentle has a winning strategy, depending on who starts. These invariants could be infinite, thus we start by giving sufficient conditions to have infinite game distinguishing numbers. We also show that for graphs with cyclic automorphisms group of prime odd order, both game invariants are finite. After that, we define a class of graphs, the involutive graphs, for which the game distinguishing number can be quadratically bounded above by the classical distinguishing number. The definition of this class is closely related to imprimitive actions whose blocks have size $2$. Then, we apply results on involutive graphs to compute the exact value of these invariants for hypercubes and even cycles. Finally, we study odd cycles, for which we are able to compute the exact value when their order is not prime. In the prime order case, we give an upper bound of $3$.]]> 0 <![CDATA[Let $S_n$ denote the symmetric group. For any $\sigma \in S_n$, we let $\mathrm{des}(\sigma)$ denote the number of descents of $\sigma$, $\mathrm{inv}(\sigma)$ denote the number of inversions of $\sigma$, and $\mathrm{LRmin}(\sigma)$ denote the number of left-to-right minima of $\sigma$. For any sequence of statistics $\mathrm{stat}_1, \ldots \mathrm{stat}_k$ on permutations, we say two permutations $\alpha$ and $\beta$ in $S_j$ are $(\mathrm{stat}_1, \ldots \mathrm{stat}_k)$-c-Wilf equivalent if the generating function of $\prod_{i=1}^k x_i^{\mathrm{stat}_i}$ over all permutations which have no consecutive occurrences of $\alpha$ equals the generating function of $\prod_{i=1}^k x_i^{\mathrm{stat}_i}$ over all permutations which have no consecutive occurrences of $\beta$. We give many examples of pairs of permutations $\alpha$ and $\beta$ in $S_j$ which are $\mathrm{des}$-c-Wilf equivalent, $(\mathrm{des},\mathrm{inv})$-c-Wilf equivalent, and $(\mathrm{des},\mathrm{inv},\mathrm{LRmin})$-c-Wilf equivalent. For example, we will show that if $\alpha$ and $\beta$ are minimally overlapping permutations in $S_j$ which start with 1 and end with the same element and $\mathrm{des}(\alpha) = \mathrm{des}(\beta)$ and $\mathrm{inv}(\alpha) = \mathrm{inv}(\beta)$, then $\alpha$ and $\beta$ are $(\mathrm{des},\mathrm{inv})$-c-Wilf equivalent.]]> Thu, 02 Mar 2017 14:13:07 +0000 https://doi.org/10.46298/dmtcs.1312 https://doi.org/10.46298/dmtcs.1312 Bach, Quang T. Remmel, Jeffrey B. Bach, Quang T. Remmel, Jeffrey B. <![CDATA[Let $S_n$ denote the symmetric group. For any $\sigma \in S_n$, we let $\mathrm{des}(\sigma)$ denote the number of descents of $\sigma$, $\mathrm{inv}(\sigma)$ denote the number of inversions of $\sigma$, and $\mathrm{LRmin}(\sigma)$ denote the number of left-to-right minima of $\sigma$. For any sequence of statistics $\mathrm{stat}_1, \ldots \mathrm{stat}_k$ on permutations, we say two permutations $\alpha$ and $\beta$ in $S_j$ are $(\mathrm{stat}_1, \ldots \mathrm{stat}_k)$-c-Wilf equivalent if the generating function of $\prod_{i=1}^k x_i^{\mathrm{stat}_i}$ over all permutations which have no consecutive occurrences of $\alpha$ equals the generating function of $\prod_{i=1}^k x_i^{\mathrm{stat}_i}$ over all permutations which have no consecutive occurrences of $\beta$. We give many examples of pairs of permutations $\alpha$ and $\beta$ in $S_j$ which are $\mathrm{des}$-c-Wilf equivalent, $(\mathrm{des},\mathrm{inv})$-c-Wilf equivalent, and $(\mathrm{des},\mathrm{inv},\mathrm{LRmin})$-c-Wilf equivalent. For example, we will show that if $\alpha$ and $\beta$ are minimally overlapping permutations in $S_j$ which start with 1 and end with the same element and $\mathrm{des}(\alpha) = \mathrm{des}(\beta)$ and $\mathrm{inv}(\alpha) = \mathrm{inv}(\beta)$, then $\alpha$ and $\beta$ are $(\mathrm{des},\mathrm{inv})$-c-Wilf equivalent.]]> 0 <![CDATA[We study the iteration of the process "a particle jumps to the right" in permutations. We prove that the set of permutations obtained in this model after a given number of iterations from the identity is a class of pattern avoiding permutations. We characterize the elements of the basis of this class and we enumerate these "forbidden minimal patterns" by giving their bivariate exponential generating function: we achieve this via a catalytic variable, the number of left-to-right maxima. We show that this generating function is a D-finite function satisfying a nice differential equation of order~2. We give some congruence properties for the coefficients of this generating function, and we show that their asymptotics involves a rather unusual algebraic exponent (the golden ratio $(1+\sqrt 5)/2$) and some unusual closed-form constants. We end by proving a limit law: a forbidden pattern of length $n$ has typically $(\ln n) /\sqrt{5}$ left-to-right maxima, with Gaussian fluctuations.]]> Fri, 10 Feb 2017 08:11:59 +0000 https://doi.org/10.46298/dmtcs.1344 https://doi.org/10.46298/dmtcs.1344 Banderier, Cyril Baril, Jean-Luc Santos, Céline Moreira Dos Banderier, Cyril Baril, Jean-Luc Santos, Céline Moreira Dos <![CDATA[We study the iteration of the process "a particle jumps to the right" in permutations. We prove that the set of permutations obtained in this model after a given number of iterations from the identity is a class of pattern avoiding permutations. We characterize the elements of the basis of this class and we enumerate these "forbidden minimal patterns" by giving their bivariate exponential generating function: we achieve this via a catalytic variable, the number of left-to-right maxima. We show that this generating function is a D-finite function satisfying a nice differential equation of order~2. We give some congruence properties for the coefficients of this generating function, and we show that their asymptotics involves a rather unusual algebraic exponent (the golden ratio $(1+\sqrt 5)/2$) and some unusual closed-form constants. We end by proving a limit law: a forbidden pattern of length $n$ has typically $(\ln n) /\sqrt{5}$ left-to-right maxima, with Gaussian fluctuations.]]> 0 <![CDATA[Given a set $Y$ of decreasing plane trees and a permutation $\pi$, how many trees in $Y$ have $\pi$ as their postorder? Using combinatorial and geometric constructions, we provide a method for answering this question for certain sets $Y$ and all permutations $\pi$. We then provide applications of our results to the study of the deterministic stack-sorting algorithm.]]> Mon, 06 Feb 2017 16:17:25 +0000 https://doi.org/10.23638/DMTCS-19-1-3 https://doi.org/10.23638/DMTCS-19-1-3 Defant, Colin Defant, Colin <![CDATA[Given a set $Y$ of decreasing plane trees and a permutation $\pi$, how many trees in $Y$ have $\pi$ as their postorder? Using combinatorial and geometric constructions, we provide a method for answering this question for certain sets $Y$ and all permutations $\pi$. We then provide applications of our results to the study of the deterministic stack-sorting algorithm.]]> 0 <![CDATA[We study Markov chains for $\alpha$-orientations of plane graphs, these are orientations where the outdegree of each vertex is prescribed by the value of a given function $\alpha$. The set of $\alpha$-orientations of a plane graph has a natural distributive lattice structure. The moves of the up-down Markov chain on this distributive lattice corresponds to reversals of directed facial cycles in the $\alpha$-orientation. We have a positive and several negative results regarding the mixing time of such Markov chains. A 2-orientation of a plane quadrangulation is an orientation where every inner vertex has outdegree 2. We show that there is a class of plane quadrangulations such that the up-down Markov chain on the 2-orientations of these quadrangulations is slowly mixing. On the other hand the chain is rapidly mixing on 2-orientations of quadrangulations with maximum degree at most 4. Regarding examples for slow mixing we also revisit the case of 3-orientations of triangulations which has been studied before by Miracle et al.. Our examples for slow mixing are simpler and have a smaller maximum degree, Finally we present the first example of a function $\alpha$ and a class of plane triangulations of constant maximum degree such that the up-down Markov chain on the $\alpha$-orientations of these graphs is slowly mixing.]]> Fri, 03 Feb 2017 10:43:52 +0000 https://doi.org/10.46298/dmtcs.1376 https://doi.org/10.46298/dmtcs.1376 Felsner, Stefan Heldt, Daniel Felsner, Stefan Heldt, Daniel <![CDATA[We study Markov chains for $\alpha$-orientations of plane graphs, these are orientations where the outdegree of each vertex is prescribed by the value of a given function $\alpha$. The set of $\alpha$-orientations of a plane graph has a natural distributive lattice structure. The moves of the up-down Markov chain on this distributive lattice corresponds to reversals of directed facial cycles in the $\alpha$-orientation. We have a positive and several negative results regarding the mixing time of such Markov chains. A 2-orientation of a plane quadrangulation is an orientation where every inner vertex has outdegree 2. We show that there is a class of plane quadrangulations such that the up-down Markov chain on the 2-orientations of these quadrangulations is slowly mixing. On the other hand the chain is rapidly mixing on 2-orientations of quadrangulations with maximum degree at most 4. Regarding examples for slow mixing we also revisit the case of 3-orientations of triangulations which has been studied before by Miracle et al.. Our examples for slow mixing are simpler and have a smaller maximum degree, Finally we present the first example of a function $\alpha$ and a class of plane triangulations of constant maximum degree such that the up-down Markov chain on the $\alpha$-orientations of these graphs is slowly mixing.]]> 0 <![CDATA[The Permutation Pattern Matching problem, asking whether a pattern permutation $\pi$ is contained in a permutation $\tau$, is known to be NP-complete. In this paper we present two polynomial time algorithms for special cases. The first algorithm is applicable if both $\pi$ and $\tau$ are $321$-avoiding; the second is applicable if $\pi$ and $\tau$ are skew-merged. Both algorithms have a runtime of $O(kn)$, where $k$ is the length of $\pi$ and $n$ the length of $\tau$.]]> Wed, 21 Dec 2016 14:02:07 +0000 https://doi.org/10.46298/dmtcs.1308 https://doi.org/10.46298/dmtcs.1308 Albert, Michael H. Lackner, Marie-Louise Lackner, Martin Vatter, Vincent Albert, Michael H. Lackner, Marie-Louise Lackner, Martin Vatter, Vincent <![CDATA[The Permutation Pattern Matching problem, asking whether a pattern permutation $\pi$ is contained in a permutation $\tau$, is known to be NP-complete. In this paper we present two polynomial time algorithms for special cases. The first algorithm is applicable if both $\pi$ and $\tau$ are $321$-avoiding; the second is applicable if $\pi$ and $\tau$ are skew-merged. Both algorithms have a runtime of $O(kn)$, where $k$ is the length of $\pi$ and $n$ the length of $\tau$.]]> 0 <![CDATA[In this paper, we confirm conjectures of Laborde-Zubieta on the enumeration of corners in tree-like tableaux and in symmetric tree-like tableaux. In the process, we also enumerate corners in (type $B$) permutation tableaux and (symmetric) alternative tableaux. The proof is based on Corteel and Nadeau's bijection between permutation tableaux and permutations. It allows us to interpret the number of corners as a statistic over permutations that is easier to count. The type $B$ case uses the bijection of Corteel and Kim between type $B$ permutation tableaux and signed permutations. Moreover, we give a bijection between corners and runs of size 1 in permutations, which gives an alternative proof of the enumeration of corners. Finally, we introduce conjectural polynomial analogues of these enumerations, and explain the implications on the PASEP.]]> Fri, 02 Dec 2016 13:27:20 +0000 https://doi.org/10.46298/dmtcs.1408 https://doi.org/10.46298/dmtcs.1408 Gao, Alice L. L. Gao, Emily X. L. Laborde-Zubieta, Patxi Sun, Brian Y. Gao, Alice L. L. Gao, Emily X. L. Laborde-Zubieta, Patxi Sun, Brian Y. <![CDATA[In this paper, we confirm conjectures of Laborde-Zubieta on the enumeration of corners in tree-like tableaux and in symmetric tree-like tableaux. In the process, we also enumerate corners in (type $B$) permutation tableaux and (symmetric) alternative tableaux. The proof is based on Corteel and Nadeau's bijection between permutation tableaux and permutations. It allows us to interpret the number of corners as a statistic over permutations that is easier to count. The type $B$ case uses the bijection of Corteel and Kim between type $B$ permutation tableaux and signed permutations. Moreover, we give a bijection between corners and runs of size 1 in permutations, which gives an alternative proof of the enumeration of corners. Finally, we introduce conjectural polynomial analogues of these enumerations, and explain the implications on the PASEP.]]> 0 <![CDATA[We study two different objects attached to an arbitrary quadrangulation of a regular polygon. The first one is a poset, closely related to the Stokes polytopes introduced by Baryshnikov. The second one is a set of some paths configurations inside the quadrangulation, satisfying some specific constraints. These objects provide a generalisation of the existing combinatorics of cluster algebras and nonnesting partitions of type A.]]> Fri, 02 Dec 2016 13:12:07 +0000 https://doi.org/10.46298/dmtcs.1382 https://doi.org/10.46298/dmtcs.1382 Chapoton, Frédéric Chapoton, Frédéric <![CDATA[We study two different objects attached to an arbitrary quadrangulation of a regular polygon. The first one is a poset, closely related to the Stokes polytopes introduced by Baryshnikov. The second one is a set of some paths configurations inside the quadrangulation, satisfying some specific constraints. These objects provide a generalisation of the existing combinatorics of cluster algebras and nonnesting partitions of type A.]]> 0 <![CDATA[A narrow connection between infinite binary words rich in classical palindromes and infinite binary words rich simultaneously in palindromes and pseudopalindromes (the so-called $H$-rich words) is demonstrated. The correspondence between rich and $H$-rich words is based on the operation $S$ acting over words over the alphabet $\{0,1\}$ and defined by $S(u_0u_1u_2\ldots) = v_1v_2v_3\ldots$, where $v_i= u_{i-1} + u_i \mod 2$. The operation $S$ enables us to construct a new class of rich words and a new class of $H$-rich words. Finally, the operation $S$ is considered on the multiliteral alphabet $\mathbb{Z}_m$ as well and applied to the generalized Thue--Morse words. As a byproduct, new binary rich and $H$-rich words are obtained by application of $S$ on the generalized Thue--Morse words over the alphabet $\mathbb{Z}_4$.]]> Tue, 22 Nov 2016 14:00:59 +0000 https://doi.org/10.46298/dmtcs.655 https://doi.org/10.46298/dmtcs.655 Pelantová, Edita Starosta, Štěpán Pelantová, Edita Starosta, Štěpán <![CDATA[A narrow connection between infinite binary words rich in classical palindromes and infinite binary words rich simultaneously in palindromes and pseudopalindromes (the so-called $H$-rich words) is demonstrated. The correspondence between rich and $H$-rich words is based on the operation $S$ acting over words over the alphabet $\{0,1\}$ and defined by $S(u_0u_1u_2\ldots) = v_1v_2v_3\ldots$, where $v_i= u_{i-1} + u_i \mod 2$. The operation $S$ enables us to construct a new class of rich words and a new class of $H$-rich words. Finally, the operation $S$ is considered on the multiliteral alphabet $\mathbb{Z}_m$ as well and applied to the generalized Thue--Morse words. As a byproduct, new binary rich and $H$-rich words are obtained by application of $S$ on the generalized Thue--Morse words over the alphabet $\mathbb{Z}_4$.]]> 0 <![CDATA[The objective of this paper is to find in a setting of n sequential observations of objects a good online policy to select the k bestof these n uniquely rankable objects. This focus is motivated by the fact that it is hard to find closed form solutions of optimalstrategies for general k and n. Selection is without recall, and the idea is to investigate threshold functions which maintain allpresent information, that is thresholds which are functions of all selections made so far. Our main interest lies in the asymptoticbehaviour of these thresholds as n -> infinity and in the corresponding asymptotic performance of the threshold algorithm.]]> Fri, 28 Oct 2016 10:01:41 +0000 https://doi.org/10.46298/dmtcs.1291 https://doi.org/10.46298/dmtcs.1291 Bruss, F. Thomas Louchard, Guy Bruss, F. Thomas Louchard, Guy <![CDATA[The objective of this paper is to find in a setting of n sequential observations of objects a good online policy to select the k bestof these n uniquely rankable objects. This focus is motivated by the fact that it is hard to find closed form solutions of optimalstrategies for general k and n. Selection is without recall, and the idea is to investigate threshold functions which maintain allpresent information, that is thresholds which are functions of all selections made so far. Our main interest lies in the asymptoticbehaviour of these thresholds as n -> infinity and in the corresponding asymptotic performance of the threshold algorithm.]]> 0 <![CDATA[A right ideal (left ideal, two-sided ideal) is a non-empty language $L$ over an alphabet $\Sigma$ such that $L=L\Sigma^*$ ($L=\Sigma^*L$, $L=\Sigma^*L\Sigma^*$). Let $k=3$ for right ideals, 4 for left ideals and 5 for two-sided ideals. We show that there exist sequences ($L_n \mid n \ge k $) of right, left, and two-sided regular ideals, where $L_n$ has quotient complexity (state complexity) $n$, such that $L_n$ is most complex in its class under the following measures of complexity: the size of the syntactic semigroup, the quotient complexities of the left quotients of $L_n$, the number of atoms (intersections of complemented and uncomplemented left quotients), the quotient complexities of the atoms, and the quotient complexities of reversal, star, product (concatenation), and all binary boolean operations. In that sense, these ideals are "most complex" languages in their classes, or "universal witnesses" to the complexity of the various operations.]]> Mon, 17 Oct 2016 09:42:14 +0000 https://doi.org/10.46298/dmtcs.1343 https://doi.org/10.46298/dmtcs.1343 Brzozowski, Janusz Davies, Sylvie Liu, Bo Yang Victor Brzozowski, Janusz Davies, Sylvie Liu, Bo Yang Victor <![CDATA[A right ideal (left ideal, two-sided ideal) is a non-empty language $L$ over an alphabet $\Sigma$ such that $L=L\Sigma^*$ ($L=\Sigma^*L$, $L=\Sigma^*L\Sigma^*$). Let $k=3$ for right ideals, 4 for left ideals and 5 for two-sided ideals. We show that there exist sequences ($L_n \mid n \ge k $) of right, left, and two-sided regular ideals, where $L_n$ has quotient complexity (state complexity) $n$, such that $L_n$ is most complex in its class under the following measures of complexity: the size of the syntactic semigroup, the quotient complexities of the left quotients of $L_n$, the number of atoms (intersections of complemented and uncomplemented left quotients), the quotient complexities of the atoms, and the quotient complexities of reversal, star, product (concatenation), and all binary boolean operations. In that sense, these ideals are "most complex" languages in their classes, or "universal witnesses" to the complexity of the various operations.]]> 0 <![CDATA[We prove geometric Ramsey-type statements on collections of lines in 3-space. These statements give guarantees on the size of a clique or an independent set in (hyper)graphs induced by incidence relations between lines, points, and reguli in 3-space. Among other things, we prove that: (1) The intersection graph of n lines in R^3 has a clique or independent set of size Omega(n^{1/3}). (2) Every set of n lines in R^3 has a subset of n^{1/2} lines that are all stabbed by one line, or a subset of Omega((n/log n)^{1/5}) such that no 6-subset is stabbed by one line. (3) Every set of n lines in general position in R^3 has a subset of Omega(n^{2/3}) lines that all lie on a regulus, or a subset of Omega(n^{1/3}) lines such that no 4-subset is contained in a regulus. The proofs of these statements all follow from geometric incidence bounds -- such as the Guth-Katz bound on point-line incidences in R^3 -- combined with Tur\'an-type results on independent sets in sparse graphs and hypergraphs. Although similar Ramsey-type statements can be proved using existing generic algebraic frameworks, the lower bounds we get are much larger than what can be obtained with these methods. The proofs directly yield polynomial-time algorithms for finding subsets of the claimed size.]]> Mon, 19 Sep 2016 08:08:44 +0000 https://doi.org/10.46298/dmtcs.1367 https://doi.org/10.46298/dmtcs.1367 Cardinal, Jean Payne, Michael S. Solomon, Noam Cardinal, Jean Payne, Michael S. Solomon, Noam <![CDATA[We prove geometric Ramsey-type statements on collections of lines in 3-space. These statements give guarantees on the size of a clique or an independent set in (hyper)graphs induced by incidence relations between lines, points, and reguli in 3-space. Among other things, we prove that: (1) The intersection graph of n lines in R^3 has a clique or independent set of size Omega(n^{1/3}). (2) Every set of n lines in R^3 has a subset of n^{1/2} lines that are all stabbed by one line, or a subset of Omega((n/log n)^{1/5}) such that no 6-subset is stabbed by one line. (3) Every set of n lines in general position in R^3 has a subset of Omega(n^{2/3}) lines that all lie on a regulus, or a subset of Omega(n^{1/3}) lines such that no 4-subset is contained in a regulus. The proofs of these statements all follow from geometric incidence bounds -- such as the Guth-Katz bound on point-line incidences in R^3 -- combined with Tur\'an-type results on independent sets in sparse graphs and hypergraphs. Although similar Ramsey-type statements can be proved using existing generic algebraic frameworks, the lower bounds we get are much larger than what can be obtained with these methods. The proofs directly yield polynomial-time algorithms for finding subsets of the claimed size.]]> 0 <![CDATA[In 2000 Klazar introduced a new notion of pattern avoidance in the context of set partitions of $[n]=\{1,\ldots, n\}$. The purpose of the present paper is to undertake a study of the concept of Wilf-equivalence based on Klazar's notion. We determine all Wilf-equivalences for partitions with exactly two blocks, one of which is a singleton block, and we conjecture that, for $n\geq 4$, these are all the Wilf-equivalences except for those arising from complementation. If $\tau$ is a partition of $[k]$ and $\Pi_n(\tau)$ denotes the set of all partitions of $[n]$ that avoid $\tau$, we establish inequalities between $|\Pi_n(\tau_1)|$ and $|\Pi_n(\tau_2)|$ for several choices of $\tau_1$ and $\tau_2$, and we prove that if $\tau_2$ is the partition of $[k]$ with only one block, then $|\Pi_n(\tau_1)| <|\Pi_n(\tau_2)|$ for all $n>k$ and all partitions $\tau_1$ of $[k]$ with exactly two blocks. We conjecture that this result holds for all partitions $\tau_1$ of $[k]$. Finally, we enumerate $\Pi_n(\tau)$ for all partitions $\tau$ of $[4]$.]]> Wed, 07 Sep 2016 07:52:25 +0000 https://doi.org/10.46298/dmtcs.1327 https://doi.org/10.46298/dmtcs.1327 Bloom, Jonathan Saracino, Dan Bloom, Jonathan Saracino, Dan <![CDATA[In 2000 Klazar introduced a new notion of pattern avoidance in the context of set partitions of $[n]=\{1,\ldots, n\}$. The purpose of the present paper is to undertake a study of the concept of Wilf-equivalence based on Klazar's notion. We determine all Wilf-equivalences for partitions with exactly two blocks, one of which is a singleton block, and we conjecture that, for $n\geq 4$, these are all the Wilf-equivalences except for those arising from complementation. If $\tau$ is a partition of $[k]$ and $\Pi_n(\tau)$ denotes the set of all partitions of $[n]$ that avoid $\tau$, we establish inequalities between $|\Pi_n(\tau_1)|$ and $|\Pi_n(\tau_2)|$ for several choices of $\tau_1$ and $\tau_2$, and we prove that if $\tau_2$ is the partition of $[k]$ with only one block, then $|\Pi_n(\tau_1)| <|\Pi_n(\tau_2)|$ for all $n>k$ and all partitions $\tau_1$ of $[k]$ with exactly two blocks. We conjecture that this result holds for all partitions $\tau_1$ of $[k]$. Finally, we enumerate $\Pi_n(\tau)$ for all partitions $\tau$ of $[4]$.]]> 0 <![CDATA[The generalized Fibonacci cube $Q_h(f)$ is the graph obtained from the $h$-cube $Q_h$ by removing all vertices that contain a given binary string $f$ as a substring. In particular, the vertex set of the 3rd order generalized Fibonacci cube $Q_h(111)$ is the set of all binary strings $b_1b_2 \ldots b_h$ containing no three consecutive 1's. We present a new characterization of the 3rd order generalized Fibonacci cubes based on their recursive structure. The characterization is the basis for an algorithm which recognizes these graphs in linear time.]]> Fri, 02 Sep 2016 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2165 https://doi.org/10.46298/dmtcs.2165 Rho, Yoomi Vesel, Aleksander Rho, Yoomi Vesel, Aleksander <![CDATA[The generalized Fibonacci cube $Q_h(f)$ is the graph obtained from the $h$-cube $Q_h$ by removing all vertices that contain a given binary string $f$ as a substring. In particular, the vertex set of the 3rd order generalized Fibonacci cube $Q_h(111)$ is the set of all binary strings $b_1b_2 \ldots b_h$ containing no three consecutive 1's. We present a new characterization of the 3rd order generalized Fibonacci cubes based on their recursive structure. The characterization is the basis for an algorithm which recognizes these graphs in linear time.]]> 0 <![CDATA[Ruskey and Savage in 1993 asked whether every matching in a hypercube can be extended to a Hamiltonian cycle. A positive answer is known for perfect matchings, but the general case has been resolved only for matchings of linear size. In this paper we show that there is a quadratic function $q(n)$ such that every matching in the $n$-dimensional hypercube of size at most $q(n)$ may be extended to a cycle which covers at least $\frac34$ of the vertices.]]> Thu, 01 Sep 2016 15:18:52 +0000 https://doi.org/10.46298/dmtcs.1336 https://doi.org/10.46298/dmtcs.1336 Dvořák, Tomáš Dvořák, Tomáš <![CDATA[Ruskey and Savage in 1993 asked whether every matching in a hypercube can be extended to a Hamiltonian cycle. A positive answer is known for perfect matchings, but the general case has been resolved only for matchings of linear size. In this paper we show that there is a quadratic function $q(n)$ such that every matching in the $n$-dimensional hypercube of size at most $q(n)$ may be extended to a cycle which covers at least $\frac34$ of the vertices.]]> 0 <![CDATA[A subgraph of a vertex-colored graph is said to be tropical whenever it contains each color of the graph. In this work we study the problem of finding a minimal connected tropical subgraph. We first show that this problem is NP-Hard for trees, interval graphs and split graphs, but polynomial when the number of colors is logarithmic in terms of the order of the graph (i.e. FPT). We then provide upper bounds for the order of the minimal connected tropical subgraph under various conditions. We finally study the problem of finding a connected tropical subgraph in a randomly vertex-colored random graph.]]> Sat, 06 Aug 2016 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2151 https://doi.org/10.46298/dmtcs.2151 Anglès d'Auriac, Jean-Alexandre Cohen, Nathann El Mafthoui, Hakim Harutyunyan, Ararat Legay, Sylvain Manoussakis, Yannis Anglès d'Auriac, Jean-Alexandre Cohen, Nathann El Mafthoui, Hakim Harutyunyan, Ararat Legay, Sylvain Manoussakis, Yannis <![CDATA[A subgraph of a vertex-colored graph is said to be tropical whenever it contains each color of the graph. In this work we study the problem of finding a minimal connected tropical subgraph. We first show that this problem is NP-Hard for trees, interval graphs and split graphs, but polynomial when the number of colors is logarithmic in terms of the order of the graph (i.e. FPT). We then provide upper bounds for the order of the minimal connected tropical subgraph under various conditions. We finally study the problem of finding a connected tropical subgraph in a randomly vertex-colored random graph.]]> 0 <![CDATA[A graph $G$ is a $2$-tree if $G=K_3$, or $G$ has a vertex $v$ of degree 2, whose neighbors are adjacent, and $G-v$ is a 2-tree. Clearly, if $G$ is a 2-tree on $n$ vertices, then $|E(G)|=2n-3$. A non-increasing sequence $\pi =(d_1, \ldots ,d_n)$ of nonnegative integers is a graphic sequence if it is realizable by a simple graph $G$ on $n$ vertices. Yin and Li (Acta Mathematica Sinica, English Series, 25(2009)795–802) proved that if $k \geq 2$, $n \geq \frac{9}{2}k^2 + \frac{19}{2}k$ and $\pi =(d_1, \ldots ,d_n)$ is a graphic sequence with $\sum \limits_{i=1}^n d_i > (k-2)n$, then $\pi$ has a realization containing every tree on $k$ vertices as a subgraph. Moreover, the lower bound $(k-2)n$ is the best possible. This is a variation of a conjecture due to Erdős and Sós. In this paper, we investigate an analogue extremal problem for 2-trees and prove that if $k \geq 3$, $n \geq 2k^2-k$ and $\pi =(d_1, \ldots ,d_n)$ is a graphic sequence with $\sum \limits_{i=1}^n d_i > \frac{4kn}{3} - \frac{5n}{3}$ then $\pi$ has a realization containing every 2-tree on $k$ vertices as a subgraph. We also show that the lower bound $\frac{4kn}{3} - \frac{5n}{3}$ is almost the best possible.]]> Fri, 05 Aug 2016 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2152 https://doi.org/10.46298/dmtcs.2152 Zeng, De-Yan Yin, Jian-Hua Zeng, De-Yan Yin, Jian-Hua <![CDATA[A graph $G$ is a $2$-tree if $G=K_3$, or $G$ has a vertex $v$ of degree 2, whose neighbors are adjacent, and $G-v$ is a 2-tree. Clearly, if $G$ is a 2-tree on $n$ vertices, then $|E(G)|=2n-3$. A non-increasing sequence $\pi =(d_1, \ldots ,d_n)$ of nonnegative integers is a graphic sequence if it is realizable by a simple graph $G$ on $n$ vertices. Yin and Li (Acta Mathematica Sinica, English Series, 25(2009)795–802) proved that if $k \geq 2$, $n \geq \frac{9}{2}k^2 + \frac{19}{2}k$ and $\pi =(d_1, \ldots ,d_n)$ is a graphic sequence with $\sum \limits_{i=1}^n d_i > (k-2)n$, then $\pi$ has a realization containing every tree on $k$ vertices as a subgraph. Moreover, the lower bound $(k-2)n$ is the best possible. This is a variation of a conjecture due to Erdős and Sós. In this paper, we investigate an analogue extremal problem for 2-trees and prove that if $k \geq 3$, $n \geq 2k^2-k$ and $\pi =(d_1, \ldots ,d_n)$ is a graphic sequence with $\sum \limits_{i=1}^n d_i > \frac{4kn}{3} - \frac{5n}{3}$ then $\pi$ has a realization containing every 2-tree on $k$ vertices as a subgraph. We also show that the lower bound $\frac{4kn}{3} - \frac{5n}{3}$ is almost the best possible.]]> 0 <![CDATA[For every interval map with finitely many periodic points of periods 1 and 2, we associate a word by taking the periods of these points from left to right. It is natural to ask which words arise in this manner. In this paper we give two different characterizations of the language obtained in this way.]]> Fri, 05 Aug 2016 22:00:00 +0000 https://doi.org/10.46298/dmtcs.3197 https://doi.org/10.46298/dmtcs.3197 Archana, M Kannan, V Archana, M Kannan, V <![CDATA[For every interval map with finitely many periodic points of periods 1 and 2, we associate a word by taking the periods of these points from left to right. It is natural to ask which words arise in this manner. In this paper we give two different characterizations of the language obtained in this way.]]> 0 <![CDATA[Assume that $n, \delta ,k$ are integers with $0 \leq k < \delta < n$. Given a graph $G=(V,E)$ with $|V|=n$. The symbol $G-F, F \subseteq V$, denotes the graph with $V(G-F)=V-F$, and $E(G-F)$ obtained by $E$ after deleting the edges with at least one endvertex in $F$. $G$ is called $k$-vertex fault traceable, $k$-vertex fault Hamiltonian, or $k$-vertex fault Hamiltonian-connected if $G-F$ remains traceable, Hamiltonian, and Hamiltonian-connected for all $F$ with $0 \leq |F| \leq k$, respectively. The notations $h_1(n, \delta ,k)$, $h_2(n, \delta ,k)$, and $h_3(n, \delta ,k)$ denote the minimum number of edges required to guarantee an $n$-vertex graph with minimum degree $\delta (G) \geq \delta$ to be $k$-vertex fault traceable, $k$-vertex fault Hamiltonian, and $k$-vertex fault Hamiltonian-connected, respectively. In this paper, we establish a theorem which uses the degree sequence of a given graph to characterize the $k$-vertex fault traceability/hamiltonicity/Hamiltonian-connectivity, respectively. Then we use this theorem to obtain the formulas for $h_i(n, \delta ,k)$ for $1 \leq i \leq 3$, which improves and extends the known results for $k=0$.]]> Mon, 01 Aug 2016 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2149 https://doi.org/10.46298/dmtcs.2149 Chen, Shih-Yan Kao, Shin-Shin Su, Hsun Chen, Shih-Yan Kao, Shin-Shin Su, Hsun <![CDATA[Assume that $n, \delta ,k$ are integers with $0 \leq k < \delta < n$. Given a graph $G=(V,E)$ with $|V|=n$. The symbol $G-F, F \subseteq V$, denotes the graph with $V(G-F)=V-F$, and $E(G-F)$ obtained by $E$ after deleting the edges with at least one endvertex in $F$. $G$ is called $k$-vertex fault traceable, $k$-vertex fault Hamiltonian, or $k$-vertex fault Hamiltonian-connected if $G-F$ remains traceable, Hamiltonian, and Hamiltonian-connected for all $F$ with $0 \leq |F| \leq k$, respectively. The notations $h_1(n, \delta ,k)$, $h_2(n, \delta ,k)$, and $h_3(n, \delta ,k)$ denote the minimum number of edges required to guarantee an $n$-vertex graph with minimum degree $\delta (G) \geq \delta$ to be $k$-vertex fault traceable, $k$-vertex fault Hamiltonian, and $k$-vertex fault Hamiltonian-connected, respectively. In this paper, we establish a theorem which uses the degree sequence of a given graph to characterize the $k$-vertex fault traceability/hamiltonicity/Hamiltonian-connectivity, respectively. Then we use this theorem to obtain the formulas for $h_i(n, \delta ,k)$ for $1 \leq i \leq 3$, which improves and extends the known results for $k=0$.]]> 0 <![CDATA[We investigate a family of algorithms minimizing energetic effort in random networks computing aggregative functions. In contrast to previously considered models, our results minimize maximal energetic effort over all stations instead of the average usage of energy. Such approach seems to be much more suitable for some kinds of networks, in particular ad hoc radio networks, wherein we need all stations functioning and replacing batteries after the deployment is not feasible. We analyze also the latency of proposed energy-optimal algorithms. We model a network by placing randomly and independently $n$ points in a $d$-dimensional cube of side-length $n^{1/d}$. We place an edge between vertices that interact with each other. We analyze properties of the resulting graphs in order to obtain estimates on energetic effort and latency of proposed algorithms.]]> Sun, 24 Jul 2016 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2160 https://doi.org/10.46298/dmtcs.2160 Klonowski, Marek Sulkowska, Małgorzata Klonowski, Marek Sulkowska, Małgorzata <![CDATA[We investigate a family of algorithms minimizing energetic effort in random networks computing aggregative functions. In contrast to previously considered models, our results minimize maximal energetic effort over all stations instead of the average usage of energy. Such approach seems to be much more suitable for some kinds of networks, in particular ad hoc radio networks, wherein we need all stations functioning and replacing batteries after the deployment is not feasible. We analyze also the latency of proposed energy-optimal algorithms. We model a network by placing randomly and independently $n$ points in a $d$-dimensional cube of side-length $n^{1/d}$. We place an edge between vertices that interact with each other. We analyze properties of the resulting graphs in order to obtain estimates on energetic effort and latency of proposed algorithms.]]> 0 <![CDATA[The vertices of the Knödel graph $W_{\Delta, n}$ on $n \geq 2$ vertices, $n$ even, and of maximum degree $\Delta, 1 \leq \Delta \leq \lfloor log_2(n) \rfloor$, are the pairs $(i,j)$ with $i=1,2$ and $0 \leq j \leq \frac{n}{2} -1$. For $0 \leq j \leq \frac{n}{2} -1$, there is an edge between vertex $(1,j)$ and every vertex $(2,j + 2^k - 1 (mod \frac{n}{2}))$, for $k=0,1,2, \ldots , \Delta -1$. Existence of a Hamilton cycle decomposition of $W_{k, 2k}, k \geq 6$ is not yet known, see Discrete Appl. Math. 137 (2004) 173-195. In this paper, it is shown that the $k$-regular Knödel graph $W_{k,2k}, k \geq 6$ has $ \lfloor \frac{k}{2} \rfloor - 1$ edge disjoint Hamilton cycles.]]> Thu, 21 Jul 2016 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2148 https://doi.org/10.46298/dmtcs.2148 Paulraja, Palanivel Subramania Nadar Sampath Kumar, S Paulraja, Palanivel Subramania Nadar Sampath Kumar, S <![CDATA[The vertices of the Knödel graph $W_{\Delta, n}$ on $n \geq 2$ vertices, $n$ even, and of maximum degree $\Delta, 1 \leq \Delta \leq \lfloor log_2(n) \rfloor$, are the pairs $(i,j)$ with $i=1,2$ and $0 \leq j \leq \frac{n}{2} -1$. For $0 \leq j \leq \frac{n}{2} -1$, there is an edge between vertex $(1,j)$ and every vertex $(2,j + 2^k - 1 (mod \frac{n}{2}))$, for $k=0,1,2, \ldots , \Delta -1$. Existence of a Hamilton cycle decomposition of $W_{k, 2k}, k \geq 6$ is not yet known, see Discrete Appl. Math. 137 (2004) 173-195. In this paper, it is shown that the $k$-regular Knödel graph $W_{k,2k}, k \geq 6$ has $ \lfloor \frac{k}{2} \rfloor - 1$ edge disjoint Hamilton cycles.]]> 0 <![CDATA[We investigate pattern avoidance in permutations satisfying some additional restrictions. These are naturally considered in terms of avoiding patterns in linear extensions of certain forest-like partially ordered sets, which we call binary shrub forests. In this context, we enumerate forests avoiding patterns of length three. In four of the five non-equivalent cases, we present explicit enumerations by exhibiting bijections with certain lattice paths bounded above by the line $y=\ell x$, for some $\ell\in\mathbb{Q}^+$, one of these being the celebrated Duchon's club paths with $\ell=2/3$. In the remaining case, we use the machinery of analytic combinatorics to determine the minimal polynomial of its generating function, and deduce its growth rate.]]> Thu, 21 Jul 2016 10:09:41 +0000 https://doi.org/10.46298/dmtcs.1322 https://doi.org/10.46298/dmtcs.1322 Bevan, David Levin, Derek Nugent, Peter Pantone, Jay Pudwell, Lara Riehl, Manda Tlachac, ML Bevan, David Levin, Derek Nugent, Peter Pantone, Jay Pudwell, Lara Riehl, Manda Tlachac, ML <![CDATA[We investigate pattern avoidance in permutations satisfying some additional restrictions. These are naturally considered in terms of avoiding patterns in linear extensions of certain forest-like partially ordered sets, which we call binary shrub forests. In this context, we enumerate forests avoiding patterns of length three. In four of the five non-equivalent cases, we present explicit enumerations by exhibiting bijections with certain lattice paths bounded above by the line $y=\ell x$, for some $\ell\in\mathbb{Q}^+$, one of these being the celebrated Duchon's club paths with $\ell=2/3$. In the remaining case, we use the machinery of analytic combinatorics to determine the minimal polynomial of its generating function, and deduce its growth rate.]]> 0 <![CDATA[If $\mathcal{P}$ is a given graph property, we say that a graph $G$ is locally $\mathcal{P}$ if $\langle N(v) \rangle$ has property $\mathcal{P}$ for every $v \in V(G)$ where $\langle N(v) \rangle$ is the induced graph on the open neighbourhood of the vertex $v$. Pareek and Skupien (C. M. Pareek and Z. Skupien , On the smallest non-Hamiltonian locally Hamiltonian graph, J. Univ. Kuwait (Sci.), 10:9 - 17, 1983) posed the following two questions. Question 1 Is 9 the smallest order of a connected nontraceable locally traceable graph? Question 2 Is 14 the smallest order of a connected nontraceable locally hamiltonian graph? We answer the second question in the affirmative, but show that the correct number for the first question is 10. We develop a technique to construct connected locally hamiltonian and locally traceable graphs that are not traceable. We use this technique to construct such graphs with various prescribed properties.]]> Thu, 30 Jun 2016 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2144 https://doi.org/10.46298/dmtcs.2144 De Wet, Johan Van Aardt, Susan De Wet, Johan Van Aardt, Susan <![CDATA[If $\mathcal{P}$ is a given graph property, we say that a graph $G$ is locally $\mathcal{P}$ if $\langle N(v) \rangle$ has property $\mathcal{P}$ for every $v \in V(G)$ where $\langle N(v) \rangle$ is the induced graph on the open neighbourhood of the vertex $v$. Pareek and Skupien (C. M. Pareek and Z. Skupien , On the smallest non-Hamiltonian locally Hamiltonian graph, J. Univ. Kuwait (Sci.), 10:9 - 17, 1983) posed the following two questions. Question 1 Is 9 the smallest order of a connected nontraceable locally traceable graph? Question 2 Is 14 the smallest order of a connected nontraceable locally hamiltonian graph? We answer the second question in the affirmative, but show that the correct number for the first question is 10. We develop a technique to construct connected locally hamiltonian and locally traceable graphs that are not traceable. We use this technique to construct such graphs with various prescribed properties.]]> 0 <![CDATA[Network models allow one to deal with massive data sets using some standard concepts from graph theory. Understanding and investigating the structural properties of a certain data set is a crucial task in many practical applications of network optimization. Recently, labeled network optimization over colored graphs has been extensively studied. Given a (not necessarily properly) edge-colored graph $G=(V,E)$, a subgraph $H$ is said to be monochromatic if all its edges have the same color, and called multicolored if all its edges have distinct colors. The monochromatic clique and multicolored cycle partition problems have important applications in the problems of network optimization arising in information science and operations research. We investigate the computational complexity of the problems of determining the minimum number of monochromatic cliques or multicolored cycles that, respectively, partition $V(G)$. We show that the minimum monochromatic clique partition problem is APX-hard on monochromatic-diamond-free graphs, and APX-complete on monochromatic-diamond-free graphs in which the size of a maximum monochromatic clique is bounded by a constant. We also show that the minimum multicolored cycle partition problem is NP-complete, even if the input graph $G$ is triangle-free. Moreover, for the weighted version of the minimum monochromatic clique partition problem on monochromatic-diamond-free graphs, we derive an approximation algorithm with (tight) approximation guarantee ln $|V(G)|+1$.]]> Thu, 30 Jun 2016 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2159 https://doi.org/10.46298/dmtcs.2159 Zhang, Xiaoyan Zhang, Zan-Bo Broersma, Hajo Wen, Xuelian Zhang, Xiaoyan Zhang, Zan-Bo Broersma, Hajo Wen, Xuelian <![CDATA[Network models allow one to deal with massive data sets using some standard concepts from graph theory. Understanding and investigating the structural properties of a certain data set is a crucial task in many practical applications of network optimization. Recently, labeled network optimization over colored graphs has been extensively studied. Given a (not necessarily properly) edge-colored graph $G=(V,E)$, a subgraph $H$ is said to be monochromatic if all its edges have the same color, and called multicolored if all its edges have distinct colors. The monochromatic clique and multicolored cycle partition problems have important applications in the problems of network optimization arising in information science and operations research. We investigate the computational complexity of the problems of determining the minimum number of monochromatic cliques or multicolored cycles that, respectively, partition $V(G)$. We show that the minimum monochromatic clique partition problem is APX-hard on monochromatic-diamond-free graphs, and APX-complete on monochromatic-diamond-free graphs in which the size of a maximum monochromatic clique is bounded by a constant. We also show that the minimum multicolored cycle partition problem is NP-complete, even if the input graph $G$ is triangle-free. Moreover, for the weighted version of the minimum monochromatic clique partition problem on monochromatic-diamond-free graphs, we derive an approximation algorithm with (tight) approximation guarantee ln $|V(G)|+1$.]]> 0 <![CDATA[We have extended classical pattern avoidance to a new structure: multiple task-precedence posets whose Hasse diagrams have three levels, which we will call diamonds. The vertices of each diamond are assigned labels which are compatible with the poset. A corresponding permutation is formed by reading these labels by increasing levels, and then from left to right. We used Sage to form enumerative conjectures for the associated permutations avoiding collections of patterns of length three, which we then proved. We have discovered a bijection between diamonds avoiding 132 and certain generalized Dyck paths. We have also found the generating function for descents, and therefore the number of avoiders, in these permutations for the majority of collections of patterns of length three. An interesting application of this work (and the motivating example) can be found when task-precedence posets represent warehouse package fulfillment by robots, in which case avoidance of both 231 and 321 ensures we never stack two heavier packages on top of a lighter package.]]> Fri, 24 Jun 2016 13:02:33 +0000 https://doi.org/10.46298/dmtcs.1324 https://doi.org/10.46298/dmtcs.1324 Paukner, Mitchell Pepin, Lucy Riehl, Manda Wieser, Jarred Paukner, Mitchell Pepin, Lucy Riehl, Manda Wieser, Jarred <![CDATA[We have extended classical pattern avoidance to a new structure: multiple task-precedence posets whose Hasse diagrams have three levels, which we will call diamonds. The vertices of each diamond are assigned labels which are compatible with the poset. A corresponding permutation is formed by reading these labels by increasing levels, and then from left to right. We used Sage to form enumerative conjectures for the associated permutations avoiding collections of patterns of length three, which we then proved. We have discovered a bijection between diamonds avoiding 132 and certain generalized Dyck paths. We have also found the generating function for descents, and therefore the number of avoiders, in these permutations for the majority of collections of patterns of length three. An interesting application of this work (and the motivating example) can be found when task-precedence posets represent warehouse package fulfillment by robots, in which case avoidance of both 231 and 321 ensures we never stack two heavier packages on top of a lighter package.]]> 0 <![CDATA[In this paper, we present two new results of layered permutation densities. The first one generalizes theorems from H\"{a}st\"{o} (2003) and Warren (2004) to compute the permutation packing of permutations whose layer sequence is~$(1^a,\ell_1,\ell_2,\ldots,\ell_k)$ with~$2^a-a-1\geq k$ (and similar permutations). As a second result, we prove that the minimum density of monotone sequences of length~$k+1$ in an arbitrarily large layered permutation is asymptotically~$1/k^k$. This value is compatible with a conjecture from Myers (2003) for the problem without the layered restriction (the same problem where the monotone sequences have different lengths is also studied).]]> Thu, 23 Jun 2016 11:06:48 +0000 https://doi.org/10.46298/dmtcs.1313 https://doi.org/10.46298/dmtcs.1313 Bastos, Josefran de Oliveira Coregliano, Leonardo Nagami Bastos, Josefran de Oliveira Coregliano, Leonardo Nagami <![CDATA[In this paper, we present two new results of layered permutation densities. The first one generalizes theorems from H\"{a}st\"{o} (2003) and Warren (2004) to compute the permutation packing of permutations whose layer sequence is~$(1^a,\ell_1,\ell_2,\ldots,\ell_k)$ with~$2^a-a-1\geq k$ (and similar permutations). As a second result, we prove that the minimum density of monotone sequences of length~$k+1$ in an arbitrarily large layered permutation is asymptotically~$1/k^k$. This value is compatible with a conjecture from Myers (2003) for the problem without the layered restriction (the same problem where the monotone sequences have different lengths is also studied).]]> 0 <![CDATA[The partial sum of the states of a Markov chain or more generally a Markov source is asymptotically normally distributed under suitable conditions. One of these conditions is that the variance is unbounded. A simple combinatorial characterization of Markov sources which satisfy this condition is given in terms of cycles of the underlying graph of the Markov chain. Also Markov sources with higher dimensional alphabets are considered. Furthermore, the case of an unbounded covariance between two coordinates of the Markov source is combinatorically characterized. If the covariance is bounded, then the two coordinates are asymptotically independent. The results are illustrated by several examples, like the number of specific blocks in $0$-$1$-sequences and the Hamming weight of the width-$w$ non-adjacent form.]]> Thu, 23 Jun 2016 10:59:03 +0000 https://doi.org/10.46298/dmtcs.1341 https://doi.org/10.46298/dmtcs.1341 Kropf, Sara Kropf, Sara <![CDATA[The partial sum of the states of a Markov chain or more generally a Markov source is asymptotically normally distributed under suitable conditions. One of these conditions is that the variance is unbounded. A simple combinatorial characterization of Markov sources which satisfy this condition is given in terms of cycles of the underlying graph of the Markov chain. Also Markov sources with higher dimensional alphabets are considered. Furthermore, the case of an unbounded covariance between two coordinates of the Markov source is combinatorically characterized. If the covariance is bounded, then the two coordinates are asymptotically independent. The results are illustrated by several examples, like the number of specific blocks in $0$-$1$-sequences and the Hamming weight of the width-$w$ non-adjacent form.]]> 0 <![CDATA[An additive labeling of a graph $G$ is a function $\ell :V(G) \rightarrow \mathbb{N}$, such that for every two adjacent vertices $v$ and $u$ of $G$, $\Sigma_{w \sim v} \ell (w) \neq \Sigma_{w \sim u} \ell (w)$ ($x \sim y$ means that $x$ is joined to $y$). The additive number of $G$, denoted by $\eta (G)$, is the minimum number $k$ such that $G$ has a additive labeling $\ell : V(G) \rightarrow \mathbb{N}_k$. The additive choosability of a graph $G$, denoted by $\eta_\ell (G)$, is the smallest number $k$ such that $G$ has an additive labeling for any assignment of lists of size $k$ to the vertices of $G$, such that the label of each vertex belongs to its own list. Seamone in his PhD thesis conjectured that for every graph $G$, $\eta(G)= \eta_\ell (G)$. We give a negative answer to this conjecture and we show that for every $k$ there is a graph $G$ such that $\eta_\ell (G) - \eta(G) \geq k$. A $(0,1)$-additive labeling of a graph $G$ is a function $\ell :V(G) \rightarrow \{0,1 \}$, such that for every two adjacent vertices $v$ and $u$ of $G$, $\Sigma_{w \sim v} \ell (w) \neq \Sigma_{w \sim u} \ell (w)$. A graph may lack any $(0,1)$-additive labeling. We show that it is NP-complete to decide whether a $(0,1)$-additive labeling exists for some families of graphs such as perfect graphs and planar triangle-free graphs. For a graph $G$ with some $(0,1)$-additive labelings, the $(0,1)$-additive number of $G$ is defined as $\sigma_1 (G) = \mathrm{min}_{\ell \in \Gamma} \Sigma_{v \in V (G)} \ell (v)$ where $\Gamma$ is the set of $(0,1)$-additive labelings of $G$. We prove that given a planar graph that admits a $(0,1)$-additive labeling, for all $\epsilon > 0$ , approximating the $(0,1)$-additive number within $n^{1-\epsilon}$ is NP-hard.]]> Thu, 16 Jun 2016 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2145 https://doi.org/10.46298/dmtcs.2145 Ahadi, Arash Dehghan, Ali Ahadi, Arash Dehghan, Ali <![CDATA[An additive labeling of a graph $G$ is a function $\ell :V(G) \rightarrow \mathbb{N}$, such that for every two adjacent vertices $v$ and $u$ of $G$, $\Sigma_{w \sim v} \ell (w) \neq \Sigma_{w \sim u} \ell (w)$ ($x \sim y$ means that $x$ is joined to $y$). The additive number of $G$, denoted by $\eta (G)$, is the minimum number $k$ such that $G$ has a additive labeling $\ell : V(G) \rightarrow \mathbb{N}_k$. The additive choosability of a graph $G$, denoted by $\eta_\ell (G)$, is the smallest number $k$ such that $G$ has an additive labeling for any assignment of lists of size $k$ to the vertices of $G$, such that the label of each vertex belongs to its own list. Seamone in his PhD thesis conjectured that for every graph $G$, $\eta(G)= \eta_\ell (G)$. We give a negative answer to this conjecture and we show that for every $k$ there is a graph $G$ such that $\eta_\ell (G) - \eta(G) \geq k$. A $(0,1)$-additive labeling of a graph $G$ is a function $\ell :V(G) \rightarrow \{0,1 \}$, such that for every two adjacent vertices $v$ and $u$ of $G$, $\Sigma_{w \sim v} \ell (w) \neq \Sigma_{w \sim u} \ell (w)$. A graph may lack any $(0,1)$-additive labeling. We show that it is NP-complete to decide whether a $(0,1)$-additive labeling exists for some families of graphs such as perfect graphs and planar triangle-free graphs. For a graph $G$ with some $(0,1)$-additive labelings, the $(0,1)$-additive number of $G$ is defined as $\sigma_1 (G) = \mathrm{min}_{\ell \in \Gamma} \Sigma_{v \in V (G)} \ell (v)$ where $\Gamma$ is the set of $(0,1)$-additive labelings of $G$. We prove that given a planar graph that admits a $(0,1)$-additive labeling, for all $\epsilon > 0$ , approximating the $(0,1)$-additive number within $n^{1-\epsilon}$ is NP-hard.]]> 0 <![CDATA[The irregularity of a graph $G$ is defined as the sum of weights $|d(u)-d(v)|$ of all edges $uv$ of $G$, where $d(u)$ and $d(v)$ are the degrees of the vertices $u$ and $v$ in $G$, respectively. In this paper, some structural properties on trees with maximum (or minimum) irregularity among trees with given degree sequence and trees with given branching number are explored, respectively. Moreover, the corresponding trees with maximum (or minimum) irregularity are also found, respectively.]]> Tue, 07 Jun 2016 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2146 https://doi.org/10.46298/dmtcs.2146 Jianxi, Li Liu, Yang Shiu, Wai Jianxi, Li Liu, Yang Shiu, Wai <![CDATA[The irregularity of a graph $G$ is defined as the sum of weights $|d(u)-d(v)|$ of all edges $uv$ of $G$, where $d(u)$ and $d(v)$ are the degrees of the vertices $u$ and $v$ in $G$, respectively. In this paper, some structural properties on trees with maximum (or minimum) irregularity among trees with given degree sequence and trees with given branching number are explored, respectively. Moreover, the corresponding trees with maximum (or minimum) irregularity are also found, respectively.]]> 0 <![CDATA[Epigroups are semigroups equipped with an additional unary operation called pseudoinversion. Each finite semigroup can be considered as an epigroup. We prove the following theorem announced by Zhil'tsov in 2000: the equational theory of the class of all epigroups coincides with the equational theory of the class of all finite epigroups and is decidable. We show that the theory is not finitely based but provide a transparent infinite basis for it.]]> Mon, 06 Jun 2016 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2155 https://doi.org/10.46298/dmtcs.2155 Mikhaylova, Inna Mikhaylova, Inna <![CDATA[Epigroups are semigroups equipped with an additional unary operation called pseudoinversion. Each finite semigroup can be considered as an epigroup. We prove the following theorem announced by Zhil'tsov in 2000: the equational theory of the class of all epigroups coincides with the equational theory of the class of all finite epigroups and is decidable. We show that the theory is not finitely based but provide a transparent infinite basis for it.]]> 0 <![CDATA[A graph is an efficient open domination graph if there exists a subset of vertices whose open neighborhoods partition its vertex set. We characterize those graphs $G$ for which the Cartesian product $G \Box H$ is an efficient open domination graph when $H$ is a complete graph of order at least 3 or a complete bipartite graph. The characterization is based on the existence of a certain type of weak partition of $V(G)$. For the class of trees when $H$ is complete of order at least 3, the characterization is constructive. In addition, a special type of efficient open domination graph is characterized among Cartesian products $G \Box H$ when $H$ is a 5-cycle or a 4-cycle.]]> Thu, 02 Jun 2016 14:21:12 +0000 https://doi.org/10.46298/dmtcs.1277 https://doi.org/10.46298/dmtcs.1277 Šumenjak, Tadeja Kraner Peterin, Iztok Rall, Douglas F. Tepeh, Aleksandra Šumenjak, Tadeja Kraner Peterin, Iztok Rall, Douglas F. Tepeh, Aleksandra <![CDATA[A graph is an efficient open domination graph if there exists a subset of vertices whose open neighborhoods partition its vertex set. We characterize those graphs $G$ for which the Cartesian product $G \Box H$ is an efficient open domination graph when $H$ is a complete graph of order at least 3 or a complete bipartite graph. The characterization is based on the existence of a certain type of weak partition of $V(G)$. For the class of trees when $H$ is complete of order at least 3, the characterization is constructive. In addition, a special type of efficient open domination graph is characterized among Cartesian products $G \Box H$ when $H$ is a 5-cycle or a 4-cycle.]]> 0 <![CDATA[Caffrey, Egge, Michel, Rubin and Ver Steegh recently introduced snow leopard permutations, which are the anti-Baxter permutations that are compatible with the doubly alternating Baxter permutations. Among other things, they showed that these permutations preserve parity, and that the number of snow leopard permutations of length $2n-1$ is the Catalan number $C_n$. In this paper we investigate the permutations that the snow leopard permutations induce on their even and odd entries; we call these the even threads and the odd threads, respectively. We give recursive bijections between these permutations and certain families of Catalan paths. We characterize the odd (resp. even) threads which form the other half of a snow leopard permutation whose even (resp. odd) thread is layered in terms of pattern avoidance, and we give a constructive bijection between the set of permutations of length $n$ which are both even threads and odd threads and the set of peakless Motzkin paths of length $n+1$.]]> Wed, 01 Jun 2016 07:43:53 +0000 https://doi.org/10.46298/dmtcs.1279 https://doi.org/10.46298/dmtcs.1279 Egge, Eric S. Rubin, Kailee Egge, Eric S. Rubin, Kailee <![CDATA[Caffrey, Egge, Michel, Rubin and Ver Steegh recently introduced snow leopard permutations, which are the anti-Baxter permutations that are compatible with the doubly alternating Baxter permutations. Among other things, they showed that these permutations preserve parity, and that the number of snow leopard permutations of length $2n-1$ is the Catalan number $C_n$. In this paper we investigate the permutations that the snow leopard permutations induce on their even and odd entries; we call these the even threads and the odd threads, respectively. We give recursive bijections between these permutations and certain families of Catalan paths. We characterize the odd (resp. even) threads which form the other half of a snow leopard permutation whose even (resp. odd) thread is layered in terms of pattern avoidance, and we give a constructive bijection between the set of permutations of length $n$ which are both even threads and odd threads and the set of peakless Motzkin paths of length $n+1$.]]> 0 <![CDATA[For a language $L$, we consider its cyclic closure, and more generally the language $C^{k}(L)$, which consists of all words obtained by partitioning words from $L$ into $k$ factors and permuting them. We prove that the classes of ET0L and EDT0L languages are closed under the operators $C^k$. This both sharpens and generalises Brandstädt's result that if $L$ is context-free then $C^{k}(L)$ is context-sensitive and not context-free in general for $k \geq 3$. We also show that the cyclic closure of an indexed language is indexed.]]> Mon, 30 May 2016 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2164 https://doi.org/10.46298/dmtcs.2164 Brough, Tara Ciobanu, Laura Elder, Murray Zetzsche, Georg Brough, Tara Ciobanu, Laura Elder, Murray Zetzsche, Georg <![CDATA[For a language $L$, we consider its cyclic closure, and more generally the language $C^{k}(L)$, which consists of all words obtained by partitioning words from $L$ into $k$ factors and permuting them. We prove that the classes of ET0L and EDT0L languages are closed under the operators $C^k$. This both sharpens and generalises Brandstädt's result that if $L$ is context-free then $C^{k}(L)$ is context-sensitive and not context-free in general for $k \geq 3$. We also show that the cyclic closure of an indexed language is indexed.]]> 0 <![CDATA[A composition $\pi = \pi_1 \pi_2 \cdots \pi_m$ of a positive integer $n$ is an ordered collection of one or more positive integers whose sum is $n$. The number of summands, namely $m$, is called the number of parts of $\pi$. Using linear algebra, we determine formulas for generating functions that count compositions of $n$ with $m$ parts, according to the number of occurrences of the subword pattern $\tau$, and according to the sum, over all occurrences of $\tau$, of the first integers in their respective occurrences, where $\tau$ is any pattern of length three with exactly 2 distinct letters.]]> Mon, 30 May 2016 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2156 https://doi.org/10.46298/dmtcs.2156 Shabani, Armend Gjergji, Rexhep Shabani, Armend Gjergji, Rexhep <![CDATA[A composition $\pi = \pi_1 \pi_2 \cdots \pi_m$ of a positive integer $n$ is an ordered collection of one or more positive integers whose sum is $n$. The number of summands, namely $m$, is called the number of parts of $\pi$. Using linear algebra, we determine formulas for generating functions that count compositions of $n$ with $m$ parts, according to the number of occurrences of the subword pattern $\tau$, and according to the sum, over all occurrences of $\tau$, of the first integers in their respective occurrences, where $\tau$ is any pattern of length three with exactly 2 distinct letters.]]> 0 <![CDATA[A permutation $\tau$ in the symmetric group $S_j$ is minimally overlapping if any two consecutive occurrences of $\tau$ in a permutation $\sigma$ can share at most one element. B\'ona \cite{B} showed that the proportion of minimal overlapping patterns in $S_j$ is at least $3 -e$. Given a permutation $\sigma$, we let $\text{Des}(\sigma)$ denote the set of descents of $\sigma$. We study the class of permutations $\sigma \in S_{kn}$ whose descent set is contained in the set $\{k,2k, \ldots (n-1)k\}$. For example, up-down permutations in $S_{2n}$ are the set of permutations whose descent equal $\sigma$ such that $\text{Des}(\sigma) = \{2,4, \ldots, 2n-2\}$. There are natural analogues of the minimal overlapping permutations for such classes of permutations and we study the proportion of minimal overlapping patterns for each such class. We show that the proportion of minimal overlapping permutations in such classes approaches $1$ as $k$ goes to infinity. We also study the proportion of minimal overlapping patterns in standard Young tableaux of shape $(n^k)$.]]> Fri, 20 May 2016 16:16:41 +0000 https://doi.org/10.46298/dmtcs.1315 https://doi.org/10.46298/dmtcs.1315 Pan, Ran Remmel, Jeffrey B. Pan, Ran Remmel, Jeffrey B. <![CDATA[A permutation $\tau$ in the symmetric group $S_j$ is minimally overlapping if any two consecutive occurrences of $\tau$ in a permutation $\sigma$ can share at most one element. B\'ona \cite{B} showed that the proportion of minimal overlapping patterns in $S_j$ is at least $3 -e$. Given a permutation $\sigma$, we let $\text{Des}(\sigma)$ denote the set of descents of $\sigma$. We study the class of permutations $\sigma \in S_{kn}$ whose descent set is contained in the set $\{k,2k, \ldots (n-1)k\}$. For example, up-down permutations in $S_{2n}$ are the set of permutations whose descent equal $\sigma$ such that $\text{Des}(\sigma) = \{2,4, \ldots, 2n-2\}$. There are natural analogues of the minimal overlapping permutations for such classes of permutations and we study the proportion of minimal overlapping patterns for each such class. We show that the proportion of minimal overlapping permutations in such classes approaches $1$ as $k$ goes to infinity. We also study the proportion of minimal overlapping patterns in standard Young tableaux of shape $(n^k)$.]]> 0 <![CDATA[For planar graphs, we consider the problems of list edge coloring and list total coloring. Edge coloring is the problem of coloring the edges while ensuring that two edges that are adjacent receive different colors. Total coloring is the problem of coloring the edges and the vertices while ensuring that two edges that are adjacent, two vertices that are adjacent, or a vertex and an edge that are incident receive different colors. In their list extensions, instead of having the same set of colors for the whole graph, every vertex or edge is assigned some set of colors and has to be colored from it. A graph is minimally edge or total choosable if it is list $\Delta$-edge-colorable or list $(\Delta +1)$-total-colorable, respectively, where $\Delta$ is the maximum degree in the graph. It is already known that planar graphs with $\Delta \geq 8$ and no triangle adjacent to a $C_4$ are minimally edge and total choosable (Li Xu 2011), and that planar graphs with $\Delta \geq 7$ and no triangle sharing a vertex with a $C_4$ or no triangle adjacent to a $C_k (\forall 3 \leq k \leq 6)$ are minimally total colorable (Wang Wu 2011). We strengthen here these results and prove that planar graphs with $\Delta \geq 7$ and no triangle adjacent to a $C_4$ are minimally edge and total choosable.]]> Wed, 11 May 2016 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2147 https://doi.org/10.46298/dmtcs.2147 Bonamy, Marthe Lévêque, Benjamin Pinlou, Alexandre Bonamy, Marthe Lévêque, Benjamin Pinlou, Alexandre <![CDATA[For planar graphs, we consider the problems of list edge coloring and list total coloring. Edge coloring is the problem of coloring the edges while ensuring that two edges that are adjacent receive different colors. Total coloring is the problem of coloring the edges and the vertices while ensuring that two edges that are adjacent, two vertices that are adjacent, or a vertex and an edge that are incident receive different colors. In their list extensions, instead of having the same set of colors for the whole graph, every vertex or edge is assigned some set of colors and has to be colored from it. A graph is minimally edge or total choosable if it is list $\Delta$-edge-colorable or list $(\Delta +1)$-total-colorable, respectively, where $\Delta$ is the maximum degree in the graph. It is already known that planar graphs with $\Delta \geq 8$ and no triangle adjacent to a $C_4$ are minimally edge and total choosable (Li Xu 2011), and that planar graphs with $\Delta \geq 7$ and no triangle sharing a vertex with a $C_4$ or no triangle adjacent to a $C_k (\forall 3 \leq k \leq 6)$ are minimally total colorable (Wang Wu 2011). We strengthen here these results and prove that planar graphs with $\Delta \geq 7$ and no triangle adjacent to a $C_4$ are minimally edge and total choosable.]]> 0 <![CDATA[The new finite state machine package in the mathematics software system SageMath is presented and illustrated by many examples. Several combinatorial problems, in particular digit problems, are introduced, modeled by automata and transducers and solved using SageMath. In particular, we compute the asymptotic Hamming weight of a non-adjacent-form-like digit expansion, which was not known before.]]> Tue, 10 May 2016 08:00:47 +0000 https://doi.org/10.46298/dmtcs.1352 https://doi.org/10.46298/dmtcs.1352 Heuberger, Clemens Krenn, Daniel Kropf, Sara Heuberger, Clemens Krenn, Daniel Kropf, Sara <![CDATA[The new finite state machine package in the mathematics software system SageMath is presented and illustrated by many examples. Several combinatorial problems, in particular digit problems, are introduced, modeled by automata and transducers and solved using SageMath. In particular, we compute the asymptotic Hamming weight of a non-adjacent-form-like digit expansion, which was not known before.]]> 0 <![CDATA[The notion of Shared Risk Link Groups (SRLG) captures survivability issues when a set of links of a network may fail simultaneously. The theory of survivable network design relies on basic combinatorial objects that are rather easy to compute in the classical graph models: shortest paths, minimum cuts, or pairs of disjoint paths. In the SRLG context, the optimization criterion for these objects is no longer the number of edges they use, but the number of SRLGs involved. Unfortunately, computing these combinatorial objects is NP-hard and hard to approximate with this objective in general. Nevertheless some objects can be computed in polynomial time when the SRLGs satisfy certain structural properties of locality which correspond to practical ones, namely the star property (all links affected by a given SRLG are incident to a unique node) and the span 1 property (the links affected by a given SRLG form a connected component of the network). The star property is defined in a multi-colored model where a link can be affected by several SRLGs while the span property is defined only in a mono-colored model where a link can be affected by at most one SRLG. In this paper, we extend these notions to characterize new cases in which these optimization problems can be solved in polynomial time. We also investigate the computational impact of the transformation from the multi-colored model to the mono-colored one. Experimental results are presented to validate the proposed algorithms and principles.]]> Tue, 03 May 2016 09:30:11 +0000 https://doi.org/10.46298/dmtcs.1297 https://doi.org/10.46298/dmtcs.1297 Coudert, David Pérennes, Stéphane Rivano, Hervé Voge, Marie-Emilie Coudert, David Pérennes, Stéphane Rivano, Hervé Voge, Marie-Emilie <![CDATA[The notion of Shared Risk Link Groups (SRLG) captures survivability issues when a set of links of a network may fail simultaneously. The theory of survivable network design relies on basic combinatorial objects that are rather easy to compute in the classical graph models: shortest paths, minimum cuts, or pairs of disjoint paths. In the SRLG context, the optimization criterion for these objects is no longer the number of edges they use, but the number of SRLGs involved. Unfortunately, computing these combinatorial objects is NP-hard and hard to approximate with this objective in general. Nevertheless some objects can be computed in polynomial time when the SRLGs satisfy certain structural properties of locality which correspond to practical ones, namely the star property (all links affected by a given SRLG are incident to a unique node) and the span 1 property (the links affected by a given SRLG form a connected component of the network). The star property is defined in a multi-colored model where a link can be affected by several SRLGs while the span property is defined only in a mono-colored model where a link can be affected by at most one SRLG. In this paper, we extend these notions to characterize new cases in which these optimization problems can be solved in polynomial time. We also investigate the computational impact of the transformation from the multi-colored model to the mono-colored one. Experimental results are presented to validate the proposed algorithms and principles.]]> 0 <![CDATA[An arc colored eulerian multidigraph with $l$ colors is rainbow eulerian if there is an eulerian circuit in which a sequence of $l$ colors repeats. The digraph product that refers the title was introduced by Figueroa-Centeno et al. as follows: let $D$ be a digraph and let $\Gamma$ be a family of digraphs such that $V(F)=V$ for every $F\in \Gamma$. Consider any function $h:E(D) \longrightarrow \Gamma$. Then the product $D \otimes_h \Gamma$ is the digraph with vertex set $V(D) \times V$ and $((a,x),(b,y)) \in E(D \otimes_h \Gamma)$ if and only if $(a,b) \in E(D)$ and $(x,y) \in E(h (a,b))$. In this paper we use rainbow eulerian multidigraphs and permutations as a way to characterize the $\otimes_h$-product of oriented cycles. We study the behavior of the $\otimes_h$-product when applied to digraphs with unicyclic components. The results obtained allow us to get edge-magic labelings of graphs formed by the union of unicyclic components and with different magic sums.]]> Wed, 20 Apr 2016 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2153 https://doi.org/10.46298/dmtcs.2153 López, Susana Muntaner-Batle, Francesc-Antoni López, Susana Muntaner-Batle, Francesc-Antoni <![CDATA[An arc colored eulerian multidigraph with $l$ colors is rainbow eulerian if there is an eulerian circuit in which a sequence of $l$ colors repeats. The digraph product that refers the title was introduced by Figueroa-Centeno et al. as follows: let $D$ be a digraph and let $\Gamma$ be a family of digraphs such that $V(F)=V$ for every $F\in \Gamma$. Consider any function $h:E(D) \longrightarrow \Gamma$. Then the product $D \otimes_h \Gamma$ is the digraph with vertex set $V(D) \times V$ and $((a,x),(b,y)) \in E(D \otimes_h \Gamma)$ if and only if $(a,b) \in E(D)$ and $(x,y) \in E(h (a,b))$. In this paper we use rainbow eulerian multidigraphs and permutations as a way to characterize the $\otimes_h$-product of oriented cycles. We study the behavior of the $\otimes_h$-product when applied to digraphs with unicyclic components. The results obtained allow us to get edge-magic labelings of graphs formed by the union of unicyclic components and with different magic sums.]]> 0 <![CDATA[In this work we present a decentralized deployment algorithm for wireless mobile sensor networks focused on deployment Efficiency, connectivity Maintenance and network Reparation (EMR). We assume that a group of mobile sensors is placed in the area of interest to be covered, without any prior knowledge of the environment. The goal of the algorithm is to maximize the covered area and cope with sudden sensor failures. By relying on the locally available information regarding the environment and neighborhood, and without the need for any kind of synchronization in the network, each sensor iteratively chooses the next-step movement location so as to form a hexagonal lattice grid. Relying on the graph of wireless mobile sensors, we are able to provide the properties regarding the quality of coverage, the connectivity of the graph and the termination of the algorithm. We run extensive simulations to provide compactness properties of the deployment and evaluate the robustness against sensor failures. We show through the analysis and the simulations that EMR algorithm is robust to node failures and can restore the lattice grid. We also show that even after a failure, EMR algorithm call still provide a compact deployment in a reasonable time.]]> Wed, 20 Apr 2016 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2163 https://doi.org/10.46298/dmtcs.2163 Erdelj, Milan Mitton, Nathalie Razafindralambo, Tahiry Erdelj, Milan Mitton, Nathalie Razafindralambo, Tahiry <![CDATA[In this work we present a decentralized deployment algorithm for wireless mobile sensor networks focused on deployment Efficiency, connectivity Maintenance and network Reparation (EMR). We assume that a group of mobile sensors is placed in the area of interest to be covered, without any prior knowledge of the environment. The goal of the algorithm is to maximize the covered area and cope with sudden sensor failures. By relying on the locally available information regarding the environment and neighborhood, and without the need for any kind of synchronization in the network, each sensor iteratively chooses the next-step movement location so as to form a hexagonal lattice grid. Relying on the graph of wireless mobile sensors, we are able to provide the properties regarding the quality of coverage, the connectivity of the graph and the termination of the algorithm. We run extensive simulations to provide compactness properties of the deployment and evaluate the robustness against sensor failures. We show through the analysis and the simulations that EMR algorithm is robust to node failures and can restore the lattice grid. We also show that even after a failure, EMR algorithm call still provide a compact deployment in a reasonable time.]]> 0 <![CDATA[We find surprisingly simple formulas for the limiting probability that the rank of a randomly selected vertex in a randomly selected phylogenetic tree or generalized phylogenetic tree is a given integer.]]> Mon, 11 Apr 2016 19:55:16 +0000 https://doi.org/10.46298/dmtcs.653 https://doi.org/10.46298/dmtcs.653 Bóna, Miklós Bóna, Miklós <![CDATA[We find surprisingly simple formulas for the limiting probability that the rank of a randomly selected vertex in a randomly selected phylogenetic tree or generalized phylogenetic tree is a given integer.]]> 0 <![CDATA[In this paper, we study the behaviour of the generalized power domination number of a graph by small changes on the graph, namely edge and vertex deletion and edge contraction. We prove optimal bounds for $\gamma_{p,k}(G-e)$, $\gamma_{p,k}(G/e)$ and for $\gamma_{p,k}(G-v)$ in terms of $\gamma_{p,k}(G)$, and give examples for which these bounds are tight. We characterize all graphs for which $\gamma_{p,k}(G-e) = \gamma_{p,k}(G)+1$ for any edge $e$. We also consider the behaviour of the propagation radius of graphs by similar modifications.]]> Sun, 03 Apr 2016 21:25:50 +0000 https://doi.org/10.46298/dmtcs.1290 https://doi.org/10.46298/dmtcs.1290 Dorbec, Paul Varghese, Seethu Vijayakumar, Ambat Dorbec, Paul Varghese, Seethu Vijayakumar, Ambat <![CDATA[In this paper, we study the behaviour of the generalized power domination number of a graph by small changes on the graph, namely edge and vertex deletion and edge contraction. We prove optimal bounds for $\gamma_{p,k}(G-e)$, $\gamma_{p,k}(G/e)$ and for $\gamma_{p,k}(G-v)$ in terms of $\gamma_{p,k}(G)$, and give examples for which these bounds are tight. We characterize all graphs for which $\gamma_{p,k}(G-e) = \gamma_{p,k}(G)+1$ for any edge $e$. We also consider the behaviour of the propagation radius of graphs by similar modifications.]]> 0 <![CDATA[Permutations that avoid given patterns have been studied in great depth for their connections to other fields of mathematics, computer science, and biology. From a combinatorial perspective, permutation patterns have served as a unifying interpretation that relates a vast array of combinatorial structures. In this paper, we introduce the notion of patterns in inversion sequences. A sequence $(e_1,e_2,\ldots,e_n)$ is an inversion sequence if $0 \leq e_i \pi_i \}|$. This correspondence makes it a natural extension to study patterns in inversion sequences much in the same way that patterns have been studied in permutations. This paper, the first of two on patterns in inversion sequences, focuses on the enumeration of inversion sequences that avoid words of length three. Our results connect patterns in inversion sequences to a number of well-known numerical sequences including Fibonacci numbers, Bell numbers, Schr\"oder numbers, and Euler up/down numbers.]]> Thu, 31 Mar 2016 09:47:19 +0000 https://doi.org/10.46298/dmtcs.1323 https://doi.org/10.46298/dmtcs.1323 Corteel, Sylvie Martinez, Megan A. Savage, Carla D. Weselcouch, Michael Corteel, Sylvie Martinez, Megan A. Savage, Carla D. Weselcouch, Michael <![CDATA[Permutations that avoid given patterns have been studied in great depth for their connections to other fields of mathematics, computer science, and biology. From a combinatorial perspective, permutation patterns have served as a unifying interpretation that relates a vast array of combinatorial structures. In this paper, we introduce the notion of patterns in inversion sequences. A sequence $(e_1,e_2,\ldots,e_n)$ is an inversion sequence if $0 \leq e_i \pi_i \}|$. This correspondence makes it a natural extension to study patterns in inversion sequences much in the same way that patterns have been studied in permutations. This paper, the first of two on patterns in inversion sequences, focuses on the enumeration of inversion sequences that avoid words of length three. Our results connect patterns in inversion sequences to a number of well-known numerical sequences including Fibonacci numbers, Bell numbers, Schr\"oder numbers, and Euler up/down numbers.]]> 0 <![CDATA[Closed monopolies in graphs have a quite long range of applications in several problems related to overcoming failures, since they frequently have some common approaches around the notion of majorities, for instance to consensus problems, diagnosis problems or voting systems. We introduce here open $k$-monopolies in graphs which are closely related to different parameters in graphs. Given a graph $G=(V,E)$ and $X\subseteq V$, if $\delta_X(v)$ is the number of neighbors $v$ has in $X$, $k$ is an integer and $t$ is a positive integer, then we establish in this article a connection between the following three concepts: - Given a nonempty set $M\subseteq V$ a vertex $v$ of $G$ is said to be $k$-controlled by $M$ if $\delta_M(v)\ge \frac{\delta_V(v)}{2}+k$. The set $M$ is called an open $k$-monopoly for $G$ if it $k$-controls every vertex $v$ of $G$. - A function $f: V\rightarrow \{-1,1\}$ is called a signed total $t$-dominating function for $G$ if $f(N(v))=\sum_{v\in N(v)}f(v)\geq t$ for all $v\in V$. - A nonempty set $S\subseteq V$ is a global (defensive and offensive) $k$-alliance in $G$ if $\delta_S(v)\ge \delta_{V-S}(v)+k$ holds for every $v\in V$. In this article we prove that the problem of computing the minimum cardinality of an open $0$-monopoly in a graph is NP-complete even restricted to bipartite or chordal graphs. In addition we present some general bounds for the minimum cardinality of open $k$-monopolies and we derive some exact values.]]> Tue, 29 Mar 2016 19:20:25 +0000 https://doi.org/10.46298/dmtcs.654 https://doi.org/10.46298/dmtcs.654 Kuziak, Dorota Peterin, Iztok Yero, Ismael G. Kuziak, Dorota Peterin, Iztok Yero, Ismael G. <![CDATA[Closed monopolies in graphs have a quite long range of applications in several problems related to overcoming failures, since they frequently have some common approaches around the notion of majorities, for instance to consensus problems, diagnosis problems or voting systems. We introduce here open $k$-monopolies in graphs which are closely related to different parameters in graphs. Given a graph $G=(V,E)$ and $X\subseteq V$, if $\delta_X(v)$ is the number of neighbors $v$ has in $X$, $k$ is an integer and $t$ is a positive integer, then we establish in this article a connection between the following three concepts: - Given a nonempty set $M\subseteq V$ a vertex $v$ of $G$ is said to be $k$-controlled by $M$ if $\delta_M(v)\ge \frac{\delta_V(v)}{2}+k$. The set $M$ is called an open $k$-monopoly for $G$ if it $k$-controls every vertex $v$ of $G$. - A function $f: V\rightarrow \{-1,1\}$ is called a signed total $t$-dominating function for $G$ if $f(N(v))=\sum_{v\in N(v)}f(v)\geq t$ for all $v\in V$. - A nonempty set $S\subseteq V$ is a global (defensive and offensive) $k$-alliance in $G$ if $\delta_S(v)\ge \delta_{V-S}(v)+k$ holds for every $v\in V$. In this article we prove that the problem of computing the minimum cardinality of an open $0$-monopoly in a graph is NP-complete even restricted to bipartite or chordal graphs. In addition we present some general bounds for the minimum cardinality of open $k$-monopolies and we derive some exact values.]]> 0 <![CDATA[Let $\mathcal{C}$ be a permutation class that does not contain all layered permutations or all colayered permutations. We prove that there is a constant $c$ such that every permutation in $\mathcal{C}$ of length $n$ contains a monotone subsequence of length $cn$.]]> Thu, 24 Mar 2016 16:45:14 +0000 https://doi.org/10.46298/dmtcs.1328 https://doi.org/10.46298/dmtcs.1328 Vatter, Vincent Vatter, Vincent <![CDATA[Let $\mathcal{C}$ be a permutation class that does not contain all layered permutations or all colayered permutations. We prove that there is a constant $c$ such that every permutation in $\mathcal{C}$ of length $n$ contains a monotone subsequence of length $cn$.]]> 0 <![CDATA[When nodes can repeatedly update their behavior (as in agent-based models from computational social science or repeated-game play settings) the problem of optimal network seeding becomes very complex. For a popular spreading-phenomena model of binary-behavior updating based on thresholds of adoption among neighbors, we consider several planning problems in the design of \textit{Sticky Interventions}: when adoption decisions are reversible, the planner aims to find a Seed Set where temporary intervention leads to long-term behavior change. We prove that completely converting a network at minimum cost is $\Omega(\ln (OPT) )$-hard to approximate and that maximizing conversion subject to a budget is $(1-\frac{1}{e})$-hard to approximate. Optimization heuristics which rely on many objective function evaluations may still be practical, particularly in relatively-sparse networks: we prove that the long-term impact of a Seed Set can be evaluated in $O(|E|^2)$ operations. For a more descriptive model variant in which some neighbors may be more influential than others, we show that under integer edge weights from $\{0,1,2,...,k\}$ objective function evaluation requires only $O(k|E|^2)$ operations. These operation bounds are based on improvements we give for bounds on time-steps-to-convergence under discrete-time reversible-threshold updates in networks.]]> Thu, 17 Mar 2016 10:49:04 +0000 https://doi.org/10.46298/dmtcs.1241 https://doi.org/10.46298/dmtcs.1241 Spencer, Gwen Spencer, Gwen <![CDATA[When nodes can repeatedly update their behavior (as in agent-based models from computational social science or repeated-game play settings) the problem of optimal network seeding becomes very complex. For a popular spreading-phenomena model of binary-behavior updating based on thresholds of adoption among neighbors, we consider several planning problems in the design of \textit{Sticky Interventions}: when adoption decisions are reversible, the planner aims to find a Seed Set where temporary intervention leads to long-term behavior change. We prove that completely converting a network at minimum cost is $\Omega(\ln (OPT) )$-hard to approximate and that maximizing conversion subject to a budget is $(1-\frac{1}{e})$-hard to approximate. Optimization heuristics which rely on many objective function evaluations may still be practical, particularly in relatively-sparse networks: we prove that the long-term impact of a Seed Set can be evaluated in $O(|E|^2)$ operations. For a more descriptive model variant in which some neighbors may be more influential than others, we show that under integer edge weights from $\{0,1,2,...,k\}$ objective function evaluation requires only $O(k|E|^2)$ operations. These operation bounds are based on improvements we give for bounds on time-steps-to-convergence under discrete-time reversible-threshold updates in networks.]]> 0 <![CDATA[We study the configuration space of rectangulations and convex subdivisions of $n$ points in the plane. It is shown that a sequence of $O(n\log n)$ elementary flip and rotate operations can transform any rectangulation to any other rectangulation on the same set of $n$ points. This bound is the best possible for some point sets, while $\Theta(n)$ operations are sufficient and necessary for others. Some of our bounds generalize to convex subdivisions of $n$ points in the plane.]]> Thu, 17 Mar 2016 09:57:41 +0000 https://doi.org/10.46298/dmtcs.646 https://doi.org/10.46298/dmtcs.646 Ackerman, Eyal Allen, Michelle M. Barequet, Gill Löffler, Maarten Mermelstein, Joshua Souvaine, Diane L. Tóth, Csaba D. Ackerman, Eyal Allen, Michelle M. Barequet, Gill Löffler, Maarten Mermelstein, Joshua Souvaine, Diane L. Tóth, Csaba D. <![CDATA[We study the configuration space of rectangulations and convex subdivisions of $n$ points in the plane. It is shown that a sequence of $O(n\log n)$ elementary flip and rotate operations can transform any rectangulation to any other rectangulation on the same set of $n$ points. This bound is the best possible for some point sets, while $\Theta(n)$ operations are sufficient and necessary for others. Some of our bounds generalize to convex subdivisions of $n$ points in the plane.]]> 0 <![CDATA[Word $W$ is an instance of word $V$ provided there is a homomorphism $\phi$ mapping letters to nonempty words so that $\phi(V) = W$. For example, taking $\phi$ such that $\phi(c)=fr$, $\phi(o)=e$ and $\phi(l)=zer$, we see that "freezer" is an instance of "cool". Let $\mathbb{I}_n(V,[q])$ be the probability that a random length $n$ word on the alphabet $[q] = \{1,2,\cdots q\}$ is an instance of $V$. Having previously shown that $\lim_{n \rightarrow \infty} \mathbb{I}_n(V,[q])$ exists, we now calculate this limit for two Zimin words, $Z_2 = aba$ and $Z_3 = abacaba$.]]> Thu, 17 Mar 2016 08:46:11 +0000 https://doi.org/10.46298/dmtcs.1302 https://doi.org/10.46298/dmtcs.1302 Cooper, Joshua Rorabaugh, Danny Cooper, Joshua Rorabaugh, Danny <![CDATA[Word $W$ is an instance of word $V$ provided there is a homomorphism $\phi$ mapping letters to nonempty words so that $\phi(V) = W$. For example, taking $\phi$ such that $\phi(c)=fr$, $\phi(o)=e$ and $\phi(l)=zer$, we see that "freezer" is an instance of "cool". Let $\mathbb{I}_n(V,[q])$ be the probability that a random length $n$ word on the alphabet $[q] = \{1,2,\cdots q\}$ is an instance of $V$. Having previously shown that $\lim_{n \rightarrow \infty} \mathbb{I}_n(V,[q])$ exists, we now calculate this limit for two Zimin words, $Z_2 = aba$ and $Z_3 = abacaba$.]]> 0 <![CDATA[We consider implicit signatures over finite semigroups determined by sets of pseudonatural numbers. We prove that, under relatively simple hypotheses on a pseudovariety V of semigroups, the finitely generated free algebra for the largest such signature is closed under taking factors within the free pro-V semigroup on the same set of generators. Furthermore, we show that the natural analogue of the Pin-Reutenauer descriptive procedure for the closure of a rational language in the free group with respect to the profinite topology holds for the pseudovariety of all finite semigroups. As an application, we establish that a pseudovariety enjoys this property if and only if it is full.]]> Tue, 15 Mar 2016 13:04:03 +0000 https://doi.org/10.46298/dmtcs.650 https://doi.org/10.46298/dmtcs.650 Almeida, J. Costa, J. C. Zeitoun, M. Almeida, J. Costa, J. C. Zeitoun, M. <![CDATA[We consider implicit signatures over finite semigroups determined by sets of pseudonatural numbers. We prove that, under relatively simple hypotheses on a pseudovariety V of semigroups, the finitely generated free algebra for the largest such signature is closed under taking factors within the free pro-V semigroup on the same set of generators. Furthermore, we show that the natural analogue of the Pin-Reutenauer descriptive procedure for the closure of a rational language in the free group with respect to the profinite topology holds for the pseudovariety of all finite semigroups. As an application, we establish that a pseudovariety enjoys this property if and only if it is full.]]> 0 <![CDATA[In this paper, we explore completely regular codes in the Hamming graphs and related graphs. Experimental evidence suggests that many completely regular codes have the property that the eigenvalues of the code are in arithmetic progression. In order to better understand these "arithmetic completely regular codes", we focus on cartesian products of completely regular codes and products of their corresponding coset graphs in the additive case. Employing earlier results, we are then able to prove a theorem which nearly classifies these codes in the case where the graph admits a completely regular partition into such codes (e.g, the cosets of some additive completely regular code). Connections to the theory of distance-regular graphs are explored and several open questions are posed.]]> Fri, 26 Feb 2016 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2150 https://doi.org/10.46298/dmtcs.2150 Koolen, Jacobus Sun Lee, Woo Martin, William Tanaka, Hajime Koolen, Jacobus Sun Lee, Woo Martin, William Tanaka, Hajime <![CDATA[In this paper, we explore completely regular codes in the Hamming graphs and related graphs. Experimental evidence suggests that many completely regular codes have the property that the eigenvalues of the code are in arithmetic progression. In order to better understand these "arithmetic completely regular codes", we focus on cartesian products of completely regular codes and products of their corresponding coset graphs in the additive case. Employing earlier results, we are then able to prove a theorem which nearly classifies these codes in the case where the graph admits a completely regular partition into such codes (e.g, the cosets of some additive completely regular code). Connections to the theory of distance-regular graphs are explored and several open questions are posed.]]> 0 <![CDATA[We endow the set of isomorphism classes of matroids with a new Hopf algebra structure, in which the coproduct is implemented via the combinatorial operations of restriction and deletion. We also initiate the investigation of dendriform coalgebra structures on matroids and introduce a monomial invariant which satisfy a convolution identity with respect to restriction and deletion.]]> Thu, 25 Feb 2016 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2157 https://doi.org/10.46298/dmtcs.2157 Hoang-Nghia, Nguyen Tanasa, Adrian Tollu, Christophe Hoang-Nghia, Nguyen Tanasa, Adrian Tollu, Christophe <![CDATA[We endow the set of isomorphism classes of matroids with a new Hopf algebra structure, in which the coproduct is implemented via the combinatorial operations of restriction and deletion. We also initiate the investigation of dendriform coalgebra structures on matroids and introduce a monomial invariant which satisfy a convolution identity with respect to restriction and deletion.]]> 0 <![CDATA[A graph is locally irregular if every two adjacent vertices have distinct degrees. Recently, Baudon et al. introduced the notion of decomposition into locally irregular subgraphs. They conjectured that for almost every graph $G$, there exists a minimum integer $\chi^{\prime}_{\mathrm{irr}}(G)$ such that $G$ admits an edge-partition into $\chi^{\prime}_{\mathrm{irr}}(G)$ classes, each of which induces a locally irregular graph. In particular, they conjectured that $\chi^{\prime}_{\mathrm{irr}}(G) \leq 3$ for every $G$, unless $G$ belongs to a well-characterized family of non-decomposable graphs. This conjecture is far from being settled, as notably (1) no constant upper bound on$\chi^{\prime}_{\mathrm{irr}}(G)$ is known for $G$ bipartite, and (2) no satisfactory general upper bound on $\chi^{\prime}_{\mathrm{irr}}(G)$ is known. We herein investigate the consequences on this question of allowing a decomposition to include regular components as well. As a main result, we prove that every bipartite graph admits such a decomposition into at most $6$ subgraphs. This result implies that every graph $G$ admits a decomposition into at most $6(\lfloor \mathrm{log} \chi (G) \rfloor +1)$ subgraphs whose components are regular or locally irregular.]]> Tue, 16 Feb 2016 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2154 https://doi.org/10.46298/dmtcs.2154 Bensmail, Julien Stevens, Brett Bensmail, Julien Stevens, Brett <![CDATA[A graph is locally irregular if every two adjacent vertices have distinct degrees. Recently, Baudon et al. introduced the notion of decomposition into locally irregular subgraphs. They conjectured that for almost every graph $G$, there exists a minimum integer $\chi^{\prime}_{\mathrm{irr}}(G)$ such that $G$ admits an edge-partition into $\chi^{\prime}_{\mathrm{irr}}(G)$ classes, each of which induces a locally irregular graph. In particular, they conjectured that $\chi^{\prime}_{\mathrm{irr}}(G) \leq 3$ for every $G$, unless $G$ belongs to a well-characterized family of non-decomposable graphs. This conjecture is far from being settled, as notably (1) no constant upper bound on$\chi^{\prime}_{\mathrm{irr}}(G)$ is known for $G$ bipartite, and (2) no satisfactory general upper bound on $\chi^{\prime}_{\mathrm{irr}}(G)$ is known. We herein investigate the consequences on this question of allowing a decomposition to include regular components as well. As a main result, we prove that every bipartite graph admits such a decomposition into at most $6$ subgraphs. This result implies that every graph $G$ admits a decomposition into at most $6(\lfloor \mathrm{log} \chi (G) \rfloor +1)$ subgraphs whose components are regular or locally irregular.]]> 0 <![CDATA[An oriented graph $\overrightarrow{G}$ is said weak (resp. strong) if, for every pair $\{ u,v \}$ of vertices of $\overrightarrow{G}$, there are directed paths joining $u$ and $v$ in either direction (resp. both directions). In case, for every pair of vertices, some of these directed paths have length at most $k$, we call $\overrightarrow{G}$ $k$-weak (resp. $k$-strong). We consider several problems asking whether an undirected graph $G$ admits orientations satisfying some connectivity and distance properties. As a main result, we show that deciding whether $G$ admits a $k$-weak orientation is NP-complete for every $k \geq 2$. This notably implies the NP-completeness of several problems asking whether $G$ is an extremal graph (in terms of needed colours) for some vertex-colouring problems.]]> Tue, 16 Feb 2016 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2161 https://doi.org/10.46298/dmtcs.2161 Bensmail, Julien Duvignau, Romaric Kirgizov, Sergey Bensmail, Julien Duvignau, Romaric Kirgizov, Sergey <![CDATA[An oriented graph $\overrightarrow{G}$ is said weak (resp. strong) if, for every pair $\{ u,v \}$ of vertices of $\overrightarrow{G}$, there are directed paths joining $u$ and $v$ in either direction (resp. both directions). In case, for every pair of vertices, some of these directed paths have length at most $k$, we call $\overrightarrow{G}$ $k$-weak (resp. $k$-strong). We consider several problems asking whether an undirected graph $G$ admits orientations satisfying some connectivity and distance properties. As a main result, we show that deciding whether $G$ admits a $k$-weak orientation is NP-complete for every $k \geq 2$. This notably implies the NP-completeness of several problems asking whether $G$ is an extremal graph (in terms of needed colours) for some vertex-colouring problems.]]> 0 <![CDATA[In this note, we prove that all $2 \times 2$ monotone grid classes are finitely based, i.e., defined by a finite collection of minimal forbidden permutations. This follows from a slightly more general result about certain $2 \times 2$ (generalized) grid classes having two monotone cells in the same row.]]> Thu, 11 Feb 2016 10:31:02 +0000 https://doi.org/10.46298/dmtcs.1325 https://doi.org/10.46298/dmtcs.1325 Albert, Michael Brignall, Robert Albert, Michael Brignall, Robert <![CDATA[In this note, we prove that all $2 \times 2$ monotone grid classes are finitely based, i.e., defined by a finite collection of minimal forbidden permutations. This follows from a slightly more general result about certain $2 \times 2$ (generalized) grid classes having two monotone cells in the same row.]]> 0 <![CDATA[We explore the classical pattern avoidance question in the case of irreducible permutations, i.e., those in which there is no index $i$ such that $\sigma (i+1) - \sigma (i)=1$. The problem is addressed completely in the case of avoiding one or two patterns of length three, and several well known sequences are encountered in the process, such as Catalan, Motzkin, Fibonacci, Tribonacci, Padovan and Binary numbers. Also, we present constructive bijections between the set of Motzkin paths of length $n-1$ and the sets of irreducible permutations of length $n$ (respectively fixed point free irreducible involutions of length $2n$) avoiding a pattern $\alpha$ for $\alpha \in \{132,213,321\}$. This induces two new bijections between the set of Dyck paths and some restricted sets of permutations.]]> Sun, 17 Jan 2016 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2158 https://doi.org/10.46298/dmtcs.2158 Baril, Jean-Luc Baril, Jean-Luc <![CDATA[We explore the classical pattern avoidance question in the case of irreducible permutations, i.e., those in which there is no index $i$ such that $\sigma (i+1) - \sigma (i)=1$. The problem is addressed completely in the case of avoiding one or two patterns of length three, and several well known sequences are encountered in the process, such as Catalan, Motzkin, Fibonacci, Tribonacci, Padovan and Binary numbers. Also, we present constructive bijections between the set of Motzkin paths of length $n-1$ and the sets of irreducible permutations of length $n$ (respectively fixed point free irreducible involutions of length $2n$) avoiding a pattern $\alpha$ for $\alpha \in \{132,213,321\}$. This induces two new bijections between the set of Dyck paths and some restricted sets of permutations.]]> 0 <![CDATA[Let $G$ be a graph and $\mathcal{S}$ be a subset of $Z$. A vertex-coloring $\mathcal{S}$-edge-weighting of $G$ is an assignment of weights by the elements of $\mathcal{S}$ to each edge of $G$ so that adjacent vertices have different sums of incident edges weights. It was proved that every 3-connected bipartite graph admits a vertex-coloring $\mathcal{S}$-edge-weighting for $\mathcal{S} = \{1,2 \}$ (H. Lu, Q. Yu and C. Zhang, Vertex-coloring 2-edge-weighting of graphs, European J. Combin., 32 (2011), 22-27). In this paper, we show that every 2-connected and 3-edge-connected bipartite graph admits a vertex-coloring $\mathcal{S}$-edge-weighting for $\mathcal{S} \in \{ \{ 0,1 \} , \{1,2 \} \}$. These bounds we obtain are tight, since there exists a family of infinite bipartite graphs which are 2-connected and do not admit vertex-coloring $\mathcal{S}$-edge-weightings for $\mathcal{S} \in \{ \{ 0,1 \} , \{1,2 \} \}$.]]> Mon, 04 Jan 2016 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2162 https://doi.org/10.46298/dmtcs.2162 Lu, Hongliang Lu, Hongliang <![CDATA[Let $G$ be a graph and $\mathcal{S}$ be a subset of $Z$. A vertex-coloring $\mathcal{S}$-edge-weighting of $G$ is an assignment of weights by the elements of $\mathcal{S}$ to each edge of $G$ so that adjacent vertices have different sums of incident edges weights. It was proved that every 3-connected bipartite graph admits a vertex-coloring $\mathcal{S}$-edge-weighting for $\mathcal{S} = \{1,2 \}$ (H. Lu, Q. Yu and C. Zhang, Vertex-coloring 2-edge-weighting of graphs, European J. Combin., 32 (2011), 22-27). In this paper, we show that every 2-connected and 3-edge-connected bipartite graph admits a vertex-coloring $\mathcal{S}$-edge-weighting for $\mathcal{S} \in \{ \{ 0,1 \} , \{1,2 \} \}$. These bounds we obtain are tight, since there exists a family of infinite bipartite graphs which are 2-connected and do not admit vertex-coloring $\mathcal{S}$-edge-weightings for $\mathcal{S} \in \{ \{ 0,1 \} , \{1,2 \} \}$.]]> 0 <![CDATA[In this article, we introduce the notion of the double competition multigraph of a digraph. We give characterizations of the double competition multigraphs of arbitrary digraphs, loopless digraphs, reflexive digraphs, and acyclic digraphs in terms of edge clique partitions of the multigraphs.]]> Wed, 30 Dec 2015 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2133 https://doi.org/10.46298/dmtcs.2133 Sano, Yoshio Park, Jeongmi Sano, Yoshio Park, Jeongmi <![CDATA[In this article, we introduce the notion of the double competition multigraph of a digraph. We give characterizations of the double competition multigraphs of arbitrary digraphs, loopless digraphs, reflexive digraphs, and acyclic digraphs in terms of edge clique partitions of the multigraphs.]]> 0 <![CDATA[A permutation $σ$ contains the permutation $τ$ if there is a subsequence of $σ$ order isomorphic to $τ$. A permutation $σ$ is $τ$-avoiding if it does not contain the permutation $τ$. For any $n$, the popularity of a permutation $τ$, denoted $A$_$n$($τ$), is the number of copies of $τ$ contained in the set of all 132-avoiding permutations of length $n$. Rudolph conjectures that for permutations $τ$ and $μ$ of the same length, $A$_$n$($τ$) ≤ $A$_$n$($μ$) for all $n$ if and only if the spine structure of $τ$ is less than or equal to the spine structure of $μ$ in refinement order. We prove one direction of this conjecture, by showing that if the spine structure of $τ$ is less than or equal to the spine structure of $μ$, then $A$_$n$($τ$) ≤ $A$_$n$($μ$) for all $n$. We disprove the opposite direction by giving a counterexample, and hence disprove the conjecture.]]> Mon, 14 Dec 2015 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2141 https://doi.org/10.46298/dmtcs.2141 Aisbett, Natalie Aisbett, Natalie <![CDATA[A permutation $σ$ contains the permutation $τ$ if there is a subsequence of $σ$ order isomorphic to $τ$. A permutation $σ$ is $τ$-avoiding if it does not contain the permutation $τ$. For any $n$, the popularity of a permutation $τ$, denoted $A$_$n$($τ$), is the number of copies of $τ$ contained in the set of all 132-avoiding permutations of length $n$. Rudolph conjectures that for permutations $τ$ and $μ$ of the same length, $A$_$n$($τ$) ≤ $A$_$n$($μ$) for all $n$ if and only if the spine structure of $τ$ is less than or equal to the spine structure of $μ$ in refinement order. We prove one direction of this conjecture, by showing that if the spine structure of $τ$ is less than or equal to the spine structure of $μ$, then $A$_$n$($τ$) ≤ $A$_$n$($μ$) for all $n$. We disprove the opposite direction by giving a counterexample, and hence disprove the conjecture.]]> 0 <![CDATA[We consider three problems related to dynamics of one-tape Turing machines: Existence of blocking configurations, surjectivity in the trace, and entropy positiveness. In order to address them, a reversible two-counter machine is simulated by a reversible Turing machine on the right side of its tape. By completing the machine in different ways, we prove that none of the former problems is decidable. In particular, the problems about blocking configurations and entropy are shown to be undecidable for the class of reversible Turing machines.]]> Tue, 01 Dec 2015 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2137 https://doi.org/10.46298/dmtcs.2137 Gajardo, Anahí Ollinger, Nicolas Torres-Avilés, Rodrigo Gajardo, Anahí Ollinger, Nicolas Torres-Avilés, Rodrigo <![CDATA[We consider three problems related to dynamics of one-tape Turing machines: Existence of blocking configurations, surjectivity in the trace, and entropy positiveness. In order to address them, a reversible two-counter machine is simulated by a reversible Turing machine on the right side of its tape. By completing the machine in different ways, we prove that none of the former problems is decidable. In particular, the problems about blocking configurations and entropy are shown to be undecidable for the class of reversible Turing machines.]]> 0 <![CDATA[We prove that the game colouring number of the $m$-th power of a forest of maximum degree $\Delta\ge3$ is bounded from above by \[\frac{(\Delta-1)^m-1}{\Delta-2}+2^m+1,\] which improves the best known bound by an asymptotic factor of 2.]]> Tue, 24 Nov 2015 19:53:58 +0000 https://doi.org/10.46298/dmtcs.648 https://doi.org/10.46298/dmtcs.648 Andres, Stephan Dominique Hochstättler, Winfried Andres, Stephan Dominique Hochstättler, Winfried <![CDATA[We prove that the game colouring number of the $m$-th power of a forest of maximum degree $\Delta\ge3$ is bounded from above by \[\frac{(\Delta-1)^m-1}{\Delta-2}+2^m+1,\] which improves the best known bound by an asymptotic factor of 2.]]> 0 <![CDATA[A spanning connectedness property is one which involves the robust existence of a spanning subgraph which is of some special form, say a Hamiltonian cycle in which a sequence of vertices appear in an arbitrarily given ordering, or a Hamiltonian path in the subgraph obtained by deleting any three vertices, or three internally-vertex-disjoint paths with any given endpoints such that the three paths meet every vertex of the graph and cover the edges of an almost arbitrarily given linear forest of a certain fixed size. Let π = π1 · · · πn be an ordering of the vertices of an n-vertex graph G. For any positive integer k ≤ n − 1, we call π a k-thick Hamiltonian vertex ordering of G provided it holds for all i ∈ {1,. .. , n − 1} that πiπi+1 ∈ E(G) and the number of neighbors of πi among {πi+1,. .. , πn} is at least min{n − i, k}; For any nonnegative integer k, we say that π is a −k-thick Hamiltonian vertex ordering of G provided |{i : πiπi+1 / ∈ E(G), 1 ≤ i ≤ n − 1}| ≤ k + 1. Our main discovery is that the existence of a thick Hamiltonian vertex ordering will guarantee that the graph has various kinds of spanning connectedness properties and that for interval graphs, quite a few seemingly unrelated spanning connectedness properties are equivalent to the existence of a thick Hamiltonian vertex ordering. Due to the connection between Hamiltonian thickness and spanning connectedness properties, we can present several linear time algorithms for associated problems. This paper suggests that much work in graph theory may have a spanning version which deserves further study, and that the Hamiltonian thickness may be a useful concept in understanding many spanning connectedness properties.]]> Sun, 22 Nov 2015 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2082 https://doi.org/10.46298/dmtcs.2082 Li, Peng Wu, Yaokun Li, Peng Wu, Yaokun <![CDATA[A spanning connectedness property is one which involves the robust existence of a spanning subgraph which is of some special form, say a Hamiltonian cycle in which a sequence of vertices appear in an arbitrarily given ordering, or a Hamiltonian path in the subgraph obtained by deleting any three vertices, or three internally-vertex-disjoint paths with any given endpoints such that the three paths meet every vertex of the graph and cover the edges of an almost arbitrarily given linear forest of a certain fixed size. Let π = π1 · · · πn be an ordering of the vertices of an n-vertex graph G. For any positive integer k ≤ n − 1, we call π a k-thick Hamiltonian vertex ordering of G provided it holds for all i ∈ {1,. .. , n − 1} that πiπi+1 ∈ E(G) and the number of neighbors of πi among {πi+1,. .. , πn} is at least min{n − i, k}; For any nonnegative integer k, we say that π is a −k-thick Hamiltonian vertex ordering of G provided |{i : πiπi+1 / ∈ E(G), 1 ≤ i ≤ n − 1}| ≤ k + 1. Our main discovery is that the existence of a thick Hamiltonian vertex ordering will guarantee that the graph has various kinds of spanning connectedness properties and that for interval graphs, quite a few seemingly unrelated spanning connectedness properties are equivalent to the existence of a thick Hamiltonian vertex ordering. Due to the connection between Hamiltonian thickness and spanning connectedness properties, we can present several linear time algorithms for associated problems. This paper suggests that much work in graph theory may have a spanning version which deserves further study, and that the Hamiltonian thickness may be a useful concept in understanding many spanning connectedness properties.]]> 0 <![CDATA[We introduce a new graph parameter that measures fractional covering of a graph by cuts. Besides being interesting in its own right, it is useful for study of homomorphisms and tension-continuous mappings. We study the relations with chromatic number, bipartite density, and other graph parameters. We find the value of our parameter for a family of graphs based on hypercubes. These graphs play for our parameter the role that cliques play for the chromatic number and Kneser graphs for the fractional chromatic number. The fact that the defined parameter attains on these graphs the correct value suggests that our definition is a natural one. In the proof we use the eigenvalue bound for maximum cut and a recent result of Engström, Färnqvist, Jonsson, and Thapper [An approximability-related parameter on graphs – properties and applications, DMTCS vol. 17:1, 2015, 33–66]. We also provide a polynomial time approximation algorithm based on semidefinite programming and in particular on vector chromatic number (defined by Karger, Motwani and Sudan [Approximate graph coloring by semidefinite programming, J. ACM 45 (1998), no. 2, 246–265]).]]> Tue, 17 Nov 2015 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2134 https://doi.org/10.46298/dmtcs.2134 Šámal, Robert Šámal, Robert <![CDATA[We introduce a new graph parameter that measures fractional covering of a graph by cuts. Besides being interesting in its own right, it is useful for study of homomorphisms and tension-continuous mappings. We study the relations with chromatic number, bipartite density, and other graph parameters. We find the value of our parameter for a family of graphs based on hypercubes. These graphs play for our parameter the role that cliques play for the chromatic number and Kneser graphs for the fractional chromatic number. The fact that the defined parameter attains on these graphs the correct value suggests that our definition is a natural one. In the proof we use the eigenvalue bound for maximum cut and a recent result of Engström, Färnqvist, Jonsson, and Thapper [An approximability-related parameter on graphs – properties and applications, DMTCS vol. 17:1, 2015, 33–66]. We also provide a polynomial time approximation algorithm based on semidefinite programming and in particular on vector chromatic number (defined by Karger, Motwani and Sudan [Approximate graph coloring by semidefinite programming, J. ACM 45 (1998), no. 2, 246–265]).]]> 0 <![CDATA[In this paper we present an analysis of some generalization of the classic urn and balls model. In our model each urn has a fixed capacity and initially is filled with white balls. Black balls are added to the system of connected urns and gradually displace white balls. We show a general form of formulas for the expected numbers of black balls in a given urn and we analyze some special cases (parallel and serial configurations). We are mainly interested in a counterpart of the Coupon Collector Problem for the model considered. The primary motivation for our research is the formal analysis of the mix networks (introduced by D. Chaum) and its immunity to so-called flooding (blending) attacks.]]> Thu, 15 Oct 2015 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2143 https://doi.org/10.46298/dmtcs.2143 Klonowski, Marek Cichoń, Jacek Kapelko, Rafał Klonowski, Marek Cichoń, Jacek Kapelko, Rafał <![CDATA[In this paper we present an analysis of some generalization of the classic urn and balls model. In our model each urn has a fixed capacity and initially is filled with white balls. Black balls are added to the system of connected urns and gradually displace white balls. We show a general form of formulas for the expected numbers of black balls in a given urn and we analyze some special cases (parallel and serial configurations). We are mainly interested in a counterpart of the Coupon Collector Problem for the model considered. The primary motivation for our research is the formal analysis of the mix networks (introduced by D. Chaum) and its immunity to so-called flooding (blending) attacks.]]> 0 <![CDATA[The vertex corona of a vertex of some tiling is the vertex together with the adjacent tiles. A tiling where all vertex coronae are congruent is called monocoronal. We provide a classification of monocoronal tilings in the Euclidean plane and derive a list of all possible symmetry groups of monocoronal tilings. In particular, any monocoronal tiling with respect to direct congruence is crystallographic, whereas any monocoronal tiling with respect to congruence (reflections allowed) is either crystallographic or it has a one-dimensional translation group. Furthermore, bounds on the number of the dimensions of the translation group of monocoronal tilings in higher dimensional Euclidean space are obtained.]]> Sun, 04 Oct 2015 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2142 https://doi.org/10.46298/dmtcs.2142 Frettlöh, Dirk Garber, Alexey Frettlöh, Dirk Garber, Alexey <![CDATA[The vertex corona of a vertex of some tiling is the vertex together with the adjacent tiles. A tiling where all vertex coronae are congruent is called monocoronal. We provide a classification of monocoronal tilings in the Euclidean plane and derive a list of all possible symmetry groups of monocoronal tilings. In particular, any monocoronal tiling with respect to direct congruence is crystallographic, whereas any monocoronal tiling with respect to congruence (reflections allowed) is either crystallographic or it has a one-dimensional translation group. Furthermore, bounds on the number of the dimensions of the translation group of monocoronal tilings in higher dimensional Euclidean space are obtained.]]> 0 <![CDATA[The successive discrete structures generated by a sequential algorithm from random input constitute a Markov chain that may exhibit long term dependence on its first few input values. Using examples from random graph theory and search algorithms we show how such persistence of randomness can be detected and quantified with techniques from discrete potential theory. We also show that this approach can be used to obtain strong limit theorems in cases where previously only distributional convergence was known.]]> Thu, 01 Oct 2015 07:37:31 +0000 https://doi.org/10.46298/dmtcs.644 https://doi.org/10.46298/dmtcs.644 Grübel, Rudolf Grübel, Rudolf <![CDATA[The successive discrete structures generated by a sequential algorithm from random input constitute a Markov chain that may exhibit long term dependence on its first few input values. Using examples from random graph theory and search algorithms we show how such persistence of randomness can be detected and quantified with techniques from discrete potential theory. We also show that this approach can be used to obtain strong limit theorems in cases where previously only distributional convergence was known.]]> 0 <![CDATA[Vincular or dashed patterns resemble classical patterns except that some of the letters within an occurrence are required to be adjacent. We prove several infinite families of Wilf-equivalences for $k$-ary words involving vincular patterns containing a single dash, which explain the majority of the equivalences witnessed for such patterns of length four. When combined with previous results, numerical evidence, and some arguments in specific cases, we obtain the complete Wilf-classification for all vincular patterns of length four containing a single dash. In some cases, our proof shows further that the equivalence holds for multiset permutations since it is seen to respect the number of occurrences of each letter within a word. Some related enumerative results are provided for patterns $τ$ of length four, among them generating function formulas for the number of members of [$k$]^$n$ avoiding any $τ$ of the form 11$a-b$.]]> Tue, 15 Sep 2015 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2140 https://doi.org/10.46298/dmtcs.2140 Mansour, Toufik Shattuck, Mark Mansour, Toufik Shattuck, Mark <![CDATA[Vincular or dashed patterns resemble classical patterns except that some of the letters within an occurrence are required to be adjacent. We prove several infinite families of Wilf-equivalences for $k$-ary words involving vincular patterns containing a single dash, which explain the majority of the equivalences witnessed for such patterns of length four. When combined with previous results, numerical evidence, and some arguments in specific cases, we obtain the complete Wilf-classification for all vincular patterns of length four containing a single dash. In some cases, our proof shows further that the equivalence holds for multiset permutations since it is seen to respect the number of occurrences of each letter within a word. Some related enumerative results are provided for patterns $τ$ of length four, among them generating function formulas for the number of members of [$k$]^$n$ avoiding any $τ$ of the form 11$a-b$.]]> 0 <![CDATA[Promise problems were mainly studied in quantum automata theory. Here we focus on state complexity of classical automata for promise problems. First, it was known that there is a family of unary promise problems solvable by quantum automata by using a single qubit, but the number of states required by corresponding one-way deterministic automata cannot be bounded by a constant. For this family, we show that even two-way nondeterminism does not help to save a single state. By comparing this with the corresponding state complexity of alternating machines, we then get a tight exponential gap between two-way nondeterministic and one-way alternating automata solving unary promise problems. Second, despite of the existing quadratic gap between Las Vegas realtime probabilistic automata and one-way deterministic automata for language recognition, we show that, by turning to promise problems, the tight gap becomes exponential. Last, we show that the situation is different for one-way probabilistic automata with two-sided bounded-error. We present a family of unary promise problems that is very easy for these machines; solvable with only two states, but the number of states in two-way alternating or any simpler automata is not limited by a constant. Moreover, we show that one-way bounded-error probabilistic automata can solve promise problems not solvable at all by any other classical model.]]> Tue, 15 Sep 2015 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2138 https://doi.org/10.46298/dmtcs.2138 Geffert, Viliam Yakaryilmaz, Abuzer Geffert, Viliam Yakaryilmaz, Abuzer <![CDATA[Promise problems were mainly studied in quantum automata theory. Here we focus on state complexity of classical automata for promise problems. First, it was known that there is a family of unary promise problems solvable by quantum automata by using a single qubit, but the number of states required by corresponding one-way deterministic automata cannot be bounded by a constant. For this family, we show that even two-way nondeterminism does not help to save a single state. By comparing this with the corresponding state complexity of alternating machines, we then get a tight exponential gap between two-way nondeterministic and one-way alternating automata solving unary promise problems. Second, despite of the existing quadratic gap between Las Vegas realtime probabilistic automata and one-way deterministic automata for language recognition, we show that, by turning to promise problems, the tight gap becomes exponential. Last, we show that the situation is different for one-way probabilistic automata with two-sided bounded-error. We present a family of unary promise problems that is very easy for these machines; solvable with only two states, but the number of states in two-way alternating or any simpler automata is not limited by a constant. Moreover, we show that one-way bounded-error probabilistic automata can solve promise problems not solvable at all by any other classical model.]]> 0 <![CDATA[We consider the rank reduction problem for matroids: Given a matroid $M$ and an integer $k$, find a minimum size subset of elements of $M$ whose removal reduces the rank of $M$ by at least $k$. When $M$ is a graphical matroid this problem is the minimum $k$-cut problem, which admits a 2-approximation algorithm. In this paper we show that the rank reduction problem for transversal matroids is essentially at least as hard to approximate as the densest $k$-subgraph problem. We also prove that, while the problem is easily solvable in polynomial time for partition matroids, it is NP-hard when considering the intersection of two partition matroids. Our proof shows, in particular, that the maximum vertex cover problem is NP-hard on bipartite graphs, which answers an open problem of B. Simeone.]]> Tue, 15 Sep 2015 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2135 https://doi.org/10.46298/dmtcs.2135 Joret, Gwenaël Vetta, Adrian Joret, Gwenaël Vetta, Adrian <![CDATA[We consider the rank reduction problem for matroids: Given a matroid $M$ and an integer $k$, find a minimum size subset of elements of $M$ whose removal reduces the rank of $M$ by at least $k$. When $M$ is a graphical matroid this problem is the minimum $k$-cut problem, which admits a 2-approximation algorithm. In this paper we show that the rank reduction problem for transversal matroids is essentially at least as hard to approximate as the densest $k$-subgraph problem. We also prove that, while the problem is easily solvable in polynomial time for partition matroids, it is NP-hard when considering the intersection of two partition matroids. Our proof shows, in particular, that the maximum vertex cover problem is NP-hard on bipartite graphs, which answers an open problem of B. Simeone.]]> 0 <![CDATA[Given a set $P$ of $n$ points in the plane, where $n$ is even, we consider the following question: How many plane perfect matchings can be packed into $P$? For points in general position we prove the lower bound of ⌊log₂$n$⌋$-1$. For some special configurations of point sets, we give the exact answer. We also consider some restricted variants of this problem.]]> Mon, 14 Sep 2015 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2132 https://doi.org/10.46298/dmtcs.2132 Biniaz, Ahmad Bose, Prosenjit Maheshwari, Anil Smid, Michiel Biniaz, Ahmad Bose, Prosenjit Maheshwari, Anil Smid, Michiel <![CDATA[Given a set $P$ of $n$ points in the plane, where $n$ is even, we consider the following question: How many plane perfect matchings can be packed into $P$? For points in general position we prove the lower bound of ⌊log₂$n$⌋$-1$. For some special configurations of point sets, we give the exact answer. We also consider some restricted variants of this problem.]]> 0 <![CDATA[In this article we deal with the problems of finding the disimplicial arcs of a digraph and recognizing some interesting graph classes defined by their existence. A diclique of a digraph is a pair $V$ → $W$ of sets of vertices such that $v$ → $w$ is an arc for every $v$ ∈ $V$ and $w$ ∈ $W$. An arc $v$ → $w$ is disimplicial when it belongs to a unique maximal diclique. We show that the problem of finding the disimplicial arcs is equivalent, in terms of time and space complexity, to that of locating the transitive vertices. As a result, an efficient algorithm to find the bisimplicial edges of bipartite graphs is obtained. Then, we develop simple algorithms to build disimplicial elimination schemes, which can be used to generate bisimplicial elimination schemes for bipartite graphs. Finally, we study two classes related to perfect disimplicial elimination digraphs, namely weakly diclique irreducible digraphs and diclique irreducible digraphs. The former class is associated to finite posets, while the latter corresponds to dedekind complete finite posets.]]> Mon, 14 Sep 2015 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2131 https://doi.org/10.46298/dmtcs.2131 Eguia, Martiniano Soulignac, Francisco, Eguia, Martiniano Soulignac, Francisco, <![CDATA[In this article we deal with the problems of finding the disimplicial arcs of a digraph and recognizing some interesting graph classes defined by their existence. A diclique of a digraph is a pair $V$ → $W$ of sets of vertices such that $v$ → $w$ is an arc for every $v$ ∈ $V$ and $w$ ∈ $W$. An arc $v$ → $w$ is disimplicial when it belongs to a unique maximal diclique. We show that the problem of finding the disimplicial arcs is equivalent, in terms of time and space complexity, to that of locating the transitive vertices. As a result, an efficient algorithm to find the bisimplicial edges of bipartite graphs is obtained. Then, we develop simple algorithms to build disimplicial elimination schemes, which can be used to generate bisimplicial elimination schemes for bipartite graphs. Finally, we study two classes related to perfect disimplicial elimination digraphs, namely weakly diclique irreducible digraphs and diclique irreducible digraphs. The former class is associated to finite posets, while the latter corresponds to dedekind complete finite posets.]]> 0 <![CDATA[Parameterized algorithms are often used to efficiently solve NP-hard problems on graphs. In this context, vertex cover is used as a powerful parameter for dealing with graph problems which are hard to solve even when parameterized by tree-width; however, the drawback of vertex cover is that bounding it severely restricts admissible graph classes. We introduce a generalization of vertex cover called twin-cover and show that FPT algorithms exist for a wide range of difficult problems when parameterized by twin-cover. The advantage of twin-cover over vertex cover is that it imposes a lesser restriction on the graph structure and attains low values even on dense graphs. Apart from introducing the parameter itself, this article provides a number of new FPT algorithms parameterized by twin-cover with a special emphasis on solving problems which are not in FPT even when parameterized by tree-width. It also shows that MS1 model checking can be done in elementary FPT time parameterized by twin-cover and discusses the field of kernelization.]]> Sun, 13 Sep 2015 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2136 https://doi.org/10.46298/dmtcs.2136 Ganian, Robert Ganian, Robert <![CDATA[Parameterized algorithms are often used to efficiently solve NP-hard problems on graphs. In this context, vertex cover is used as a powerful parameter for dealing with graph problems which are hard to solve even when parameterized by tree-width; however, the drawback of vertex cover is that bounding it severely restricts admissible graph classes. We introduce a generalization of vertex cover called twin-cover and show that FPT algorithms exist for a wide range of difficult problems when parameterized by twin-cover. The advantage of twin-cover over vertex cover is that it imposes a lesser restriction on the graph structure and attains low values even on dense graphs. Apart from introducing the parameter itself, this article provides a number of new FPT algorithms parameterized by twin-cover with a special emphasis on solving problems which are not in FPT even when parameterized by tree-width. It also shows that MS1 model checking can be done in elementary FPT time parameterized by twin-cover and discusses the field of kernelization.]]> 0 <![CDATA[The colouring number col($G$) of a graph $G$ is the smallest integer $k$ for which there is an ordering of the vertices of $G$ such that when removing the vertices of $G$ in the specified order no vertex of degree more than $k-1$ in the remaining graph is removed at any step. An edge $e$ of a graph $G$ is said to be double-col-critical if the colouring number of $G-V(e)$ is at most the colouring number of $G$ minus 2. A connected graph G is said to be double-col-critical if each edge of $G$ is double-col-critical. We characterise the double-col-critical graphs with colouring number at most 5. In addition, we prove that every 4-col-critical non-complete graph has at most half of its edges being double-col-critical, and that the extremal graphs are precisely the odd wheels on at least six vertices. We observe that for any integer $k$ greater than 4 and any positive number $ε$, there is a $k$-col-critical graph with the ratio of double-col-critical edges between $1- ε$ and 1.]]> Tue, 08 Sep 2015 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2129 https://doi.org/10.46298/dmtcs.2129 Kriesell, Matthias Pedersen, Anders Kriesell, Matthias Pedersen, Anders <![CDATA[The colouring number col($G$) of a graph $G$ is the smallest integer $k$ for which there is an ordering of the vertices of $G$ such that when removing the vertices of $G$ in the specified order no vertex of degree more than $k-1$ in the remaining graph is removed at any step. An edge $e$ of a graph $G$ is said to be double-col-critical if the colouring number of $G-V(e)$ is at most the colouring number of $G$ minus 2. A connected graph G is said to be double-col-critical if each edge of $G$ is double-col-critical. We characterise the double-col-critical graphs with colouring number at most 5. In addition, we prove that every 4-col-critical non-complete graph has at most half of its edges being double-col-critical, and that the extremal graphs are precisely the odd wheels on at least six vertices. We observe that for any integer $k$ greater than 4 and any positive number $ε$, there is a $k$-col-critical graph with the ratio of double-col-critical edges between $1- ε$ and 1.]]> 0 <![CDATA[Let G denote a multigraph with edge set E(G), let µ(G) denote the maximum edge multiplicity in G, and let Pk denote the path on k vertices. Heinrich et al.(1999) showed that P4 decomposes a connected 4-regular graph G if and only if |E(G)| is divisible by 3. We show that P4 decomposes a connected 4-regular multigraph G with µ(G) ≤2 if and only if no 3 vertices of G induce more than 4 edges and |E(G)| is divisible by 3. Oksimets (2003) proved that for all integers k ≥3, P4 decomposes a connected 2k-regular graph G if and only if |E(G)| is divisible by 3. We prove that for all integers k ≥2, the problem of determining if P4 decomposes a (2k + 1)-regular graph is NP-Complete. El-Zanati et al.(2014) showed that for all integers k ≥1, every 6k-regular multigraph with µ(G) ≤2k has a P4-decomposition. We show that unless P = NP, this result is best possible with respect to µ(G) by proving that for all integers k ≥3 the problem of determining if P4 decomposes a 2k-regular multigraph with µ(G) ≤⌊2k / 3 ⌋+ 1 is NP-Complete.]]> Tue, 08 Sep 2015 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2128 https://doi.org/10.46298/dmtcs.2128 Diwan, Ajit Dion, Justine Mendell, David Plantholt, Michael Tipnis, Shailesh Diwan, Ajit Dion, Justine Mendell, David Plantholt, Michael Tipnis, Shailesh <![CDATA[Let G denote a multigraph with edge set E(G), let µ(G) denote the maximum edge multiplicity in G, and let Pk denote the path on k vertices. Heinrich et al.(1999) showed that P4 decomposes a connected 4-regular graph G if and only if |E(G)| is divisible by 3. We show that P4 decomposes a connected 4-regular multigraph G with µ(G) ≤2 if and only if no 3 vertices of G induce more than 4 edges and |E(G)| is divisible by 3. Oksimets (2003) proved that for all integers k ≥3, P4 decomposes a connected 2k-regular graph G if and only if |E(G)| is divisible by 3. We prove that for all integers k ≥2, the problem of determining if P4 decomposes a (2k + 1)-regular graph is NP-Complete. El-Zanati et al.(2014) showed that for all integers k ≥1, every 6k-regular multigraph with µ(G) ≤2k has a P4-decomposition. We show that unless P = NP, this result is best possible with respect to µ(G) by proving that for all integers k ≥3 the problem of determining if P4 decomposes a 2k-regular multigraph with µ(G) ≤⌊2k / 3 ⌋+ 1 is NP-Complete.]]> 0 <![CDATA[While the game chromatic number of a forest is known to be at most 4, no simple criteria are known for determining the game chromatic number of a forest. We first state necessary and sufficient conditions for forests with game chromatic number 2 and then investigate the differences between forests with game chromatic number 3 and 4. In doing so, we present a minimal example of a forest with game chromatic number 4, criteria for determining in polynomial time the game chromatic number of a forest without vertices of degree 3, and an example of a forest with maximum degree 3 and game chromatic number 4. This gives partial progress on the open question of the computational complexity of the game chromatic number of a forest.]]> Sun, 16 Aug 2015 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2130 https://doi.org/10.46298/dmtcs.2130 Dunn, Charles Larsen, Victor Lindke, Kira Retter, Troy Toci, Dustin Dunn, Charles Larsen, Victor Lindke, Kira Retter, Troy Toci, Dustin <![CDATA[While the game chromatic number of a forest is known to be at most 4, no simple criteria are known for determining the game chromatic number of a forest. We first state necessary and sufficient conditions for forests with game chromatic number 2 and then investigate the differences between forests with game chromatic number 3 and 4. In doing so, we present a minimal example of a forest with game chromatic number 4, criteria for determining in polynomial time the game chromatic number of a forest without vertices of degree 3, and an example of a forest with maximum degree 3 and game chromatic number 4. This gives partial progress on the open question of the computational complexity of the game chromatic number of a forest.]]> 0 <![CDATA[An $r$-coloring of a knot diagram is an assignment of integers modulo $r$ to the arcs of the diagram such that at each crossing, twice the the number assigned to the over-arc equals the sum of the numbers assigned to the under-arcs, modulo $r$. The number of $r$-colorings is a knot invariant i.e., for each knot, it does not depend on the diagram we are using for counting them. In this article we calculate the number of $r$-colorings for the so-called Turk's Head Knots, for each modulus $r$. Furthermore, it is also known that whenever a knot admits an $r$-coloring using more than one color then all other diagrams of the same knot admit such $r$-colorings (called non-trivial $r$-colorings). This leads to the question of what is the minimum number of colors it takes to assemble such an $r$-coloring for the knot at issue. In this article we also estimate and sometimes calculate exactly what is the minimum numbers of colors for each of the Turk's Head Knots, for each relevant modulus $r$.]]> Mon, 13 Jul 2015 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2139 https://doi.org/10.46298/dmtcs.2139 Lopes, Pedro Matias, João Lopes, Pedro Matias, João <![CDATA[An $r$-coloring of a knot diagram is an assignment of integers modulo $r$ to the arcs of the diagram such that at each crossing, twice the the number assigned to the over-arc equals the sum of the numbers assigned to the under-arcs, modulo $r$. The number of $r$-colorings is a knot invariant i.e., for each knot, it does not depend on the diagram we are using for counting them. In this article we calculate the number of $r$-colorings for the so-called Turk's Head Knots, for each modulus $r$. Furthermore, it is also known that whenever a knot admits an $r$-coloring using more than one color then all other diagrams of the same knot admit such $r$-colorings (called non-trivial $r$-colorings). This leads to the question of what is the minimum number of colors it takes to assemble such an $r$-coloring for the knot at issue. In this article we also estimate and sometimes calculate exactly what is the minimum numbers of colors for each of the Turk's Head Knots, for each relevant modulus $r$.]]> 0 <![CDATA[We introduce and study a combinatorial optimization problem motivated by the question in the title. In the simple case where you use all objects in your room equally often, we investigate asymptotics of the optimal time to clean up in terms of the number of objects in your room. In particular, we prove a logarithmic upper bound, solve an approximate version of this problem, and conjecture a precise logarithmic asymptotic.]]> Sun, 28 Jun 2015 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2109 https://doi.org/10.46298/dmtcs.2109 Martin, Kimball Shankar, Krishnan Martin, Kimball Shankar, Krishnan <![CDATA[We introduce and study a combinatorial optimization problem motivated by the question in the title. In the simple case where you use all objects in your room equally often, we investigate asymptotics of the optimal time to clean up in terms of the number of objects in your room. In particular, we prove a logarithmic upper bound, solve an approximate version of this problem, and conjecture a precise logarithmic asymptotic.]]> 0 <![CDATA[In this paper we study those generic intervals in the Bruhat order of the symmetric group that are isomorphic to the principal order ideal of a permutation w, and consider when the minimum and maximum elements of those intervals are related by a certain property of their reduced words. We show that the property does not hold when w is a decomposable permutation, and that the property always holds when w is the longest permutation.]]> Thu, 11 Jun 2015 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2110 https://doi.org/10.46298/dmtcs.2110 Tenner, Bridget Eileen Tenner, Bridget Eileen <![CDATA[In this paper we study those generic intervals in the Bruhat order of the symmetric group that are isomorphic to the principal order ideal of a permutation w, and consider when the minimum and maximum elements of those intervals are related by a certain property of their reduced words. We show that the property does not hold when w is a decomposable permutation, and that the property always holds when w is the longest permutation.]]> 0 <![CDATA[A method is described for constructing, with computer assistance, planar substitution tilings that have n-fold rotational symmetry. This method uses as prototiles the set of rhombs with angles that are integer multiples of pi/n, and includes various special cases that have already been constructed by hand for low values of n. An example constructed by this method for n = 11 is exhibited; this is the first substitution tiling with elevenfold symmetry appearing in the literature.]]> Thu, 11 Jun 2015 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2108 https://doi.org/10.46298/dmtcs.2108 Maloney, Gregory R. Maloney, Gregory R. <![CDATA[A method is described for constructing, with computer assistance, planar substitution tilings that have n-fold rotational symmetry. This method uses as prototiles the set of rhombs with angles that are integer multiples of pi/n, and includes various special cases that have already been constructed by hand for low values of n. An example constructed by this method for n = 11 is exhibited; this is the first substitution tiling with elevenfold symmetry appearing in the literature.]]> 0 <![CDATA[A k-total-coloring of G is an assignment of k colors to the edges and vertices of G, so that adjacent and incident elements have different colors. The total chromatic number of G, denoted by χT(G), is the least k for which G has a k-total-coloring. It was proved by Rosenfeld that the total chromatic number of a cubic graph is either 4 or 5. Cubic graphs with χT = 4 are said to be Type 1, and cubic graphs with χT = 5 are said to be Type 2. Snarks are cyclically 4-edge-connected cubic graphs that do not allow a 3-edge-coloring. In 2003, Cavicchioli et al. asked for a Type 2 snark with girth at least 5. As neither Type 2 cubic graphs with girth at least 5 nor Type 2 snarks are known, this is taking two steps at once, and the two requirements of being a snark and having girth at least 5 should better be treated independently. In this paper we will show that the property of being a snark can be combined with being Type 2. We will give a construction that gives Type 2 snarks for each even vertex number n≥40. We will also give the result of a computer search showing that among all Type 2 cubic graphs on up to 32 vertices, all but three contain an induced chordless cycle of length 4. These three exceptions contain triangles. The question of the existence of a Type 2 cubic graph with girth at least 5 remains open.]]> Thu, 28 May 2015 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2111 https://doi.org/10.46298/dmtcs.2111 Brinkmann, Gunnar Preissmann, Myriam Sasaki, Diana Brinkmann, Gunnar Preissmann, Myriam Sasaki, Diana <![CDATA[A k-total-coloring of G is an assignment of k colors to the edges and vertices of G, so that adjacent and incident elements have different colors. The total chromatic number of G, denoted by χT(G), is the least k for which G has a k-total-coloring. It was proved by Rosenfeld that the total chromatic number of a cubic graph is either 4 or 5. Cubic graphs with χT = 4 are said to be Type 1, and cubic graphs with χT = 5 are said to be Type 2. Snarks are cyclically 4-edge-connected cubic graphs that do not allow a 3-edge-coloring. In 2003, Cavicchioli et al. asked for a Type 2 snark with girth at least 5. As neither Type 2 cubic graphs with girth at least 5 nor Type 2 snarks are known, this is taking two steps at once, and the two requirements of being a snark and having girth at least 5 should better be treated independently. In this paper we will show that the property of being a snark can be combined with being Type 2. We will give a construction that gives Type 2 snarks for each even vertex number n≥40. We will also give the result of a computer search showing that among all Type 2 cubic graphs on up to 32 vertices, all but three contain an induced chordless cycle of length 4. These three exceptions contain triangles. The question of the existence of a Type 2 cubic graph with girth at least 5 remains open.]]> 0 <![CDATA[This paper deals with the calculation of the Hausdorff measure of regular ω-languages, that is, subsets of the Cantor space definable by finite automata. Using methods for decomposing regular ω-languages into disjoint unions of parts of simple structure we derive two sufficient conditions under which ω-languages with a closure definable by a finite automaton have the same Hausdorff measure as this closure. The first of these condition is related to the homogeneity of the local behaviour of the Hausdorff dimension of the underlying set, and the other with a certain topological density of the set in its closure.]]> Wed, 20 May 2015 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2112 https://doi.org/10.46298/dmtcs.2112 Staiger, Ludwig Staiger, Ludwig <![CDATA[This paper deals with the calculation of the Hausdorff measure of regular ω-languages, that is, subsets of the Cantor space definable by finite automata. Using methods for decomposing regular ω-languages into disjoint unions of parts of simple structure we derive two sufficient conditions under which ω-languages with a closure definable by a finite automaton have the same Hausdorff measure as this closure. The first of these condition is related to the homogeneity of the local behaviour of the Hausdorff dimension of the underlying set, and the other with a certain topological density of the set in its closure.]]> 0 <![CDATA[The prisms over cubic graphs are 4-regular graphs. The prisms over 3-connected cubic graphs are Hamiltonian. In 1986 Brian Alspach and Moshe Rosenfeld conjectured that these prisms are Hamiltonian decomposable. In this paper we present a short survey of the status of this conjecture, various constructions proving that certain families of prisms over 3-connected cubic graphs are Hamiltonian decomposable. Among others, we prove that the prisms over cubic Halin graphs, cubic generalized Halin graphs of order 4k + 2 and other infinite sequences of cubic graphs are Hamiltonian decomposable.]]> Sun, 10 May 2015 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2079 https://doi.org/10.46298/dmtcs.2079 Rosenfeld, Moshe Xiang, Ziqing Rosenfeld, Moshe Xiang, Ziqing <![CDATA[The prisms over cubic graphs are 4-regular graphs. The prisms over 3-connected cubic graphs are Hamiltonian. In 1986 Brian Alspach and Moshe Rosenfeld conjectured that these prisms are Hamiltonian decomposable. In this paper we present a short survey of the status of this conjecture, various constructions proving that certain families of prisms over 3-connected cubic graphs are Hamiltonian decomposable. Among others, we prove that the prisms over cubic Halin graphs, cubic generalized Halin graphs of order 4k + 2 and other infinite sequences of cubic graphs are Hamiltonian decomposable.]]> 0 <![CDATA[Let G=(V,E) be a simple undirected graph. We call any subset C⊆V an identifying code if the sets I(v)={c∈C | {v,c}∈E or v=c } are distinct and non-empty for all vertices v∈V. A graph is called twin-free if there is an identifying code in the graph. The identifying code with minimum size in a twin-free graph G is called the optimal identifying code and the size of such a code is denoted by γ(G). Let GS denote the induced subgraph of G where the vertex set S⊂V is deleted. We provide a tight upper bound for γ(GS)-γ(G) when both graphs are twin-free and |V| is large enough with respect to |S|. Moreover, we prove tight upper bound when G is a bipartite graph and |S|=1.]]> Tue, 05 May 2015 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2107 https://doi.org/10.46298/dmtcs.2107 Pelto, Mikko Pelto, Mikko <![CDATA[Let G=(V,E) be a simple undirected graph. We call any subset C⊆V an identifying code if the sets I(v)={c∈C | {v,c}∈E or v=c } are distinct and non-empty for all vertices v∈V. A graph is called twin-free if there is an identifying code in the graph. The identifying code with minimum size in a twin-free graph G is called the optimal identifying code and the size of such a code is denoted by γ(G). Let GS denote the induced subgraph of G where the vertex set S⊂V is deleted. We provide a tight upper bound for γ(GS)-γ(G) when both graphs are twin-free and |V| is large enough with respect to |S|. Moreover, we prove tight upper bound when G is a bipartite graph and |S|=1.]]> 0 <![CDATA[The state-of-the-art telecommunication technologies have widely been adapted for sensing the traffic related information and collection of it. Vehicular Ad-Hoc Networks (VANETs) have emerged as a novel technology for revolutionizing the driving experiences of human. The most effective and widely recognized way for mutual authentication among entities in VANETs is digital signature scheme. The new and attractive paradigm which eliminates the use of certificates in public key cryptography and solves the key escrow problem in identity based cryptography is certificateless cryptography. A new certificateless aggregate signature scheme is proposed in the paper for VANETs with constant pairing computations. Assuming the hardness of computational Diffie-Hellman Problem, the scheme is proved to be existentially unforgeable in the random oracle model against adaptive chosen-message attacks.]]> Sun, 03 May 2015 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2106 https://doi.org/10.46298/dmtcs.2106 Malhi, Avleen Kaur Batra, Shalini Malhi, Avleen Kaur Batra, Shalini <![CDATA[The state-of-the-art telecommunication technologies have widely been adapted for sensing the traffic related information and collection of it. Vehicular Ad-Hoc Networks (VANETs) have emerged as a novel technology for revolutionizing the driving experiences of human. The most effective and widely recognized way for mutual authentication among entities in VANETs is digital signature scheme. The new and attractive paradigm which eliminates the use of certificates in public key cryptography and solves the key escrow problem in identity based cryptography is certificateless cryptography. A new certificateless aggregate signature scheme is proposed in the paper for VANETs with constant pairing computations. Assuming the hardness of computational Diffie-Hellman Problem, the scheme is proved to be existentially unforgeable in the random oracle model against adaptive chosen-message attacks.]]> 0 <![CDATA[We provide a counterexample to a lemma used in a recent tentative improvement of the Pin-Frankl bound for synchronizing automata. This example naturally leads us to formulate an open question, whose answer could fix the line of the proof, and improve the bound.]]> Sun, 26 Apr 2015 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2101 https://doi.org/10.46298/dmtcs.2101 Gonze, François Jungers, Raphaël M. Trahtman, Avraham N. Gonze, François Jungers, Raphaël M. Trahtman, Avraham N. <![CDATA[We provide a counterexample to a lemma used in a recent tentative improvement of the Pin-Frankl bound for synchronizing automata. This example naturally leads us to formulate an open question, whose answer could fix the line of the proof, and improve the bound.]]> 0 <![CDATA[In this paper we devise some output sensitive algorithms for a problem where a set of points and a positive integer, m, are given and the goal is to cover a maximal number of these points with m disks. We introduce a parameter, ρ, as the maximum number of points that one disk can cover and we analyse the algorithms based on this parameter. At first, we solve the problem for m=1 in O(nρ) time, which improves the previous O(n2) time algorithm for this problem. Then we solve the problem for m=2 in O(nρ + 3 log ρ) time, which improves the previous O(n3 log n) algorithm for this problem. Our algorithms outperform the previous algorithms because ρ is much smaller than n in many cases. Finally, we extend the algorithm for any value of m and solve the problem in O(mnρ + (mρ)2m - 1 log mρ) time. The previous algorithm for this problem runs in O(n2m - 1 log n) time and our algorithm usually runs faster than the previous algorithm because mρ is smaller than n in many cases. We obtain output sensitive algorithms by confining the areas that we should search for the result. The techniques used in this paper may be applicable in other covering problems to obtain faster algorithms.]]> Sun, 26 Apr 2015 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2102 https://doi.org/10.46298/dmtcs.2102 Ghasemalizadeh, Hossein Razzazi, Mohammadreza Ghasemalizadeh, Hossein Razzazi, Mohammadreza <![CDATA[In this paper we devise some output sensitive algorithms for a problem where a set of points and a positive integer, m, are given and the goal is to cover a maximal number of these points with m disks. We introduce a parameter, ρ, as the maximum number of points that one disk can cover and we analyse the algorithms based on this parameter. At first, we solve the problem for m=1 in O(nρ) time, which improves the previous O(n2) time algorithm for this problem. Then we solve the problem for m=2 in O(nρ + 3 log ρ) time, which improves the previous O(n3 log n) algorithm for this problem. Our algorithms outperform the previous algorithms because ρ is much smaller than n in many cases. Finally, we extend the algorithm for any value of m and solve the problem in O(mnρ + (mρ)2m - 1 log mρ) time. The previous algorithm for this problem runs in O(n2m - 1 log n) time and our algorithm usually runs faster than the previous algorithm because mρ is smaller than n in many cases. We obtain output sensitive algorithms by confining the areas that we should search for the result. The techniques used in this paper may be applicable in other covering problems to obtain faster algorithms.]]> 0 <![CDATA[First, we close the multi-parameter analysis of a canonical problem concerning short reset words (SYN) initiated by Fernau et al. (2013). Namely, we prove that the problem, parameterized by the number of states, does not admit a polynomial kernel unless the polynomial hierarchy collapses. Second, we consider a related canonical problem concerning synchronizing road colorings (SRCP). Here we give a similar complete multi-parameter analysis. Namely, we show that the problem, parameterized by the number of states, admits a polynomial kernel and we close the previous research of restrictions to particular values of both the alphabet size and the maximum length of a reset word.]]> Tue, 21 Apr 2015 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2103 https://doi.org/10.46298/dmtcs.2103 Vorel, Vojtěch Roman, Adam Vorel, Vojtěch Roman, Adam <![CDATA[First, we close the multi-parameter analysis of a canonical problem concerning short reset words (SYN) initiated by Fernau et al. (2013). Namely, we prove that the problem, parameterized by the number of states, does not admit a polynomial kernel unless the polynomial hierarchy collapses. Second, we consider a related canonical problem concerning synchronizing road colorings (SRCP). Here we give a similar complete multi-parameter analysis. Namely, we show that the problem, parameterized by the number of states, admits a polynomial kernel and we close the previous research of restrictions to particular values of both the alphabet size and the maximum length of a reset word.]]> 0 <![CDATA[Let G be a graph with no isolated vertex. In this paper, we study a parameter that is a relaxation of arguably the most important domination parameter, namely the total domination number, γt(G). A set S of vertices in G is a disjunctive total dominating set of G if every vertex is adjacent to a vertex of S or has at least two vertices in S at distance 2 from it. The disjunctive total domination number, γdt(G), is the minimum cardinality of such a set. We observe that γdt(G) ≤γt(G). Let G be a connected graph on n vertices with minimum degree δ. It is known [J. Graph Theory 35 (2000), 21 13;45] that if δ≥2 and n ≥11, then γt(G) ≤4n/7. Further [J. Graph Theory 46 (2004), 207 13;210] if δ≥3, then γt(G) ≤n/2. We prove that if δ≥2 and n ≥8, then γdt(G) ≤n/2 and we characterize the extremal graphs.]]> Tue, 21 Apr 2015 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2104 https://doi.org/10.46298/dmtcs.2104 Henning, Michael A. Naicker, Viroshan Henning, Michael A. Naicker, Viroshan <![CDATA[Let G be a graph with no isolated vertex. In this paper, we study a parameter that is a relaxation of arguably the most important domination parameter, namely the total domination number, γt(G). A set S of vertices in G is a disjunctive total dominating set of G if every vertex is adjacent to a vertex of S or has at least two vertices in S at distance 2 from it. The disjunctive total domination number, γdt(G), is the minimum cardinality of such a set. We observe that γdt(G) ≤γt(G). Let G be a connected graph on n vertices with minimum degree δ. It is known [J. Graph Theory 35 (2000), 21 13;45] that if δ≥2 and n ≥11, then γt(G) ≤4n/7. Further [J. Graph Theory 46 (2004), 207 13;210] if δ≥3, then γt(G) ≤n/2. We prove that if δ≥2 and n ≥8, then γdt(G) ≤n/2 and we characterize the extremal graphs.]]> 0 <![CDATA[In this paper we discuss how to assess the performance of algorithms for optimisation problems in a way that balances solution quality and time. We propose measures of cost-effectiveness for such algorithms. These measures give the gain in solution quality per time unit over a sequence of inputs, and give a basis for deciding which algorithm to use when aiming for best accumulated solution quality for a given time investment over such an input sequence. Cost-effectiveness measures can be defined for both average-case and worst-case performance. We apply these ideas to three problems: maximum matching, graph colouring and Kolmogorov complexity. For the latter, we propose a cost-effectiveness measure for the time-bounded complexity Kτ(x), and argue that it can be used to measure the cost-effectiveness both of finding a short program to output x and of generating x from such a program. Under mild assumptions, we show that (roughly speaking) if the time-bounded complexity Kτ(x) is to be a cost-effective approximation to K(x) then τ(n)=O(n2).]]> Fri, 27 Mar 2015 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2114 https://doi.org/10.46298/dmtcs.2114 Farr, Graham Farr, Graham <![CDATA[In this paper we discuss how to assess the performance of algorithms for optimisation problems in a way that balances solution quality and time. We propose measures of cost-effectiveness for such algorithms. These measures give the gain in solution quality per time unit over a sequence of inputs, and give a basis for deciding which algorithm to use when aiming for best accumulated solution quality for a given time investment over such an input sequence. Cost-effectiveness measures can be defined for both average-case and worst-case performance. We apply these ideas to three problems: maximum matching, graph colouring and Kolmogorov complexity. For the latter, we propose a cost-effectiveness measure for the time-bounded complexity Kτ(x), and argue that it can be used to measure the cost-effectiveness both of finding a short program to output x and of generating x from such a program. Under mild assumptions, we show that (roughly speaking) if the time-bounded complexity Kτ(x) is to be a cost-effective approximation to K(x) then τ(n)=O(n2).]]> 0 <![CDATA[We study the enumeration of Hamiltonian cycles on the thin grid cylinder graph $C_m \times P_{n+1}$. We distinguish two types of Hamiltonian cycles, and denote their numbers $h_m^A(n)$ and $h_m^B(n)$. For fixed $m$, both of them satisfy linear homogeneous recurrence relations with constant coefficients, and we derive their generating functions and other related results for $m\leq10$. The computational data we gathered suggests that $h^A_m(n)\sim h^B_m(n)$ when $m$ is even.]]> Fri, 27 Mar 2015 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2113 https://doi.org/10.46298/dmtcs.2113 Bodroža-Pantić, Olga Kwong, Harris Pantić, Milan Bodroža-Pantić, Olga Kwong, Harris Pantić, Milan <![CDATA[We study the enumeration of Hamiltonian cycles on the thin grid cylinder graph $C_m \times P_{n+1}$. We distinguish two types of Hamiltonian cycles, and denote their numbers $h_m^A(n)$ and $h_m^B(n)$. For fixed $m$, both of them satisfy linear homogeneous recurrence relations with constant coefficients, and we derive their generating functions and other related results for $m\leq10$. The computational data we gathered suggests that $h^A_m(n)\sim h^B_m(n)$ when $m$ is even.]]> 0 <![CDATA[Ruskey and Savage conjectured that in the d-dimensional hypercube, every matching M can be extended to a Hamiltonian cycle. Fink verified this for every perfect matching M, remarkably even if M contains external edges. We prove that this property also holds for sparse spanning regular subgraphs of the cubes: for every d ≥7 and every k, where 7 ≤k ≤d, the d-dimensional hypercube contains a k-regular spanning subgraph such that every perfect matching (possibly with external edges) can be extended to a Hamiltonian cycle. We do not know if this result can be extended to k=4,5,6. It cannot be extended to k=3. Indeed, there are only three 3-regular graphs such that every perfect matching (possibly with external edges) can be extended to a Hamiltonian cycle, namely the complete graph on 4 vertices, the complete bipartite 3-regular graph on 6 vertices and the 3-cube on 8 vertices. Also, we do not know if there are graphs of girth at least 5 with this matching-extendability property.]]> Fri, 27 Mar 2015 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2105 https://doi.org/10.46298/dmtcs.2105 Alahmadi, Adel Aldred, Robert E. L. Alkenani, Ahmad Hijazi, Rola Solé, P. Thomassen, Carsten Alahmadi, Adel Aldred, Robert E. L. Alkenani, Ahmad Hijazi, Rola Solé, P. Thomassen, Carsten <![CDATA[Ruskey and Savage conjectured that in the d-dimensional hypercube, every matching M can be extended to a Hamiltonian cycle. Fink verified this for every perfect matching M, remarkably even if M contains external edges. We prove that this property also holds for sparse spanning regular subgraphs of the cubes: for every d ≥7 and every k, where 7 ≤k ≤d, the d-dimensional hypercube contains a k-regular spanning subgraph such that every perfect matching (possibly with external edges) can be extended to a Hamiltonian cycle. We do not know if this result can be extended to k=4,5,6. It cannot be extended to k=3. Indeed, there are only three 3-regular graphs such that every perfect matching (possibly with external edges) can be extended to a Hamiltonian cycle, namely the complete graph on 4 vertices, the complete bipartite 3-regular graph on 6 vertices and the 3-cube on 8 vertices. Also, we do not know if there are graphs of girth at least 5 with this matching-extendability property.]]> 0 <![CDATA[Given a class G of graphs, probe G graphs are defined as follows. A graph G is probe G if there exists a partition of its vertices into a set of probe vertices and a stable set of nonprobe vertices in such a way that non-edges of G, whose endpoints are nonprobe vertices, can be added so that the resulting graph belongs to G. We investigate probe 2-clique graphs and probe diamond-free graphs. For probe 2-clique graphs, we present a polynomial-time recognition algorithm. Probe diamond-free graphs are characterized by minimal forbidden induced subgraphs. As a by-product, it is proved that the class of probe block graphs is the intersection between the classes of chordal graphs and probe diamond-free graphs.]]> Fri, 27 Mar 2015 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2122 https://doi.org/10.46298/dmtcs.2122 Bonomo, Flavia Figueiredo, Celina M. H., Duran, Guillermo Grippo, Luciano N. Safe, Martín D. Szwarcfiter, Jayme L. Bonomo, Flavia Figueiredo, Celina M. H., Duran, Guillermo Grippo, Luciano N. Safe, Martín D. Szwarcfiter, Jayme L. <![CDATA[Given a class G of graphs, probe G graphs are defined as follows. A graph G is probe G if there exists a partition of its vertices into a set of probe vertices and a stable set of nonprobe vertices in such a way that non-edges of G, whose endpoints are nonprobe vertices, can be added so that the resulting graph belongs to G. We investigate probe 2-clique graphs and probe diamond-free graphs. For probe 2-clique graphs, we present a polynomial-time recognition algorithm. Probe diamond-free graphs are characterized by minimal forbidden induced subgraphs. As a by-product, it is proved that the class of probe block graphs is the intersection between the classes of chordal graphs and probe diamond-free graphs.]]> 0 <![CDATA[We introduce the concept of guarded subgraph of a graph, which as a condition lies between convex and 2-isometric subgraphs and is not comparable to isometric subgraphs. Some basic metric properties of guarded subgraphs are obtained, and then this concept is applied to the domination game. In this game two players, Dominator and Staller, alternate choosing vertices of a graph, one at a time, such that each chosen vertex enlarges the set of vertices dominated so far. The aim of Dominator is that the graph is dominated in as few steps as possible, while the aim of Staller is just the opposite. The game domination number is the number of vertices chosen when Dominator starts the game and both players play optimally. The main result of this paper is that the game domination number of a graph is not smaller than the game domination number of any guarded subgraph. Several applications of this result are presented.]]> Thu, 26 Mar 2015 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2123 https://doi.org/10.46298/dmtcs.2123 Brešar, Boštjan Klavžar, Sandi Košmrlj, Gasper Rall, Doug F. Brešar, Boštjan Klavžar, Sandi Košmrlj, Gasper Rall, Doug F. <![CDATA[We introduce the concept of guarded subgraph of a graph, which as a condition lies between convex and 2-isometric subgraphs and is not comparable to isometric subgraphs. Some basic metric properties of guarded subgraphs are obtained, and then this concept is applied to the domination game. In this game two players, Dominator and Staller, alternate choosing vertices of a graph, one at a time, such that each chosen vertex enlarges the set of vertices dominated so far. The aim of Dominator is that the graph is dominated in as few steps as possible, while the aim of Staller is just the opposite. The game domination number is the number of vertices chosen when Dominator starts the game and both players play optimally. The main result of this paper is that the game domination number of a graph is not smaller than the game domination number of any guarded subgraph. Several applications of this result are presented.]]> 0 <![CDATA[In this document, we study the scope of the following graph model: each vertex is assigned to a box in ℝd and to a representative element that belongs to that box. Two vertices are connected by an edge if and only if its respective boxes contain the opposite representative element. We focus our study on the case where boxes (and therefore representative elements) associated to vertices are spread in ℝ. We give both, a combinatorial and an intersection characterization of the model. Based on these characterizations, we determine graph families that contain the model (e. g., boxicity 2 graphs) and others that the new model contains (e. g., rooted directed path). We also study the particular case where each representative element is the center of its respective box. In this particular case, we provide constructive representations for interval, block and outerplanar graphs. Finally, we show that the general and the particular model are not equivalent by constructing a graph family that separates the two cases.]]> Thu, 26 Mar 2015 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2121 https://doi.org/10.46298/dmtcs.2121 Soto, Mauricio Thraves-Caro, Christopher Soto, Mauricio Thraves-Caro, Christopher <![CDATA[In this document, we study the scope of the following graph model: each vertex is assigned to a box in ℝd and to a representative element that belongs to that box. Two vertices are connected by an edge if and only if its respective boxes contain the opposite representative element. We focus our study on the case where boxes (and therefore representative elements) associated to vertices are spread in ℝ. We give both, a combinatorial and an intersection characterization of the model. Based on these characterizations, we determine graph families that contain the model (e. g., boxicity 2 graphs) and others that the new model contains (e. g., rooted directed path). We also study the particular case where each representative element is the center of its respective box. In this particular case, we provide constructive representations for interval, block and outerplanar graphs. Finally, we show that the general and the particular model are not equivalent by constructing a graph family that separates the two cases.]]> 0 <![CDATA[In this paper, we study (1 : b) Avoider-Enforcer games played on the edge set of the complete graph on n vertices. For every constant k≥3 we analyse the k-star game, where Avoider tries to avoid claiming k edges incident to the same vertex. We consider both versions of Avoider-Enforcer games — the strict and the monotone — and for each provide explicit winning strategies for both players. We determine the order of magnitude of the threshold biases fmonF, f-F and f+F, where F is the hypergraph of the game.]]> Wed, 25 Mar 2015 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2124 https://doi.org/10.46298/dmtcs.2124 Grzesik, Andrzej Mikalački, Mirjana Nagy, Zoltán Lóránt Naor, Alon Patkós, Balázs Skerman, Fiona Grzesik, Andrzej Mikalački, Mirjana Nagy, Zoltán Lóránt Naor, Alon Patkós, Balázs Skerman, Fiona <![CDATA[In this paper, we study (1 : b) Avoider-Enforcer games played on the edge set of the complete graph on n vertices. For every constant k≥3 we analyse the k-star game, where Avoider tries to avoid claiming k edges incident to the same vertex. We consider both versions of Avoider-Enforcer games — the strict and the monotone — and for each provide explicit winning strategies for both players. We determine the order of magnitude of the threshold biases fmonF, f-F and f+F, where F is the hypergraph of the game.]]> 0 <![CDATA[We provide a rather general asymptotic scheme for combinatorial parameters that asymptotically follow a discrete double-exponential distribution. It is based on analysing generating functions Gh(z) whose dominant singularities converge to a certain value at an exponential rate. This behaviour is typically found by means of a bootstrapping approach. Our scheme is illustrated by a number of classical and new examples, such as the longest run in words or compositions, patterns in Dyck and Motzkin paths, or the maximum degree in planted plane trees.]]> Tue, 17 Mar 2015 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2125 https://doi.org/10.46298/dmtcs.2125 Prodinger, Helmut Wagner, Stephan Prodinger, Helmut Wagner, Stephan <![CDATA[We provide a rather general asymptotic scheme for combinatorial parameters that asymptotically follow a discrete double-exponential distribution. It is based on analysing generating functions Gh(z) whose dominant singularities converge to a certain value at an exponential rate. This behaviour is typically found by means of a bootstrapping approach. Our scheme is illustrated by a number of classical and new examples, such as the longest run in words or compositions, patterns in Dyck and Motzkin paths, or the maximum degree in planted plane trees.]]> 0 <![CDATA[A subset X of the vertex set of a graph G is a secure dominating set of G if X is a dominating set of G and if, for each vertex u not in X, there is a neighbouring vertex v of u in X such that the swap set (X-v)∪u is again a dominating set of G. The secure domination number of G is the cardinality of a smallest secure dominating set of G. A graph G is p-stable if the largest arbitrary subset of edges whose removal from G does not increase the secure domination number of the resulting graph, has cardinality p. In this paper we study the problem of computing p-stable graphs for all admissible values of p and determine the exact values of p for which members of various infinite classes of graphs are p-stable. We also consider the problem of determining analytically the largest value ωn of p for which a graph of order n can be p-stable. We conjecture that ωn=n-2 and motivate this conjecture.]]> Sun, 15 Mar 2015 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2120 https://doi.org/10.46298/dmtcs.2120 Burger, Anton Pierre Villiers, Alewyn Petrus, Vuuren, Jan Harm, Burger, Anton Pierre Villiers, Alewyn Petrus, Vuuren, Jan Harm, <![CDATA[A subset X of the vertex set of a graph G is a secure dominating set of G if X is a dominating set of G and if, for each vertex u not in X, there is a neighbouring vertex v of u in X such that the swap set (X-v)∪u is again a dominating set of G. The secure domination number of G is the cardinality of a smallest secure dominating set of G. A graph G is p-stable if the largest arbitrary subset of edges whose removal from G does not increase the secure domination number of the resulting graph, has cardinality p. In this paper we study the problem of computing p-stable graphs for all admissible values of p and determine the exact values of p for which members of various infinite classes of graphs are p-stable. We also consider the problem of determining analytically the largest value ωn of p for which a graph of order n can be p-stable. We conjecture that ωn=n-2 and motivate this conjecture.]]> 0 <![CDATA[We show that if the two parts of a finite bipartite graph have the same degree sequence, then there is a bipartite graph, with the same degree sequences, which is symmetric, in that it has an involutive graph automorphism that interchanges its two parts. To prove this, we study the relationship between symmetric bipartite graphs and graphs with loops.]]> Wed, 11 Feb 2015 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2119 https://doi.org/10.46298/dmtcs.2119 Cairns, Grant Mendan, Stacey Cairns, Grant Mendan, Stacey <![CDATA[We show that if the two parts of a finite bipartite graph have the same degree sequence, then there is a bipartite graph, with the same degree sequences, which is symmetric, in that it has an involutive graph automorphism that interchanges its two parts. To prove this, we study the relationship between symmetric bipartite graphs and graphs with loops.]]> 0 <![CDATA[We describe new classification results in the theory of generalized quadrangles (= Tits-buildings of rank 2 and type B2), more precisely in the (large) subtheory of skew translation generalized quadrangles (``STGQs''). Some of these involve, and solve, long-standing open problems.]]> Wed, 11 Feb 2015 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2116 https://doi.org/10.46298/dmtcs.2116 Thas, Koen Thas, Koen <![CDATA[We describe new classification results in the theory of generalized quadrangles (= Tits-buildings of rank 2 and type B2), more precisely in the (large) subtheory of skew translation generalized quadrangles (``STGQs''). Some of these involve, and solve, long-standing open problems.]]> 0 <![CDATA[If f is a binary word and d a positive integer, then the generalized Fibonacci cube Qd(f) is the graph obtained from the d-cube Qd by removing all the vertices that contain f as a factor, while the generalized Lucas cube Qd(lucas(f)) is the graph obtained from Qd by removing all the vertices that have a circulation containing f as a factor. The Fibonacci cube Γd and the Lucas cube Λd are the graphs Qd(11) and Qd(lucas(11)), respectively. It is proved that the connectivity and the edge-connectivity of Γd as well as of Λd are equal to ⌊ d+2 / 3⌋. Connected generalized Lucas cubes are characterized and generalized Fibonacci cubes are proved to be 2-connected. It is asked whether the connectivity equals minimum degree also for all generalized Fibonacci/Lucas cubes. It was checked by computer that the answer is positive for all f and all d≤9.]]> Wed, 11 Feb 2015 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2115 https://doi.org/10.46298/dmtcs.2115 Azarija, Jernej Klavžar, Sandi Lee, Jaehun Rho, Yoomi Azarija, Jernej Klavžar, Sandi Lee, Jaehun Rho, Yoomi <![CDATA[If f is a binary word and d a positive integer, then the generalized Fibonacci cube Qd(f) is the graph obtained from the d-cube Qd by removing all the vertices that contain f as a factor, while the generalized Lucas cube Qd(lucas(f)) is the graph obtained from Qd by removing all the vertices that have a circulation containing f as a factor. The Fibonacci cube Γd and the Lucas cube Λd are the graphs Qd(11) and Qd(lucas(11)), respectively. It is proved that the connectivity and the edge-connectivity of Γd as well as of Λd are equal to ⌊ d+2 / 3⌋. Connected generalized Lucas cubes are characterized and generalized Fibonacci cubes are proved to be 2-connected. It is asked whether the connectivity equals minimum degree also for all generalized Fibonacci/Lucas cubes. It was checked by computer that the answer is positive for all f and all d≤9.]]> 0 <![CDATA[A k-edge-weighting of a graph G is a function w:E(G)→{1,…,k}. An edge-weighting naturally induces a vertex coloring c, where for every vertex v∈V(G), c(v)=∑e∼vw(e). If the induced coloring c is a proper vertex coloring, then w is called a vertex-coloring k-edge-weighting (VC k-EW). Karoński et al. (J. Combin. Theory Ser. B, 91 (2004) 151 13;157) conjectured that every graph admits a VC 3-EW. This conjecture is known as the 1-2-3-conjecture. In this paper, first, we study the vertex-coloring edge-weighting of the Cartesian product of graphs. We prove that if the 1-2-3-conjecture holds for two graphs G and H, then it also holds for G□H. Also we prove that the Cartesian product of connected bipartite graphs admits a VC 2-EW. Moreover, we present several sufficient conditions for a graph to admit a VC 2-EW. Finally, we explore some bipartite graphs which do not admit a VC 2-EW.]]> Mon, 09 Feb 2015 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2117 https://doi.org/10.46298/dmtcs.2117 Davoodi, Akbar Omoomi, Behnaz Davoodi, Akbar Omoomi, Behnaz <![CDATA[A k-edge-weighting of a graph G is a function w:E(G)→{1,…,k}. An edge-weighting naturally induces a vertex coloring c, where for every vertex v∈V(G), c(v)=∑e∼vw(e). If the induced coloring c is a proper vertex coloring, then w is called a vertex-coloring k-edge-weighting (VC k-EW). Karoński et al. (J. Combin. Theory Ser. B, 91 (2004) 151 13;157) conjectured that every graph admits a VC 3-EW. This conjecture is known as the 1-2-3-conjecture. In this paper, first, we study the vertex-coloring edge-weighting of the Cartesian product of graphs. We prove that if the 1-2-3-conjecture holds for two graphs G and H, then it also holds for G□H. Also we prove that the Cartesian product of connected bipartite graphs admits a VC 2-EW. Moreover, we present several sufficient conditions for a graph to admit a VC 2-EW. Finally, we explore some bipartite graphs which do not admit a VC 2-EW.]]> 0 <![CDATA[We introduce a binary parameter on optimisation problems called separation. The parameter is used to relate the approximation ratios of different optimisation problems; in other words, we can convert approximability (and non-approximability) result for one problem into (non)-approximability results for other problems. Our main application is the problem (weighted) maximum H-colourable subgraph (Max H-Col), which is a restriction of the general maximum constraint satisfaction problem (Max CSP) to a single, binary, and symmetric relation. Using known approximation ratios for Max k-cut, we obtain general asymptotic approximability results for Max H-Col for an arbitrary graph H. For several classes of graphs, we provide near-optimal results under the unique games conjecture. We also investigate separation as a graph parameter. In this vein, we study its properties on circular complete graphs. Furthermore, we establish a close connection to work by Šámal on cubical colourings of graphs. This connection shows that our parameter is closely related to a special type of chromatic number. We believe that this insight may turn out to be crucial for understanding the behaviour of the parameter, and in the longer term, for understanding the approximability of optimisation problems such as Max H-Col.]]> Wed, 04 Feb 2015 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2118 https://doi.org/10.46298/dmtcs.2118 Engström, Robert Färnqvist, Tommy Jonsson, Peter Thapper, Johan Engström, Robert Färnqvist, Tommy Jonsson, Peter Thapper, Johan <![CDATA[We introduce a binary parameter on optimisation problems called separation. The parameter is used to relate the approximation ratios of different optimisation problems; in other words, we can convert approximability (and non-approximability) result for one problem into (non)-approximability results for other problems. Our main application is the problem (weighted) maximum H-colourable subgraph (Max H-Col), which is a restriction of the general maximum constraint satisfaction problem (Max CSP) to a single, binary, and symmetric relation. Using known approximation ratios for Max k-cut, we obtain general asymptotic approximability results for Max H-Col for an arbitrary graph H. For several classes of graphs, we provide near-optimal results under the unique games conjecture. We also investigate separation as a graph parameter. In this vein, we study its properties on circular complete graphs. Furthermore, we establish a close connection to work by Šámal on cubical colourings of graphs. This connection shows that our parameter is closely related to a special type of chromatic number. We believe that this insight may turn out to be crucial for understanding the behaviour of the parameter, and in the longer term, for understanding the approximability of optimisation problems such as Max H-Col.]]> 0 <![CDATA[A classic theorem of Dirac from 1952 states that every graph with minimum degree at least n=2 contains a Hamiltonian cycle. In 1963, P´osa conjectured that every graph with minimum degree at least 2n=3 contains the square of a Hamiltonian cycle. In 1960, Ore relaxed the degree condition in the Dirac’s theorem by proving that every graph with deg(u) + deg(v) ≥ n for every uv =2 E(G) contains a Hamiltonian cycle. Recently, Chˆau proved an Ore-type version of P´osa’s conjecture for graphs on n ≥ n0 vertices using the regularity–blow-up method; consequently the n0 is very large (involving a tower function). Here we present another proof that avoids the use of the regularity lemma. Aside from the fact that our proof holds for much smaller n0, we believe that our method of proof will be of independent interest.]]> Sun, 01 Feb 2015 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2127 https://doi.org/10.46298/dmtcs.2127 DeBiasio, Louis Faizullah, Safi Khan, Imdadullah DeBiasio, Louis Faizullah, Safi Khan, Imdadullah <![CDATA[A classic theorem of Dirac from 1952 states that every graph with minimum degree at least n=2 contains a Hamiltonian cycle. In 1963, P´osa conjectured that every graph with minimum degree at least 2n=3 contains the square of a Hamiltonian cycle. In 1960, Ore relaxed the degree condition in the Dirac’s theorem by proving that every graph with deg(u) + deg(v) ≥ n for every uv =2 E(G) contains a Hamiltonian cycle. Recently, Chˆau proved an Ore-type version of P´osa’s conjecture for graphs on n ≥ n0 vertices using the regularity–blow-up method; consequently the n0 is very large (involving a tower function). Here we present another proof that avoids the use of the regularity lemma. Aside from the fact that our proof holds for much smaller n0, we believe that our method of proof will be of independent interest.]]> 0 <![CDATA[We present a randomized algorithm to compute a clique of maximum size in the visibility graph G of the vertices of a simple polygon P. The input of the problem consists of the visibility graph G, a Hamiltonian cycle describing the boundary of P, and a parameter δ∈(0,1) controlling the probability of error of the algorithm. The algorithm does not require the coordinates of the vertices of P. With probability at least 1-δ the algorithm runs in O( |E(G)|2 / ω(G) log(1/δ)) time and returns a maximum clique, where ω(G) is the number of vertices in a maximum clique in G. A deterministic variant of the algorithm takes O(|E(G)|2) time and always outputs a maximum size clique. This compares well to the best previous algorithm by Ghosh et al. (2007) for the problem, which is deterministic and runs in O(|V(G)|2 |E(G)|) time.]]> Tue, 20 Jan 2015 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2126 https://doi.org/10.46298/dmtcs.2126 Cabello, Sergio Saumell, Maria Cabello, Sergio Saumell, Maria <![CDATA[We present a randomized algorithm to compute a clique of maximum size in the visibility graph G of the vertices of a simple polygon P. The input of the problem consists of the visibility graph G, a Hamiltonian cycle describing the boundary of P, and a parameter δ∈(0,1) controlling the probability of error of the algorithm. The algorithm does not require the coordinates of the vertices of P. With probability at least 1-δ the algorithm runs in O( |E(G)|2 / ω(G) log(1/δ)) time and returns a maximum clique, where ω(G) is the number of vertices in a maximum clique in G. A deterministic variant of the algorithm takes O(|E(G)|2) time and always outputs a maximum size clique. This compares well to the best previous algorithm by Ghosh et al. (2007) for the problem, which is deterministic and runs in O(|V(G)|2 |E(G)|) time.]]> 0 <![CDATA[This paper contains two results. First, I propose a $q$-generalization of a certain sequence of positive integers, related to Catalan numbers, introduced by Zeilberger, see Lassalle (2010). These $q$-integers are palindromic polynomials in $q$ with positive integer coefficients. The positivity depends on the positivity of a certain difference of products of $q$-binomial coefficients.To this end, I introduce a new inversion/major statistics on lattice walks. The difference in $q$-binomial coefficients is then seen as a generating function of weighted walks that remain in the upper half-plan.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2528 https://doi.org/10.46298/dmtcs.2528 Tevlin, Lenny Tevlin, Lenny <![CDATA[This paper contains two results. First, I propose a $q$-generalization of a certain sequence of positive integers, related to Catalan numbers, introduced by Zeilberger, see Lassalle (2010). These $q$-integers are palindromic polynomials in $q$ with positive integer coefficients. The positivity depends on the positivity of a certain difference of products of $q$-binomial coefficients.To this end, I introduce a new inversion/major statistics on lattice walks. The difference in $q$-binomial coefficients is then seen as a generating function of weighted walks that remain in the upper half-plan.]]> 0 <![CDATA[The $n$-dimensional associahedron is a polytope whose vertices correspond to triangulations of a convex $(n + 3)$-gon and whose edges are flips between them. It was recently shown that the diameter of this polytope is $2n - 4$ as soon as $n > 9$. We study the diameters of the graphs of relevant generalizations of the associahedron: on the one hand the generalized associahedra arising from cluster algebras, and on the other hand the graph associahedra and nestohedra. Related to the diameter, we investigate the non-leaving-face property for these polytopes, which asserts that every geodesic connecting two vertices in the graph of the polytope stays in the minimal face containing both.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2540 https://doi.org/10.46298/dmtcs.2540 Ceballos, C. Manneville, T. Pilaud, V. Pournin, L. Ceballos, C. Manneville, T. Pilaud, V. Pournin, L. <![CDATA[The $n$-dimensional associahedron is a polytope whose vertices correspond to triangulations of a convex $(n + 3)$-gon and whose edges are flips between them. It was recently shown that the diameter of this polytope is $2n - 4$ as soon as $n > 9$. We study the diameters of the graphs of relevant generalizations of the associahedron: on the one hand the generalized associahedra arising from cluster algebras, and on the other hand the graph associahedra and nestohedra. Related to the diameter, we investigate the non-leaving-face property for these polytopes, which asserts that every geodesic connecting two vertices in the graph of the polytope stays in the minimal face containing both.]]> 0 <![CDATA[We introduce an axiom system for a collection of matchings that describes the triangulation of product of simplices.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2539 https://doi.org/10.46298/dmtcs.2539 Oh, Suho Yoo, Hwanchul Oh, Suho Yoo, Hwanchul <![CDATA[We introduce an axiom system for a collection of matchings that describes the triangulation of product of simplices.]]> 0 <![CDATA[We construct a poset from a simple acyclic digraph together with a valuation on its vertices, and we compute the values of its Möbius function. We show that the weak order on Coxeter groups $A$_$n-1$, $B$_$n$, $Ã$_$n$, and the flag weak order on the wreath product ℤ_$r$ ≀ $S$_$n$ introduced by Adin, Brenti and Roichman (2012), are special instances of our construction. We conclude by briefly explaining how to use our work to define quasi-symmetric functions, with a special emphasis on the $A$_$n-1$ case, in which case we obtain the classical Stanley symmetric function.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2538 https://doi.org/10.46298/dmtcs.2538 Viard, Francois Viard, Francois <![CDATA[We construct a poset from a simple acyclic digraph together with a valuation on its vertices, and we compute the values of its Möbius function. We show that the weak order on Coxeter groups $A$_$n-1$, $B$_$n$, $Ã$_$n$, and the flag weak order on the wreath product ℤ_$r$ ≀ $S$_$n$ introduced by Adin, Brenti and Roichman (2012), are special instances of our construction. We conclude by briefly explaining how to use our work to define quasi-symmetric functions, with a special emphasis on the $A$_$n-1$ case, in which case we obtain the classical Stanley symmetric function.]]> 0 <![CDATA[We consider generalizations of juggling Markov chains introduced by Ayyer, Bouttier, Corteel and Nunzi. We first study multispecies generalizations of all the finite models therein, namely the MJMC, the add-drop and the annihilation models. We then consider the case of several jugglers exchanging balls. In all cases, we give explicit product formulas for the stationary probability and closed-form expressions for the normalization factor if known.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2537 https://doi.org/10.46298/dmtcs.2537 Ayyer, Arvind Bouttier, Jérémie Linusson, Svante Nunzi, François Ayyer, Arvind Bouttier, Jérémie Linusson, Svante Nunzi, François <![CDATA[We consider generalizations of juggling Markov chains introduced by Ayyer, Bouttier, Corteel and Nunzi. We first study multispecies generalizations of all the finite models therein, namely the MJMC, the add-drop and the annihilation models. We then consider the case of several jugglers exchanging balls. In all cases, we give explicit product formulas for the stationary probability and closed-form expressions for the normalization factor if known.]]> 0 <![CDATA[We extend the Marcus-Schaeffer bijection between orientable rooted bipartite quadrangulations (equivalently: rooted maps) and orientable labeled one-face maps to the case of all surfaces, orientable or non-orientable. This general construction requires new ideas and is more delicate than the special orientable case, but carries the same information. It thus gives a uniform combinatorial interpretation of the counting exponent $\frac{5(h-1)}{2}$ for both orientable and non-orientable maps of Euler characteristic $2-2h$ and of the algebraicity of their generating functions. It also shows the universality of the renormalization factor $n$^¼ for the metric of maps, on all surfaces: the renormalized profile and radius in a uniform random pointed bipartite quadrangulation of size $n$ on any fixed surface converge in distribution. Finally, it also opens the way to the study of Brownian surfaces for any compact 2-dimensional manifold.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2536 https://doi.org/10.46298/dmtcs.2536 Chapuy, Guillaume Dołęga, Maciej Chapuy, Guillaume Dołęga, Maciej <![CDATA[We extend the Marcus-Schaeffer bijection between orientable rooted bipartite quadrangulations (equivalently: rooted maps) and orientable labeled one-face maps to the case of all surfaces, orientable or non-orientable. This general construction requires new ideas and is more delicate than the special orientable case, but carries the same information. It thus gives a uniform combinatorial interpretation of the counting exponent $\frac{5(h-1)}{2}$ for both orientable and non-orientable maps of Euler characteristic $2-2h$ and of the algebraicity of their generating functions. It also shows the universality of the renormalization factor $n$^¼ for the metric of maps, on all surfaces: the renormalized profile and radius in a uniform random pointed bipartite quadrangulation of size $n$ on any fixed surface converge in distribution. Finally, it also opens the way to the study of Brownian surfaces for any compact 2-dimensional manifold.]]> 0 <![CDATA[For any finite path $v$ on the square lattice consisting of north and east unit steps, we construct a poset Tam$(v)$ that consists of all the paths lying weakly above $v$ with the same endpoints as $v$. For particular choices of $v$, we recover the traditional Tamari lattice and the $m$-Tamari lattice. In particular this solves the problem of extending the $m$-Tamari lattice to any pair $(a; b)$ of relatively prime numbers in the context of the so-called rational Catalan combinatorics.For that purpose we introduce the notion of canopy of a binary tree and explicit a bijection between pairs $(u; v)$ of paths in Tam$(v)$ and binary trees with canopy $v$. Let $(\overleftarrow{v})$ be the path obtained from $v$ by reading the unit steps of $v$ in reverse order and exchanging east and north steps. We show that the poset Tam$(v)$ is isomorphic to the dual of the poset Tam$(\overleftarrow{v})$ and that Tam$(v)$ is isomorphic to the set of binary trees having the canopy $v$, which is an interval of the ordinary Tamari lattice. Thus the usual Tamari lattice is partitioned into (smaller) lattices Tam$(v)$, where the $v$’s are all the paths of length $n-1$ on the square lattice.We explain possible connections between the poset Tam$(v)$ and (the combinatorics of) the generalized diagonal coinvariant spaces of the symmetric group.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2535 https://doi.org/10.46298/dmtcs.2535 Préville-Ratelle, Louis-François Viennot, Xavier Préville-Ratelle, Louis-François Viennot, Xavier <![CDATA[For any finite path $v$ on the square lattice consisting of north and east unit steps, we construct a poset Tam$(v)$ that consists of all the paths lying weakly above $v$ with the same endpoints as $v$. For particular choices of $v$, we recover the traditional Tamari lattice and the $m$-Tamari lattice. In particular this solves the problem of extending the $m$-Tamari lattice to any pair $(a; b)$ of relatively prime numbers in the context of the so-called rational Catalan combinatorics.For that purpose we introduce the notion of canopy of a binary tree and explicit a bijection between pairs $(u; v)$ of paths in Tam$(v)$ and binary trees with canopy $v$. Let $(\overleftarrow{v})$ be the path obtained from $v$ by reading the unit steps of $v$ in reverse order and exchanging east and north steps. We show that the poset Tam$(v)$ is isomorphic to the dual of the poset Tam$(\overleftarrow{v})$ and that Tam$(v)$ is isomorphic to the set of binary trees having the canopy $v$, which is an interval of the ordinary Tamari lattice. Thus the usual Tamari lattice is partitioned into (smaller) lattices Tam$(v)$, where the $v$’s are all the paths of length $n-1$ on the square lattice.We explain possible connections between the poset Tam$(v)$ and (the combinatorics of) the generalized diagonal coinvariant spaces of the symmetric group.]]> 0 <![CDATA[We present a generalization of the Tamari lattice to parabolic quotients of the symmetric group. More precisely, we generalize the notions of 231-avoiding permutations, noncrossing set partitions, and nonnesting set partitions to parabolic quotients, and show bijectively that these sets are equinumerous. Furthermore, the restriction of weak order on the parabolic quotient to the parabolic 231-avoiding permutations is a lattice quotient. Lastly, we suggest how to extend these constructions to all Coxeter groups.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2534 https://doi.org/10.46298/dmtcs.2534 Mühle, Henri Williams, Nathan Mühle, Henri Williams, Nathan <![CDATA[We present a generalization of the Tamari lattice to parabolic quotients of the symmetric group. More precisely, we generalize the notions of 231-avoiding permutations, noncrossing set partitions, and nonnesting set partitions to parabolic quotients, and show bijectively that these sets are equinumerous. Furthermore, the restriction of weak order on the parabolic quotient to the parabolic 231-avoiding permutations is a lattice quotient. Lastly, we suggest how to extend these constructions to all Coxeter groups.]]> 0 <![CDATA[Cambrian trees are oriented and labeled trees which fulfill local conditions around each node generalizing the conditions for classical binary search trees. Based on the bijective correspondence between signed permutations and leveled Cambrian trees, we define the Cambrian Hopf algebra generalizing J.-L. Loday and M. Ronco’s algebra on binary trees. We describe combinatorially the products and coproducts of both the Cambrian algebra and its dual in terms of operations on Cambrian trees. Finally, we define multiplicative bases of the Cambrian algebra and study structural and combinatorial properties of their indecomposable elements.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2533 https://doi.org/10.46298/dmtcs.2533 Chatel, G. Pilaud, V. Chatel, G. Pilaud, V. <![CDATA[Cambrian trees are oriented and labeled trees which fulfill local conditions around each node generalizing the conditions for classical binary search trees. Based on the bijective correspondence between signed permutations and leveled Cambrian trees, we define the Cambrian Hopf algebra generalizing J.-L. Loday and M. Ronco’s algebra on binary trees. We describe combinatorially the products and coproducts of both the Cambrian algebra and its dual in terms of operations on Cambrian trees. Finally, we define multiplicative bases of the Cambrian algebra and study structural and combinatorial properties of their indecomposable elements.]]> 0 <![CDATA[We present here a family of posets which generalizes both partition and pointed partition posets. After a short description of these new posets, we show that they are Cohen-Macaulay, compute their Moebius numbers and their characteristic polynomials. The characteristic polynomials are obtained using a combinatorial interpretation of the incidence Hopf algebra associated to these posets.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2532 https://doi.org/10.46298/dmtcs.2532 Delcroix-Oger, Bérénice Delcroix-Oger, Bérénice <![CDATA[We present here a family of posets which generalizes both partition and pointed partition posets. After a short description of these new posets, we show that they are Cohen-Macaulay, compute their Moebius numbers and their characteristic polynomials. The characteristic polynomials are obtained using a combinatorial interpretation of the incidence Hopf algebra associated to these posets.]]> 0 <![CDATA[We compute, for each genus $g$ ≥ 0, the generating function $L$_$g$ ≡ $L$_$g$($t$;$p$₁,$p$₂,...) of (labelled) bipartite maps on the orientable surface of genus $g$, with control on all face degrees. We exhibit an explicit change of variables such that for each $g$, $L$_$g$ is a rational function in the new variables, computable by an explicit recursion on the genus. The same holds for the generating function $L$_$g$ of rooted bipartite maps. The form of the result is strikingly similar to the Goulden/Jackson/Vakil and Goulden/Guay-Paquet/Novak formulas for the generating functions of classical and monotone Hurwitz numbers respectively, which suggests stronger links between these models. Our result strengthens recent results of Kazarian and Zograf, who studied the case where the number of faces is bounded, in the equivalent formalism of dessins d’enfants. Our proofs borrow some ideas from Eynard’s “topological recursion” that he applied in particular to even-faced maps (unconventionally called “bipartite maps” in his work). However, the present paper requires no previous knowledge of this topic and comes with elementary (complex-analysis-free) proofs written in the perspective of formal power series.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2531 https://doi.org/10.46298/dmtcs.2531 Chapuy, Guillaume Fang, Wenjie Chapuy, Guillaume Fang, Wenjie <![CDATA[We compute, for each genus $g$ ≥ 0, the generating function $L$_$g$ ≡ $L$_$g$($t$;$p$₁,$p$₂,...) of (labelled) bipartite maps on the orientable surface of genus $g$, with control on all face degrees. We exhibit an explicit change of variables such that for each $g$, $L$_$g$ is a rational function in the new variables, computable by an explicit recursion on the genus. The same holds for the generating function $L$_$g$ of rooted bipartite maps. The form of the result is strikingly similar to the Goulden/Jackson/Vakil and Goulden/Guay-Paquet/Novak formulas for the generating functions of classical and monotone Hurwitz numbers respectively, which suggests stronger links between these models. Our result strengthens recent results of Kazarian and Zograf, who studied the case where the number of faces is bounded, in the equivalent formalism of dessins d’enfants. Our proofs borrow some ideas from Eynard’s “topological recursion” that he applied in particular to even-faced maps (unconventionally called “bipartite maps” in his work). However, the present paper requires no previous knowledge of this topic and comes with elementary (complex-analysis-free) proofs written in the perspective of formal power series.]]> 0 <![CDATA[Tableau sequences of bounded height have been central to the analysis of $k$-noncrossing set partitions and matchings. We show here that families of sequences that end with a row shape are particularly compelling and lead to some interesting connections. First, we prove that hesitating tableaux of height at most two ending with a row shape are counted by Baxter numbers. This permits us to define three new Baxter classes which, remarkably, do not obviously possess the antipodal symmetry of other known Baxter classes. Oscillating tableau of height bounded by $k$ ending in a row are in bijection with Young tableaux of bounded height 2$k$. We discuss this recent result, and somegenerating function implications. Many of our proofs are analytic in nature, so there are intriguing combinatorial bijections to be found.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2530 https://doi.org/10.46298/dmtcs.2530 Burrill, Sophie Melczer, Stephen Mishna, Marni Burrill, Sophie Melczer, Stephen Mishna, Marni <![CDATA[Tableau sequences of bounded height have been central to the analysis of $k$-noncrossing set partitions and matchings. We show here that families of sequences that end with a row shape are particularly compelling and lead to some interesting connections. First, we prove that hesitating tableaux of height at most two ending with a row shape are counted by Baxter numbers. This permits us to define three new Baxter classes which, remarkably, do not obviously possess the antipodal symmetry of other known Baxter classes. Oscillating tableau of height bounded by $k$ ending in a row are in bijection with Young tableaux of bounded height 2$k$. We discuss this recent result, and somegenerating function implications. Many of our proofs are analytic in nature, so there are intriguing combinatorial bijections to be found.]]> 0 <![CDATA[In 1968 and 1969, Andrews proved two partition theorems of the Rogers-Ramanujan type which generalise Schur’s celebrated partition identity (1926). Andrews’ two generalisations of Schur’s theorem went on to become two of the most influential results in the theory of partitions, finding applications in combinatorics, representation theory and quantum algebra. In this paper we generalise both of Andrews’ theorems to overpartitions. The proofs use a new technique which consists in going back and forth from $q$-difference equations on generating functions to recurrence equations on their coefficients.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2529 https://doi.org/10.46298/dmtcs.2529 Dousse, Jehanne Dousse, Jehanne <![CDATA[In 1968 and 1969, Andrews proved two partition theorems of the Rogers-Ramanujan type which generalise Schur’s celebrated partition identity (1926). Andrews’ two generalisations of Schur’s theorem went on to become two of the most influential results in the theory of partitions, finding applications in combinatorics, representation theory and quantum algebra. In this paper we generalise both of Andrews’ theorems to overpartitions. The proofs use a new technique which consists in going back and forth from $q$-difference equations on generating functions to recurrence equations on their coefficients.]]> 0 <![CDATA[The Stanley chromatic polynomial of a graph $G$ is a symmetric function generalization of the chromatic polynomial, and has interesting combinatorial properties. We apply the ideas of Khovanov homology to construct a homology $H$_*($G$) of graded $S_n$-modules, whose graded Frobenius series $Frob_G(q,t)$ reduces to the chromatic symmetric function at $q=t=1$. We also obtain analogues of several familiar properties of the chromatic symmetric polynomials in terms of homology.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2527 https://doi.org/10.46298/dmtcs.2527 Sazdanović, Radmila Yip, Martha Sazdanović, Radmila Yip, Martha <![CDATA[The Stanley chromatic polynomial of a graph $G$ is a symmetric function generalization of the chromatic polynomial, and has interesting combinatorial properties. We apply the ideas of Khovanov homology to construct a homology $H$_*($G$) of graded $S_n$-modules, whose graded Frobenius series $Frob_G(q,t)$ reduces to the chromatic symmetric function at $q=t=1$. We also obtain analogues of several familiar properties of the chromatic symmetric polynomials in terms of homology.]]> 0 <![CDATA[We prove the affine Pieri rule for the cohomology of the affine flag variety conjectured by Lam, Lapointe, Morse and Shimozono. We study the cap operator on the affine nilHecke ring that is motivated by Kostant and Kumar’s work on the equivariant cohomology of the affine flag variety. We show that the cap operators for Pieri elements are the same as Pieri operators defined by Berg, Saliola and Serrano. This establishes the affine Pieri rule.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2517 https://doi.org/10.46298/dmtcs.2517 Lee, Seung Jin Lee, Seung Jin <![CDATA[We prove the affine Pieri rule for the cohomology of the affine flag variety conjectured by Lam, Lapointe, Morse and Shimozono. We study the cap operator on the affine nilHecke ring that is motivated by Kostant and Kumar’s work on the equivariant cohomology of the affine flag variety. We show that the cap operators for Pieri elements are the same as Pieri operators defined by Berg, Saliola and Serrano. This establishes the affine Pieri rule.]]> 0 <![CDATA[Plethysm coefficients are important structural constants in the theory of symmetric functions and in the representations theory of symmetric groups and general linear groups. In 1950, Foulkes observed stability properties: some sequences of plethysm coefficients are eventually constants. Such stability properties were proven by Brion with geometric techniques and by Thibon and Carré by means of vertex operators. In this paper we present a newapproach to prove such stability properties. This new proofs are purely combinatorial and follow the same scheme. We decompose plethysm coefficients in terms of other plethysm coefficients (related to the complete homogeneous basis of symmetric functions). We show that these other plethysm coefficients count integer points in polytopes and we prove stability for them by exhibiting bijections between the corresponding sets of integer points of each polytope.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2526 https://doi.org/10.46298/dmtcs.2526 Colmenarejo, Laura Colmenarejo, Laura <![CDATA[Plethysm coefficients are important structural constants in the theory of symmetric functions and in the representations theory of symmetric groups and general linear groups. In 1950, Foulkes observed stability properties: some sequences of plethysm coefficients are eventually constants. Such stability properties were proven by Brion with geometric techniques and by Thibon and Carré by means of vertex operators. In this paper we present a newapproach to prove such stability properties. This new proofs are purely combinatorial and follow the same scheme. We decompose plethysm coefficients in terms of other plethysm coefficients (related to the complete homogeneous basis of symmetric functions). We show that these other plethysm coefficients count integer points in polytopes and we prove stability for them by exhibiting bijections between the corresponding sets of integer points of each polytope.]]> 0 <![CDATA[In this paper we give a sufficient condition for a general stability of Kronecker coefficients, which we call additive stability. Its main ingredient is the property of a matrix of being additive. This notion seems to be an important one: it appears in Discrete Tomography as a sufficient condition to uniqueness; it also appears in Manivel’s study of asymptotic properties of plethysm through Borel-Weil theory. The proof sketched here combines several results of the author on integer matrices motivated by Discrete Tomography with a new idea of Stembridge, that permits to bound some sequences of Kronecker coefficients. The advantage of additivity with respect to the previous approach by Stembridge is that it is very easy to produce new examples of additive matrices and, therefore, to produce many new examples of stability of Kronecker coefficients. We also show that Murnaghan’s stability property and other instances of stability discovered previously by the author are special cases of additive stability. Besides, our approach permits us to disprove a recent conjecture of Stembridge and to give a new characterization of additivity.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2525 https://doi.org/10.46298/dmtcs.2525 Vallejo, Ernesto Vallejo, Ernesto <![CDATA[In this paper we give a sufficient condition for a general stability of Kronecker coefficients, which we call additive stability. Its main ingredient is the property of a matrix of being additive. This notion seems to be an important one: it appears in Discrete Tomography as a sufficient condition to uniqueness; it also appears in Manivel’s study of asymptotic properties of plethysm through Borel-Weil theory. The proof sketched here combines several results of the author on integer matrices motivated by Discrete Tomography with a new idea of Stembridge, that permits to bound some sequences of Kronecker coefficients. The advantage of additivity with respect to the previous approach by Stembridge is that it is very easy to produce new examples of additive matrices and, therefore, to produce many new examples of stability of Kronecker coefficients. We also show that Murnaghan’s stability property and other instances of stability discovered previously by the author are special cases of additive stability. Besides, our approach permits us to disprove a recent conjecture of Stembridge and to give a new characterization of additivity.]]> 0 <![CDATA[We extend a $T$-path expansion formula for arcs on an unpunctured surface to the case of arcs on a once-punctured polygon and use this formula to give a combinatorial proof that cluster monomials form the atomic basis of a cluster algebra of type $D$.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2524 https://doi.org/10.46298/dmtcs.2524 Gunawan, Emily Musiker, Gregg Gunawan, Emily Musiker, Gregg <![CDATA[We extend a $T$-path expansion formula for arcs on an unpunctured surface to the case of arcs on a once-punctured polygon and use this formula to give a combinatorial proof that cluster monomials form the atomic basis of a cluster algebra of type $D$.]]> 0 <![CDATA[We introduce a new combinatorial structure: the metasylvester lattice on decreasing trees. It appears in the context of the $m$-Tamari lattices and other related $m$-generalizations. The metasylvester congruence has been recently introduced by Novelli and Thibon. We show that it defines a sublattice of the $m$-permutations where elements can be represented by decreasing labelled trees: the metasylvester lattice. We study the combinatorial properties of this new structure. In particular, we give different realizations of the lattice. The $m$-Tamari lattice is by definition a sublattice of our newly defined metasylvester lattice. It leads us to a new realization of the $m$-Tamari lattice, using certain chains of the classical Tamari lattice.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2523 https://doi.org/10.46298/dmtcs.2523 Pons, Viviane Pons, Viviane <![CDATA[We introduce a new combinatorial structure: the metasylvester lattice on decreasing trees. It appears in the context of the $m$-Tamari lattices and other related $m$-generalizations. The metasylvester congruence has been recently introduced by Novelli and Thibon. We show that it defines a sublattice of the $m$-permutations where elements can be represented by decreasing labelled trees: the metasylvester lattice. We study the combinatorial properties of this new structure. In particular, we give different realizations of the lattice. The $m$-Tamari lattice is by definition a sublattice of our newly defined metasylvester lattice. It leads us to a new realization of the $m$-Tamari lattice, using certain chains of the classical Tamari lattice.]]> 0 <![CDATA[We prove an explicit closed formula, written as a sum of Pfaffians, which describes each equivariant Schubert class for the Grassmannian of isotropic subspaces in a symplectic vector space]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2522 https://doi.org/10.46298/dmtcs.2522 Ikeda, Takeshi Matsumura, Tomoo Ikeda, Takeshi Matsumura, Tomoo <![CDATA[We prove an explicit closed formula, written as a sum of Pfaffians, which describes each equivariant Schubert class for the Grassmannian of isotropic subspaces in a symplectic vector space]]> 0 <![CDATA[We construct universal geometric coefficients for the cluster algebra associated to the four-punctured sphere and obtain, as a by-product, the $g$ -vectors of cluster variables. We also construct the rational part of the mutation fan. These constructions rely on a classification of the allowable curves (the curves which can appear in quasi-laminations). The classification allows us to prove the Null Tangle Property for the four-punctured sphere, thus adding this surface to a short list of surfaces for which this property is known. The Null Tangle Property then implies that the shear coordinates of allowable curves are the universal coefficients. We compute these shear coordinates to obtain universal geometric coefficients.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2521 https://doi.org/10.46298/dmtcs.2521 Barnard, Emily Meehan, Emily Polster, Shira Reading, Nathan Barnard, Emily Meehan, Emily Polster, Shira Reading, Nathan <![CDATA[We construct universal geometric coefficients for the cluster algebra associated to the four-punctured sphere and obtain, as a by-product, the $g$ -vectors of cluster variables. We also construct the rational part of the mutation fan. These constructions rely on a classification of the allowable curves (the curves which can appear in quasi-laminations). The classification allows us to prove the Null Tangle Property for the four-punctured sphere, thus adding this surface to a short list of surfaces for which this property is known. The Null Tangle Property then implies that the shear coordinates of allowable curves are the universal coefficients. We compute these shear coordinates to obtain universal geometric coefficients.]]> 0 <![CDATA[We introduce a rich family of generalizations of the pentagram map sharing the property that each generates an infinite configuration of points and lines with four points on each line. These systems all have a description as $Y$ -mutations in a cluster algebra and hence establish new connections between cluster theory and projective geometry.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2520 https://doi.org/10.46298/dmtcs.2520 Glick, Max Pylyavskyy, Pavlo Glick, Max Pylyavskyy, Pavlo <![CDATA[We introduce a rich family of generalizations of the pentagram map sharing the property that each generates an infinite configuration of points and lines with four points on each line. These systems all have a description as $Y$ -mutations in a cluster algebra and hence establish new connections between cluster theory and projective geometry.]]> 0 <![CDATA[We consider several counting problems related to Coxeter-Catalan combinatorics and conjecture that the problems all have the same answer, which we call the $W$ -biCatalan number. We prove the conjecture in many cases.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2519 https://doi.org/10.46298/dmtcs.2519 Barnard, Emily Reading, Nathan Barnard, Emily Reading, Nathan <![CDATA[We consider several counting problems related to Coxeter-Catalan combinatorics and conjecture that the problems all have the same answer, which we call the $W$ -biCatalan number. We prove the conjecture in many cases.]]> 0 <![CDATA[The totally nonnegative Grassmannian is the set of $k$-dimensional subspaces $V$ of ℝ^$n$ whose nonzero Plücker coordinates (i.e. $k × k$ minors of a $k × n$ matrix whose rows span $V$) all have the same sign. Total positivity has been much studied in the past two decades from an algebraic, combinatorial, and topological perspective, but first arose in the theory of oscillations in analysis. It was in the latter context that Gantmakher and Krein (1950) and Schoenberg and Whitney (1951) independently showed that a subspace $V$ is totally nonnegative iff every vector in $V$, when viewed as a sequence of $n$ numbers and ignoring any zeros, changes sign fewer than $k$ times. We generalize this result, showing that the vectors in $V$ change sign fewer than $l$ times iff certain sequences of the Plücker coordinates of some generic perturbation of $V$ change sign fewer than $l − k &plus; 1$ times. We give an algorithm which constructs such a generic perturbation. Also, we determine the positroid cell of each totally nonnegative $V$ from sign patterns of vectors in $V$. These results generalize to oriented matroids.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2518 https://doi.org/10.46298/dmtcs.2518 Karp, Steven N. Karp, Steven N. <![CDATA[The totally nonnegative Grassmannian is the set of $k$-dimensional subspaces $V$ of ℝ^$n$ whose nonzero Plücker coordinates (i.e. $k × k$ minors of a $k × n$ matrix whose rows span $V$) all have the same sign. Total positivity has been much studied in the past two decades from an algebraic, combinatorial, and topological perspective, but first arose in the theory of oscillations in analysis. It was in the latter context that Gantmakher and Krein (1950) and Schoenberg and Whitney (1951) independently showed that a subspace $V$ is totally nonnegative iff every vector in $V$, when viewed as a sequence of $n$ numbers and ignoring any zeros, changes sign fewer than $k$ times. We generalize this result, showing that the vectors in $V$ change sign fewer than $l$ times iff certain sequences of the Plücker coordinates of some generic perturbation of $V$ change sign fewer than $l − k &plus; 1$ times. We give an algorithm which constructs such a generic perturbation. Also, we determine the positroid cell of each totally nonnegative $V$ from sign patterns of vectors in $V$. These results generalize to oriented matroids.]]> 0 <![CDATA[Let $u$ and $v$ be permutations on $n$ letters, with $u$ ≤ $v$ in Bruhat order. A Bruhat interval polytope $Q_{u,v}$ is the convex hull of all permutation vectors $z=(z(1),z(2),...,z(n))$ with $u$ ≤ $z$ ≤ $v$. Note that when $u=e$ and $v=w_0$ are the shortest and longest elements of the symmetric group, $Q_{e,w_0}$ is the classical permutohedron. Bruhat interval polytopes were studied recently in the 2013 paper “The full Kostant-Toda hierarchy on the positive flag variety” by Kodama and the second author, in the context of the Toda lattice and the moment map on the flag variety. In this paper we study combinatorial aspects of Bruhat interval polytopes. For example, we give an inequality description and a dimension formula for Bruhat interval polytopes, and prove that every face of a Bruhat interval polytope is a Bruhat interval polytope. A key tool in the proof of the latter statement is a generalization of the well-known lifting property for Coxeter groups. Motivated by the relationship between the lifting property and $R$-polynomials, we also give a generalization of the standard recurrence for $R$-polynomials.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2507 https://doi.org/10.46298/dmtcs.2507 Tsukerman, Emmanuel Williams, Lauren Tsukerman, Emmanuel Williams, Lauren <![CDATA[Let $u$ and $v$ be permutations on $n$ letters, with $u$ ≤ $v$ in Bruhat order. A Bruhat interval polytope $Q_{u,v}$ is the convex hull of all permutation vectors $z=(z(1),z(2),...,z(n))$ with $u$ ≤ $z$ ≤ $v$. Note that when $u=e$ and $v=w_0$ are the shortest and longest elements of the symmetric group, $Q_{e,w_0}$ is the classical permutohedron. Bruhat interval polytopes were studied recently in the 2013 paper “The full Kostant-Toda hierarchy on the positive flag variety” by Kodama and the second author, in the context of the Toda lattice and the moment map on the flag variety. In this paper we study combinatorial aspects of Bruhat interval polytopes. For example, we give an inequality description and a dimension formula for Bruhat interval polytopes, and prove that every face of a Bruhat interval polytope is a Bruhat interval polytope. A key tool in the proof of the latter statement is a generalization of the well-known lifting property for Coxeter groups. Motivated by the relationship between the lifting property and $R$-polynomials, we also give a generalization of the standard recurrence for $R$-polynomials.]]> 0 <![CDATA[We introduce the Dyck path triangulation of the cartesian product of two simplices $\Delta_{n-1}\times\Delta_{n-1}$. The maximal simplices of this triangulation are given by Dyck paths, and its construction naturally generalizes to produce triangulations of $\Delta_{r\ n-1}\times\Delta_{n-1}$ using rational Dyck paths. Our study of the Dyck path triangulation is motivated by extendability problems of partial triangulations of products of two simplices. We show that whenever$m\geq k>n$, any triangulations of $\Delta_{m-1}^{(k-1)}\times\Delta_{n-1}$ extends to a unique triangulation of $\Delta_{m-1}\times\Delta_{n-1}$. Moreover, with an explicit construction, we prove that the bound $k>n$ is optimal. We also exhibit interpretations of our results in the language of tropical oriented matroids, which are analogous to classical results in oriented matroid theory.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2516 https://doi.org/10.46298/dmtcs.2516 Ceballos, Cesar Padrol, Arnau Sarmiento, Camilo Ceballos, Cesar Padrol, Arnau Sarmiento, Camilo <![CDATA[We introduce the Dyck path triangulation of the cartesian product of two simplices $\Delta_{n-1}\times\Delta_{n-1}$. The maximal simplices of this triangulation are given by Dyck paths, and its construction naturally generalizes to produce triangulations of $\Delta_{r\ n-1}\times\Delta_{n-1}$ using rational Dyck paths. Our study of the Dyck path triangulation is motivated by extendability problems of partial triangulations of products of two simplices. We show that whenever$m\geq k>n$, any triangulations of $\Delta_{m-1}^{(k-1)}\times\Delta_{n-1}$ extends to a unique triangulation of $\Delta_{m-1}\times\Delta_{n-1}$. Moreover, with an explicit construction, we prove that the bound $k>n$ is optimal. We also exhibit interpretations of our results in the language of tropical oriented matroids, which are analogous to classical results in oriented matroid theory.]]> 0 <![CDATA[We define a $K$ -theoretic analogue of Fomin’s dual graded graphs, which we call dual filtered graphs. The key formula in the definition is $DU - UD = D + I$. Our major examples are $K$ -theoretic analogues of Young’s lattice, the binary tree, and the graph determined by the Poirier-Reutenauer Hopf algebra. Most of our examples arise via two constructions, which we call the Pieri construction and the Möbius construction. The Pieri construction is closely related to the construction of dual graded graphs from a graded Hopf algebra, as described in Bergeron-Lam-Li, Nzeutchap, and Lam-Shimozono. The Möbius construction is more mysterious but also potentially more important, as it corresponds to natural insertion algorithms.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2515 https://doi.org/10.46298/dmtcs.2515 Patrias, Rebecca Pylyavskyy, Pavlo Patrias, Rebecca Pylyavskyy, Pavlo <![CDATA[We define a $K$ -theoretic analogue of Fomin’s dual graded graphs, which we call dual filtered graphs. The key formula in the definition is $DU - UD = D + I$. Our major examples are $K$ -theoretic analogues of Young’s lattice, the binary tree, and the graph determined by the Poirier-Reutenauer Hopf algebra. Most of our examples arise via two constructions, which we call the Pieri construction and the Möbius construction. The Pieri construction is closely related to the construction of dual graded graphs from a graded Hopf algebra, as described in Bergeron-Lam-Li, Nzeutchap, and Lam-Shimozono. The Möbius construction is more mysterious but also potentially more important, as it corresponds to natural insertion algorithms.]]> 0 <![CDATA[We present new combinatorial methods for solving algebraic problems such as computing the Hilbert series of a free $P$-algebra over one generator, or proving the freeness of a $P$-algebra. In particular, we apply these methods to the cases of dendriform algebras, quadrialgebras and tridendriform algebras, which leads us to prove a conjecture of Aguiar and Loday about the freeness of the quadrialgebra generated by the permutation 12.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2514 https://doi.org/10.46298/dmtcs.2514 Vong, Vincent Vong, Vincent <![CDATA[We present new combinatorial methods for solving algebraic problems such as computing the Hilbert series of a free $P$-algebra over one generator, or proving the freeness of a $P$-algebra. In particular, we apply these methods to the cases of dendriform algebras, quadrialgebras and tridendriform algebras, which leads us to prove a conjecture of Aguiar and Loday about the freeness of the quadrialgebra generated by the permutation 12.]]> 0 <![CDATA[Exceptional sequences are certain ordered sequences of quiver representations. We use noncrossing edge-labeled trees in a disk with boundary vertices (expanding on T. Araya’s work) to classify exceptional sequences of representations of $Q$, the linearly ordered quiver with $n$ vertices. We also show how to use variations of this model to classify $c$-matrices of $Q$, to interpret exceptional sequences as linear extensions, and to give a simple bijection between exceptional sequences and certain chains in the lattice of noncrossing partitions. In the case of $c$-matrices, we also give an interpretation of $c$-matrix mutation in terms of our noncrossing trees with directed edges.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2513 https://doi.org/10.46298/dmtcs.2513 Garver, Alexander Matherne, Jacob P. Garver, Alexander Matherne, Jacob P. <![CDATA[Exceptional sequences are certain ordered sequences of quiver representations. We use noncrossing edge-labeled trees in a disk with boundary vertices (expanding on T. Araya’s work) to classify exceptional sequences of representations of $Q$, the linearly ordered quiver with $n$ vertices. We also show how to use variations of this model to classify $c$-matrices of $Q$, to interpret exceptional sequences as linear extensions, and to give a simple bijection between exceptional sequences and certain chains in the lattice of noncrossing partitions. In the case of $c$-matrices, we also give an interpretation of $c$-matrix mutation in terms of our noncrossing trees with directed edges.]]> 0 <![CDATA[We present complete simplicial fan realizations of any spherical subword complex of type $A_n$ for $n\leq 3$. This provides complete simplicial fan realizations of simplicial multi-associahedra $\Delta_{2k+4,k}$, whose facets are in correspondence with $k$-triangulations of a convex $(2k+4)$-gon. This solves the first open case of the problem of finding fan realizations where polytopality is not known. The techniques presented in this paper work for all finite Coxeter groups and we hope that they will be useful to construct fans realizing subword complexes in general. In particular, we present fan realizations of two previously unknown cases of subword complexes of type $A_4$, namely the multi-associahedra $\Delta_{9,2}$ and $\Delta_{11,3}$.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2512 https://doi.org/10.46298/dmtcs.2512 Bergeron, Nantel Ceballos, Cesar Labbé, Jean-Philippe Bergeron, Nantel Ceballos, Cesar Labbé, Jean-Philippe <![CDATA[We present complete simplicial fan realizations of any spherical subword complex of type $A_n$ for $n\leq 3$. This provides complete simplicial fan realizations of simplicial multi-associahedra $\Delta_{2k+4,k}$, whose facets are in correspondence with $k$-triangulations of a convex $(2k+4)$-gon. This solves the first open case of the problem of finding fan realizations where polytopality is not known. The techniques presented in this paper work for all finite Coxeter groups and we hope that they will be useful to construct fans realizing subword complexes in general. In particular, we present fan realizations of two previously unknown cases of subword complexes of type $A_4$, namely the multi-associahedra $\Delta_{9,2}$ and $\Delta_{11,3}$.]]> 0 <![CDATA[Recently, Diaconis, Ram and I created Markov chains out of the coproduct-then-product operator on combinatorial Hopf algebras. These chains model the breaking and recombining of combinatorial objects. Our motivating example was the riffle-shuffling of a deck of cards, for which this Hopf algebra connection allowed explicit computation of all the eigenfunctions. The present note replaces in this construction the coproduct-then-product map with convolutions of projections to the graded subspaces, effectively allowing us to dictate the distribution of sizes of the pieces in the breaking step of the previous chains. An important example is removing one “vertex” and reattaching it, in analogy with top-to-random shuffling. This larger family of Markov chains all admit analysis by Hopf-algebraic techniques. There are simple combinatorial expressions for their stationary distributions and for their eigenvalues and multiplicities and, in some cases, the eigenfunctions are also calculable.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2511 https://doi.org/10.46298/dmtcs.2511 Pang, C. Y. Amy Pang, C. Y. Amy <![CDATA[Recently, Diaconis, Ram and I created Markov chains out of the coproduct-then-product operator on combinatorial Hopf algebras. These chains model the breaking and recombining of combinatorial objects. Our motivating example was the riffle-shuffling of a deck of cards, for which this Hopf algebra connection allowed explicit computation of all the eigenfunctions. The present note replaces in this construction the coproduct-then-product map with convolutions of projections to the graded subspaces, effectively allowing us to dictate the distribution of sizes of the pieces in the breaking step of the previous chains. An important example is removing one “vertex” and reattaching it, in analogy with top-to-random shuffling. This larger family of Markov chains all admit analysis by Hopf-algebraic techniques. There are simple combinatorial expressions for their stationary distributions and for their eigenvalues and multiplicities and, in some cases, the eigenfunctions are also calculable.]]> 0 <![CDATA[Based on the Hermite–Biehler theorem, we simultaneously prove the real-rootedness of Eulerian polynomials of type $D$ and the real-rootedness of affine Eulerian polynomials of type $B$, which were first obtained by Savage and Visontai by using the theory of $s$-Eulerian polynomials. We also confirm Hyatt’s conjectures on the inter-lacing property of half Eulerian polynomials. Borcea and Brändén’s work on the characterization of linear operators preserving Hurwitz stability is critical to this approach.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2510 https://doi.org/10.46298/dmtcs.2510 Yang, Arthur L.B. Zhang, Philip B. Yang, Arthur L.B. Zhang, Philip B. <![CDATA[Based on the Hermite–Biehler theorem, we simultaneously prove the real-rootedness of Eulerian polynomials of type $D$ and the real-rootedness of affine Eulerian polynomials of type $B$, which were first obtained by Savage and Visontai by using the theory of $s$-Eulerian polynomials. We also confirm Hyatt’s conjectures on the inter-lacing property of half Eulerian polynomials. Borcea and Brändén’s work on the characterization of linear operators preserving Hurwitz stability is critical to this approach.]]> 0 <![CDATA[A combinatorial expression for the coefficient of the Schur function $s_{\lambda}$ in the expansion of the plethysm $p_{n/d}^d \circ s_{\mu}$ is given for all $d$ dividing $n$ for the cases in which $n=2$ or $\lambda$ is rectangular. In these cases, the coefficient $\langle p_{n/d}^d \circ s_{\mu}, s_{\lambda} \rangle$ is shown to count, up to sign, the number of fixed points of an $\langle s_{\mu}^n, s_{\lambda} \rangle$-element set under the $d^e$ power of an order $n$ cyclic action. If $n=2$, the action is the Schützenberger involution on semistandard Young tableaux (also known as evacuation), and, if $\lambda$ is rectangular, the action is a certain power of Schützenberger and Shimozono's jeu-de-taquin promotion.This work extends results of Stembridge and Rhoades linking fixed points of the Schützenberger actions to ribbon tableaux enumeration. The conclusion for the case $n=2$ is equivalent to the domino tableaux rule of Carré and Leclerc for discriminating between the symmetric and antisymmetric parts of the square of a Schur function.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2509 https://doi.org/10.46298/dmtcs.2509 Rush, David B Rush, David B <![CDATA[A combinatorial expression for the coefficient of the Schur function $s_{\lambda}$ in the expansion of the plethysm $p_{n/d}^d \circ s_{\mu}$ is given for all $d$ dividing $n$ for the cases in which $n=2$ or $\lambda$ is rectangular. In these cases, the coefficient $\langle p_{n/d}^d \circ s_{\mu}, s_{\lambda} \rangle$ is shown to count, up to sign, the number of fixed points of an $\langle s_{\mu}^n, s_{\lambda} \rangle$-element set under the $d^e$ power of an order $n$ cyclic action. If $n=2$, the action is the Schützenberger involution on semistandard Young tableaux (also known as evacuation), and, if $\lambda$ is rectangular, the action is a certain power of Schützenberger and Shimozono's jeu-de-taquin promotion.This work extends results of Stembridge and Rhoades linking fixed points of the Schützenberger actions to ribbon tableaux enumeration. The conclusion for the case $n=2$ is equivalent to the domino tableaux rule of Carré and Leclerc for discriminating between the symmetric and antisymmetric parts of the square of a Schur function.]]> 0 <![CDATA[An important problem from invariant theory is to describe the subspace of a tensor power of a representation invariant under the action of the group. According to Weyl's classic, the first main (later: 'fundamental') theorem of invariant theory states that all invariants are expressible in terms of a finite number among them, whereas a second main theorem determines the relations between those basic invariants.Here we present a transparent, combinatorial proof of a second fundamental theorem for the defining representation of the symplectic group $Sp(2n)$. Our formulation is completely explicit and provides a very precise link to $(n+1)$-noncrossing perfect matchings, going beyond a dimension count. As a corollary, we obtain an instance of the cyclic sieving phenomenon.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2508 https://doi.org/10.46298/dmtcs.2508 Rubey, Martin Westbury, Bruce W. Rubey, Martin Westbury, Bruce W. <![CDATA[An important problem from invariant theory is to describe the subspace of a tensor power of a representation invariant under the action of the group. According to Weyl's classic, the first main (later: 'fundamental') theorem of invariant theory states that all invariants are expressible in terms of a finite number among them, whereas a second main theorem determines the relations between those basic invariants.Here we present a transparent, combinatorial proof of a second fundamental theorem for the defining representation of the symplectic group $Sp(2n)$. Our formulation is completely explicit and provides a very precise link to $(n+1)$-noncrossing perfect matchings, going beyond a dimension count. As a corollary, we obtain an instance of the cyclic sieving phenomenon.]]> 0 <![CDATA[This extended abstract presents some recent (exact and asymptotic) enumerative results concerning rhombustilings of hexagons that have had symmetrically distributed inward pointing triangles of side length 2 removedfrom their interiors. These results form part of a larger article that is currently available online (arXiv:1501.05772).]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2473 https://doi.org/10.46298/dmtcs.2473 Gilmore, Tomack Gilmore, Tomack <![CDATA[This extended abstract presents some recent (exact and asymptotic) enumerative results concerning rhombustilings of hexagons that have had symmetrically distributed inward pointing triangles of side length 2 removedfrom their interiors. These results form part of a larger article that is currently available online (arXiv:1501.05772).]]> 0 <![CDATA[We use the modules introduced by Kraśkiewicz and Pragacz (1987, 2004) to show some positivity propertiesof Schubert polynomials. We give a new proof to the classical fact that the product of two Schubert polynomialsis Schubert-positive, and also show a new result that the plethystic composition of a Schur function with a Schubertpolynomial is Schubert-positive. The present submission is an extended abstract on these results and the full versionof this work will be published elsewhere.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2483 https://doi.org/10.46298/dmtcs.2483 Watanabe, Masaki Watanabe, Masaki <![CDATA[We use the modules introduced by Kraśkiewicz and Pragacz (1987, 2004) to show some positivity propertiesof Schubert polynomials. We give a new proof to the classical fact that the product of two Schubert polynomialsis Schubert-positive, and also show a new result that the plethystic composition of a Schur function with a Schubertpolynomial is Schubert-positive. The present submission is an extended abstract on these results and the full versionof this work will be published elsewhere.]]> 0 <![CDATA[We introduce genomic tableaux, with applications to Schubert calculus. We report a combinatorial rule for structure coefficients in the torus-equivariant $K$-theory of Grassmannians for the basis of Schubert structure sheaves. This rule is positive in the sense of [Anderson-Griffeth-Miller ’11]. We thereby deduce an earlier conjecture of [Thomas-Yong ’13] for the coefficients. Moreover, our rule specializes to give a new Schubert calculus rule in the (non-equivariant) $K$-theory of Grassmannians. From this perspective, we also obtain a new rule for $K$-theoretic Schubert structure constants of maximal orthogonal Grassmannians, and give conjectural bounds on such constants for Lagrangian Grassmannians.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2482 https://doi.org/10.46298/dmtcs.2482 Pechenik, Oliver Yong, Alexander Pechenik, Oliver Yong, Alexander <![CDATA[We introduce genomic tableaux, with applications to Schubert calculus. We report a combinatorial rule for structure coefficients in the torus-equivariant $K$-theory of Grassmannians for the basis of Schubert structure sheaves. This rule is positive in the sense of [Anderson-Griffeth-Miller ’11]. We thereby deduce an earlier conjecture of [Thomas-Yong ’13] for the coefficients. Moreover, our rule specializes to give a new Schubert calculus rule in the (non-equivariant) $K$-theory of Grassmannians. From this perspective, we also obtain a new rule for $K$-theoretic Schubert structure constants of maximal orthogonal Grassmannians, and give conjectural bounds on such constants for Lagrangian Grassmannians.]]> 0 <![CDATA[Using the powerful machinery available for reduced words of type $B$, we demonstrate a bijection between centrally symmetric $k$-triangulations of a $2(n + k)$-gon and plane partitions of height at most $k$ in a square of size $n$. This bijection can be viewed as the type $B$ analogue of a bijection for $k$-triangulations due to L. Serrano and C. Stump.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2481 https://doi.org/10.46298/dmtcs.2481 Hamaker, Zachary Williams, Nathan Hamaker, Zachary Williams, Nathan <![CDATA[Using the powerful machinery available for reduced words of type $B$, we demonstrate a bijection between centrally symmetric $k$-triangulations of a $2(n + k)$-gon and plane partitions of height at most $k$ in a square of size $n$. This bijection can be viewed as the type $B$ analogue of a bijection for $k$-triangulations due to L. Serrano and C. Stump.]]> 0 <![CDATA[Let $A$ be an $n$-element set. Let $\mathscr{L} ie_2(A)$ be the multilinear part of the free Lie algebra on $A$ with a pair of compatible Lie brackets, and $\mathscr{L} ie_2(A, i)$ the subspace of $\mathscr{L} ie_2(A)$ generated by all the monomials in $\mathscr{L} ie_2(A)$ with $i$ brackets of one type. The author and Dotsenko-Khoroshkin show that the dimension of $\mathscr{L} ie_2(A, i)$ is the size of $R_{A,i}$, the set of rooted trees on $A$ with $i$ decreasing edges. There are three families of bases known for $\mathscr{L} ie_2(A, i)$ the comb basis, the Lyndon basis, and the Liu-Lyndon basis. Recently, González D'León and Wachs, in their study of (co)homology of the poset of weighted partitions (which has close connection to $\mathscr{L} ie_2(A, i)$), asked whether there are nice bijections between $R_{A,i}$ and the comb basis or the Lyndon basis. We give a natural definition for " nice bijections " , and conjecture that there is a unique nice bijection between $R_{A,i}$ and the comb basis. We show the conjecture is true for the extreme cases where $i=0$, $n−1$.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2480 https://doi.org/10.46298/dmtcs.2480 Liu, Fu Liu, Fu <![CDATA[Let $A$ be an $n$-element set. Let $\mathscr{L} ie_2(A)$ be the multilinear part of the free Lie algebra on $A$ with a pair of compatible Lie brackets, and $\mathscr{L} ie_2(A, i)$ the subspace of $\mathscr{L} ie_2(A)$ generated by all the monomials in $\mathscr{L} ie_2(A)$ with $i$ brackets of one type. The author and Dotsenko-Khoroshkin show that the dimension of $\mathscr{L} ie_2(A, i)$ is the size of $R_{A,i}$, the set of rooted trees on $A$ with $i$ decreasing edges. There are three families of bases known for $\mathscr{L} ie_2(A, i)$ the comb basis, the Lyndon basis, and the Liu-Lyndon basis. Recently, González D'León and Wachs, in their study of (co)homology of the poset of weighted partitions (which has close connection to $\mathscr{L} ie_2(A, i)$), asked whether there are nice bijections between $R_{A,i}$ and the comb basis or the Lyndon basis. We give a natural definition for " nice bijections " , and conjecture that there is a unique nice bijection between $R_{A,i}$ and the comb basis. We show the conjecture is true for the extreme cases where $i=0$, $n−1$.]]> 0 <![CDATA[Shapiro and Chekhov (2011) have introduced the notion of generalised cluster algebra; we focus on an example in type $C_n$. On the other hand, Chari and Pressley (1997), as well as Frenkel and Mukhin (2002), have studied the restricted integral form $U^{\mathtt{res}}_ε (\widehat{\mathfrak{g}})$ of a quantum affine algebra $U_q(\widehat{\mathfrak{g}})$ where $q=ε$ is a root of unity. Our main result states that the Grothendieck ring of a tensor subcategory $C_{ε^\mathbb{z}}$ of representations of $U^{\mathtt{res}}_ε (L\mathfrak{sl}_2)$ is a generalised cluster algebra of type $C_{l−1}$, where $l$ is the order of $ε^2$. We also state a conjecture for $U^{\mathtt{res}}_ε (L\mathfrak{sl}_3)$, and sketch a proof for $l=2$.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2479 https://doi.org/10.46298/dmtcs.2479 Gleitz, Anne-Sophie Gleitz, Anne-Sophie <![CDATA[Shapiro and Chekhov (2011) have introduced the notion of generalised cluster algebra; we focus on an example in type $C_n$. On the other hand, Chari and Pressley (1997), as well as Frenkel and Mukhin (2002), have studied the restricted integral form $U^{\mathtt{res}}_ε (\widehat{\mathfrak{g}})$ of a quantum affine algebra $U_q(\widehat{\mathfrak{g}})$ where $q=ε$ is a root of unity. Our main result states that the Grothendieck ring of a tensor subcategory $C_{ε^\mathbb{z}}$ of representations of $U^{\mathtt{res}}_ε (L\mathfrak{sl}_2)$ is a generalised cluster algebra of type $C_{l−1}$, where $l$ is the order of $ε^2$. We also state a conjecture for $U^{\mathtt{res}}_ε (L\mathfrak{sl}_3)$, and sketch a proof for $l=2$.]]> 0 <![CDATA[Given a permutation $\pi=\pi_1\pi_2\cdots \pi_n \in S_n$, we say an index $i$ is a peak if $\pi_{i-1} < \pi_i > \pi_{i+1}$. Let $P(\pi)$ denote the set of peaks of $\pi$. Given any set $S$ of positive integers, define ${P_S(n)=\{\pi\in S_n:P(\pi)=S\}}$. Billey-Burdzy-Sagan showed that for all fixed subsets of positive integers $S$ and sufficiently large $n$, $|P_S(n)|=p_S(n)2^{n-|S|-1}$ for some polynomial $p_S(x)$ depending on $S$. They conjectured that the coefficients of $p_S(x)$ expanded in a binomial coefficient basis centered at $max(S)$ are all positive. We show that this is a consequence of a stronger conjecture that bounds the modulus of the roots of $p_S(x)$. Furthermore, we give an efficient explicit formula for peak polynomials in the binomial basis centered at $0$, which we use to identify many integer roots of peak polynomials along with certain inequalities and identities.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2478 https://doi.org/10.46298/dmtcs.2478 Billey, Sara Fahrbach, Matthew Talmage, Alan Billey, Sara Fahrbach, Matthew Talmage, Alan <![CDATA[Given a permutation $\pi=\pi_1\pi_2\cdots \pi_n \in S_n$, we say an index $i$ is a peak if $\pi_{i-1} < \pi_i > \pi_{i+1}$. Let $P(\pi)$ denote the set of peaks of $\pi$. Given any set $S$ of positive integers, define ${P_S(n)=\{\pi\in S_n:P(\pi)=S\}}$. Billey-Burdzy-Sagan showed that for all fixed subsets of positive integers $S$ and sufficiently large $n$, $|P_S(n)|=p_S(n)2^{n-|S|-1}$ for some polynomial $p_S(x)$ depending on $S$. They conjectured that the coefficients of $p_S(x)$ expanded in a binomial coefficient basis centered at $max(S)$ are all positive. We show that this is a consequence of a stronger conjecture that bounds the modulus of the roots of $p_S(x)$. Furthermore, we give an efficient explicit formula for peak polynomials in the binomial basis centered at $0$, which we use to identify many integer roots of peak polynomials along with certain inequalities and identities.]]> 0 <![CDATA[It was proved by Rubey that the number of fillings with zeros and ones of a given moon polyomino thatdo not contain a northeast chain of a fixed size depends only on the set of column lengths of the polyomino. Rubey’sproof uses an adaption of jeu de taquin and promotion for arbitrary fillings of moon polyominoes and deduces theresult for 01-fillings via a variation of the pigeonhole principle. In this paper we present the first completely bijectiveproof of this result by considering fillings of almost-moon polyominoes, which are moon polyominoes after removingone of the rows. More precisely, we construct a simple bijection which preserves the size of the largest northeast chainof the fillings when two adjacent rows of the polyomino are exchanged. This bijection also preserves the column sumof the fillings. In addition, we also present a simple bijection that preserves the size of the largest northeast chains, therow sum and the column sum if every row of the filling has at most one 1. Thereby, we not only provide a bijectiveproof of Rubey’s result but also two refinements of it.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2477 https://doi.org/10.46298/dmtcs.2477 Poznanović, Svetlana Yan, Catherine H. Poznanović, Svetlana Yan, Catherine H. <![CDATA[It was proved by Rubey that the number of fillings with zeros and ones of a given moon polyomino thatdo not contain a northeast chain of a fixed size depends only on the set of column lengths of the polyomino. Rubey’sproof uses an adaption of jeu de taquin and promotion for arbitrary fillings of moon polyominoes and deduces theresult for 01-fillings via a variation of the pigeonhole principle. In this paper we present the first completely bijectiveproof of this result by considering fillings of almost-moon polyominoes, which are moon polyominoes after removingone of the rows. More precisely, we construct a simple bijection which preserves the size of the largest northeast chainof the fillings when two adjacent rows of the polyomino are exchanged. This bijection also preserves the column sumof the fillings. In addition, we also present a simple bijection that preserves the size of the largest northeast chains, therow sum and the column sum if every row of the filling has at most one 1. Thereby, we not only provide a bijectiveproof of Rubey’s result but also two refinements of it.]]> 0 <![CDATA[In the past decade, the use of ordinal patterns in the analysis of time series and dynamical systems has become an important tool. Ordinal patterns (otherwise known as a permutation patterns) are found in time series by taking $n$ data points at evenly-spaced time intervals and mapping them to a length-$n$ permutation determined by relative ordering. The frequency with which certain patterns occur is a useful statistic for such series. However, the behavior of the frequency of pattern occurrence is unstudied for most models. We look at the frequency of pattern occurrence in random walks in discrete time, and we define a natural equivalence relation on permutations under which equivalent patterns appear with equal frequency, regardless of probability distribution. We characterize these equivalence classes applying combinatorial methods.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2476 https://doi.org/10.46298/dmtcs.2476 Elizalde, Sergi Martinez, Megan Elizalde, Sergi Martinez, Megan <![CDATA[In the past decade, the use of ordinal patterns in the analysis of time series and dynamical systems has become an important tool. Ordinal patterns (otherwise known as a permutation patterns) are found in time series by taking $n$ data points at evenly-spaced time intervals and mapping them to a length-$n$ permutation determined by relative ordering. The frequency with which certain patterns occur is a useful statistic for such series. However, the behavior of the frequency of pattern occurrence is unstudied for most models. We look at the frequency of pattern occurrence in random walks in discrete time, and we define a natural equivalence relation on permutations under which equivalent patterns appear with equal frequency, regardless of probability distribution. We characterize these equivalence classes applying combinatorial methods.]]> 0 <![CDATA[We introduce the Tesler polytope $Tes_n(a)$, whose integer points are the Tesler matrices of size n with hook sums $a_1,a_2,...,a_n in Z_{\geq 0}$. We show that $Tes_n(a)$ is a flow polytope and therefore the number of Tesler matrices is counted by the type $A_n$ Kostant partition function evaluated at $(a_1,a_2,...,a_n,-\sum_{i=1}^n a_i)$. We describe the faces of this polytope in terms of "Tesler tableaux" and characterize when the polytope is simple. We prove that the h-vector of $Tes_n(a)$ when all $a_i>0$ is given by the Mahonian numbers and calculate the volume of $Tes_n(1,1,...,1)$ to be a product of consecutive Catalan numbers multiplied by the number of standard Young tableaux of staircase shape.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2475 https://doi.org/10.46298/dmtcs.2475 Mészáros, Karola Morales, Alejandro H. Rhoades, Brendon Mészáros, Karola Morales, Alejandro H. Rhoades, Brendon <![CDATA[We introduce the Tesler polytope $Tes_n(a)$, whose integer points are the Tesler matrices of size n with hook sums $a_1,a_2,...,a_n in Z_{\geq 0}$. We show that $Tes_n(a)$ is a flow polytope and therefore the number of Tesler matrices is counted by the type $A_n$ Kostant partition function evaluated at $(a_1,a_2,...,a_n,-\sum_{i=1}^n a_i)$. We describe the faces of this polytope in terms of "Tesler tableaux" and characterize when the polytope is simple. We prove that the h-vector of $Tes_n(a)$ when all $a_i>0$ is given by the Mahonian numbers and calculate the volume of $Tes_n(1,1,...,1)$ to be a product of consecutive Catalan numbers multiplied by the number of standard Young tableaux of staircase shape.]]> 0 <![CDATA[We study tilings with lozenges of a domain with free boundary conditions on one side. These correspondto boxed symmetric plane partitions. We show that the positions of the horizontal lozenges near the left flatboundary, in the limit, have the same joint distribution as the eigenvalues from a Gaussian Unitary Ensemble (theGUE-corners/minors process). We also prove the existence of a limit shape of the height function (the symmetricplane partition). We also consider domains where the sides converge to $\infty$ at different rates and recover again theGUE-corners process.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2474 https://doi.org/10.46298/dmtcs.2474 Panova, Greta Panova, Greta <![CDATA[We study tilings with lozenges of a domain with free boundary conditions on one side. These correspondto boxed symmetric plane partitions. We show that the positions of the horizontal lozenges near the left flatboundary, in the limit, have the same joint distribution as the eigenvalues from a Gaussian Unitary Ensemble (theGUE-corners/minors process). We also prove the existence of a limit shape of the height function (the symmetricplane partition). We also consider domains where the sides converge to $\infty$ at different rates and recover again theGUE-corners process.]]> 0 <![CDATA[To each finite subset of a discrete grid $\mathbb{N}×\mathbb{N}$ (a diagram), one can associate a subvariety of a complex Grassmannian (a diagram variety), and a representation of a symmetric group (a Specht module). Liu has conjectured that the cohomology class of a diagram variety is represented by the Frobenius characteristic of the corresponding Specht module. We give a counterexample to this conjecture.However, we show that for the diagram variety of a permutation diagram, Liu's conjectured cohomology class $\sigma$ is at least an upper bound on the actual class $\tau$, in the sense that $\sigma - \tau$ is a nonnegative linear combination of Schubert classes. To do this, we consider a degeneration of Coskun's rank varieties which contains the appropriate diagram variety as a component. Rank varieties are instances of Knutson-Lam-Speyer's positroid varieties, whose cohomology classes are represented by affine Stanley symmetric functions. We show that the cohomology class of a rank variety is in fact represented by an ordinary Stanley symmetric function.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2462 https://doi.org/10.46298/dmtcs.2462 Pawlowski, Brendan Pawlowski, Brendan <![CDATA[To each finite subset of a discrete grid $\mathbb{N}×\mathbb{N}$ (a diagram), one can associate a subvariety of a complex Grassmannian (a diagram variety), and a representation of a symmetric group (a Specht module). Liu has conjectured that the cohomology class of a diagram variety is represented by the Frobenius characteristic of the corresponding Specht module. We give a counterexample to this conjecture.However, we show that for the diagram variety of a permutation diagram, Liu's conjectured cohomology class $\sigma$ is at least an upper bound on the actual class $\tau$, in the sense that $\sigma - \tau$ is a nonnegative linear combination of Schubert classes. To do this, we consider a degeneration of Coskun's rank varieties which contains the appropriate diagram variety as a component. Rank varieties are instances of Knutson-Lam-Speyer's positroid varieties, whose cohomology classes are represented by affine Stanley symmetric functions. We show that the cohomology class of a rank variety is in fact represented by an ordinary Stanley symmetric function.]]> 0 <![CDATA[Let $I_n$ be the set of involutions in the symmetric group $S_n$, and for $A \subseteq \{0,1,\ldots,n\}$, let \[ F_n^A=\{\sigma \in I_n \mid \text{$\sigma$ has $a$ fixed points for some $a \in A$}\}. \] We give a complete characterisation of the sets $A$ for which $F_n^A$, with the order induced by the Bruhat order on $S_n$, is a graded poset. In particular, we prove that $F_n^{\{1\}}$ (i.e., the set of involutions with exactly one fixed point) is graded, which settles a conjecture of Hultman in the affirmative. When $F_n^A$ is graded, we give its rank function. We also give a short new proof of the EL-shellability of $F_n^{\{0\}}$ (i.e., the set of fixed point-free involutions), which was recently proved by Can, Cherniavsky, and Twelbeck.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2472 https://doi.org/10.46298/dmtcs.2472 Hansson, Mikael Hansson, Mikael <![CDATA[Let $I_n$ be the set of involutions in the symmetric group $S_n$, and for $A \subseteq \{0,1,\ldots,n\}$, let \[ F_n^A=\{\sigma \in I_n \mid \text{$\sigma$ has $a$ fixed points for some $a \in A$}\}. \] We give a complete characterisation of the sets $A$ for which $F_n^A$, with the order induced by the Bruhat order on $S_n$, is a graded poset. In particular, we prove that $F_n^{\{1\}}$ (i.e., the set of involutions with exactly one fixed point) is graded, which settles a conjecture of Hultman in the affirmative. When $F_n^A$ is graded, we give its rank function. We also give a short new proof of the EL-shellability of $F_n^{\{0\}}$ (i.e., the set of fixed point-free involutions), which was recently proved by Can, Cherniavsky, and Twelbeck.]]> 0 <![CDATA[We give an exact enumerative formula for the minimal acyclic deterministic finite automata. This formula is obtained from a bijection between a family of generalized parking functions and the transitions functions of acyclic automata.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2471 https://doi.org/10.46298/dmtcs.2471 Priez, Jean-Baptiste Priez, Jean-Baptiste <![CDATA[We give an exact enumerative formula for the minimal acyclic deterministic finite automata. This formula is obtained from a bijection between a family of generalized parking functions and the transitions functions of acyclic automata.]]> 0 <![CDATA[We provide a new description of the Pieri rule of the homology of the affine Grassmannian and an affineanalogue of the charge statistics in terms of bounded partitions. This makes it possible to extend the formulation ofthe Kostka–Foulkes polynomials in terms of solvable lattice models by Nakayashiki and Yamada to the affine setting.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2470 https://doi.org/10.46298/dmtcs.2470 Morse, Jennifer Schilling, Anne Morse, Jennifer Schilling, Anne <![CDATA[We provide a new description of the Pieri rule of the homology of the affine Grassmannian and an affineanalogue of the charge statistics in terms of bounded partitions. This makes it possible to extend the formulation ofthe Kostka–Foulkes polynomials in terms of solvable lattice models by Nakayashiki and Yamada to the affine setting.]]> 0 <![CDATA[The structure of zero and nonzero minors in the Grassmannian leads to rich combinatorics of matroids. In this paper, we investigate an even richer structure of possible equalities and inequalities between the minors in the positive Grassmannian. It was previously shown that arrangements of equal minors of largest value are in bijection with the simplices in a certain triangulation of the hypersimplex that was studied by Stanley, Sturmfels, Lam and Postnikov. Here we investigate the entire set of arrangements and its relations with this triangulation. First, we show that second largest minors correspond to the facets of the simplices. We then introduce the notion of cubical distance on the dual graph of the triangulation, and study its relations with the arrangement of t-th largest minors. Finally, we show that arrangements of largest minors induce a structure of partially ordered sets on the entire collection of minors. We use the Lam and Postnikov circuit triangulation of the hypersimplex to describe a 2-dimensional grid structure of this poset.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2469 https://doi.org/10.46298/dmtcs.2469 Farber, Miriam Mandelshtam, Yelena Farber, Miriam Mandelshtam, Yelena <![CDATA[The structure of zero and nonzero minors in the Grassmannian leads to rich combinatorics of matroids. In this paper, we investigate an even richer structure of possible equalities and inequalities between the minors in the positive Grassmannian. It was previously shown that arrangements of equal minors of largest value are in bijection with the simplices in a certain triangulation of the hypersimplex that was studied by Stanley, Sturmfels, Lam and Postnikov. Here we investigate the entire set of arrangements and its relations with this triangulation. First, we show that second largest minors correspond to the facets of the simplices. We then introduce the notion of cubical distance on the dual graph of the triangulation, and study its relations with the arrangement of t-th largest minors. Finally, we show that arrangements of largest minors induce a structure of partially ordered sets on the entire collection of minors. We use the Lam and Postnikov circuit triangulation of the hypersimplex to describe a 2-dimensional grid structure of this poset.]]> 0 <![CDATA[We develop a diagrammatic categorification of the polynomial ring $\mathbb{Z} [x]$, based on a geometrically-defined graded algebra and show how to lift various operations on polynomials to the categorified setting. Our categorification satisfies a version of the Bernstein-Gelfand-Gelfand reciprocity property, with indecomposable projective modules corresponding to $x^n$ and standard modules to $(x -1)^n$ in the Grothendieck ring. This construction generalizes tocategorification of various orthogonal polynomials.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2468 https://doi.org/10.46298/dmtcs.2468 Khovanov, Mikhail Sazdanovic, Radmila Khovanov, Mikhail Sazdanovic, Radmila <![CDATA[We develop a diagrammatic categorification of the polynomial ring $\mathbb{Z} [x]$, based on a geometrically-defined graded algebra and show how to lift various operations on polynomials to the categorified setting. Our categorification satisfies a version of the Bernstein-Gelfand-Gelfand reciprocity property, with indecomposable projective modules corresponding to $x^n$ and standard modules to $(x -1)^n$ in the Grothendieck ring. This construction generalizes tocategorification of various orthogonal polynomials.]]> 0 <![CDATA[We derive combinatorial identities for variables satisfying specific sets of commutation relations. The identities thus obtained extend corresponding ones for $q$-commuting variables $x$ and $y$ satisfying $yx=qxy$. In particular, we obtain weight-dependent binomial theorems, functional equations for generalized exponential functions, we propose a derivative of noncommuting variables, and finally utilize one of the considered weight functions to extend rook theory. This leads us to an extension of the $q$-Stirling numbers of the second kind, and of the $q$-Lah numbers.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2467 https://doi.org/10.46298/dmtcs.2467 Schlosser, Michael Yoo, Meesue Schlosser, Michael Yoo, Meesue <![CDATA[We derive combinatorial identities for variables satisfying specific sets of commutation relations. The identities thus obtained extend corresponding ones for $q$-commuting variables $x$ and $y$ satisfying $yx=qxy$. In particular, we obtain weight-dependent binomial theorems, functional equations for generalized exponential functions, we propose a derivative of noncommuting variables, and finally utilize one of the considered weight functions to extend rook theory. This leads us to an extension of the $q$-Stirling numbers of the second kind, and of the $q$-Lah numbers.]]> 0 <![CDATA[Given an irreducible well-generated complex reflection group $W$ with Coxeter number $h$, we call a Coxeter element any regular element (in the sense of Springer) of order $h$ in $W$; this is a slight extension of the most common notion of Coxeter element. We show that the class of these Coxeter elements forms a single orbit in $W$ under the action of reflection automorphisms. For Coxeter and Shephard groups, this implies that an element $c$ is a Coxeter element if and only if there exists a simple system $S$ of reflections such that $c$ is the product of the generators in $S$. We moreover deduce multiple further implications of this property. In particular, we obtain that all noncrossing partition lattices of $W$ associated to different Coxeter elements are isomorphic. We also prove that there is a simply transitive action of the Galois group of the field of definition of $W$ on the set of conjugacy classes of Coxeter elements. Finally, we extend several of these properties to Springer's regular elements of arbitrary order.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2466 https://doi.org/10.46298/dmtcs.2466 Reiner, Victor Ripoll, Vivien Stump, Christian Reiner, Victor Ripoll, Vivien Stump, Christian <![CDATA[Given an irreducible well-generated complex reflection group $W$ with Coxeter number $h$, we call a Coxeter element any regular element (in the sense of Springer) of order $h$ in $W$; this is a slight extension of the most common notion of Coxeter element. We show that the class of these Coxeter elements forms a single orbit in $W$ under the action of reflection automorphisms. For Coxeter and Shephard groups, this implies that an element $c$ is a Coxeter element if and only if there exists a simple system $S$ of reflections such that $c$ is the product of the generators in $S$. We moreover deduce multiple further implications of this property. In particular, we obtain that all noncrossing partition lattices of $W$ associated to different Coxeter elements are isomorphic. We also prove that there is a simply transitive action of the Galois group of the field of definition of $W$ on the set of conjugacy classes of Coxeter elements. Finally, we extend several of these properties to Springer's regular elements of arbitrary order.]]> 0 <![CDATA[We present a new method to obtain the generating functions for directed convex polyominoes according to several different statistics including: width, height, size of last column/row and number of corners. This method can be used to study different families of directed convex polyominoes: symmetric polyominoes, parallelogram polyominoes. In this paper, we apply our method to determine the generating function for directed $k$-convex polyominoes.We show it is a rational function and we study its asymptotic behavior.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2465 https://doi.org/10.46298/dmtcs.2465 Boussicault, Adrien Rinaldi, Simone Socci, Samanta Boussicault, Adrien Rinaldi, Simone Socci, Samanta <![CDATA[We present a new method to obtain the generating functions for directed convex polyominoes according to several different statistics including: width, height, size of last column/row and number of corners. This method can be used to study different families of directed convex polyominoes: symmetric polyominoes, parallelogram polyominoes. In this paper, we apply our method to determine the generating function for directed $k$-convex polyominoes.We show it is a rational function and we study its asymptotic behavior.]]> 0 <![CDATA[We show that several of the main structural constants for symmetric functions (Littlewood-Richardsoncoefficients, Kronecker coefficients, plethysm coefficients, and the Kostka–Foulkes polynomials) share invarianceproperties related to the operations of taking complements with respect to rectangles and adding rectangles.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2464 https://doi.org/10.46298/dmtcs.2464 Briand, Emmanuel Orellana, Rosa Rosas, Mercedes Briand, Emmanuel Orellana, Rosa Rosas, Mercedes <![CDATA[We show that several of the main structural constants for symmetric functions (Littlewood-Richardsoncoefficients, Kronecker coefficients, plethysm coefficients, and the Kostka–Foulkes polynomials) share invarianceproperties related to the operations of taking complements with respect to rectangles and adding rectangles.]]> 0 <![CDATA[We extend the classification of nearest neighbour walks in the quarter plane to models in which multiplicities are attached to each direction in the step set. Our study leads to a small number of infinite families that completely characterize all the models whose associated group is D4, D6, or D8. These families cover all the models with multiplicites 0, 1, 2, or 3, which were experimentally found to be D-finite — with three noteworthy exceptions.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2463 https://doi.org/10.46298/dmtcs.2463 Kauers, Manuel Yatchak, Rika Kauers, Manuel Yatchak, Rika <![CDATA[We extend the classification of nearest neighbour walks in the quarter plane to models in which multiplicities are attached to each direction in the step set. Our study leads to a small number of infinite families that completely characterize all the models whose associated group is D4, D6, or D8. These families cover all the models with multiplicites 0, 1, 2, or 3, which were experimentally found to be D-finite — with three noteworthy exceptions.]]> 0 <![CDATA[In the present paper, the relation between the dominant regions in the $m$-Shi arrangement of types $B_n/C_n$, and those of the $m$-Shi arrangement of type $A_{n-1}$ is investigated. More precisely, it is shown explicitly how the sets $R^m(B_n)$ and $R^m(C_n)$, of dominant regions of the $m$-Shi arrangement of types $B_n$ and $C_n$ respectively, can be projected to the set $R^m(A_{n-1})$ of dominant regions of the $m$-Shi arrangement of type $A_{n-1}$. This is done by using two different viewpoints for the representative alcoves of these regions: the Shi tableaux and the abacus diagrams. Moreover, bijections between the sets $R^m(B_n)$, $R^m(C_n)$, and lattice paths inside a rectangle $n\times{mn}$ are provided.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2495 https://doi.org/10.46298/dmtcs.2495 Kallipoliti, Myrto Tzanaki, Eleni Kallipoliti, Myrto Tzanaki, Eleni <![CDATA[In the present paper, the relation between the dominant regions in the $m$-Shi arrangement of types $B_n/C_n$, and those of the $m$-Shi arrangement of type $A_{n-1}$ is investigated. More precisely, it is shown explicitly how the sets $R^m(B_n)$ and $R^m(C_n)$, of dominant regions of the $m$-Shi arrangement of types $B_n$ and $C_n$ respectively, can be projected to the set $R^m(A_{n-1})$ of dominant regions of the $m$-Shi arrangement of type $A_{n-1}$. This is done by using two different viewpoints for the representative alcoves of these regions: the Shi tableaux and the abacus diagrams. Moreover, bijections between the sets $R^m(B_n)$, $R^m(C_n)$, and lattice paths inside a rectangle $n\times{mn}$ are provided.]]> 0 <![CDATA[We consider a Hopf algebra of simplicial complexes and provide a cancellation-free formula for its antipode. We then obtain a family of combinatorial Hopf algebras by defining a family of characters on this Hopf algebra. The characters of these Hopf algebras give rise to symmetric functions that encode information about colorings of simplicial complexes and their $f$-vectors. We also use characters to give a generalization of Stanley’s $(-1)$-color theorem.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2506 https://doi.org/10.46298/dmtcs.2506 Benedetti, Carolina Hallam, Joshua Machacek, John Benedetti, Carolina Hallam, Joshua Machacek, John <![CDATA[We consider a Hopf algebra of simplicial complexes and provide a cancellation-free formula for its antipode. We then obtain a family of combinatorial Hopf algebras by defining a family of characters on this Hopf algebra. The characters of these Hopf algebras give rise to symmetric functions that encode information about colorings of simplicial complexes and their $f$-vectors. We also use characters to give a generalization of Stanley’s $(-1)$-color theorem.]]> 0 <![CDATA[There are few general results about the coefficients of Ehrhart polynomials. We present a conjecture about their positivity for a certain family of polytopes known as generalized permutohedra. We have verified the conjecture for small dimensions combining perturbation methods with a new valuation on the algebra of rational pointed polyhedral cones constructed by Berline and Vergne.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2505 https://doi.org/10.46298/dmtcs.2505 Castillo, Federico Liu, Fu Castillo, Federico Liu, Fu <![CDATA[There are few general results about the coefficients of Ehrhart polynomials. We present a conjecture about their positivity for a certain family of polytopes known as generalized permutohedra. We have verified the conjecture for small dimensions combining perturbation methods with a new valuation on the algebra of rational pointed polyhedral cones constructed by Berline and Vergne.]]> 0 <![CDATA[We give a recursive definition of generalized parking functions that allows them to be viewed as a species. From there we compute a non-commutative characteristic of the generalized parking function module and deduce some enumeration formulas of structures and isomorphism types. We give as well an interpretation in several bases of non commutative symmetric functions. Finally, we investigate an inclusion-exclusion formula given by Kung and Yan.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2504 https://doi.org/10.46298/dmtcs.2504 Priez, Jean-Baptiste Virmaux, Aladin Priez, Jean-Baptiste Virmaux, Aladin <![CDATA[We give a recursive definition of generalized parking functions that allows them to be viewed as a species. From there we compute a non-commutative characteristic of the generalized parking function module and deduce some enumeration formulas of structures and isomorphism types. We give as well an interpretation in several bases of non commutative symmetric functions. Finally, we investigate an inclusion-exclusion formula given by Kung and Yan.]]> 0 <![CDATA[The notion of the negative $q$-binomial was recently introduced by Fu, Reiner, Stanton and Thiem. Mirroring the negative $q$-binomial, we show the classical $q$ -Stirling numbers of the second kind can be expressed as a pair of statistics on a subset of restricted growth words. The resulting expressions are polynomials in $q$ and $(1+q)$. We extend this enumerative result via a decomposition of the Stirling poset, as well as a homological version of Stembridge’s $q=-1$ phenomenon. A parallel enumerative, poset theoretic and homological study for the $q$-Stirling numbers of the first kind is done beginning with de Médicis and Leroux’s rook placement formulation. Letting $t=1+q$ we give a bijective combinatorial argument à la Viennot showing the $(q; t)$-Stirling numbers of the first and second kind are orthogonal.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2503 https://doi.org/10.46298/dmtcs.2503 Cai, Yue Readdy, Margaret Cai, Yue Readdy, Margaret <![CDATA[The notion of the negative $q$-binomial was recently introduced by Fu, Reiner, Stanton and Thiem. Mirroring the negative $q$-binomial, we show the classical $q$ -Stirling numbers of the second kind can be expressed as a pair of statistics on a subset of restricted growth words. The resulting expressions are polynomials in $q$ and $(1+q)$. We extend this enumerative result via a decomposition of the Stirling poset, as well as a homological version of Stembridge’s $q=-1$ phenomenon. A parallel enumerative, poset theoretic and homological study for the $q$-Stirling numbers of the first kind is done beginning with de Médicis and Leroux’s rook placement formulation. Letting $t=1+q$ we give a bijective combinatorial argument à la Viennot showing the $(q; t)$-Stirling numbers of the first and second kind are orthogonal.]]> 0 <![CDATA[An important combinatorial result in equivariant cohomology and $K$-theory Schubert calculus is represented by the formulas of Billey and Graham-Willems for the localization of Schubert classes at torus fixed points. These formulas work uniformly in all Lie types, and are based on the concept of a root polynomial. We define formal root polynomials associated with an arbitrary formal group law (and thus a generalized cohomology theory). We usethese polynomials to simplify the approach of Billey and Graham-Willems, as well as to generalize it to connective $K$-theory and elliptic cohomology. Another result is concerned with defining a Schubert basis in elliptic cohomology (i.e., classes independent of a reduced word), using the Kazhdan-Lusztig basis of the corresponding Hecke algebra.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2502 https://doi.org/10.46298/dmtcs.2502 Lenart, Cristian Zainoulline, Kirill Lenart, Cristian Zainoulline, Kirill <![CDATA[An important combinatorial result in equivariant cohomology and $K$-theory Schubert calculus is represented by the formulas of Billey and Graham-Willems for the localization of Schubert classes at torus fixed points. These formulas work uniformly in all Lie types, and are based on the concept of a root polynomial. We define formal root polynomials associated with an arbitrary formal group law (and thus a generalized cohomology theory). We usethese polynomials to simplify the approach of Billey and Graham-Willems, as well as to generalize it to connective $K$-theory and elliptic cohomology. Another result is concerned with defining a Schubert basis in elliptic cohomology (i.e., classes independent of a reduced word), using the Kazhdan-Lusztig basis of the corresponding Hecke algebra.]]> 0 <![CDATA[We analyze a general model of weighted graphs, introduced by de Panafieu and Ravelomanana (2014) and similar to the inhomogeneous graph model of Söderberg (2002). We investigate the sum of the weights of those graphs and their structure. Those results allow us to give a new proof in a more general setting of a theorem of Wright (1972) on the enumeration of properly colored graphs. We also discuss applications related to social networks]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2501 https://doi.org/10.46298/dmtcs.2501 De Panafieu, Élie De Panafieu, Élie <![CDATA[We analyze a general model of weighted graphs, introduced by de Panafieu and Ravelomanana (2014) and similar to the inhomogeneous graph model of Söderberg (2002). We investigate the sum of the weights of those graphs and their structure. Those results allow us to give a new proof in a more general setting of a theorem of Wright (1972) on the enumeration of properly colored graphs. We also discuss applications related to social networks]]> 0 <![CDATA[We introduce a new statistic, skip, on rational $(3,n)$-Dyck paths and define a marked rank word for each path when $n$ is not a multiple of 3. If a triple of valid statistics (area; skip; dinv) are given, we have an algorithm to construct the marked rank word corresponding to the triple. By considering all valid triples we give an explicit formula for the $(m,n)$-rational $q; t$-Catalan polynomials when $m=3$. Then there is a natural bijection on the triples of statistics (area; skip; dinv) which exchanges the statistics area and dinv while fixing the skip. Thus we prove the $q; t$-symmetry of $(m,n)$-rational $q; t$-Catalan polynomials for $m=3$..]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2500 https://doi.org/10.46298/dmtcs.2500 Kaliszewski, Ryan Li, Huilan Kaliszewski, Ryan Li, Huilan <![CDATA[We introduce a new statistic, skip, on rational $(3,n)$-Dyck paths and define a marked rank word for each path when $n$ is not a multiple of 3. If a triple of valid statistics (area; skip; dinv) are given, we have an algorithm to construct the marked rank word corresponding to the triple. By considering all valid triples we give an explicit formula for the $(m,n)$-rational $q; t$-Catalan polynomials when $m=3$. Then there is a natural bijection on the triples of statistics (area; skip; dinv) which exchanges the statistics area and dinv while fixing the skip. Thus we prove the $q; t$-symmetry of $(m,n)$-rational $q; t$-Catalan polynomials for $m=3$..]]> 0 <![CDATA[If $f(x)$ is an invertible power series we may form the symmetric function $f(f^{-1}(x_1)+f^{-1}(x_2)+...)$ which is called a formal group law. We give a number of examples of power series $f(x)$ that are ordinary generating functions for combinatorial objects with a recursive structure, each of which is associated with a certain hypergraph. In each case, we show that the corresponding formal group law is the sum of the chromatic symmetric functions of these hypergraphs by finding a combinatorial interpretation for $f^{-1}(x)$. We conjecture that the chromatic symmetric functions arising in this way are Schur-positive.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2499 https://doi.org/10.46298/dmtcs.2499 Taylor, Jair Taylor, Jair <![CDATA[If $f(x)$ is an invertible power series we may form the symmetric function $f(f^{-1}(x_1)+f^{-1}(x_2)+...)$ which is called a formal group law. We give a number of examples of power series $f(x)$ that are ordinary generating functions for combinatorial objects with a recursive structure, each of which is associated with a certain hypergraph. In each case, we show that the corresponding formal group law is the sum of the chromatic symmetric functions of these hypergraphs by finding a combinatorial interpretation for $f^{-1}(x)$. We conjecture that the chromatic symmetric functions arising in this way are Schur-positive.]]> 0 <![CDATA[In 2008, Han rediscovered an expansion of powers of Dedekind $\eta$ function due to Nekrasov and Okounkov by using Macdonald's identity in type $\widetilde{A}$. In this paper, we obtain new combinatorial expansions of powers of $\eta$, in terms of partition hook lengths, by using Macdonald's identity in type $\widetilde{C}$ and a new bijection. As applications, we derive a symplectic hook formula and a relation between Macdonald's identities in types $\widetilde{C}$, $\widetilde{B}$, and $\widetilde{BC}$.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2498 https://doi.org/10.46298/dmtcs.2498 Pétréolle, Mathias Pétréolle, Mathias <![CDATA[In 2008, Han rediscovered an expansion of powers of Dedekind $\eta$ function due to Nekrasov and Okounkov by using Macdonald's identity in type $\widetilde{A}$. In this paper, we obtain new combinatorial expansions of powers of $\eta$, in terms of partition hook lengths, by using Macdonald's identity in type $\widetilde{C}$ and a new bijection. As applications, we derive a symplectic hook formula and a relation between Macdonald's identities in types $\widetilde{C}$, $\widetilde{B}$, and $\widetilde{BC}$.]]> 0 <![CDATA[The Schur functions in superspace $s_\Lambda$ and $\overline{s}_\Lambda$ are the limits $q=t= 0$ and $q=t=\infty$ respectively of the Macdonald polynomials in superspace. We present the elementary properties of the bases $s_\Lambda$ and $\overline{s}_\Lambda$ (which happen to be essentially dual) such as Pieri rules, dualities, monomial expansions, tableaux generating functions, and Cauchy identities.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2497 https://doi.org/10.46298/dmtcs.2497 Jones, Miles Eli Lapointe, Luc Jones, Miles Eli Lapointe, Luc <![CDATA[The Schur functions in superspace $s_\Lambda$ and $\overline{s}_\Lambda$ are the limits $q=t= 0$ and $q=t=\infty$ respectively of the Macdonald polynomials in superspace. We present the elementary properties of the bases $s_\Lambda$ and $\overline{s}_\Lambda$ (which happen to be essentially dual) such as Pieri rules, dualities, monomial expansions, tableaux generating functions, and Cauchy identities.]]> 0 <![CDATA[We introduce type $C$ parking functions, encoded as vertically labelled lattice paths and endowed with a statistic dinv'. We define a bijection from type $C$ parking functions to regions of the Shi arrangement of type $C$, encoded as diagonally labelled ballot paths and endowed with a natural statistic area'. This bijection is a natural analogue of the zeta map of Haglund and Loehr and maps dinv' to area'. We give three different descriptions of it.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2496 https://doi.org/10.46298/dmtcs.2496 Sulzgruber, Robin Thiel, Marko Sulzgruber, Robin Thiel, Marko <![CDATA[We introduce type $C$ parking functions, encoded as vertically labelled lattice paths and endowed with a statistic dinv'. We define a bijection from type $C$ parking functions to regions of the Shi arrangement of type $C$, encoded as diagonally labelled ballot paths and endowed with a natural statistic area'. This bijection is a natural analogue of the zeta map of Haglund and Loehr and maps dinv' to area'. We give three different descriptions of it.]]> 0 <![CDATA[Alignments, crossings and inversions of signed permutations are realized in the corresponding permutation tableaux of type $B$, and the cycles of signed permutations are understood in the corresponding bare tableaux of type $B$. We find the relation between the number of alignments, crossings and other statistics of signed permutations, and also characterize the covering relation in weak Bruhat order on Coxeter system of type $B$ in terms of permutation tableaux of type $B$.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2484 https://doi.org/10.46298/dmtcs.2484 Cho, Soojin Park, Kyoungsuk Cho, Soojin Park, Kyoungsuk <![CDATA[Alignments, crossings and inversions of signed permutations are realized in the corresponding permutation tableaux of type $B$, and the cycles of signed permutations are understood in the corresponding bare tableaux of type $B$. We find the relation between the number of alignments, crossings and other statistics of signed permutations, and also characterize the covering relation in weak Bruhat order on Coxeter system of type $B$ in terms of permutation tableaux of type $B$.]]> 0 <![CDATA[We give a statistic preserving bijection from rigged configurations to a tensor product of Kirillov–Reshetikhin crystals $\otimes_{i=1}^{N}B^{1,s_i}$ in type $D_4^{(3)}$ by using virtualization into type $D_4^{(1)}$. We consider a special case of this bijection with $B=B^{1,s}$, and we obtain the so-called Kirillov–Reshetikhin tableaux model for the Kirillov–Reshetikhin crystal.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2494 https://doi.org/10.46298/dmtcs.2494 Scrimshaw, Travis Scrimshaw, Travis <![CDATA[We give a statistic preserving bijection from rigged configurations to a tensor product of Kirillov–Reshetikhin crystals $\otimes_{i=1}^{N}B^{1,s_i}$ in type $D_4^{(3)}$ by using virtualization into type $D_4^{(1)}$. We consider a special case of this bijection with $B=B^{1,s}$, and we obtain the so-called Kirillov–Reshetikhin tableaux model for the Kirillov–Reshetikhin crystal.]]> 0 <![CDATA[We give a new representation-theoretic proof of the branching rule for Macdonald polynomials using the Etingof-Kirillov Jr. expression for Macdonald polynomials as traces of intertwiners of $U_q(gl_n)$. In the Gelfand-Tsetlin basis, we show that diagonal matrix elements of such intertwiners are given by application of Macdonald's operators to a simple kernel. An essential ingredient in the proof is a map between spherical parts of double affine Hecke algebras of different ranks based upon the Dunkl-Kasatani conjecture.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2493 https://doi.org/10.46298/dmtcs.2493 Sun, Yi Sun, Yi <![CDATA[We give a new representation-theoretic proof of the branching rule for Macdonald polynomials using the Etingof-Kirillov Jr. expression for Macdonald polynomials as traces of intertwiners of $U_q(gl_n)$. In the Gelfand-Tsetlin basis, we show that diagonal matrix elements of such intertwiners are given by application of Macdonald's operators to a simple kernel. An essential ingredient in the proof is a map between spherical parts of double affine Hecke algebras of different ranks based upon the Dunkl-Kasatani conjecture.]]> 0 <![CDATA[We explore the enumeration of some natural classes of graded posets, including $(2 + 2)$-avoiding graded posets, $(3 + 1)$-avoiding graded posets, $(2 + 2)$- and $(3 + 1)$-avoiding graded posets, and the set of all graded posets. As part of this story, we discuss a situation when we can switch between enumeration of labeled and unlabeled objects with ease, which helps us generalize a result by Postnikov and Stanley from the theory of hyperplane arrangements, answer a question posed by Stanley, and see an old result of Klarner in a new light.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2492 https://doi.org/10.46298/dmtcs.2492 Zhang, Yan X Zhang, Yan X <![CDATA[We explore the enumeration of some natural classes of graded posets, including $(2 + 2)$-avoiding graded posets, $(3 + 1)$-avoiding graded posets, $(2 + 2)$- and $(3 + 1)$-avoiding graded posets, and the set of all graded posets. As part of this story, we discuss a situation when we can switch between enumeration of labeled and unlabeled objects with ease, which helps us generalize a result by Postnikov and Stanley from the theory of hyperplane arrangements, answer a question posed by Stanley, and see an old result of Klarner in a new light.]]> 0 <![CDATA[Kirillov-Reshetikhin (KR) crystals are colored directed graphs encoding the structure of certain finite-dimensional representations of affine Lie algebras. A tensor product of column shape KR crystals has recently been realized in a uniform way, for all untwisted affine types, in terms of the quantum alcove model. We enhance this model by using it to give a uniform realization of the combinatorial $R$-matrix, i.e., the unique affine crystal isomorphism permuting factors in a tensor product of KR crystals. In other words, we are generalizing to all Lie types Schützenberger’s sliding game (jeu de taquin) for Young tableaux, which realizes the combinatorial $R$-matrix in type $A$. We also show that the quantum alcove model does not depend on the choice of a sequence of alcoves]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2491 https://doi.org/10.46298/dmtcs.2491 Lenart, Cristian Lubovsky, Arthur Lenart, Cristian Lubovsky, Arthur <![CDATA[Kirillov-Reshetikhin (KR) crystals are colored directed graphs encoding the structure of certain finite-dimensional representations of affine Lie algebras. A tensor product of column shape KR crystals has recently been realized in a uniform way, for all untwisted affine types, in terms of the quantum alcove model. We enhance this model by using it to give a uniform realization of the combinatorial $R$-matrix, i.e., the unique affine crystal isomorphism permuting factors in a tensor product of KR crystals. In other words, we are generalizing to all Lie types Schützenberger’s sliding game (jeu de taquin) for Young tableaux, which realizes the combinatorial $R$-matrix in type $A$. We also show that the quantum alcove model does not depend on the choice of a sequence of alcoves]]> 0 <![CDATA[A parametrization of a positroid variety $\Pi$ of dimension $d$ is a regular map $(\mathbb{C}^{\times})^{d} \rightarrow \Pi$ which is birational onto a dense subset of $\Pi$. There are several remarkable combinatorial constructions which yield parametrizations of positroid varieties. We investigate the relationship between two families of such parametrizations, and prove they are essentially the same. Our first family is defined in terms of Postnikov’s boundary measurement map, and the domain of each parametrization is the space of edge weights of a planar network. We focus on a special class of planar networks called bridge graphs, which have applications to particle physics. Our second family arises from Marsh and Rietsch’s parametrizations of Deodhar components of the flag variety, which are indexed by certain subexpressions of reduced words. Projecting to the Grassmannian gives a family of parametrizations for each positroid variety. We show that each Deodhar parametrization for a positroid variety corresponds to a bridge graph, while each parametrization from a bridge graph agrees with some projected Deodhar parametrization.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2490 https://doi.org/10.46298/dmtcs.2490 Karpman, Rachel Karpman, Rachel <![CDATA[A parametrization of a positroid variety $\Pi$ of dimension $d$ is a regular map $(\mathbb{C}^{\times})^{d} \rightarrow \Pi$ which is birational onto a dense subset of $\Pi$. There are several remarkable combinatorial constructions which yield parametrizations of positroid varieties. We investigate the relationship between two families of such parametrizations, and prove they are essentially the same. Our first family is defined in terms of Postnikov’s boundary measurement map, and the domain of each parametrization is the space of edge weights of a planar network. We focus on a special class of planar networks called bridge graphs, which have applications to particle physics. Our second family arises from Marsh and Rietsch’s parametrizations of Deodhar components of the flag variety, which are indexed by certain subexpressions of reduced words. Projecting to the Grassmannian gives a family of parametrizations for each positroid variety. We show that each Deodhar parametrization for a positroid variety corresponds to a bridge graph, while each parametrization from a bridge graph agrees with some projected Deodhar parametrization.]]> 0 <![CDATA[We generalize previous definitions of Tesler matrices to allow negative matrix entries and non-positive hook sums. Our main result is an algebraic interpretation of a certain weighted sum over these matrices. Our interpretation uses virtual Hilbert series, a new class of symmetric function specializations which are defined by their values on (modified) Macdonald polynomials. As a result of this interpretation, we obtain a Tesler matrix expression for the Hall inner product $\langle \Delta_f e_n, p_{1^{n}}\rangle$, where $\Delta_f$ is a symmetric function operator from the theory of diagonal harmonics. We use our Tesler matrix expression, along with various facts about Tesler matrices, to provide simple formulas for $\langle \Delta_{e_1} e_n, p_{1^{n}}\rangle$ and $\langle \Delta_{e_k} e_n, p_{1^{n}}\rangle \mid_{t=0}$ involving $q; t$-binomial coefficients and ordered set partitions, respectively.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2489 https://doi.org/10.46298/dmtcs.2489 Wilson, Andrew Timothy Wilson, Andrew Timothy <![CDATA[We generalize previous definitions of Tesler matrices to allow negative matrix entries and non-positive hook sums. Our main result is an algebraic interpretation of a certain weighted sum over these matrices. Our interpretation uses virtual Hilbert series, a new class of symmetric function specializations which are defined by their values on (modified) Macdonald polynomials. As a result of this interpretation, we obtain a Tesler matrix expression for the Hall inner product $\langle \Delta_f e_n, p_{1^{n}}\rangle$, where $\Delta_f$ is a symmetric function operator from the theory of diagonal harmonics. We use our Tesler matrix expression, along with various facts about Tesler matrices, to provide simple formulas for $\langle \Delta_{e_1} e_n, p_{1^{n}}\rangle$ and $\langle \Delta_{e_k} e_n, p_{1^{n}}\rangle \mid_{t=0}$ involving $q; t$-binomial coefficients and ordered set partitions, respectively.]]> 0 <![CDATA[In this extended abstract we consider mixed volumes of combinations of hypersimplices. These numbers, called mixed Eulerian numbers, were first considered by A. Postnikov and were shown to satisfy many properties related to Eulerian numbers, Catalan numbers, binomial coefficients, etc. We give a general combinatorial interpretation for mixed Eulerian numbers and prove the above properties combinatorially. In particular, we show that each mixed Eulerian number enumerates a certain set of permutations in $S_n$. We also prove several new properties of mixed Eulerian numbers using our methods. Finally, we consider a type $B$ analogue of mixed Eulerian numbers and give an analogous combinatorial interpretation for these numbers.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2488 https://doi.org/10.46298/dmtcs.2488 Liu, Gaku Liu, Gaku <![CDATA[In this extended abstract we consider mixed volumes of combinations of hypersimplices. These numbers, called mixed Eulerian numbers, were first considered by A. Postnikov and were shown to satisfy many properties related to Eulerian numbers, Catalan numbers, binomial coefficients, etc. We give a general combinatorial interpretation for mixed Eulerian numbers and prove the above properties combinatorially. In particular, we show that each mixed Eulerian number enumerates a certain set of permutations in $S_n$. We also prove several new properties of mixed Eulerian numbers using our methods. Finally, we consider a type $B$ analogue of mixed Eulerian numbers and give an analogous combinatorial interpretation for these numbers.]]> 0 <![CDATA[Triangular fully packed loop configurations (TFPLs) came up in the study of fully packed loop configurations on a square (FPLs) corresponding to link patterns with a large number of nested arches. To a TFPL is assigned a triple $(u,v;w)$ of $01$-words encoding its boundary conditions. A necessary condition for the boundary $(u,v;w)$ of a TFPL is $\lvert \lambda(u) \rvert +\lvert \lambda(v) \rvert \leq \lvert \lambda(w) \rvert$, where $\lambda(u)$ denotes the Young diagram associated with the $01$-word $u$. Wieland gyration, on the other hand, was invented to show the rotational invariance of the numbers $A_\pi$ of FPLs corresponding to a given link pattern $\pi$. Later, Wieland drift was defined as the natural adaption of Wieland gyration to TFPLs. The main contribution of this article is a linear expression for the number of TFPLs with boundary $(u,v;w)$ where $\lvert \lambda (w) \rvert - \lvert\lambda (u) \rvert - \lvert \lambda (v)\rvert \leq 2$ in terms of numbers of stable TFPLs that is TFPLs invariant under Wieland drift. These stable TFPLs have boundary $(u^{+},v^{+};w)$ for words $u^{+}$ and $v^{+}$ such that $\lambda (u) \subseteq \lambda (u^{+})$ and $\lambda (v) \subseteq \lambda (v^{+})$.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2487 https://doi.org/10.46298/dmtcs.2487 Beil, Sabine Beil, Sabine <![CDATA[Triangular fully packed loop configurations (TFPLs) came up in the study of fully packed loop configurations on a square (FPLs) corresponding to link patterns with a large number of nested arches. To a TFPL is assigned a triple $(u,v;w)$ of $01$-words encoding its boundary conditions. A necessary condition for the boundary $(u,v;w)$ of a TFPL is $\lvert \lambda(u) \rvert +\lvert \lambda(v) \rvert \leq \lvert \lambda(w) \rvert$, where $\lambda(u)$ denotes the Young diagram associated with the $01$-word $u$. Wieland gyration, on the other hand, was invented to show the rotational invariance of the numbers $A_\pi$ of FPLs corresponding to a given link pattern $\pi$. Later, Wieland drift was defined as the natural adaption of Wieland gyration to TFPLs. The main contribution of this article is a linear expression for the number of TFPLs with boundary $(u,v;w)$ where $\lvert \lambda (w) \rvert - \lvert\lambda (u) \rvert - \lvert \lambda (v)\rvert \leq 2$ in terms of numbers of stable TFPLs that is TFPLs invariant under Wieland drift. These stable TFPLs have boundary $(u^{+},v^{+};w)$ for words $u^{+}$ and $v^{+}$ such that $\lambda (u) \subseteq \lambda (u^{+})$ and $\lambda (v) \subseteq \lambda (v^{+})$.]]> 0 <![CDATA[We study the growth rate of the hard squares lattice gas, equivalent to the number of independent sets on the square lattice, and two related models — non-attacking kings and read-write isolated memory. We use an assortment of techniques from combinatorics, statistical mechanics and linear algebra to prove upper bounds on these growth rates. We start from Calkin and Wilf’s transfer matrix eigenvalue bound, then bound that with the Collatz-Wielandt formula from linear algebra. To obtain an approximate eigenvector, we use an ansatz from Baxter’s corner transfer matrix formalism, optimised with Nishino and Okunishi’s corner transfer matrix renormalisation group method. This results in an upper bound algorithm which no longer requires exponential memory and so is much faster to calculate than a direct evaluation of the Calkin-Wilf bound. Furthermore, it is extremely parallelisable and so allows us to make dramatic improvements to the previous best known upper bounds. In all cases we reduce the gap between upper and lower bounds by 4-6 orders of magnitude.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2486 https://doi.org/10.46298/dmtcs.2486 Chan, Yao-Ban Chan, Yao-Ban <![CDATA[We study the growth rate of the hard squares lattice gas, equivalent to the number of independent sets on the square lattice, and two related models — non-attacking kings and read-write isolated memory. We use an assortment of techniques from combinatorics, statistical mechanics and linear algebra to prove upper bounds on these growth rates. We start from Calkin and Wilf’s transfer matrix eigenvalue bound, then bound that with the Collatz-Wielandt formula from linear algebra. To obtain an approximate eigenvector, we use an ansatz from Baxter’s corner transfer matrix formalism, optimised with Nishino and Okunishi’s corner transfer matrix renormalisation group method. This results in an upper bound algorithm which no longer requires exponential memory and so is much faster to calculate than a direct evaluation of the Calkin-Wilf bound. Furthermore, it is extremely parallelisable and so allows us to make dramatic improvements to the previous best known upper bounds. In all cases we reduce the gap between upper and lower bounds by 4-6 orders of magnitude.]]> 0 <![CDATA[The Ish arrangement was introduced by Armstrong to give a new interpretation of the $q; t$-Catalan numbers of Garsia and Haiman. Armstrong and Rhoades showed that there are some striking similarities between the Shi arrangement and the Ish arrangement and posed some problems. One of them is whether the Ish arrangement is a free arrangement or not. In this paper, we verify that the Ish arrangement is supersolvable and hence free. Moreover, we give a necessary and sufficient condition for the deleted Ish arrangement to be free]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2485 https://doi.org/10.46298/dmtcs.2485 Abe, Takuro Suyama, Daisuke Tsujie, Shuhei Abe, Takuro Suyama, Daisuke Tsujie, Shuhei <![CDATA[The Ish arrangement was introduced by Armstrong to give a new interpretation of the $q; t$-Catalan numbers of Garsia and Haiman. Armstrong and Rhoades showed that there are some striking similarities between the Shi arrangement and the Ish arrangement and posed some problems. One of them is whether the Ish arrangement is a free arrangement or not. In this paper, we verify that the Ish arrangement is supersolvable and hence free. Moreover, we give a necessary and sufficient condition for the deleted Ish arrangement to be free]]> 0 <![CDATA[We study the uniform random graph $\mathsf{C}_n$ with $n$ vertices drawn from a subcritical class of connected graphs. Our main result is that the rescaled graph $\mathsf{C}_n / \sqrt{n}$ converges to the Brownian Continuum Random Tree $\mathcal{T}_{\mathsf{e}}$ multiplied by a constant scaling factor that depends on the class under consideration. In addition, we provide subgaussian tail bounds for the diameter $\text{D}(\mathsf{C}_n)$ and height $\text{H}(\mathsf{C}_n^\bullet)$ of the rooted random graph $\mathsf{C}_n^\bullet$. We give analytic expressions for the scaling factor of several classes, including for example the prominent class of outerplanar graphs. Our methods also enable us to study first passage percolation on $\mathsf{C}_n$, where we show the convergence to $\mathcal{T}_{\mathsf{e}}$ under an appropriate rescaling.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2461 https://doi.org/10.46298/dmtcs.2461 Panagiotou, Konstantinos Stufler, Benedikt Weller, Kerstin Panagiotou, Konstantinos Stufler, Benedikt Weller, Kerstin <![CDATA[We study the uniform random graph $\mathsf{C}_n$ with $n$ vertices drawn from a subcritical class of connected graphs. Our main result is that the rescaled graph $\mathsf{C}_n / \sqrt{n}$ converges to the Brownian Continuum Random Tree $\mathcal{T}_{\mathsf{e}}$ multiplied by a constant scaling factor that depends on the class under consideration. In addition, we provide subgaussian tail bounds for the diameter $\text{D}(\mathsf{C}_n)$ and height $\text{H}(\mathsf{C}_n^\bullet)$ of the rooted random graph $\mathsf{C}_n^\bullet$. We give analytic expressions for the scaling factor of several classes, including for example the prominent class of outerplanar graphs. Our methods also enable us to study first passage percolation on $\mathsf{C}_n$, where we show the convergence to $\mathcal{T}_{\mathsf{e}}$ under an appropriate rescaling.]]> 0 <![CDATA[The Tamari order is a central object in algebraic combinatorics and many other areas. Defined as the transitive closure of an associativity law, the Tamari order possesses a surprisingly rich structure: it is a congruence-uniform lattice. In this work, we consider a larger class of posets, the Grassmann-Tamari orders, which arise as an ordering on the facets of the non-crossing complex introduced by Pylyavskyy, Petersen, and Speyer. We prove that the Grassmann-Tamari orders are congruence-uniform lattices, which resolves a conjecture of Santos, Stump, and Welker. Towards this goal, we define a closure operator on sets of paths inside a rectangle, and prove that the biclosed sets of paths, ordered by inclusion, form a congruence-uniform lattice. We then prove that the Grassmann-Tamari order is a quotient lattice of the corresponding lattice of biclosed sets.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2460 https://doi.org/10.46298/dmtcs.2460 McConville, Thomas McConville, Thomas <![CDATA[The Tamari order is a central object in algebraic combinatorics and many other areas. Defined as the transitive closure of an associativity law, the Tamari order possesses a surprisingly rich structure: it is a congruence-uniform lattice. In this work, we consider a larger class of posets, the Grassmann-Tamari orders, which arise as an ordering on the facets of the non-crossing complex introduced by Pylyavskyy, Petersen, and Speyer. We prove that the Grassmann-Tamari orders are congruence-uniform lattices, which resolves a conjecture of Santos, Stump, and Welker. Towards this goal, we define a closure operator on sets of paths inside a rectangle, and prove that the biclosed sets of paths, ordered by inclusion, form a congruence-uniform lattice. We then prove that the Grassmann-Tamari order is a quotient lattice of the corresponding lattice of biclosed sets.]]> 0 <![CDATA[We give a new formula for the weighted high-dimensional tree-numbers of matroid complexes. This formula is derived from our result that the spectra of the weighted combinatorial Laplacians of matroid complexes consist of polynomials in the weights. In the formula, Crapo’s $\beta$-invariant appears as the key factor relating weighted combinatorial Laplacians and weighted tree-numbers for matroid complexes.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2459 https://doi.org/10.46298/dmtcs.2459 Kook, Woong Lee, Kang-Ju Kook, Woong Lee, Kang-Ju <![CDATA[We give a new formula for the weighted high-dimensional tree-numbers of matroid complexes. This formula is derived from our result that the spectra of the weighted combinatorial Laplacians of matroid complexes consist of polynomials in the weights. In the formula, Crapo’s $\beta$-invariant appears as the key factor relating weighted combinatorial Laplacians and weighted tree-numbers for matroid complexes.]]> 0 <![CDATA[The goal of this paper is to provide a combinatorial expression for the steady state probabilities of the twospecies PASEP. In this model, there are two species of particles, one “heavy” and one “light”, on a one-dimensional finite lattice with open boundaries. Both particles can hop into adjacent holes to the right and left at rates 1 and $q$. Moreover, when the heavy and light particles are adjacent to each other, they can switch places as if the light particle were a hole. Additionally, the heavy particle can hop in and out at the boundary of the lattice. Our first result is a combinatorial interpretation for the stationary distribution at $q=0$ in terms of certain multi-Catalan tableaux. We provide an explicit determinantal formula for the steady state probabilities, as well as some general enumerative results for this case. We also describe a Markov process on these tableaux that projects to the two-species PASEP, and hence directly explains the connection between the two. Finally, we extend our formula for the stationary distribution to the $q=1$ case, using certain two-species alternative tableaux.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2458 https://doi.org/10.46298/dmtcs.2458 Mandelshtam, Olya Mandelshtam, Olya <![CDATA[The goal of this paper is to provide a combinatorial expression for the steady state probabilities of the twospecies PASEP. In this model, there are two species of particles, one “heavy” and one “light”, on a one-dimensional finite lattice with open boundaries. Both particles can hop into adjacent holes to the right and left at rates 1 and $q$. Moreover, when the heavy and light particles are adjacent to each other, they can switch places as if the light particle were a hole. Additionally, the heavy particle can hop in and out at the boundary of the lattice. Our first result is a combinatorial interpretation for the stationary distribution at $q=0$ in terms of certain multi-Catalan tableaux. We provide an explicit determinantal formula for the steady state probabilities, as well as some general enumerative results for this case. We also describe a Markov process on these tableaux that projects to the two-species PASEP, and hence directly explains the connection between the two. Finally, we extend our formula for the stationary distribution to the $q=1$ case, using certain two-species alternative tableaux.]]> 0 <![CDATA[The depth statistic was defined for every Coxeter group in terms of factorizations of its elements into product of reflections. Essentially, the depth gives the minimal path cost in the Bruaht graph, where the edges have prescribed weights. We present an algorithm for calculating the depth of a signed permutation which yields a simple formula for this statistic. We use our algorithm to characterize signed permutations having depth equal to length. These are the fully commutative top-and-bottom elements defined by Stembridge. We finally give a characterization of the signed permutations in which the reflection length coincides with both the depth and the length.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2457 https://doi.org/10.46298/dmtcs.2457 Bagno, Eli Biagioli, Riccardo Novick, Mordechai Bagno, Eli Biagioli, Riccardo Novick, Mordechai <![CDATA[The depth statistic was defined for every Coxeter group in terms of factorizations of its elements into product of reflections. Essentially, the depth gives the minimal path cost in the Bruaht graph, where the edges have prescribed weights. We present an algorithm for calculating the depth of a signed permutation which yields a simple formula for this statistic. We use our algorithm to characterize signed permutations having depth equal to length. These are the fully commutative top-and-bottom elements defined by Stembridge. We finally give a characterization of the signed permutations in which the reflection length coincides with both the depth and the length.]]> 0 <![CDATA[This work enrols the research line of M. Haiman on the Operator Theorem (the old operator conjecture). This theorem states that the smallest $\mathfrak{S}_n$-module closed under taking partial derivatives and closed under the action of polarization operators that contains the Vandermonde determinant is the space of diagonal harmonics polynomials. We start generalizing the context of this theorem to the context of polynomials in $\ell$ sets of $n$ variables $x_{ij}$ with $1\le i \le \ell$ and $1 \le j \le n$. Given a $\mathfrak{S}_n$-stable family of homogeneous polynomials in the variables $x_{ij}$ the smallest vector space closed under taking partial derivatives and closed under the action of polarization operators that contains $F$ is the polarization module generated by the family $F$. These polarization modules are all representation of the direct product $\mathfrak{S}_n \times GL_\ell(\mathbb{C})$. In order to study the decomposition into irreducible submodules, we compute the graded Frobenius characteristic of these modules. For several cases of $\mathfrak{S}_n$-stable families of homogeneous polynomials in n variables, for every $n \ge 1$, we show general formulas for this graded characteristic in a global manner, independent of the value of $\ell$.]]> Wed, 31 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2456 https://doi.org/10.46298/dmtcs.2456 Blandin, Héctor Blandin, Héctor <![CDATA[This work enrols the research line of M. Haiman on the Operator Theorem (the old operator conjecture). This theorem states that the smallest $\mathfrak{S}_n$-module closed under taking partial derivatives and closed under the action of polarization operators that contains the Vandermonde determinant is the space of diagonal harmonics polynomials. We start generalizing the context of this theorem to the context of polynomials in $\ell$ sets of $n$ variables $x_{ij}$ with $1\le i \le \ell$ and $1 \le j \le n$. Given a $\mathfrak{S}_n$-stable family of homogeneous polynomials in the variables $x_{ij}$ the smallest vector space closed under taking partial derivatives and closed under the action of polarization operators that contains $F$ is the polarization module generated by the family $F$. These polarization modules are all representation of the direct product $\mathfrak{S}_n \times GL_\ell(\mathbb{C})$. In order to study the decomposition into irreducible submodules, we compute the graded Frobenius characteristic of these modules. For several cases of $\mathfrak{S}_n$-stable families of homogeneous polynomials in n variables, for every $n \ge 1$, we show general formulas for this graded characteristic in a global manner, independent of the value of $\ell$.]]> 0 <![CDATA[For a connected graph G of order |V(G)| ≥3 and a k-labelling c : E(G) →{1,2,…,k} of the edges of G, the code of a vertex v of G is the ordered k-tuple (ℓ1,ℓ2,…,ℓk), where ℓi is the number of edges incident with v that are labelled i. The k-labelling c is detectable if every two adjacent vertices of G have distinct codes. The minimum positive integer k for which G has a detectable k-labelling is the detection number det(G) of G. In this paper, we show that it is NP-complete to decide if the detection number of a cubic graph is 2. We also show that the detection number of every bipartite graph of minimum degree at least 3 is at most 2. Finally, we give some sufficient condition for a cubic graph to have detection number 3.]]> Tue, 30 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.642 https://doi.org/10.46298/dmtcs.642 Havet, Frederic Paramaguru, Nagarajan Sampathkumar, Rathinaswamy Havet, Frederic Paramaguru, Nagarajan Sampathkumar, Rathinaswamy <![CDATA[For a connected graph G of order |V(G)| ≥3 and a k-labelling c : E(G) →{1,2,…,k} of the edges of G, the code of a vertex v of G is the ordered k-tuple (ℓ1,ℓ2,…,ℓk), where ℓi is the number of edges incident with v that are labelled i. The k-labelling c is detectable if every two adjacent vertices of G have distinct codes. The minimum positive integer k for which G has a detectable k-labelling is the detection number det(G) of G. In this paper, we show that it is NP-complete to decide if the detection number of a cubic graph is 2. We also show that the detection number of every bipartite graph of minimum degree at least 3 is at most 2. Finally, we give some sufficient condition for a cubic graph to have detection number 3.]]> 0 <![CDATA[We prove that the dual graph of any arrangement of n lines in general position always contains a path of length at least n2/4. Further, we show that in every arrangement of n red and blue lines — in general position and not all of the same color — there is a simple path through at least n cells where red and blue lines are crossed alternatingly.]]> Wed, 17 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2088 https://doi.org/10.46298/dmtcs.2088 Aichholzer, Oswin Cardinal, Jean Hackl, Thomas Hurtado, Ferran Korman, Matias Pilz, Alexander Silveira, Rodrigo I. Uehara, Ryuhei Valtr, Pavel Vogtenhuber, Birgit Welzl, Emo Aichholzer, Oswin Cardinal, Jean Hackl, Thomas Hurtado, Ferran Korman, Matias Pilz, Alexander Silveira, Rodrigo I. Uehara, Ryuhei Valtr, Pavel Vogtenhuber, Birgit Welzl, Emo <![CDATA[We prove that the dual graph of any arrangement of n lines in general position always contains a path of length at least n2/4. Further, we show that in every arrangement of n red and blue lines — in general position and not all of the same color — there is a simple path through at least n cells where red and blue lines are crossed alternatingly.]]> 0 <![CDATA[Given a set S of n points in the plane, a radial ordering of S with respect to a point p (not in S) is a clockwise circular ordering of the elements in S by angle around p. If S is two-colored, a colored radial ordering is a radial ordering of S in which only the colors of the points are considered. In this paper, we obtain bounds on the number of distinct non-colored and colored radial orderings of S. We assume a strong general position on S, not three points are collinear and not three lines 14;each passing through a pair of points in S 14;intersect in a point of ℝ2 S. In the colored case, S is a set of 2n points partitioned into n red and n blue points, and n is even. We prove that: the number of distinct radial orderings of S is at most O(n4) and at least Ω(n3); the number of colored radial orderings of S is at most O(n4) and at least Ω(n); there exist sets of points with Θ(n4) colored radial orderings and sets of points with only O(n2) colored radial orderings.]]> Mon, 15 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2087 https://doi.org/10.46298/dmtcs.2087 Dıaz-Banez, José Miguel Fabila-Monroy, Ruy Pérez-Lantero, Pablo Dıaz-Banez, José Miguel Fabila-Monroy, Ruy Pérez-Lantero, Pablo <![CDATA[Given a set S of n points in the plane, a radial ordering of S with respect to a point p (not in S) is a clockwise circular ordering of the elements in S by angle around p. If S is two-colored, a colored radial ordering is a radial ordering of S in which only the colors of the points are considered. In this paper, we obtain bounds on the number of distinct non-colored and colored radial orderings of S. We assume a strong general position on S, not three points are collinear and not three lines 14;each passing through a pair of points in S 14;intersect in a point of ℝ2 S. In the colored case, S is a set of 2n points partitioned into n red and n blue points, and n is even. We prove that: the number of distinct radial orderings of S is at most O(n4) and at least Ω(n3); the number of colored radial orderings of S is at most O(n4) and at least Ω(n); there exist sets of points with Θ(n4) colored radial orderings and sets of points with only O(n2) colored radial orderings.]]> 0 <![CDATA[By the algorithm implemented in the paper by Akiyama-Lee [Adv. Math. 226(4):2855 13;2883, 2011] and some of its predecessors, we have examined the pure discreteness of the spectrum for all irreducible Pisot substitutions of trace less than or equal to 2, and some cases of planar tilings generated by boundary substitutions due to Kenyon [Geom. Func. Anal. 6:471 13;488, 1996].]]> Mon, 15 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2089 https://doi.org/10.46298/dmtcs.2089 Akiyama, Shigeki Gähler, Franz Lee, Jeong-Yup Akiyama, Shigeki Gähler, Franz Lee, Jeong-Yup <![CDATA[By the algorithm implemented in the paper by Akiyama-Lee [Adv. Math. 226(4):2855 13;2883, 2011] and some of its predecessors, we have examined the pure discreteness of the spectrum for all irreducible Pisot substitutions of trace less than or equal to 2, and some cases of planar tilings generated by boundary substitutions due to Kenyon [Geom. Func. Anal. 6:471 13;488, 1996].]]> 0 <![CDATA[Flood-it is a combinatorial game played on a colored graph G whose aim is to make the graph monochromatic using the minimum number of flooding moves, relatively to a fixed pivot. Free-Flood-it is a variant where the pivot can be freely chosen for each move of the game. The standard versions of Flood-it and Free-Flood-it are played on m ×n grids. In this paper we analyze the behavior of these games when played on other classes of graphs, such as d-boards, powers of cycles and circular grids. We describe polynomial time algorithms to play Flood-it on C2n (the second power of a cycle on n vertices), 2 ×n circular grids, and some types of d-boards (grids with a monochromatic column). We also show that Free-Flood-it is NP-hard on C2n and 2 ×n circular grids.]]> Sat, 13 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2086 https://doi.org/10.46298/dmtcs.2086 Souza, Uéverton dos Santos Protti, Fábio Silva, Maise, Souza, Uéverton dos Santos Protti, Fábio Silva, Maise, <![CDATA[Flood-it is a combinatorial game played on a colored graph G whose aim is to make the graph monochromatic using the minimum number of flooding moves, relatively to a fixed pivot. Free-Flood-it is a variant where the pivot can be freely chosen for each move of the game. The standard versions of Flood-it and Free-Flood-it are played on m ×n grids. In this paper we analyze the behavior of these games when played on other classes of graphs, such as d-boards, powers of cycles and circular grids. We describe polynomial time algorithms to play Flood-it on C2n (the second power of a cycle on n vertices), 2 ×n circular grids, and some types of d-boards (grids with a monochromatic column). We also show that Free-Flood-it is NP-hard on C2n and 2 ×n circular grids.]]> 0 <![CDATA[A graph G is arbitrarily partitionable (AP for short) if for every partition (n_1, n_2, ..., n_p) of |V(G)| there exists a partition (V_1, V_2, ..., V_p) of V(G) such that each V_i induces a connected subgraph of G with order n_i. If, additionally, k of these subgraphs (k <= p) each contains an arbitrary vertex of G prescribed beforehand, then G is arbitrarily partitionable under k prescriptions (AP+k for short). Every AP+k graph on n vertices is (k+1)-connected, and thus has at least ceil(n(k+1)/2) edges. We show that there exist AP+k graphs on n vertices and ceil(n(k+1)/2) edges for every k >= 1 and n >= k.]]> Thu, 11 Dec 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.641 https://doi.org/10.46298/dmtcs.641 Baudon, Olivier Bensmail, Julien Sopena, Eric Baudon, Olivier Bensmail, Julien Sopena, Eric <![CDATA[A graph G is arbitrarily partitionable (AP for short) if for every partition (n_1, n_2, ..., n_p) of |V(G)| there exists a partition (V_1, V_2, ..., V_p) of V(G) such that each V_i induces a connected subgraph of G with order n_i. If, additionally, k of these subgraphs (k <= p) each contains an arbitrary vertex of G prescribed beforehand, then G is arbitrarily partitionable under k prescriptions (AP+k for short). Every AP+k graph on n vertices is (k+1)-connected, and thus has at least ceil(n(k+1)/2) edges. We show that there exist AP+k graphs on n vertices and ceil(n(k+1)/2) edges for every k >= 1 and n >= k.]]> 0 <![CDATA[Dynamic Storage Allocation is a problem concerned with storing items that each have weight and time restrictions. Approximate algorithms have been constructed through online coloring of interval graphs. We present a generalization that uses online coloring of tolerance graphs. We utilize online-with-representation algorithms on tolerance graphs, which are online algorithms in which the corresponding tolerance representation of a vertex is also presented. We find linear bounds for the online-with-representation chromatic number of various classes of tolerance graphs and apply these results to a generalization of Dynamic Storage Allocation, giving us a polynomial time approximation algorithm with linear performance ratio.]]> Sun, 23 Nov 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2083 https://doi.org/10.46298/dmtcs.2083 Kierstead, H. A. Saoub, Karin R. Kierstead, H. A. Saoub, Karin R. <![CDATA[Dynamic Storage Allocation is a problem concerned with storing items that each have weight and time restrictions. Approximate algorithms have been constructed through online coloring of interval graphs. We present a generalization that uses online coloring of tolerance graphs. We utilize online-with-representation algorithms on tolerance graphs, which are online algorithms in which the corresponding tolerance representation of a vertex is also presented. We find linear bounds for the online-with-representation chromatic number of various classes of tolerance graphs and apply these results to a generalization of Dynamic Storage Allocation, giving us a polynomial time approximation algorithm with linear performance ratio.]]> 0 <![CDATA[We study a two-person game played on graphs based on the widely studied chip-firing game. Players Max and Min alternately place chips on the vertices of a graph. When a vertex accumulates as many chips as its degree, it fires, sending one chip to each neighbour; this may in turn cause other vertices to fire. The game ends when vertices continue firing forever. Min seeks to minimize the number of chips played during the game, while Max seeks to maximize it. When both players play optimally, the length of the game is the toppling number of a graph G, and is denoted by t(G). By considering strategies for both players and investigating the evolution of the game with differential equations, we provide asymptotic bounds on the toppling number of the complete graph. In particular, we prove that for sufficiently large n 0.596400 n2 < t(Kn) < 0.637152 n2. Using a fractional version of the game, we couple the toppling numbers of complete graphs and the binomial random graph G(n,p). It is shown that for pn ≥n² / √ log(n) asymptotically almost surely t(G(n,p))=(1+o(1)) p t(Kn).]]> Mon, 17 Nov 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2084 https://doi.org/10.46298/dmtcs.2084 Bonato, Anthony Kinnersley, William B. Pralat, Pawel Bonato, Anthony Kinnersley, William B. Pralat, Pawel <![CDATA[We study a two-person game played on graphs based on the widely studied chip-firing game. Players Max and Min alternately place chips on the vertices of a graph. When a vertex accumulates as many chips as its degree, it fires, sending one chip to each neighbour; this may in turn cause other vertices to fire. The game ends when vertices continue firing forever. Min seeks to minimize the number of chips played during the game, while Max seeks to maximize it. When both players play optimally, the length of the game is the toppling number of a graph G, and is denoted by t(G). By considering strategies for both players and investigating the evolution of the game with differential equations, we provide asymptotic bounds on the toppling number of the complete graph. In particular, we prove that for sufficiently large n 0.596400 n2 < t(Kn) < 0.637152 n2. Using a fractional version of the game, we couple the toppling numbers of complete graphs and the binomial random graph G(n,p). It is shown that for pn ≥n² / √ log(n) asymptotically almost surely t(G(n,p))=(1+o(1)) p t(Kn).]]> 0 <![CDATA[A coloring c of the vertices of a graph G is nonrepetitive if there exists no path v1v2\textellipsisv2l for which c(vi)=c(vl+i) for all 1<=i<=l. Given graphs G and H with |V(H)|=k, the lexicographic product G[H] is the graph obtained by substituting every vertex of G by a copy of H, and every edge of G by a copy of Kk,k. We prove that for a sufficiently long path P, a nonrepetitive coloring of P[Kk] needs at least 3k+⌊k/2⌋ colors. If k>2 then we need exactly 2k+1 colors to nonrepetitively color P[Ek], where Ek is the empty graph on k vertices. If we further require that every copy of Ek be rainbow-colored and the path P is sufficiently long, then the smallest number of colors needed for P[Ek] is at least 3k+1 and at most 3k+⌈k/2⌉. Finally, we define fractional nonrepetitive colorings of graphs and consider the connections between this notion and the above results.]]> Wed, 29 Oct 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2077 https://doi.org/10.46298/dmtcs.2077 Keszegh, Balázs Patkós, Balázs Zhu, Xuding Keszegh, Balázs Patkós, Balázs Zhu, Xuding <![CDATA[A coloring c of the vertices of a graph G is nonrepetitive if there exists no path v1v2\textellipsisv2l for which c(vi)=c(vl+i) for all 1<=i<=l. Given graphs G and H with |V(H)|=k, the lexicographic product G[H] is the graph obtained by substituting every vertex of G by a copy of H, and every edge of G by a copy of Kk,k. We prove that for a sufficiently long path P, a nonrepetitive coloring of P[Kk] needs at least 3k+⌊k/2⌋ colors. If k>2 then we need exactly 2k+1 colors to nonrepetitively color P[Ek], where Ek is the empty graph on k vertices. If we further require that every copy of Ek be rainbow-colored and the path P is sufficiently long, then the smallest number of colors needed for P[Ek] is at least 3k+1 and at most 3k+⌈k/2⌉. Finally, we define fractional nonrepetitive colorings of graphs and consider the connections between this notion and the above results.]]> 0 <![CDATA[We prove that any irreducible triangulation on n vertices has O(4.6807n) regular edge labelings and that there are irreducible triangulations on n vertices with Ω(3.0426n) regular edge labelings. Our upper bound relies on a novel application of Shearer's entropy lemma. As an example of the wider applicability of this technique, we also improve the upper bound on the number of 2-orientations of a quadrangulation to O(1.87n).]]> Sun, 19 Oct 2014 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2085 https://doi.org/10.46298/dmtcs.2085 Buchin, Kevin Speckmann, Bettina Verdonschot, Sander Buchin, Kevin Speckmann, Bettina Verdonschot, Sander <![CDATA[We prove that any irreducible triangulation on n vertices has O(4.6807n) regular edge labelings and that there are irreducible triangulations on n vertices with Ω(3.0426n) regular edge labelings. Our upper bound relies on a novel application of Shearer's entropy lemma. As an example of the wider applicability of this technique, we also improve the upper bound on the number of 2-orientations of a quadrangulation to O(1.87n).]]> 0 <![CDATA[The graph isomorphism (GI) problem asks whether two given graphs are isomorphic or not. The GI problem is quite basic and simple, however, it\textquoterights time complexity is a long standing open problem. The GI problem is clearly in NP, no polynomial time algorithm is known, and the GI problem is not NP-complete unless the polynomial hierarchy collapses. In this paper, we survey the computational complexity of the problem on some graph classes that have geometric characterizations. Sometimes the GI problem becomes polynomial time solvable when we add some restrictions on some graph classes. The properties of these graph classes on the boundary indicate us the essence of difficulty of the GI problem. We also show that the GI problem is as hard as the problem on general graphs even for grid unit intersection graphs on a torus, that partially solves an open problem.]]> Mon, 13 Oct 2014 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2076 https://doi.org/10.46298/dmtcs.2076 Uehara, Ryuhei Uehara, Ryuhei <![CDATA[The graph isomorphism (GI) problem asks whether two given graphs are isomorphic or not. The GI problem is quite basic and simple, however, it\textquoterights time complexity is a long standing open problem. The GI problem is clearly in NP, no polynomial time algorithm is known, and the GI problem is not NP-complete unless the polynomial hierarchy collapses. In this paper, we survey the computational complexity of the problem on some graph classes that have geometric characterizations. Sometimes the GI problem becomes polynomial time solvable when we add some restrictions on some graph classes. The properties of these graph classes on the boundary indicate us the essence of difficulty of the GI problem. We also show that the GI problem is as hard as the problem on general graphs even for grid unit intersection graphs on a torus, that partially solves an open problem.]]> 0 <![CDATA[Given a graph and a positive integer k, the biclique vertex-partition problem asks whether the vertex set of the graph can be partitioned into at most k bicliques (connected complete bipartite subgraphs). It is known that this problem is NP-complete for bipartite graphs. In this paper we investigate the computational complexity of this problem in special subclasses of bipartite graphs. We prove that the biclique vertex-partition problem is polynomially solvable for bipartite permutation graphs, bipartite distance-hereditary graphs and remains NP-complete for perfect elimination bipartite graphs and bipartite graphs containing no 4-cycles as induced subgraphs.]]> Wed, 24 Sep 2014 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2090 https://doi.org/10.46298/dmtcs.2090 Duginov, Oleg Duginov, Oleg <![CDATA[Given a graph and a positive integer k, the biclique vertex-partition problem asks whether the vertex set of the graph can be partitioned into at most k bicliques (connected complete bipartite subgraphs). It is known that this problem is NP-complete for bipartite graphs. In this paper we investigate the computational complexity of this problem in special subclasses of bipartite graphs. We prove that the biclique vertex-partition problem is polynomially solvable for bipartite permutation graphs, bipartite distance-hereditary graphs and remains NP-complete for perfect elimination bipartite graphs and bipartite graphs containing no 4-cycles as induced subgraphs.]]> 0 <![CDATA[Let Ck denote a cycle of length k and let Sk denote a star with k edges. For multigraphs F, G and H, an (F,G)-decomposition of H is an edge decomposition of H into copies of F and G using at least one of each. For L⊆H and R⊆rH, an (F,G)-packing (resp. (F,G)-covering) of H with leave L (resp. padding R) is an (F,G)-decomposition of H-E(L) (resp. H+E(R)). An (F,G)-packing (resp. (F,G)-covering) of H with the largest (resp. smallest) cardinality is a maximum (F,G)-packing (resp. minimum (F,G)-covering), and its cardinality is referred to as the (F,G)-packing number (resp. (F,G)-covering number) of H. In this paper, we determine the packing number and the covering number of λKn,n with Ck's and Sk's for any λ, n and k, and give the complete solution of the maximum packing and the minimum covering of λKn,n with 4-cycles and 4-stars for any λ and n with all possible leaves and paddings.]]> Wed, 24 Sep 2014 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2091 https://doi.org/10.46298/dmtcs.2091 Lee, Hung-Chih Lee, Hung-Chih <![CDATA[Let Ck denote a cycle of length k and let Sk denote a star with k edges. For multigraphs F, G and H, an (F,G)-decomposition of H is an edge decomposition of H into copies of F and G using at least one of each. For L⊆H and R⊆rH, an (F,G)-packing (resp. (F,G)-covering) of H with leave L (resp. padding R) is an (F,G)-decomposition of H-E(L) (resp. H+E(R)). An (F,G)-packing (resp. (F,G)-covering) of H with the largest (resp. smallest) cardinality is a maximum (F,G)-packing (resp. minimum (F,G)-covering), and its cardinality is referred to as the (F,G)-packing number (resp. (F,G)-covering number) of H. In this paper, we determine the packing number and the covering number of λKn,n with Ck's and Sk's for any λ, n and k, and give the complete solution of the maximum packing and the minimum covering of λKn,n with 4-cycles and 4-stars for any λ and n with all possible leaves and paddings.]]> 0 <![CDATA[Let G = (V,E) be a graph. For each e ∈E(G) and v ∈V(G), let Le and Lv, respectively, be a list of real numbers. Let w be a function on V(G) ∪E(G) such that w(e) ∈Le for each e ∈E(G) and w(v) ∈Lv for each v ∈V(G), and let cw be the vertex colouring obtained by cw(v) = w(v) + ∑ₑ ∋vw(e). A graph is (k,l)-weight choosable if there exists a weighting function w for which cw is proper whenever |Lv| ≥k and |Le| ≥l for every v ∈V(G) and e ∈E(G). A sufficient condition for a graph to be (1,l)-weight choosable was developed by Bartnicki, Grytczuk and Niwczyk (2009), based on the Combinatorial Nullstellensatz, a parameter which they call the monomial index of a graph, and matrix permanents. This paper extends their method to establish the first general upper bound on the monomial index of a graph, and thus to obtain an upper bound on l for which every admissible graph is (1,l)-weight choosable. Let ∂2(G) denote the smallest value s such that every induced subgraph of G has vertices at distance 2 whose degrees sum to at most s. We show that every admissible graph has monomial index at most ∂2(G) and hence that such graphs are (1, ∂2(G)+1)-weight choosable. While this does not improve the best known result on (1,l)-weight choosability, we show that the results can be extended to obtain improved bounds for some graph products; for instance, it is shown that G □ Kn is (1, nd+3)-weight choosable if G is d-degenerate.]]> Tue, 16 Sep 2014 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2097 https://doi.org/10.46298/dmtcs.2097 Seamone, Ben Seamone, Ben <![CDATA[Let G = (V,E) be a graph. For each e ∈E(G) and v ∈V(G), let Le and Lv, respectively, be a list of real numbers. Let w be a function on V(G) ∪E(G) such that w(e) ∈Le for each e ∈E(G) and w(v) ∈Lv for each v ∈V(G), and let cw be the vertex colouring obtained by cw(v) = w(v) + ∑ₑ ∋vw(e). A graph is (k,l)-weight choosable if there exists a weighting function w for which cw is proper whenever |Lv| ≥k and |Le| ≥l for every v ∈V(G) and e ∈E(G). A sufficient condition for a graph to be (1,l)-weight choosable was developed by Bartnicki, Grytczuk and Niwczyk (2009), based on the Combinatorial Nullstellensatz, a parameter which they call the monomial index of a graph, and matrix permanents. This paper extends their method to establish the first general upper bound on the monomial index of a graph, and thus to obtain an upper bound on l for which every admissible graph is (1,l)-weight choosable. Let ∂2(G) denote the smallest value s such that every induced subgraph of G has vertices at distance 2 whose degrees sum to at most s. We show that every admissible graph has monomial index at most ∂2(G) and hence that such graphs are (1, ∂2(G)+1)-weight choosable. While this does not improve the best known result on (1,l)-weight choosability, we show that the results can be extended to obtain improved bounds for some graph products; for instance, it is shown that G □ Kn is (1, nd+3)-weight choosable if G is d-degenerate.]]> 0 <![CDATA[We consider the P₃-convexity on simple undirected graphs, in which a set of vertices S is convex if no vertex outside S has two or more neighbors in S. The convex hull H(S) of a set S is the smallest convex set containing S as a subset. A set S is a convexly independent set if v \not ∈ H(S\setminus \v\) for all v in S. The rank \rk(G) of a graph is the size of the largest convexly independent set. In this paper we consider the complexity of determining \rk(G). We show that the problem is NP-complete even for split or bipartite graphs with small diameter. We also show how to determine \rk(G) in polynomial time for the well structured classes of graphs of trees and threshold graphs. Finally, we give a tight upper bound for \rk(G), which in turn gives a tight upper bound for the Radon number as byproduct, which is the same obtained before by Henning, Rautenbach and Schäfer. Additionally, we briefly show that the problem is NP-complete also in the monophonic convexity.]]> Sun, 31 Aug 2014 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2075 https://doi.org/10.46298/dmtcs.2075 Ramos, Igor, Santos, Vinícius F., Szwarcfiter, Jayme L. Ramos, Igor, Santos, Vinícius F., Szwarcfiter, Jayme L. <![CDATA[We consider the P₃-convexity on simple undirected graphs, in which a set of vertices S is convex if no vertex outside S has two or more neighbors in S. The convex hull H(S) of a set S is the smallest convex set containing S as a subset. A set S is a convexly independent set if v \not ∈ H(S\setminus \v\) for all v in S. The rank \rk(G) of a graph is the size of the largest convexly independent set. In this paper we consider the complexity of determining \rk(G). We show that the problem is NP-complete even for split or bipartite graphs with small diameter. We also show how to determine \rk(G) in polynomial time for the well structured classes of graphs of trees and threshold graphs. Finally, we give a tight upper bound for \rk(G), which in turn gives a tight upper bound for the Radon number as byproduct, which is the same obtained before by Henning, Rautenbach and Schäfer. Additionally, we briefly show that the problem is NP-complete also in the monophonic convexity.]]> 0 <![CDATA[We derive a quadratic-time algorithm for the genus distribution of any 3-regular, biconnected series-parallel graph, which we extend to any biconnected series-parallel graph of maximum degree at most 3. Since the biconnected components of every graph of treewidth 2 are series-parallel graphs, this yields, by use of bar-amalgamation, a quadratic-time algorithm for every graph of treewidth at most 2 and maximum degree at most 3.]]> Thu, 31 Jul 2014 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2099 https://doi.org/10.46298/dmtcs.2099 Gross, Jonathan L. Kotrbčík, Michal Sun, Timothy Gross, Jonathan L. Kotrbčík, Michal Sun, Timothy <![CDATA[We derive a quadratic-time algorithm for the genus distribution of any 3-regular, biconnected series-parallel graph, which we extend to any biconnected series-parallel graph of maximum degree at most 3. Since the biconnected components of every graph of treewidth 2 are series-parallel graphs, this yields, by use of bar-amalgamation, a quadratic-time algorithm for every graph of treewidth at most 2 and maximum degree at most 3.]]> 0 <![CDATA[In a biased weak (a,b) polyform achievement game, the maker and the breaker alternately mark a,b previously unmarked cells on an infinite board, respectively. The maker's goal is to mark a set of cells congruent to a polyform. The breaker tries to prevent the maker from achieving this goal. A winning maker strategy for the (a,b) game can be built from winning strategies for games involving fewer marks for the maker and the breaker. A new type of breaker strategy called the priority strategy is introduced. The winners are determined for all (a,b) pairs for polyiamonds and polyominoes up to size four.]]> Thu, 31 Jul 2014 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2098 https://doi.org/10.46298/dmtcs.2098 Norris, Ian Sieben, Nándor Norris, Ian Sieben, Nándor <![CDATA[In a biased weak (a,b) polyform achievement game, the maker and the breaker alternately mark a,b previously unmarked cells on an infinite board, respectively. The maker's goal is to mark a set of cells congruent to a polyform. The breaker tries to prevent the maker from achieving this goal. A winning maker strategy for the (a,b) game can be built from winning strategies for games involving fewer marks for the maker and the breaker. A new type of breaker strategy called the priority strategy is introduced. The winners are determined for all (a,b) pairs for polyiamonds and polyominoes up to size four.]]> 0 <![CDATA[We study properties of infinite permutations generated by fixed points of some uniform binary morphisms, and find the formula for their complexity.]]> Mon, 28 Jul 2014 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2096 https://doi.org/10.46298/dmtcs.2096 Valyuzhenich, Alexandr Valyuzhenich, Alexandr <![CDATA[We study properties of infinite permutations generated by fixed points of some uniform binary morphisms, and find the formula for their complexity.]]> 0 <![CDATA[End-vertices of a given graph search may have some nice properties, as for example it is well known that the last vertex of Lexicographic Breadth First Search (LBFS) in a chordal graph is simplicial, see Rose, Tarjan and Lueker 1976. Therefore it is interesting to consider if these vertices can be recognized in polynomial time or not, as first studied in Corneil, Köhler and Lanlignel 2010. A graph search is a mechanism for systematically visiting the vertices of a graph. At each step of a graph search, the key point is the choice of the next vertex to be explored. Graph searches only differ by this selection mechanism during which a tie-break rule is used. In this paper we study how the choice of the tie-break in case of equality during the search, for a given graph search including the classic ones such as BFS and DFS, can determine the complexity of the end-vertex problem. In particular we prove a counterintuitive NP-completeness result for Breadth First Search solving a problem raised in Corneil, Köhler and Lanlignel 2010.]]> Mon, 28 Jul 2014 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2081 https://doi.org/10.46298/dmtcs.2081 Charbit, Pierre Habib, Michel Mamcarz, Antoine Charbit, Pierre Habib, Michel Mamcarz, Antoine <![CDATA[End-vertices of a given graph search may have some nice properties, as for example it is well known that the last vertex of Lexicographic Breadth First Search (LBFS) in a chordal graph is simplicial, see Rose, Tarjan and Lueker 1976. Therefore it is interesting to consider if these vertices can be recognized in polynomial time or not, as first studied in Corneil, Köhler and Lanlignel 2010. A graph search is a mechanism for systematically visiting the vertices of a graph. At each step of a graph search, the key point is the choice of the next vertex to be explored. Graph searches only differ by this selection mechanism during which a tie-break rule is used. In this paper we study how the choice of the tie-break in case of equality during the search, for a given graph search including the classic ones such as BFS and DFS, can determine the complexity of the end-vertex problem. In particular we prove a counterintuitive NP-completeness result for Breadth First Search solving a problem raised in Corneil, Köhler and Lanlignel 2010.]]> 0 <![CDATA[A (pseudo-)metric D on a finite set X is said to be a \textquotelefttree metric\textquoteright if there is a finite tree with leaf set X and non-negative edge weights so that, for all x,y ∈X, D(x,y) is the path distance in the tree between x and y. It is well known that not every metric is a tree metric. However, when some such tree exists, one can always find one whose interior edges have strictly positive edge weights and that has no vertices of degree 2, any such tree is – up to canonical isomorphism – uniquely determined by D, and one does not even need all of the distances in order to fully (re-)construct the tree\textquoterights edge weights in this case. Thus, it seems of some interest to investigate which subsets of X, 2 suffice to determine (\textquoteleftlasso\textquoteright) these edge weights. In this paper, we use the results of a previous paper to discuss the structure of a matroid that can be associated with an (unweighted) X-tree T defined by the requirement that its bases are exactly the \textquotelefttight edge-weight lassos\textquoteright for T, i.e, the minimal subsets of X, 2 that lasso the edge weights of T.]]> Thu, 19 Jun 2014 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2078 https://doi.org/10.46298/dmtcs.2078 Dress, Andreas Huber, Katharina Steel, Mike Dress, Andreas Huber, Katharina Steel, Mike <![CDATA[A (pseudo-)metric D on a finite set X is said to be a \textquotelefttree metric\textquoteright if there is a finite tree with leaf set X and non-negative edge weights so that, for all x,y ∈X, D(x,y) is the path distance in the tree between x and y. It is well known that not every metric is a tree metric. However, when some such tree exists, one can always find one whose interior edges have strictly positive edge weights and that has no vertices of degree 2, any such tree is – up to canonical isomorphism – uniquely determined by D, and one does not even need all of the distances in order to fully (re-)construct the tree\textquoterights edge weights in this case. Thus, it seems of some interest to investigate which subsets of X, 2 suffice to determine (\textquoteleftlasso\textquoteright) these edge weights. In this paper, we use the results of a previous paper to discuss the structure of a matroid that can be associated with an (unweighted) X-tree T defined by the requirement that its bases are exactly the \textquotelefttight edge-weight lassos\textquoteright for T, i.e, the minimal subsets of X, 2 that lasso the edge weights of T.]]> 0 <![CDATA[The generalized list T-coloring is a common generalization of many graph coloring models, including classical coloring, L(p,q)-labeling, channel assignment and T-coloring. Every vertex from the input graph has a list of permitted labels. Moreover, every edge has a set of forbidden differences. We ask for a labeling of vertices of the input graph with natural numbers, in which every vertex gets a label from its list of permitted labels and the difference of labels of the endpoints of each edge does not belong to the set of forbidden differences of this edge. In this paper we present an exact algorithm solving this problem, running in time O*((τ+2)n), where τ is the maximum forbidden difference over all edges of the input graph and n is the number of its vertices. Moreover, we show how to improve this bound if the input graph has some special structure, e.g. a bounded maximum degree, no big induced stars or a perfect matching.]]> Wed, 18 Jun 2014 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2095 https://doi.org/10.46298/dmtcs.2095 Junosza-Szaniawski, Konstanty Rzazewski, Pawel Junosza-Szaniawski, Konstanty Rzazewski, Pawel <![CDATA[The generalized list T-coloring is a common generalization of many graph coloring models, including classical coloring, L(p,q)-labeling, channel assignment and T-coloring. Every vertex from the input graph has a list of permitted labels. Moreover, every edge has a set of forbidden differences. We ask for a labeling of vertices of the input graph with natural numbers, in which every vertex gets a label from its list of permitted labels and the difference of labels of the endpoints of each edge does not belong to the set of forbidden differences of this edge. In this paper we present an exact algorithm solving this problem, running in time O*((τ+2)n), where τ is the maximum forbidden difference over all edges of the input graph and n is the number of its vertices. Moreover, we show how to improve this bound if the input graph has some special structure, e.g. a bounded maximum degree, no big induced stars or a perfect matching.]]> 0 <![CDATA[The oriented diameter of a bridgeless graph G is min diam(H) | H is a strang orientation of G. A path in an edge-colored graph G, where adjacent edges may have the same color, is called rainbow if no two edges of the path are colored the same. The rainbow connection number rc(G) of G is the smallest integer number k for which there exists a k-edge-coloring of G such that every two distinct vertices of G are connected by a rainbow path. In this paper, we obtain upper bounds for the oriented diameter and the rainbow connection number of a graph in terms of rad(G) and η(G), where rad(G) is the radius of G and η(G) is the smallest integer number such that every edge of G is contained in a cycle of length at most η(G). We also obtain constant bounds of the oriented diameter and the rainbow connection number for a (bipartite) graph G in terms of the minimum degree of G.]]> Wed, 18 Jun 2014 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2093 https://doi.org/10.46298/dmtcs.2093 Huang, Xiaolong Li, Hengzhe Li, Xueliang Sun, Yuefang Huang, Xiaolong Li, Hengzhe Li, Xueliang Sun, Yuefang <![CDATA[The oriented diameter of a bridgeless graph G is min diam(H) | H is a strang orientation of G. A path in an edge-colored graph G, where adjacent edges may have the same color, is called rainbow if no two edges of the path are colored the same. The rainbow connection number rc(G) of G is the smallest integer number k for which there exists a k-edge-coloring of G such that every two distinct vertices of G are connected by a rainbow path. In this paper, we obtain upper bounds for the oriented diameter and the rainbow connection number of a graph in terms of rad(G) and η(G), where rad(G) is the radius of G and η(G) is the smallest integer number such that every edge of G is contained in a cycle of length at most η(G). We also obtain constant bounds of the oriented diameter and the rainbow connection number for a (bipartite) graph G in terms of the minimum degree of G.]]> 0 <![CDATA[We study partitions of the vertex set of a given graph into cells that each induce a subgraph in a given family, and for which edges can have ends in different cells only when those cells correspond to adjacent vertices of a fixed template graph H. For triangle-free templates, a general collection of graph families for which the partitioning problem can be solved in polynomial time is described. For templates with a triangle, the problem is in some cases shown to be NP-complete.]]> Wed, 18 Jun 2014 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2092 https://doi.org/10.46298/dmtcs.2092 Dukes, Peter J. Lowdon, Steve Macgillivray, Gary Dukes, Peter J. Lowdon, Steve Macgillivray, Gary <![CDATA[We study partitions of the vertex set of a given graph into cells that each induce a subgraph in a given family, and for which edges can have ends in different cells only when those cells correspond to adjacent vertices of a fixed template graph H. For triangle-free templates, a general collection of graph families for which the partitioning problem can be solved in polynomial time is described. For templates with a triangle, the problem is in some cases shown to be NP-complete.]]> 0 <![CDATA[A graph is balanced if its clique-vertex incidence matrix contains no square submatrix of odd order with exactly two ones per row and per column. There is a characterization of balanced graphs by forbidden induced subgraphs, but no characterization by mininal forbidden induced subgraphs is known, not even for the case of circular-arc graphs. A circular-arc graph is the intersection graph of a family of arcs on a circle. In this work, we characterize when a given graph G is balanced in terms of minimal forbidden induced subgraphs, by restricting the analysis to the case where G belongs to certain classes of circular-arc graphs, including Helly circular-arc graphs, claw-free circular-arc graphs, and gem-free circular-arc graphs. In the case of gem-free circular-arc graphs, analogous characterizations are derived for two superclasses of balanced graphs: clique-perfect graphs and coordinated graphs.]]> Tue, 17 Jun 2014 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2100 https://doi.org/10.46298/dmtcs.2100 Bonomo, Flavia Duran, Guillermo Safe, Martın D. Wagler, Annegret K. Bonomo, Flavia Duran, Guillermo Safe, Martın D. Wagler, Annegret K. <![CDATA[A graph is balanced if its clique-vertex incidence matrix contains no square submatrix of odd order with exactly two ones per row and per column. There is a characterization of balanced graphs by forbidden induced subgraphs, but no characterization by mininal forbidden induced subgraphs is known, not even for the case of circular-arc graphs. A circular-arc graph is the intersection graph of a family of arcs on a circle. In this work, we characterize when a given graph G is balanced in terms of minimal forbidden induced subgraphs, by restricting the analysis to the case where G belongs to certain classes of circular-arc graphs, including Helly circular-arc graphs, claw-free circular-arc graphs, and gem-free circular-arc graphs. In the case of gem-free circular-arc graphs, analogous characterizations are derived for two superclasses of balanced graphs: clique-perfect graphs and coordinated graphs.]]> 0 <![CDATA[In their 2009 paper, Corneil et al. design a linear time interval graph recognition algorithm based on six sweeps of Lexicographic Breadth-First Search (LBFS) and prove its correctness. They believe that their corresponding 5-sweep LBFS interval graph recognition algorithm is also correct. Thanks to the LBFS structure theory established mainly by Corneil et al., we are able to present a 4-sweep LBFS algorithm which determines whether or not the input graph is a unit interval graph or an interval graph. Like the algorithm of Corneil et al., our algorithm does not involve any complicated data structure and can be executed in linear time.]]> Tue, 17 Jun 2014 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2094 https://doi.org/10.46298/dmtcs.2094 Li, Peng Wu, Yaokun Li, Peng Wu, Yaokun <![CDATA[In their 2009 paper, Corneil et al. design a linear time interval graph recognition algorithm based on six sweeps of Lexicographic Breadth-First Search (LBFS) and prove its correctness. They believe that their corresponding 5-sweep LBFS interval graph recognition algorithm is also correct. Thanks to the LBFS structure theory established mainly by Corneil et al., we are able to present a 4-sweep LBFS algorithm which determines whether or not the input graph is a unit interval graph or an interval graph. Like the algorithm of Corneil et al., our algorithm does not involve any complicated data structure and can be executed in linear time.]]> 0 <![CDATA[We study a robot hand model in the framework of the theory of expansions in non-integer bases. We investigate the reachable workspace and we study some configurations enjoying form closure properties. Keywords: Robot hand, discrete control, expansions in non-integer bases, expansions in complex bases]]> Thu, 05 Jun 2014 22:00:00 +0000 https://doi.org/10.46298/dmtcs.3913 https://doi.org/10.46298/dmtcs.3913 Loreti, Paola Lai, Anna Chiara Loreti, Paola Lai, Anna Chiara <![CDATA[We study a robot hand model in the framework of the theory of expansions in non-integer bases. We investigate the reachable workspace and we study some configurations enjoying form closure properties. Keywords: Robot hand, discrete control, expansions in non-integer bases, expansions in complex bases]]> 0 <![CDATA[The generalized k-connectivity κk(G) of a graph G, first introduced by Hager, is a natural generalization of the concept of (vertex-)connectivity. Denote by G^H and G&Box;H the lexicographic product and Cartesian product of two graphs G and H, respectively. In this paper, we prove that for any two connected graphs G and H, κ3(G^H)≥ κ3(G)|V(H)|. We also give upper bounds for κ3(G&Box; H) and κ3(G^H). Moreover, all the bounds are sharp.]]> Sun, 01 Jun 2014 22:00:00 +0000 https://doi.org/10.46298/dmtcs.1266 https://doi.org/10.46298/dmtcs.1266 Li, Xueliang Mao, Yaping Li, Xueliang Mao, Yaping <![CDATA[The generalized k-connectivity κk(G) of a graph G, first introduced by Hager, is a natural generalization of the concept of (vertex-)connectivity. Denote by G^H and G&Box;H the lexicographic product and Cartesian product of two graphs G and H, respectively. In this paper, we prove that for any two connected graphs G and H, κ3(G^H)≥ κ3(G)|V(H)|. We also give upper bounds for κ3(G&Box; H) and κ3(G^H). Moreover, all the bounds are sharp.]]> 0 <![CDATA[A non-commutative, planar, Hopf algebra of planar rooted trees was defined independently by one of the authors in Foissy (2002) and by R. Holtkamp in Holtkamp (2003). In this paper we propose such a non-commutative Hopf algebra for graphs. In order to define a non-commutative product we use a quantum field theoretical (QFT) idea, namely the one of introducing discrete scales on each edge of the graph (which, within the QFT framework, corresponds to energy scales of the associated propagators). Finally, we analyze the associated quadri-coalgebra and codendrifrom structures.]]> Sun, 01 Jun 2014 22:00:00 +0000 https://doi.org/10.46298/dmtcs.1250 https://doi.org/10.46298/dmtcs.1250 Tanasa, Adrian Duchamp, Gerard Foissy, Loïc Hoang-Nghia, Nguyen Manchon, Dominique Tanasa, Adrian Duchamp, Gerard Foissy, Loïc Hoang-Nghia, Nguyen Manchon, Dominique <![CDATA[A non-commutative, planar, Hopf algebra of planar rooted trees was defined independently by one of the authors in Foissy (2002) and by R. Holtkamp in Holtkamp (2003). In this paper we propose such a non-commutative Hopf algebra for graphs. In order to define a non-commutative product we use a quantum field theoretical (QFT) idea, namely the one of introducing discrete scales on each edge of the graph (which, within the QFT framework, corresponds to energy scales of the associated propagators). Finally, we analyze the associated quadri-coalgebra and codendrifrom structures.]]> 0 <![CDATA[Let ℤ_n denote the ring of integers modulo n. A permutation of ℤ_n is a sequence of n distinct elements of ℤ_n. Addition and subtraction of two permutations is defined element-wise. In this paper we consider two extremal problems on permutations of ℤ_n, namely, the maximum size of a collection of permutations such that the sum of any two distinct permutations in the collection is again a permutation, and the maximum size of a collection of permutations such that no sum of two distinct permutations in the collection is a permutation. Let the sizes be denoted by s(n) and t(n) respectively. The case when n is even is trivial in both the cases, with s(n)=1 and t(n)=n!. For n odd, we prove (nφ(n))/2^k≤s(n)≤n!· 2^-(n-1)/2((n-1)/2)! and 2^(n-1)/2·(n-1 / 2)!≤t(n)≤ 2^k·(n-1)!/φ(n), where k is the number of distinct prime divisors of n and φ is the Euler's totient function.]]> Sat, 31 May 2014 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2074 https://doi.org/10.46298/dmtcs.2074 Chandran, L. Sunil Rajendraprasad, Deepak Singh, Nitin Chandran, L. Sunil Rajendraprasad, Deepak Singh, Nitin <![CDATA[Let ℤ_n denote the ring of integers modulo n. A permutation of ℤ_n is a sequence of n distinct elements of ℤ_n. Addition and subtraction of two permutations is defined element-wise. In this paper we consider two extremal problems on permutations of ℤ_n, namely, the maximum size of a collection of permutations such that the sum of any two distinct permutations in the collection is again a permutation, and the maximum size of a collection of permutations such that no sum of two distinct permutations in the collection is a permutation. Let the sizes be denoted by s(n) and t(n) respectively. The case when n is even is trivial in both the cases, with s(n)=1 and t(n)=n!. For n odd, we prove (nφ(n))/2^k≤s(n)≤n!· 2^-(n-1)/2((n-1)/2)! and 2^(n-1)/2·(n-1 / 2)!≤t(n)≤ 2^k·(n-1)!/φ(n), where k is the number of distinct prime divisors of n and φ is the Euler's totient function.]]> 0 <![CDATA[A composition is a sequence of positive integers, called parts, having a fixed sum. By an m-congruence succession, we will mean a pair of adjacent parts x and y within a composition such that x=y(modm). Here, we consider the problem of counting the compositions of size n according to the number of m-congruence successions, extending recent results concerning successions on subsets and permutations. A general formula is obtained, which reduces in the limiting case to the known generating function formula for the number of Carlitz compositions. Special attention is paid to the case m=2, where further enumerative results may be obtained by means of combinatorial arguments. Finally, an asymptotic estimate is provided for the number of compositions of size n having no m-congruence successions.]]> Wed, 21 May 2014 22:00:00 +0000 https://doi.org/10.46298/dmtcs.1252 https://doi.org/10.46298/dmtcs.1252 Mansour, Toufik Shattuck, Mark Wilson, Mark, Mansour, Toufik Shattuck, Mark Wilson, Mark, <![CDATA[A composition is a sequence of positive integers, called parts, having a fixed sum. By an m-congruence succession, we will mean a pair of adjacent parts x and y within a composition such that x=y(modm). Here, we consider the problem of counting the compositions of size n according to the number of m-congruence successions, extending recent results concerning successions on subsets and permutations. A general formula is obtained, which reduces in the limiting case to the known generating function formula for the number of Carlitz compositions. Special attention is paid to the case m=2, where further enumerative results may be obtained by means of combinatorial arguments. Finally, an asymptotic estimate is provided for the number of compositions of size n having no m-congruence successions.]]> 0 <![CDATA[We estimate the proportion of several classes of elements in finite classical groups which are readily recognised algorithmically, and for which some power has a large fixed point subspace and acts irreducibly on a complement of it. The estimates are used in complexity analyses of new recognition algorithms for finite classical groups in arbitrary characteristic.]]> Mon, 12 May 2014 22:00:00 +0000 https://doi.org/10.46298/dmtcs.3908 https://doi.org/10.46298/dmtcs.3908 Niemeyer, Alice, Praeger, Cheryl, Niemeyer, Alice, Praeger, Cheryl, <![CDATA[We estimate the proportion of several classes of elements in finite classical groups which are readily recognised algorithmically, and for which some power has a large fixed point subspace and acts irreducibly on a complement of it. The estimates are used in complexity analyses of new recognition algorithms for finite classical groups in arbitrary characteristic.]]> 0 <![CDATA[Let fm,n,h be the number of spanning forests with h edges in the complete bipartite graph Km,n. Kirchhoff\textquoterights Matrix Tree Theorem implies fm,n,m+n-1=mn-1 nm-1 when m ≥1 and n ≥1, since fm,n,m+n-1 is the number of spanning trees in Km,n. In this paper, we give an algorithm for computing fm,n,h for general m,n,h. We implement this algorithm and use it to compute all non-zero fm,n,h when m ≤50 and n ≤50 in under 2 days.]]> Sun, 11 May 2014 22:00:00 +0000 https://doi.org/10.46298/dmtcs.1248 https://doi.org/10.46298/dmtcs.1248 Stones, Rebecca, Stones, Rebecca, <![CDATA[Let fm,n,h be the number of spanning forests with h edges in the complete bipartite graph Km,n. Kirchhoff\textquoterights Matrix Tree Theorem implies fm,n,m+n-1=mn-1 nm-1 when m ≥1 and n ≥1, since fm,n,m+n-1 is the number of spanning trees in Km,n. In this paper, we give an algorithm for computing fm,n,h for general m,n,h. We implement this algorithm and use it to compute all non-zero fm,n,h when m ≤50 and n ≤50 in under 2 days.]]> 0 <![CDATA[Let (a1,a2,\textellipsis,an) and (b1,b2,\textellipsis,bn) be two sequences of nonnegative integers satisfying the condition that b1>=b2>=...>=bn, ai<= bi for i=1,2,\textellipsis,n and ai+bi>=ai+1+bi+1 for i=1,2,\textellipsis, n-1. In this paper, we give two different conditions, one of which is sufficient and the other one necessary, for the sequences (a1,a2,\textellipsis,an) and (b1,b2,\textellipsis,bn) such that for every (c1,c2,\textellipsis,cn) with ai<=ci<=bi for i=1,2,\textellipsis,n and ∑&limits;i=1n ci=0 (mod 2), there exists a simple graph G with vertices v1,v2,\textellipsis,vn such that dG(vi)=ci for i=1,2,\textellipsis,n. This is a variant of Niessen\textquoterights problem on degree sequences of graphs (Discrete Math., 191 (1998), 247–253).]]> Mon, 05 May 2014 22:00:00 +0000 https://doi.org/10.46298/dmtcs.1260 https://doi.org/10.46298/dmtcs.1260 Guo, Jiyun Yin, Jianhua Guo, Jiyun Yin, Jianhua <![CDATA[Let (a1,a2,\textellipsis,an) and (b1,b2,\textellipsis,bn) be two sequences of nonnegative integers satisfying the condition that b1>=b2>=...>=bn, ai<= bi for i=1,2,\textellipsis,n and ai+bi>=ai+1+bi+1 for i=1,2,\textellipsis, n-1. In this paper, we give two different conditions, one of which is sufficient and the other one necessary, for the sequences (a1,a2,\textellipsis,an) and (b1,b2,\textellipsis,bn) such that for every (c1,c2,\textellipsis,cn) with ai<=ci<=bi for i=1,2,\textellipsis,n and ∑&limits;i=1n ci=0 (mod 2), there exists a simple graph G with vertices v1,v2,\textellipsis,vn such that dG(vi)=ci for i=1,2,\textellipsis,n. This is a variant of Niessen\textquoterights problem on degree sequences of graphs (Discrete Math., 191 (1998), 247–253).]]> 0 <![CDATA[As a common generalization of bipartite and split graphs, monopolar graphs are defined in terms of the existence of certain vertex partitions. It has been shown that to determine whether a graph has such a partition is NP-complete for general graphs and polynomial for several classes of graphs. In this paper, we investigate graphs that admit a unique such partition and call them uniquely monopolar-partitionable graphs. By employing a tree trimming technique, we obtain a characterization of uniquely monopolar-partitionable block graphs. Our characterization implies a polynomial time algorithm for recognizing them.]]> Mon, 05 May 2014 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2073 https://doi.org/10.46298/dmtcs.2073 Chen, Xuegang Huang, Jing Chen, Xuegang Huang, Jing <![CDATA[As a common generalization of bipartite and split graphs, monopolar graphs are defined in terms of the existence of certain vertex partitions. It has been shown that to determine whether a graph has such a partition is NP-complete for general graphs and polynomial for several classes of graphs. In this paper, we investigate graphs that admit a unique such partition and call them uniquely monopolar-partitionable graphs. By employing a tree trimming technique, we obtain a characterization of uniquely monopolar-partitionable block graphs. Our characterization implies a polynomial time algorithm for recognizing them.]]> 0 <![CDATA[We prove a sharp Meyniel-type criterion for hamiltonicity of a balanced bipartite digraph: For a≥2, a strongly connected balanced bipartite digraph D on 2a vertices is hamiltonian if d(u)+d(v)≥3a whenever uv∉A(D) and vu∉A(D). As a consequence, we obtain a sharp sufficient condition for hamiltonicity in terms of the minimal degree: a strongly connected balanced bipartite digraph D on 2a vertices is hamiltonian if δ(D)≥3a/2.]]> Mon, 05 May 2014 22:00:00 +0000 https://doi.org/10.46298/dmtcs.1264 https://doi.org/10.46298/dmtcs.1264 Adamus, Janusz Adamus, Lech Yeo, Anders Adamus, Janusz Adamus, Lech Yeo, Anders <![CDATA[We prove a sharp Meyniel-type criterion for hamiltonicity of a balanced bipartite digraph: For a≥2, a strongly connected balanced bipartite digraph D on 2a vertices is hamiltonian if d(u)+d(v)≥3a whenever uv∉A(D) and vu∉A(D). As a consequence, we obtain a sharp sufficient condition for hamiltonicity in terms of the minimal degree: a strongly connected balanced bipartite digraph D on 2a vertices is hamiltonian if δ(D)≥3a/2.]]> 0 <![CDATA[An S-adic characterization of minimal subshifts with first difference of complexity 1 ≤ p(n + 1) − p(n) ≤ 2 S. Ferenczi proved that any minimal subshift with first difference of complexity bounded by 2 is S-adic with Card(S) ≤ 3 27. In this paper, we improve this result by giving an S-adic characterization of these subshifts with a set S of 5 morphisms, solving by this way the S-adic conjecture for this particular case.]]> Sat, 03 May 2014 22:00:00 +0000 https://doi.org/10.46298/dmtcs.1249 https://doi.org/10.46298/dmtcs.1249 Leroy, Julien Leroy, Julien <![CDATA[An S-adic characterization of minimal subshifts with first difference of complexity 1 ≤ p(n + 1) − p(n) ≤ 2 S. Ferenczi proved that any minimal subshift with first difference of complexity bounded by 2 is S-adic with Card(S) ≤ 3 27. In this paper, we improve this result by giving an S-adic characterization of these subshifts with a set S of 5 morphisms, solving by this way the S-adic conjecture for this particular case.]]> 0 <![CDATA[The embedding of finite metrics in 1 has become a fundamental tool for both combinatorial optimization and large-scale data analysis. One important application is to network flow problems as there is close relation between max-flow min-cut theorems and the minimal distortion embeddings of metrics into 1. Here we show that this theory can be generalized to a larger set of combinatorial optimization problems on both graphs and hypergraphs. This theory is not built on metrics and metric embeddings, but on diversities, a type of multi-way metric introduced recently by the authors. We explore diversity embeddings, 1 diversities, and their application to Steiner Tree Packing and Hypergraph Cut problems.]]> Wed, 23 Apr 2014 22:00:00 +0000 https://doi.org/10.46298/dmtcs.2080 https://doi.org/10.46298/dmtcs.2080 Bryant, David Tupper, Paul, Bryant, David Tupper, Paul, <![CDATA[The embedding of finite metrics in 1 has become a fundamental tool for both combinatorial optimization and large-scale data analysis. One important application is to network flow problems as there is close relation between max-flow min-cut theorems and the minimal distortion embeddings of metrics into 1. Here we show that this theory can be generalized to a larger set of combinatorial optimization problems on both graphs and hypergraphs. This theory is not built on metrics and metric embeddings, but on diversities, a type of multi-way metric introduced recently by the authors. We explore diversity embeddings, 1 diversities, and their application to Steiner Tree Packing and Hypergraph Cut problems.]]> 0 <![CDATA[The vertex cover number of a graph is the minimum number of vertices that are needed to cover all edges. When those vertices are further required to induce a connected subgraph, the corresponding number is called the connected vertex cover number, and is always greater or equal to the vertex cover number. Connected vertex covers are found in many applications, and the relationship between those two graph invariants is therefore a natural question to investigate. For that purpose, we introduce the \em Price of Connectivity, defined as the ratio between the two vertex cover numbers. We prove that the price of connectivity is at most 2 for arbitrary graphs. We further consider graph classes in which the price of connectivity of every induced subgraph is bounded by some real number t. We obtain forbidden induced subgraph characterizations for every real value t ≤q 3/2. We also investigate critical graphs for this property, namely, graphs whose price of connectivity is strictly greater than that of any proper induced subgraph. Those are the only graphs that can appear in a forbidden subgraph characterization for the hereditary property of having a price of connectivity at most t. In particular, we completely characterize the critical graphs that are also chordal. Finally, we also consider the question of computing the price of connectivity of a given graph. Unsurprisingly, the decision version of this question is NP-hard. In fact, we show that it is even complete for the class Θ₂^P = P^NP[\log], the class of decision problems that can be solved in polynomial time, provided we can make O(\log n) queries to an NP-oracle. This paves the way for a thorough investigation of the complexity of problems involving ratios of graph invariants.]]> Mon, 14 Apr 2014 22:00:00 +0000 https://doi.org/10.46298/dmtcs.1262 https://doi.org/10.46298/dmtcs.1262 Camby, Eglantine Cardinal, Jean Fiorini, Samuel Schaudt, Oliver Camby, Eglantine Cardinal, Jean Fiorini, Samuel Schaudt, Oliver <![CDATA[The vertex cover number of a graph is the minimum number of vertices that are needed to cover all edges. When those vertices are further required to induce a connected subgraph, the corresponding number is called the connected vertex cover number, and is always greater or equal to the vertex cover number. Connected vertex covers are found in many applications, and the relationship between those two graph invariants is therefore a natural question to investigate. For that purpose, we introduce the \em Price of Connectivity, defined as the ratio between the two vertex cover numbers. We prove that the price of connectivity is at most 2 for arbitrary graphs. We further consider graph classes in which the price of connectivity of every induced subgraph is bounded by some real number t. We obtain forbidden induced subgraph characterizations for every real value t ≤q 3/2. We also investigate critical graphs for this property, namely, graphs whose price of connectivity is strictly greater than that of any proper induced subgraph. Those are the only graphs that can appear in a forbidden subgraph characterization for the hereditary property of having a price of connectivity at most t. In particular, we completely characterize the critical graphs that are also chordal. Finally, we also consider the question of computing the price of connectivity of a given graph. Unsurprisingly, the decision version of this question is NP-hard. In fact, we show that it is even complete for the class Θ₂^P = P^NP[\log], the class of decision problems that can be solved in polynomial time, provided we can make O(\log n) queries to an NP-oracle. This paves the way for a thorough investigation of the complexity of problems involving ratios of graph invariants.]]> 0 <![CDATA[A graph G of order n is called arbitrarily partitionable (AP, for short) if, for every sequence τ=(n1,\textellipsis,nk) of positive integers that sum up to n, there exists a partition (V1,\textellipsis,Vk) of the vertex set V(G) such that each set Vi induces a connected subgraph of order ni. A graph G is called AP+1 if, given a vertex u∈V(G) and an index q∈ {1,\textellipsis,k}, such a partition exists with u∈Vq. We consider the Cartesian product of AP graphs. We prove that if G is AP+1 and H is traceable, then the Cartesian product G□ H is AP+1. We also prove that G□H is AP, whenever G and H are AP and the order of one of them is not greater than four.]]> Mon, 14 Apr 2014 22:00:00 +0000 https://doi.org/10.46298/dmtcs.1259 https://doi.org/10.46298/dmtcs.1259 Baudon, Olivier Bensmail, Julien Kalinowski, Rafał Marczyk, Antoni Przybyło, Jakub Wozniak, Mariusz Baudon, Olivier Bensmail, Julien Kalinowski, Rafał Marczyk, Antoni Przybyło, Jakub Wozniak, Mariusz <![CDATA[A graph G of order n is called arbitrarily partitionable (AP, for short) if, for every sequence τ=(n1,\textellipsis,nk) of positive integers that sum up to n, there exists a partition (V1,\textellipsis,Vk) of the vertex set V(G) such that each set Vi induces a connected subgraph of order ni. A graph G is called AP+1 if, given a vertex u∈V(G) and an index q∈ {1,\textellipsis,k}, such a partition exists with u∈Vq. We consider the Cartesian product of AP graphs. We prove that if G is AP+1 and H is traceable, then the Cartesian product G□ H is AP+1. We also prove that G□H is AP, whenever G and H are AP and the order of one of them is not greater than four.]]> 0 <![CDATA[In this note a new measure of irregularity of a graph G is introduced. It is named the total irregularity of a graph and is defined as irr(t)(G) - 1/2 Sigma(u, v is an element of V(G)) vertical bar d(G)(u) - d(G)(v)vertical bar, where d(G)(u) denotes the degree of a vertex u is an element of V(G). All graphs with maximal total irregularity are determined. It is also shown that among all trees of the same order the star has the maximal total irregularity.]]> Thu, 03 Apr 2014 22:00:00 +0000 https://doi.org/10.46298/dmtcs.1263 https://doi.org/10.46298/dmtcs.1263 Abdo, Hosam Brandt, Stephan Dimitrov, D. Abdo, Hosam Brandt, Stephan Dimitrov, D. <![CDATA[In this note a new measure of irregularity of a graph G is introduced. It is named the total irregularity of a graph and is defined as irr(t)(G) - 1/2 Sigma(u, v is an element of V(G)) vertical bar d(G)(u) - d(G)(v)vertical bar, where d(G)(u) denotes the degree of a vertex u is an element of V(G). All graphs with maximal total irregularity are determined. It is also shown that among all trees of the same order the star has the maximal total irregularity.]]> 0 <![CDATA[The problem of finding a spanning tree in an undirected graph with a maximum number of leaves is known to be NP-hard. We present an algorithm which finds a spanning tree with at least k leaves in time O*(3.4575k) which improves the currently best algorithm. The estimation of the running time is done by using a non-standard measure. The present paper is one of the still few examples that employ the Measure & Conquer paradigm of algorithm analysis in the area of Parameterized Algorithmics.]]> Sat, 29 Mar 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.1256 https://doi.org/10.46298/dmtcs.1256 Binkele-Raible, Daniel Fernau, Henning Binkele-Raible, Daniel Fernau, Henning <![CDATA[The problem of finding a spanning tree in an undirected graph with a maximum number of leaves is known to be NP-hard. We present an algorithm which finds a spanning tree with at least k leaves in time O*(3.4575k) which improves the currently best algorithm. The estimation of the running time is done by using a non-standard measure. The present paper is one of the still few examples that employ the Measure & Conquer paradigm of algorithm analysis in the area of Parameterized Algorithmics.]]> 0 <![CDATA[The implicit signature κ consists of the multiplication and the (ω-1)-power. We describe a procedure to transform each κ-term over a finite alphabet A into a certain canonical form and show that different canonical forms have different interpretations over some finite semigroup. The procedure of construction of the canonical forms, which is inspired in McCammond\textquoterights normal form algorithm for ω-terms interpreted over the pseudovariety A of all finite aperiodic semigroups, consists in applying elementary changes determined by an elementary set Σ of pseudoidentities. As an application, we deduce that the variety of κ-semigroups generated by the pseudovariety S of all finite semigroups is defined by the set Σ and that the free κ-semigroup generated by the alphabet A in that variety has decidable word problem. Furthermore, we show that each ω-term has a unique ω-term in canonical form with the same value over A. In particular, the canonical forms provide new, simpler, representatives for ω-terms interpreted over that pseudovariety.]]> Wed, 19 Mar 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.1247 https://doi.org/10.46298/dmtcs.1247 Costa, José Carlos Costa, José Carlos <![CDATA[The implicit signature κ consists of the multiplication and the (ω-1)-power. We describe a procedure to transform each κ-term over a finite alphabet A into a certain canonical form and show that different canonical forms have different interpretations over some finite semigroup. The procedure of construction of the canonical forms, which is inspired in McCammond\textquoterights normal form algorithm for ω-terms interpreted over the pseudovariety A of all finite aperiodic semigroups, consists in applying elementary changes determined by an elementary set Σ of pseudoidentities. As an application, we deduce that the variety of κ-semigroups generated by the pseudovariety S of all finite semigroups is defined by the set Σ and that the free κ-semigroup generated by the alphabet A in that variety has decidable word problem. Furthermore, we show that each ω-term has a unique ω-term in canonical form with the same value over A. In particular, the canonical forms provide new, simpler, representatives for ω-terms interpreted over that pseudovariety.]]> 0 <![CDATA[A strong parity vertex coloring of a 2-connected plane graph is a coloring of the vertices such that every face is incident with zero or an odd number of vertices of each color. We prove that every 2-connected loopless plane graph has a strong parity vertex coloring with 97 colors. Moreover the coloring we construct is proper. This proves a conjecture of Czap and Jendrol' [Discuss. Math. Graph Theory 29 (2009), pp. 521-543.]. We also provide examples showing that eight colors may be necessary (ten when restricted to proper colorings).]]> Wed, 19 Mar 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.1268 https://doi.org/10.46298/dmtcs.1268 Kaiser, Tomas Rucky, Ondrej Stehlik, Matej Škrekovski, Riste Kaiser, Tomas Rucky, Ondrej Stehlik, Matej Škrekovski, Riste <![CDATA[A strong parity vertex coloring of a 2-connected plane graph is a coloring of the vertices such that every face is incident with zero or an odd number of vertices of each color. We prove that every 2-connected loopless plane graph has a strong parity vertex coloring with 97 colors. Moreover the coloring we construct is proper. This proves a conjecture of Czap and Jendrol' [Discuss. Math. Graph Theory 29 (2009), pp. 521-543.]. We also provide examples showing that eight colors may be necessary (ten when restricted to proper colorings).]]> 0 <![CDATA[We investigate the computation of mappings from a set S^n to itself with "in situ programs", that is using no extra variables than the input, and performing modifications of one component at a time, hence using no extra memory. In this paper, we survey this problem introduced in previous papers by the authors, we detail its close relation with rearrangeable multicast networks, and we provide new results for both viewpoints. A bijective mapping can be computed by 2n-1 component modifications, that is by a program of length 2n-1, a result equivalent to the rearrangeability of the concatenation of two reversed butterfly networks. For a general arbitrary mapping, we give two methods to build a program with maximal length 4n-3. Equivalently, this yields rearrangeable multicast routing methods for the network formed by four successive butterflies with alternating reversions. The first method is available for any set S and practically equivalent to a known method in network theory. The second method, a refinment of the first, described when |S| is a power of 2, is new and allows more flexibility than the known method. For a linear mapping, when S is any field, or a quotient of an Euclidean domain (e.g Z/sZ for any integer s), we build a program with maximal length 2n-1. In this case the assignments are also linear, thereby particularly efficient from the algorithmic viewpoint, and giving moreover directly a program for the inverse when it exists. This yields also a new result on matrix decompositions, and a new result on the multicast properties of two successive reversed butterflies. Results of this flavour were known only for the boolean field Z/2Z.]]> Sun, 09 Mar 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.638 https://doi.org/10.46298/dmtcs.638 Gioan, Emeric Burckel, Serge Thomé, Emmanuel Gioan, Emeric Burckel, Serge Thomé, Emmanuel <![CDATA[We investigate the computation of mappings from a set S^n to itself with "in situ programs", that is using no extra variables than the input, and performing modifications of one component at a time, hence using no extra memory. In this paper, we survey this problem introduced in previous papers by the authors, we detail its close relation with rearrangeable multicast networks, and we provide new results for both viewpoints. A bijective mapping can be computed by 2n-1 component modifications, that is by a program of length 2n-1, a result equivalent to the rearrangeability of the concatenation of two reversed butterfly networks. For a general arbitrary mapping, we give two methods to build a program with maximal length 4n-3. Equivalently, this yields rearrangeable multicast routing methods for the network formed by four successive butterflies with alternating reversions. The first method is available for any set S and practically equivalent to a known method in network theory. The second method, a refinment of the first, described when |S| is a power of 2, is new and allows more flexibility than the known method. For a linear mapping, when S is any field, or a quotient of an Euclidean domain (e.g Z/sZ for any integer s), we build a program with maximal length 2n-1. In this case the assignments are also linear, thereby particularly efficient from the algorithmic viewpoint, and giving moreover directly a program for the inverse when it exists. This yields also a new result on matrix decompositions, and a new result on the multicast properties of two successive reversed butterflies. Results of this flavour were known only for the boolean field Z/2Z.]]> 0 <![CDATA[A natural generalization of graph colouring involves taking colours from a metric space and insisting that the endpoints of an edge receive colours separated by a minimum distance dictated by properties of the edge. In the q-backbone colouring problem, these minimum distances are either q or 1, depending on whether or not the edge is in the backbone. In this paper we consider the list version of this problem, with particular focus on colours in ℤp - this problem is closely related to the problem of circular choosability. We first prove that the list circular q-backbone chromatic number of a graph is bounded by a function of the list chromatic number. We then consider the more general problem in which each edge is assigned an individual distance between its endpoints, and provide bounds using the Combinatorial Nullstellensatz. Through this result and through structural approaches, we achieve good bounds when both the graph and the backbone belong to restricted families of graphs.]]> Fri, 28 Feb 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.1258 https://doi.org/10.46298/dmtcs.1258 Havet, Frédéric King, Andrew Havet, Frédéric King, Andrew <![CDATA[A natural generalization of graph colouring involves taking colours from a metric space and insisting that the endpoints of an edge receive colours separated by a minimum distance dictated by properties of the edge. In the q-backbone colouring problem, these minimum distances are either q or 1, depending on whether or not the edge is in the backbone. In this paper we consider the list version of this problem, with particular focus on colours in ℤp - this problem is closely related to the problem of circular choosability. We first prove that the list circular q-backbone chromatic number of a graph is bounded by a function of the list chromatic number. We then consider the more general problem in which each edge is assigned an individual distance between its endpoints, and provide bounds using the Combinatorial Nullstellensatz. Through this result and through structural approaches, we achieve good bounds when both the graph and the backbone belong to restricted families of graphs.]]> 0 <![CDATA[We consider compositions of n, i.e., sequences of positive integers (or parts) (σi)i=1k where σ1+σ2+...+σk=n. We define a maximum to be any part which is not less than any other part. The variable of interest is the size of the descent immediately following the first and the last maximum. Using generating functions and Mellin transforms, we obtain asymptotic expressions for the average size of these descents. Finally, we show with the use of a simple bijection between the compositions of n for n>1, that on average the descent after the last maximum is greater than the descent after the first.]]> Fri, 28 Feb 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.1251 https://doi.org/10.46298/dmtcs.1251 Blecher, Aubrey Brennan, Charlotte Knopfmacher, Arnold Blecher, Aubrey Brennan, Charlotte Knopfmacher, Arnold <![CDATA[We consider compositions of n, i.e., sequences of positive integers (or parts) (σi)i=1k where σ1+σ2+...+σk=n. We define a maximum to be any part which is not less than any other part. The variable of interest is the size of the descent immediately following the first and the last maximum. Using generating functions and Mellin transforms, we obtain asymptotic expressions for the average size of these descents. Finally, we show with the use of a simple bijection between the compositions of n for n>1, that on average the descent after the last maximum is greater than the descent after the first.]]> 0 <![CDATA[Let $G=(V,E)$ be an undirected graph without loops and multiple edges. A subset $C\subseteq V$ is called \emph{identifying} if for every vertex $x\in V$ the intersection of $C$ and the closed neighbourhood of $x$ is nonempty, and these intersections are different for different vertices $x$. Let $k$ be a positive integer. We will consider graphs where \emph{every} $k$-subset is identifying. We prove that for every $k>1$ the maximal order of such a graph is at most $2k-2.$ Constructions attaining the maximal order are given for infinitely many values of $k.$ The corresponding problem of $k$-subsets identifying any at most $\ell$ vertices is considered as well.]]> Fri, 28 Feb 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.1253 https://doi.org/10.46298/dmtcs.1253 Gravier, Sylvain Janson, Svante Laihonen, Tero Ranto, Sanna Gravier, Sylvain Janson, Svante Laihonen, Tero Ranto, Sanna <![CDATA[Let $G=(V,E)$ be an undirected graph without loops and multiple edges. A subset $C\subseteq V$ is called \emph{identifying} if for every vertex $x\in V$ the intersection of $C$ and the closed neighbourhood of $x$ is nonempty, and these intersections are different for different vertices $x$. Let $k$ be a positive integer. We will consider graphs where \emph{every} $k$-subset is identifying. We prove that for every $k>1$ the maximal order of such a graph is at most $2k-2.$ Constructions attaining the maximal order are given for infinitely many values of $k.$ The corresponding problem of $k$-subsets identifying any at most $\ell$ vertices is considered as well.]]> 0 <![CDATA[A graph G is an efficient open domination graph if there exists a subset D of V(G) for which the open neighborhoods centered in vertices of D form a partition of V(G). We completely describe efficient open domination graphs among lexicographic, strong, and disjunctive products of graphs. For the Cartesian product we give a characterization when one factor is K2.]]> Fri, 28 Feb 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.1267 https://doi.org/10.46298/dmtcs.1267 Kuziak, Dorota Peterin, Iztok Yero, Ismael Gonzalez Kuziak, Dorota Peterin, Iztok Yero, Ismael Gonzalez <![CDATA[A graph G is an efficient open domination graph if there exists a subset D of V(G) for which the open neighborhoods centered in vertices of D form a partition of V(G). We completely describe efficient open domination graphs among lexicographic, strong, and disjunctive products of graphs. For the Cartesian product we give a characterization when one factor is K2.]]> 0 <![CDATA[We study the relationship between correlated equilibria and Nash equilibria. In contrast to previous work focusing on the possible benefits of a benevolent mediator, we define and bound the Price of Mediation (PoM): the ratio of the social cost (or utility) of the worst correlated equilibrium to the social cost (or utility) of the worst Nash. We observe that in practice, the heuristics used for mediation are frequently non-optimal, and from an economic perspective mediators may be inept or self-interested. Recent results on computation of equilibria also motivate our work. We consider the Price of Mediation for general games with small numbers of players and pure strategies. For two player, two strategy games we give tight bounds in the non-negative cost model and the non-negative utility model. For larger games (either more players, or more pure strategies per player, or both) we show that the PoM can be arbitrary. We also have many results on symmetric congestion games (also known as load balancing games). We show that for general convex cost functions, the PoM can grow exponentially in the number of players. We prove that the PoM is one for linear costs and at most a small constant (but can be larger than one) for concave costs. For polynomial cost functions, we prove bounds on the PoM which are exponential in the degree.]]> Mon, 10 Feb 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.1255 https://doi.org/10.46298/dmtcs.1255 Bradonjic, Milan Ercal, Gunes Meyerson, Adam Roytman, Alan Bradonjic, Milan Ercal, Gunes Meyerson, Adam Roytman, Alan <![CDATA[We study the relationship between correlated equilibria and Nash equilibria. In contrast to previous work focusing on the possible benefits of a benevolent mediator, we define and bound the Price of Mediation (PoM): the ratio of the social cost (or utility) of the worst correlated equilibrium to the social cost (or utility) of the worst Nash. We observe that in practice, the heuristics used for mediation are frequently non-optimal, and from an economic perspective mediators may be inept or self-interested. Recent results on computation of equilibria also motivate our work. We consider the Price of Mediation for general games with small numbers of players and pure strategies. For two player, two strategy games we give tight bounds in the non-negative cost model and the non-negative utility model. For larger games (either more players, or more pure strategies per player, or both) we show that the PoM can be arbitrary. We also have many results on symmetric congestion games (also known as load balancing games). We show that for general convex cost functions, the PoM can grow exponentially in the number of players. We prove that the PoM is one for linear costs and at most a small constant (but can be larger than one) for concave costs. For polynomial cost functions, we prove bounds on the PoM which are exponential in the degree.]]> 0 <![CDATA[For a positive integer n∈ℕ and a set D⊆ ℕ, the distance graph GnD has vertex set { 0,1,\textellipsis,n-1} and two vertices i and j of GnD are adjacent exactly if |j-i|∈D. The condition gcd(D)=1 is necessary for a distance graph GnD being connected. Let D={d1,d2}⊆ℕ be such that d1>d2 and gcd(d1,d2)=1. We prove the following results. If n is sufficiently large in terms of D, then GnD has a Hamiltonian path with endvertices 0 and n-1. If d1d2 is odd, n is even and sufficiently large in terms of D, then GnD has a Hamiltonian cycle. If d1d2 is even and n is sufficiently large in terms of D, then GnD has a Hamiltonian cycle.]]> Mon, 03 Feb 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.1261 https://doi.org/10.46298/dmtcs.1261 Löwenstein, Christian Rautenbach, Dieter Soták, Roman Löwenstein, Christian Rautenbach, Dieter Soták, Roman <![CDATA[For a positive integer n∈ℕ and a set D⊆ ℕ, the distance graph GnD has vertex set { 0,1,\textellipsis,n-1} and two vertices i and j of GnD are adjacent exactly if |j-i|∈D. The condition gcd(D)=1 is necessary for a distance graph GnD being connected. Let D={d1,d2}⊆ℕ be such that d1>d2 and gcd(d1,d2)=1. We prove the following results. If n is sufficiently large in terms of D, then GnD has a Hamiltonian path with endvertices 0 and n-1. If d1d2 is odd, n is even and sufficiently large in terms of D, then GnD has a Hamiltonian cycle. If d1d2 is even and n is sufficiently large in terms of D, then GnD has a Hamiltonian cycle.]]> 0 <![CDATA[Let G be a finite connected graph. We give an asymptotically tight upper bound on the size of G in terms of order, radius and minimum degree. Our result is a strengthening of an old classical theorem of Vizing (1967) if minimum degree is prescribed.]]> Thu, 16 Jan 2014 23:00:00 +0000 https://doi.org/10.46298/dmtcs.1265 https://doi.org/10.46298/dmtcs.1265 Mukwembi, Simon Mukwembi, Simon <![CDATA[Let G be a finite connected graph. We give an asymptotically tight upper bound on the size of G in terms of order, radius and minimum degree. Our result is a strengthening of an old classical theorem of Vizing (1967) if minimum degree is prescribed.]]> 0 <![CDATA[A natural generalization of graph colouring involves taking colours from a metric space and insisting that the endpoints of an edge receive colours separated by a minimum distance dictated by properties of the edge. In the q-backbone colouring problem, these minimum distances are either q or 1, depending on whether or not the edge is in the backbone. In this paper we consider the list version of this problem, with particular focus on colours in ℤp - this problem is closely related to the problem of circular choosability. We first prove that the list circular q-backbone chromatic number of a graph is bounded by a function of the list chromatic number. We then consider the more general problem in which each edge is assigned an individual distance between its endpoints, and provide bounds using the Combinatorial Nullstellensatz. Through this result and through structural approaches, we achieve good bounds when both the graph and the backbone belong to restricted families of graphs.]]> Wed, 01 Jan 2014 07:00:00 +0000 https://doi.org/10.46298/dmtcs.639 https://doi.org/10.46298/dmtcs.639 Havet, Frederic King, Andrew D. Havet, Frederic King, Andrew D. <![CDATA[A natural generalization of graph colouring involves taking colours from a metric space and insisting that the endpoints of an edge receive colours separated by a minimum distance dictated by properties of the edge. In the q-backbone colouring problem, these minimum distances are either q or 1, depending on whether or not the edge is in the backbone. In this paper we consider the list version of this problem, with particular focus on colours in ℤp - this problem is closely related to the problem of circular choosability. We first prove that the list circular q-backbone chromatic number of a graph is bounded by a function of the list chromatic number. We then consider the more general problem in which each edge is assigned an individual distance between its endpoints, and provide bounds using the Combinatorial Nullstellensatz. Through this result and through structural approaches, we achieve good bounds when both the graph and the backbone belong to restricted families of graphs.]]> 0 <![CDATA[A strong parity vertex coloring of a 2-connected plane graph is a coloring of the vertices such that every face is incident with zero or an odd number of vertices of each color. We prove that every 2-connected loopless plane graph has a strong parity vertex coloring with 97 colors. Moreover the coloring we construct is proper. This proves a conjecture of Czap and Jendrol' [Discuss. Math. Graph Theory 29 (2009), pp. 521-543.]. We also provide examples showing that eight colors may be necessary (ten when restricted to proper colorings).]]> Wed, 01 Jan 2014 07:00:00 +0000 https://doi.org/10.46298/dmtcs.640 https://doi.org/10.46298/dmtcs.640 Kaiser, Tomas Rucky, Ondrej Stehlik, Matej Skrekovski, Riste Kaiser, Tomas Rucky, Ondrej Stehlik, Matej Skrekovski, Riste <![CDATA[A strong parity vertex coloring of a 2-connected plane graph is a coloring of the vertices such that every face is incident with zero or an odd number of vertices of each color. We prove that every 2-connected loopless plane graph has a strong parity vertex coloring with 97 colors. Moreover the coloring we construct is proper. This proves a conjecture of Czap and Jendrol' [Discuss. Math. Graph Theory 29 (2009), pp. 521-543.]. We also provide examples showing that eight colors may be necessary (ten when restricted to proper colorings).]]> 0 <![CDATA[A Peterson variety is a subvariety of the flag variety $G/B$ defined by certain linear conditions. Peterson varieties appear in the construction of the quantum cohomology of partial flag varieties and in applications to the Toda flows. Each Peterson variety has a one-dimensional torus $S^1$ acting on it. We give a basis of Peterson Schubert classes for $H_{S^1}^*(Pet)$ and identify the ring generators. In type A Harada-Tymoczko gave a positive Monk formula, and Bayegan-Harada gave Giambelli's formula for multiplication in the cohomology ring. This paper gives a Chevalley-Monk rule and Giambelli's formula for all Lie types.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2449 https://doi.org/10.46298/dmtcs.2449 Drellich, Elizabeth Drellich, Elizabeth <![CDATA[A Peterson variety is a subvariety of the flag variety $G/B$ defined by certain linear conditions. Peterson varieties appear in the construction of the quantum cohomology of partial flag varieties and in applications to the Toda flows. Each Peterson variety has a one-dimensional torus $S^1$ acting on it. We give a basis of Peterson Schubert classes for $H_{S^1}^*(Pet)$ and identify the ring generators. In type A Harada-Tymoczko gave a positive Monk formula, and Bayegan-Harada gave Giambelli's formula for multiplication in the cohomology ring. This paper gives a Chevalley-Monk rule and Giambelli's formula for all Lie types.]]> 0 <![CDATA[Given an underlying undirected simple graph, we consider the set of all acyclic orientations of its edges. Each of these orientations induces a partial order on the vertices of our graph, and therefore we can count the number of linear extensions of these posets. We want to know which choice of orientation maximizes the number of linear extensions of the corresponding poset, and this problem is solved essentially for comparability graphs and odd cycles, presenting several proofs. We then provide an inequality for general graphs and discuss further techniques.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2455 https://doi.org/10.46298/dmtcs.2455 Iriarte, Benjamin Iriarte, Benjamin <![CDATA[Given an underlying undirected simple graph, we consider the set of all acyclic orientations of its edges. Each of these orientations induces a partial order on the vertices of our graph, and therefore we can count the number of linear extensions of these posets. We want to know which choice of orientation maximizes the number of linear extensions of the corresponding poset, and this problem is solved essentially for comparability graphs and odd cycles, presenting several proofs. We then provide an inequality for general graphs and discuss further techniques.]]> 0 <![CDATA[Given an odd prime p, we give an explicit factorization over the ring of formal power series with integer coefficients for certain reducible polynomials whose constant term is of the form $p^w$ with $w>1$. Our formulas are given in terms of partial Bell polynomials and rely on the inversion formula of Lagrange.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2454 https://doi.org/10.46298/dmtcs.2454 Birmajer, Daniel Gil, Juan B. Weiner, Michael D. Birmajer, Daniel Gil, Juan B. Weiner, Michael D. <![CDATA[Given an odd prime p, we give an explicit factorization over the ring of formal power series with integer coefficients for certain reducible polynomials whose constant term is of the form $p^w$ with $w>1$. Our formulas are given in terms of partial Bell polynomials and rely on the inversion formula of Lagrange.]]> 0 <![CDATA[Define the interval rank $r_[i,j] : Gr_k(\mathbb C^n) →\mathbb{N}$ of a k-plane V as the dimension of the orthogonal projection $π _[i,j](V)$ of V to the $(j-i+1)$-dimensional subspace that uses the coordinates $i,i+1,\ldots,j$. By measuring all these ranks, we define the interval rank stratification of the Grassmannian $Gr_k(\mathbb C^n)$. It is finer than the Schubert and Richardson stratifications, and coarser than the positroid stratification studied by Lusztig, Postnikov, and others, so we call the closures of these strata interval positroid varieties. We connect Vakil's "geometric Littlewood-Richardson rule", in which he computed the homology classes of Richardson varieties (Schubert varieties intersected with opposite Schubert varieties), to Erd&odblac;s-Ko-Rado shifting, and show that all of Vakil's varieties are interval positroid varieties. We build on his work in three ways: (1) we extend it to arbitrary interval positroid varieties, (2) we use it to compute in equivariant K-theory, not just homology, and (3) we simplify Vakil's (2+1)-dimensional "checker games" to 2-dimensional diagrams we call "IP pipe dreams". The ring Symm of symmetric functions and its basis of Schur functions is well-known to be very closely related to the ring $\bigoplus_a,b H_*(Gr_a(\mathbb{C}^{(a+b)})$ and its basis of Schubert classes. We extend the latter ring to equivariant K-theory (with respect to a circle action on each $\mathbb{C}^{(a+b)}$, and compute the structure constants of this two-parameter deformation of Symm using the interval positroid technology above.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2453 https://doi.org/10.46298/dmtcs.2453 Knutson, Allen Lederer, Mathias Knutson, Allen Lederer, Mathias <![CDATA[Define the interval rank $r_[i,j] : Gr_k(\mathbb C^n) →\mathbb{N}$ of a k-plane V as the dimension of the orthogonal projection $π _[i,j](V)$ of V to the $(j-i+1)$-dimensional subspace that uses the coordinates $i,i+1,\ldots,j$. By measuring all these ranks, we define the interval rank stratification of the Grassmannian $Gr_k(\mathbb C^n)$. It is finer than the Schubert and Richardson stratifications, and coarser than the positroid stratification studied by Lusztig, Postnikov, and others, so we call the closures of these strata interval positroid varieties. We connect Vakil's "geometric Littlewood-Richardson rule", in which he computed the homology classes of Richardson varieties (Schubert varieties intersected with opposite Schubert varieties), to Erd&odblac;s-Ko-Rado shifting, and show that all of Vakil's varieties are interval positroid varieties. We build on his work in three ways: (1) we extend it to arbitrary interval positroid varieties, (2) we use it to compute in equivariant K-theory, not just homology, and (3) we simplify Vakil's (2+1)-dimensional "checker games" to 2-dimensional diagrams we call "IP pipe dreams". The ring Symm of symmetric functions and its basis of Schur functions is well-known to be very closely related to the ring $\bigoplus_a,b H_*(Gr_a(\mathbb{C}^{(a+b)})$ and its basis of Schubert classes. We extend the latter ring to equivariant K-theory (with respect to a circle action on each $\mathbb{C}^{(a+b)}$, and compute the structure constants of this two-parameter deformation of Symm using the interval positroid technology above.]]> 0 <![CDATA[The family of Buchsbaum simplicial posets generalizes the family of simplicial cell manifolds. The $h'-$vector of a simplicial complex or simplicial poset encodes the combinatorial and topological data of its face numbers and the reduced Betti numbers of its geometric realization. Novik and Swartz showed that the $h'-$vector of a Buchsbaum simplicial poset satisfies certain simple inequalities. In this paper we show that these necessary conditions are in fact sufficient to characterize the h'-vectors of Buchsbaum simplicial posets with prescribed Betti numbers.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2452 https://doi.org/10.46298/dmtcs.2452 Browder, Jonathan Klee, Steven Browder, Jonathan Klee, Steven <![CDATA[The family of Buchsbaum simplicial posets generalizes the family of simplicial cell manifolds. The $h'-$vector of a simplicial complex or simplicial poset encodes the combinatorial and topological data of its face numbers and the reduced Betti numbers of its geometric realization. Novik and Swartz showed that the $h'-$vector of a Buchsbaum simplicial poset satisfies certain simple inequalities. In this paper we show that these necessary conditions are in fact sufficient to characterize the h'-vectors of Buchsbaum simplicial posets with prescribed Betti numbers.]]> 0 <![CDATA[Knutson and Tao's work on the Horn conjectures used combinatorial invariants called hives and honeycombs to relate spectra of sums of Hermitian matrices to Littlewood-Richardson coefficients and problems in representation theory, but these relationships remained implicit. Here, let $M$ and $N$ be two $n ×n$ Hermitian matrices. We will show how to determine a hive $\mathcal{H}(M, N)={H_ijk}$ using linear algebra constructions from this matrix pair. With this construction, one may also define an explicit Littlewood-Richardson filling (enumerated by the Littlewood-Richardson coefficient $c_μν ^λ$ associated to the matrix pair). We then relate rotations of orthonormal bases of eigenvectors of $M$ and $N$ to deformations of honeycombs (and hives), which we interpret in terms of the structure of crystal graphs and Littelmann's path operators. We find that the crystal structure is determined \emphmore simply from the perspective of rotations than that of path operators.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2451 https://doi.org/10.46298/dmtcs.2451 Appleby, Glenn Whitehead, Tamsen Appleby, Glenn Whitehead, Tamsen <![CDATA[Knutson and Tao's work on the Horn conjectures used combinatorial invariants called hives and honeycombs to relate spectra of sums of Hermitian matrices to Littlewood-Richardson coefficients and problems in representation theory, but these relationships remained implicit. Here, let $M$ and $N$ be two $n ×n$ Hermitian matrices. We will show how to determine a hive $\mathcal{H}(M, N)={H_ijk}$ using linear algebra constructions from this matrix pair. With this construction, one may also define an explicit Littlewood-Richardson filling (enumerated by the Littlewood-Richardson coefficient $c_μν ^λ$ associated to the matrix pair). We then relate rotations of orthonormal bases of eigenvectors of $M$ and $N$ to deformations of honeycombs (and hives), which we interpret in terms of the structure of crystal graphs and Littelmann's path operators. We find that the crystal structure is determined \emphmore simply from the perspective of rotations than that of path operators.]]> 0 <![CDATA[We introduce a new approach to the enumeration of rational slope parking functions with respect to the area and a generalized dinv statistics, and relate the combinatorics of parking functions to that of affine permutations. We relate our construction to two previously known combinatorial constructions: Haglund's bijection ζ exchanging the pairs of statistics (area,dinv) and (bounce, area) on Dyck paths, and Pak-Stanley labeling of the regions of k-Shi hyperplane arrangements by k-parking functions. Essentially, our approach can be viewed as a generalization and a unification of these two constructions.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2450 https://doi.org/10.46298/dmtcs.2450 Gorsky, Eugene Mazin, Mikhail Vazirani, Monica Gorsky, Eugene Mazin, Mikhail Vazirani, Monica <![CDATA[We introduce a new approach to the enumeration of rational slope parking functions with respect to the area and a generalized dinv statistics, and relate the combinatorics of parking functions to that of affine permutations. We relate our construction to two previously known combinatorial constructions: Haglund's bijection ζ exchanging the pairs of statistics (area,dinv) and (bounce, area) on Dyck paths, and Pak-Stanley labeling of the regions of k-Shi hyperplane arrangements by k-parking functions. Essentially, our approach can be viewed as a generalization and a unification of these two constructions.]]> 0 <![CDATA[This paper uses the theory of dual equivalence graphs to give explicit Schur expansions to several families of symmetric functions. We begin by giving a combinatorial definition of the modified Macdonald polynomials and modified Hall-Littlewood polynomials indexed by any diagram $δ ⊂ \mathbb{Z} \times \mathbb{Z}$, written as $\widetilde H_δ (X;q,t)$ and $\widetilde P_δ (X;t)$, respectively. We then give an explicit Schur expansion of $\widetilde P_δ (X;t)$ as a sum over a subset of the Yamanouchi words, as opposed to the expansion using the charge statistic given in 1978 by Lascoux and Schüztenberger. We further define the symmetric function $R_γ ,δ (X)$ as a refinement of $\widetilde P_δ$ and similarly describe its Schur expansion. We then analysize $R_γ ,δ (X)$ to determine the leading term of its Schur expansion. To gain these results, we associate each Macdonald polynomial with a signed colored graph $\mathcal{H}_δ$ . In the case where a subgraph of $\mathcal{H}_δ$ is a dual equivalence graph, we provide the Schur expansion of its associated symmetric function, yielding several corollaries.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2437 https://doi.org/10.46298/dmtcs.2437 Roberts, Austin Roberts, Austin <![CDATA[This paper uses the theory of dual equivalence graphs to give explicit Schur expansions to several families of symmetric functions. We begin by giving a combinatorial definition of the modified Macdonald polynomials and modified Hall-Littlewood polynomials indexed by any diagram $δ ⊂ \mathbb{Z} \times \mathbb{Z}$, written as $\widetilde H_δ (X;q,t)$ and $\widetilde P_δ (X;t)$, respectively. We then give an explicit Schur expansion of $\widetilde P_δ (X;t)$ as a sum over a subset of the Yamanouchi words, as opposed to the expansion using the charge statistic given in 1978 by Lascoux and Schüztenberger. We further define the symmetric function $R_γ ,δ (X)$ as a refinement of $\widetilde P_δ$ and similarly describe its Schur expansion. We then analysize $R_γ ,δ (X)$ to determine the leading term of its Schur expansion. To gain these results, we associate each Macdonald polynomial with a signed colored graph $\mathcal{H}_δ$ . In the case where a subgraph of $\mathcal{H}_δ$ is a dual equivalence graph, we provide the Schur expansion of its associated symmetric function, yielding several corollaries.]]> 0 <![CDATA[Bott-Samelson varieties factor the flag variety $G/B$ into a product of $\mathbb{C}\mathbb{P}^1$'s with a map into $G/B$. These varieties are mostly studied in the case in which the map into $G/B$ is birational; however in this paper we study fibers of this map when it is not birational. We will see that in some cases this fiber is a toric variety. In order to do so we use the moment map of a Bott-Samelson variety to translate this problem into a purely combinatorial one in terms of a subword complex. These simplicial complexes, defined by Knutson and Miller, encode a lot of information about reduced words in a Coxeter system. Pilaud and Stump realized certain subword complexes as the dual to the boundary of a polytope which generalizes the brick polytope defined by Pilaud and Santos. For a nice family of words, the brick polytope is the generalized associahedron realized by Hohlweg and Lange. These stories connect in a nice way: the moment polytope of a fiber of the Bott-Samelson map is the Brick polytope. In particular, we give a nice description of the toric variety of the associahedron.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2448 https://doi.org/10.46298/dmtcs.2448 Escobar, Laura Escobar, Laura <![CDATA[Bott-Samelson varieties factor the flag variety $G/B$ into a product of $\mathbb{C}\mathbb{P}^1$'s with a map into $G/B$. These varieties are mostly studied in the case in which the map into $G/B$ is birational; however in this paper we study fibers of this map when it is not birational. We will see that in some cases this fiber is a toric variety. In order to do so we use the moment map of a Bott-Samelson variety to translate this problem into a purely combinatorial one in terms of a subword complex. These simplicial complexes, defined by Knutson and Miller, encode a lot of information about reduced words in a Coxeter system. Pilaud and Stump realized certain subword complexes as the dual to the boundary of a polytope which generalizes the brick polytope defined by Pilaud and Santos. For a nice family of words, the brick polytope is the generalized associahedron realized by Hohlweg and Lange. These stories connect in a nice way: the moment polytope of a fiber of the Bott-Samelson map is the Brick polytope. In particular, we give a nice description of the toric variety of the associahedron.]]> 0 <![CDATA[Many combinatorial and topological invariants of a hyperplane arrangement can be computed in terms of its Tutte polynomial. Similarly, many invariants of a hypertoric arrangement can be computed in terms of its arithmetic Tutte polynomial. We compute the arithmetic Tutte polynomials of the classical root systems $A_n, B_n, C_n$, and $D_n$ with respect to their integer, root, and weight lattices. We do it in two ways: by introducing a \emphfinite field method for arithmetic Tutte polynomials, and by enumerating signed graphs with respect to six parameters.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2447 https://doi.org/10.46298/dmtcs.2447 Ardila, Federico Castillo, Federico Henley, Michael Ardila, Federico Castillo, Federico Henley, Michael <![CDATA[Many combinatorial and topological invariants of a hyperplane arrangement can be computed in terms of its Tutte polynomial. Similarly, many invariants of a hypertoric arrangement can be computed in terms of its arithmetic Tutte polynomial. We compute the arithmetic Tutte polynomials of the classical root systems $A_n, B_n, C_n$, and $D_n$ with respect to their integer, root, and weight lattices. We do it in two ways: by introducing a \emphfinite field method for arithmetic Tutte polynomials, and by enumerating signed graphs with respect to six parameters.]]> 0 <![CDATA[We study permutation patterns from an algebraic combinatorics point of view. Using analogues of the classical shuffle and infiltration products for word, we define two new Hopf algebras of permutations related to the notion of permutation pattern. We show several remarkable properties of permutation patterns functions, as well their occurrence in other domains.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2446 https://doi.org/10.46298/dmtcs.2446 Vargas, Yannic Vargas, Yannic <![CDATA[We study permutation patterns from an algebraic combinatorics point of view. Using analogues of the classical shuffle and infiltration products for word, we define two new Hopf algebras of permutations related to the notion of permutation pattern. We show several remarkable properties of permutation patterns functions, as well their occurrence in other domains.]]> 0 <![CDATA[We define and enumerate a new class of self-avoiding walks on the square lattice, which we call weakly prudent bridges. Their definition is inspired by two previously-considered classes of self-avoiding walks, and can be viewed as a combination of those two models. We consider several methods for recursively generating these objects, each with its own advantages and disadvantages, and use these methods to solve the generating function, obtain very long series, and randomly generate walks of arbitrary size. We find that the growth constant of these walks is approximately 2.58, which is larger than that of any previously-solved class of self-avoiding walks.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2445 https://doi.org/10.46298/dmtcs.2445 Bacher, Axel Beaton, Nicholas, Bacher, Axel Beaton, Nicholas, <![CDATA[We define and enumerate a new class of self-avoiding walks on the square lattice, which we call weakly prudent bridges. Their definition is inspired by two previously-considered classes of self-avoiding walks, and can be viewed as a combination of those two models. We consider several methods for recursively generating these objects, each with its own advantages and disadvantages, and use these methods to solve the generating function, obtain very long series, and randomly generate walks of arbitrary size. We find that the growth constant of these walks is approximately 2.58, which is larger than that of any previously-solved class of self-avoiding walks.]]> 0 <![CDATA[We introduce a poset structure on the reduced galleries in a supersolvable arrangement of hyperplanes. In particular, for Coxeter groups of type A or B, we construct a poset of reduced words for the longest element whose Hasse diagram is the graph of reduced words. Using Rambau's Suspension Lemma, we show that these posets are homotopy equivalent to spheres. We furthermore conjecture that its intervals are either homotopy equivalent to spheres or are contractible. One may view this as a analogue of a result of Edelman and Walker on the homotopy type of intervals of a poset of chambers of a hyperplane arrangement.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2444 https://doi.org/10.46298/dmtcs.2444 McConville, Thomas McConville, Thomas <![CDATA[We introduce a poset structure on the reduced galleries in a supersolvable arrangement of hyperplanes. In particular, for Coxeter groups of type A or B, we construct a poset of reduced words for the longest element whose Hasse diagram is the graph of reduced words. Using Rambau's Suspension Lemma, we show that these posets are homotopy equivalent to spheres. We furthermore conjecture that its intervals are either homotopy equivalent to spheres or are contractible. One may view this as a analogue of a result of Edelman and Walker on the homotopy type of intervals of a poset of chambers of a hyperplane arrangement.]]> 0 <![CDATA[A stabilized-interval-free (SIF) permutation on [n], introduced by Callan, is a permutation that does not stabilize any proper interval of [n]. Such permutations are known to be the irreducibles in the decomposition of permutations along non-crossing partitions. That is, if $s_n$ denotes the number of SIF permutations on [n], $S(z)=1+\sum_{n\geq1} s_n z^n$, and $F(z)=1+\sum_{n\geq1} n! z^n$, then $F(z)= S(zF(z))$. This article presents, in turn, a decomposition of SIF permutations along non-crossing partitions. Specifically, by working with a convenient diagrammatic representation, given in terms of perfect matchings on alternating binary strings, we arrive at the \emphchord-connected permutations on [n], counted by $\{c_n\}_{n\geq1}$, whose generating function satisfies $S(z)= C(zS(z))$. The expressions at hand have immediate probabilistic interpretations, via the celebrated moment-cumulant formula of Speicher, in the context of the free probability theory of Voiculescu. The probability distributions that appear are the exponential and the complex Gaussian.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2443 https://doi.org/10.46298/dmtcs.2443 Blitvić, Natasha Blitvić, Natasha <![CDATA[A stabilized-interval-free (SIF) permutation on [n], introduced by Callan, is a permutation that does not stabilize any proper interval of [n]. Such permutations are known to be the irreducibles in the decomposition of permutations along non-crossing partitions. That is, if $s_n$ denotes the number of SIF permutations on [n], $S(z)=1+\sum_{n\geq1} s_n z^n$, and $F(z)=1+\sum_{n\geq1} n! z^n$, then $F(z)= S(zF(z))$. This article presents, in turn, a decomposition of SIF permutations along non-crossing partitions. Specifically, by working with a convenient diagrammatic representation, given in terms of perfect matchings on alternating binary strings, we arrive at the \emphchord-connected permutations on [n], counted by $\{c_n\}_{n\geq1}$, whose generating function satisfies $S(z)= C(zS(z))$. The expressions at hand have immediate probabilistic interpretations, via the celebrated moment-cumulant formula of Speicher, in the context of the free probability theory of Voiculescu. The probability distributions that appear are the exponential and the complex Gaussian.]]> 0 <![CDATA[The Classical Shuffle Conjecture of Haglund et al. (2005) has a symmetric function side and a combinatorial side. The combinatorial side $q,t$-enumerates parking functions in the $n ×n$ lattice. The symmetric function side may be simply expressed as $∇ e_n$ , where $∇$ is the Macdonald eigen-operator introduced by Bergeron and Garsia (1999) and $e_n$ is the elementary symmetric function. The combinatorial side has been extended to parking functions in the $m ×n$ lattice for coprime $m,n$ by Hikita (2012). Recently, Gorsky and Negut have been able to extend the Shuffle Conjecture by combining their work (2012a, 2012b, 2013) (related to work of Schiffmann and Vasserot (2011, 2013)) with Hikita's combinatorial results. We prove this new conjecture for the cases $m=2$ and $n=2$ .]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2442 https://doi.org/10.46298/dmtcs.2442 Leven, Emily Leven, Emily <![CDATA[The Classical Shuffle Conjecture of Haglund et al. (2005) has a symmetric function side and a combinatorial side. The combinatorial side $q,t$-enumerates parking functions in the $n ×n$ lattice. The symmetric function side may be simply expressed as $∇ e_n$ , where $∇$ is the Macdonald eigen-operator introduced by Bergeron and Garsia (1999) and $e_n$ is the elementary symmetric function. The combinatorial side has been extended to parking functions in the $m ×n$ lattice for coprime $m,n$ by Hikita (2012). Recently, Gorsky and Negut have been able to extend the Shuffle Conjecture by combining their work (2012a, 2012b, 2013) (related to work of Schiffmann and Vasserot (2011, 2013)) with Hikita's combinatorial results. We prove this new conjecture for the cases $m=2$ and $n=2$ .]]> 0 <![CDATA[We discuss arrangements of equal minors in totally positive matrices. More precisely, we would like to investigate the structure of possible equalities and inequalities between the minors. We show that arrangements of equals minors of largest value are in bijection with sorted sets, which earlier appeared in the context of alcoved polytopes and Gröbner bases. Maximal arrangements of this form correspond to simplices of the alcoved triangulation of the hypersimplex; and the number of such arrangements equals the Eulerian number. On the other hand, we conjecture and prove in many cases that arrangements of equal minors of smallest value are exactly the weakly separated sets. Weakly separated sets, originally introduced by Leclerc and Zelevinsky, are closely related to the \textitpositive Grassmannian and the associated cluster algebra.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2441 https://doi.org/10.46298/dmtcs.2441 Farber, Miriam Postnikov, Alexander Farber, Miriam Postnikov, Alexander <![CDATA[We discuss arrangements of equal minors in totally positive matrices. More precisely, we would like to investigate the structure of possible equalities and inequalities between the minors. We show that arrangements of equals minors of largest value are in bijection with sorted sets, which earlier appeared in the context of alcoved polytopes and Gröbner bases. Maximal arrangements of this form correspond to simplices of the alcoved triangulation of the hypersimplex; and the number of such arrangements equals the Eulerian number. On the other hand, we conjecture and prove in many cases that arrangements of equal minors of smallest value are exactly the weakly separated sets. Weakly separated sets, originally introduced by Leclerc and Zelevinsky, are closely related to the \textitpositive Grassmannian and the associated cluster algebra.]]> 0 <![CDATA[The study of rhomboid-shaped fully packed loop configurations (RFPLs) is inspired by the work of Fischer and Nadeau on triangular fully packed loop configurations (TFPLs). By using the same techniques as they did some nice combinatorics for RFPLs arise. To each RFPL and to each oriented RFPL a quadruple of binary words (α ,β ;γ ,δ ) – its so-called boundary – is assigned. There are necessary conditions for the boundary of an RFPL respectively an oriented RFPL. For instance, it has to fulfill the inequality $d(γ )+d(δ )\geq(α )+d(β )+\vert α \vert _0\vert β \vert _1$, where $\vert α \vert _i $ denotes the number of occurrences of $i=0,1$ in α and d(α ) denotes the number of inversions of α . Furthermore, the number of ordinary RFPLs with boundary (α ,β ;γ ,δ ) can be expressed in terms of oriented RFPLs with the same boundary. Finally, oriented RFPLs with boundary (α ,β ;γ ,δ ) such that $d(γ )+d(δ )=d(α )+d(β )+\vert α \vert _0\vert β \vert _1$ are considered. They are in bijection with rhomboid-shaped Knutson-Tao puzzles. Also, Littlewood-Richardson tableaux of defect d are defined. They can be understood as a generalization of Littlewood-Richardson tableaux. Those tableaux are in bijection with rhomboid-shaped Knutson-Tao puzzles.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2440 https://doi.org/10.46298/dmtcs.2440 Beil, Sabine Beil, Sabine <![CDATA[The study of rhomboid-shaped fully packed loop configurations (RFPLs) is inspired by the work of Fischer and Nadeau on triangular fully packed loop configurations (TFPLs). By using the same techniques as they did some nice combinatorics for RFPLs arise. To each RFPL and to each oriented RFPL a quadruple of binary words (α ,β ;γ ,δ ) – its so-called boundary – is assigned. There are necessary conditions for the boundary of an RFPL respectively an oriented RFPL. For instance, it has to fulfill the inequality $d(γ )+d(δ )\geq(α )+d(β )+\vert α \vert _0\vert β \vert _1$, where $\vert α \vert _i $ denotes the number of occurrences of $i=0,1$ in α and d(α ) denotes the number of inversions of α . Furthermore, the number of ordinary RFPLs with boundary (α ,β ;γ ,δ ) can be expressed in terms of oriented RFPLs with the same boundary. Finally, oriented RFPLs with boundary (α ,β ;γ ,δ ) such that $d(γ )+d(δ )=d(α )+d(β )+\vert α \vert _0\vert β \vert _1$ are considered. They are in bijection with rhomboid-shaped Knutson-Tao puzzles. Also, Littlewood-Richardson tableaux of defect d are defined. They can be understood as a generalization of Littlewood-Richardson tableaux. Those tableaux are in bijection with rhomboid-shaped Knutson-Tao puzzles.]]> 0 <![CDATA[Various authors have studied a natural operation (under various names) on the order ideals (equivalently antichains) of a finite poset, here called \emphrowmotion. For certain posets of interest, the order of this map is much smaller than one would naively expect, and the orbits exhibit unexpected properties. In very recent work (inspired by discussions with Berenstein) Einstein and Propp describe how rowmotion can be generalized: first to the piecewise-linear setting of order polytopes, then via detropicalization to the birational setting. In the latter setting, it is no longer \empha priori clear even that birational rowmotion has finite order, and for many posets the order is infinite. However, we are able to show that birational rowmotion has the same order, p+q, for the poset P=[p]×[q] (product of two chains), as ordinary rowmotion. We also show that birational (hence ordinary) rowmotion has finite order for some other classes of posets, e.g., the upper, lower, right and left halves of the poset above, and trees having all leaves on the same level. Our methods are based on those used by Volkov to resolve the type AA (rectangular) Zamolodchikov Periodicity Conjecture.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2439 https://doi.org/10.46298/dmtcs.2439 Grinberg, Darij Roby, Tom Grinberg, Darij Roby, Tom <![CDATA[Various authors have studied a natural operation (under various names) on the order ideals (equivalently antichains) of a finite poset, here called \emphrowmotion. For certain posets of interest, the order of this map is much smaller than one would naively expect, and the orbits exhibit unexpected properties. In very recent work (inspired by discussions with Berenstein) Einstein and Propp describe how rowmotion can be generalized: first to the piecewise-linear setting of order polytopes, then via detropicalization to the birational setting. In the latter setting, it is no longer \empha priori clear even that birational rowmotion has finite order, and for many posets the order is infinite. However, we are able to show that birational rowmotion has the same order, p+q, for the poset P=[p]×[q] (product of two chains), as ordinary rowmotion. We also show that birational (hence ordinary) rowmotion has finite order for some other classes of posets, e.g., the upper, lower, right and left halves of the poset above, and trees having all leaves on the same level. Our methods are based on those used by Volkov to resolve the type AA (rectangular) Zamolodchikov Periodicity Conjecture.]]> 0 <![CDATA[This paper considers the representation theory of towers of algebras of $\mathcal{J} -trivial$ monoids. Using a very general lemma on induction, we derive a combinatorial description of the algebra and coalgebra structure on the Grothendieck rings $G_0$ and $K_0$. We then apply our theory to some examples. We first retrieve the classical Krob-Thibon's categorification of the pair of Hopf algebras QSym$/NCSF$ as representation theory of the tower of 0-Hecke algebras. Considering the towers of semilattices given by the permutohedron, associahedron, and Boolean lattices, we categorify the algebra and the coalgebra structure of the Hopf algebras $FQSym , PBT$ , and $NCSF$ respectively. Lastly we completely describe the representation theory of the tower of the monoids of Non Decreasing Parking Functions.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2438 https://doi.org/10.46298/dmtcs.2438 Virmaux, Aladin Virmaux, Aladin <![CDATA[This paper considers the representation theory of towers of algebras of $\mathcal{J} -trivial$ monoids. Using a very general lemma on induction, we derive a combinatorial description of the algebra and coalgebra structure on the Grothendieck rings $G_0$ and $K_0$. We then apply our theory to some examples. We first retrieve the classical Krob-Thibon's categorification of the pair of Hopf algebras QSym$/NCSF$ as representation theory of the tower of 0-Hecke algebras. Considering the towers of semilattices given by the permutohedron, associahedron, and Boolean lattices, we categorify the algebra and the coalgebra structure of the Hopf algebras $FQSym , PBT$ , and $NCSF$ respectively. Lastly we completely describe the representation theory of the tower of the monoids of Non Decreasing Parking Functions.]]> 0 <![CDATA[Littlewood Richardson coefficients are structure constants appearing in the representation theory of the general linear groups $(GL_n)$. The main results of this paper are: 1. A strongly polynomial randomized approximation scheme for Littlewood-Richardson coefficients corresponding to indices sufficiently far from the boundary of the Littlewood Richardson cone. 2. A proof of approximate log-concavity of the above mentioned class of Littlewood-Richardson coefficients.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2403 https://doi.org/10.46298/dmtcs.2403 Narayanan, Hariharan Narayanan, Hariharan <![CDATA[Littlewood Richardson coefficients are structure constants appearing in the representation theory of the general linear groups $(GL_n)$. The main results of this paper are: 1. A strongly polynomial randomized approximation scheme for Littlewood-Richardson coefficients corresponding to indices sufficiently far from the boundary of the Littlewood Richardson cone. 2. A proof of approximate log-concavity of the above mentioned class of Littlewood-Richardson coefficients.]]> 0 <![CDATA[We study an extension of the chip-firing game. A given set of admissible moves, called Yamanouchi moves, allows the player to pass from a starting configuration $\alpha$ to a further configuration $\beta$. This can be encoded via an action of a certain group, the toppling group, associated with each connected graph. This action gives rise to a generalization of Hall-Littlewood symmetric polynomials and a new combinatorial basis for them. Moreover, it provides a general method to construct all orthogonal systems associated with a given random variable.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2413 https://doi.org/10.46298/dmtcs.2413 Cori, Robert Senato, Domenico Petrullo, Pasquale Cori, Robert Senato, Domenico Petrullo, Pasquale <![CDATA[We study an extension of the chip-firing game. A given set of admissible moves, called Yamanouchi moves, allows the player to pass from a starting configuration $\alpha$ to a further configuration $\beta$. This can be encoded via an action of a certain group, the toppling group, associated with each connected graph. This action gives rise to a generalization of Hall-Littlewood symmetric polynomials and a new combinatorial basis for them. Moreover, it provides a general method to construct all orthogonal systems associated with a given random variable.]]> 0 <![CDATA[We introduce a series of conjectured identities that deform Weyl's denominator formula and generalize Tokuyama's formula to other root systems. These conjectures generalize a number of well-known results due to Okada. We also prove a related result for $B'_n$ that generalizes a theorem of Simpson.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2412 https://doi.org/10.46298/dmtcs.2412 Hamel, Angêle, King, Ronald, Hamel, Angêle, King, Ronald, <![CDATA[We introduce a series of conjectured identities that deform Weyl's denominator formula and generalize Tokuyama's formula to other root systems. These conjectures generalize a number of well-known results due to Okada. We also prove a related result for $B'_n$ that generalizes a theorem of Simpson.]]> 0 <![CDATA[We prove an identity conjectured by Adin and Roichman involving the descent set of $\lambda$-unimodal cyclic permutations. These permutations appear in the character formulas for certain representations of the symmetric group and these formulas are usually proven algebraically. Here, we give a combinatorial proof for one such formula and discuss the consequences for the distribution of the descent set on cyclic permutations.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2411 https://doi.org/10.46298/dmtcs.2411 Archer, Kassie Archer, Kassie <![CDATA[We prove an identity conjectured by Adin and Roichman involving the descent set of $\lambda$-unimodal cyclic permutations. These permutations appear in the character formulas for certain representations of the symmetric group and these formulas are usually proven algebraically. Here, we give a combinatorial proof for one such formula and discuss the consequences for the distribution of the descent set on cyclic permutations.]]> 0 <![CDATA[The purpose of this work is to initiate a combinatorial study of the Bruhat-Chevalley ordering on certain sets of permutations obtained by omitting the parentheses from their standard cyclic notation. In particular, we show that these sets form bounded, graded, unimodal, rank-symmetric and EL-shellable posets. Moreover, we determine the homotopy types of the associated order complexes.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2410 https://doi.org/10.46298/dmtcs.2410 Can, Mahir Bilen Cherniavsky, Yonah Can, Mahir Bilen Cherniavsky, Yonah <![CDATA[The purpose of this work is to initiate a combinatorial study of the Bruhat-Chevalley ordering on certain sets of permutations obtained by omitting the parentheses from their standard cyclic notation. In particular, we show that these sets form bounded, graded, unimodal, rank-symmetric and EL-shellable posets. Moreover, we determine the homotopy types of the associated order complexes.]]> 0 <![CDATA[In the first part of this article we present a realization of the $m$-Tamari lattice $\mathcal{T}_n^{(m)}$ in terms of $m$-tuples of Dyck paths of height $n$, equipped with componentwise rotation order. For that, we define the $m$-cover poset $\mathcal{P}^{\langle m \rangle}$ of an arbitrary bounded poset $\mathcal{P}$, and show that the smallest lattice completion of the $m$-cover poset of the Tamari lattice $\mathcal{T}_n$ is isomorphic to the $m$-Tamari lattice $\mathcal{T}_n^{(m)}$. A crucial tool for the proof of this isomorphism is a decomposition of $m$-Dyck paths into $m$-tuples of classical Dyck paths, which we call the strip-decomposition. Subsequently, we characterize the cases where the $m$-cover poset of an arbitrary poset is a lattice. Finally, we show that the $m$-cover poset of the Cambrian lattice of the dihedral group is a trim lattice with cardinality equal to the generalized Fuss-Catalan number of the dihedral group.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2409 https://doi.org/10.46298/dmtcs.2409 Kallipoliti, Myrto Mühle, Henri Kallipoliti, Myrto Mühle, Henri <![CDATA[In the first part of this article we present a realization of the $m$-Tamari lattice $\mathcal{T}_n^{(m)}$ in terms of $m$-tuples of Dyck paths of height $n$, equipped with componentwise rotation order. For that, we define the $m$-cover poset $\mathcal{P}^{\langle m \rangle}$ of an arbitrary bounded poset $\mathcal{P}$, and show that the smallest lattice completion of the $m$-cover poset of the Tamari lattice $\mathcal{T}_n$ is isomorphic to the $m$-Tamari lattice $\mathcal{T}_n^{(m)}$. A crucial tool for the proof of this isomorphism is a decomposition of $m$-Dyck paths into $m$-tuples of classical Dyck paths, which we call the strip-decomposition. Subsequently, we characterize the cases where the $m$-cover poset of an arbitrary poset is a lattice. Finally, we show that the $m$-cover poset of the Cambrian lattice of the dihedral group is a trim lattice with cardinality equal to the generalized Fuss-Catalan number of the dihedral group.]]> 0 <![CDATA[The Selberg integral is an important integral first evaluated by Selberg in 1944. Stanley found a combinatorial interpretation of the Selberg integral in terms of permutations. In this paper, new combinatorial objects "Young books'' are introduced and shown to have a connection with the Selberg integral. This connection gives an enumeration formula for Young books. It is shown that special cases of Young books become standard Young tableaux of various shapes: shifted staircases, squares, certain skew shapes, and certain truncated shapes. As a consequence, enumeration formulas for standard Young tableaux of these shapes are obtained.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2408 https://doi.org/10.46298/dmtcs.2408 Kim, Jang Soo Oh, Suho Kim, Jang Soo Oh, Suho <![CDATA[The Selberg integral is an important integral first evaluated by Selberg in 1944. Stanley found a combinatorial interpretation of the Selberg integral in terms of permutations. In this paper, new combinatorial objects "Young books'' are introduced and shown to have a connection with the Selberg integral. This connection gives an enumeration formula for Young books. It is shown that special cases of Young books become standard Young tableaux of various shapes: shifted staircases, squares, certain skew shapes, and certain truncated shapes. As a consequence, enumeration formulas for standard Young tableaux of these shapes are obtained.]]> 0 <![CDATA[Curtis-Ingerman-Morrow studied the space of circular planar electrical networks, and classified all possible response matrices for such networks. Lam and Pylyavskyy found a Lie group $EL_{2n}$ whose positive part $(EL_{2n})_{\geq 0}$ naturally acts on the circular planar electrical network via some combinatorial description, where the action is inspired by the star-triangle transformation of the electrical networks. The Lie algebra $el_{2n}$ is semisimple and isomorphic to the symplectic algebra. In the end of their paper, they suggest a generalization of electrical Lie algebras to all finite Dynkin types. We give the structure of the type $B$ electrical Lie algebra $e_{b_{2n}}$. The nonnegative part $(E_{B_{2n}})_{\geq 0}$ of the corresponding Lie group conjecturally acts on a class of "mirror symmetric circular planar electrical networks". This class of networks has interesting combinatorial properties. Finally, we mention some partial results for type $C$ and $D$ electrical Lie algebras, where an analogous story needs to be developed.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2407 https://doi.org/10.46298/dmtcs.2407 Su, Yi Su, Yi <![CDATA[Curtis-Ingerman-Morrow studied the space of circular planar electrical networks, and classified all possible response matrices for such networks. Lam and Pylyavskyy found a Lie group $EL_{2n}$ whose positive part $(EL_{2n})_{\geq 0}$ naturally acts on the circular planar electrical network via some combinatorial description, where the action is inspired by the star-triangle transformation of the electrical networks. The Lie algebra $el_{2n}$ is semisimple and isomorphic to the symplectic algebra. In the end of their paper, they suggest a generalization of electrical Lie algebras to all finite Dynkin types. We give the structure of the type $B$ electrical Lie algebra $e_{b_{2n}}$. The nonnegative part $(E_{B_{2n}})_{\geq 0}$ of the corresponding Lie group conjecturally acts on a class of "mirror symmetric circular planar electrical networks". This class of networks has interesting combinatorial properties. Finally, we mention some partial results for type $C$ and $D$ electrical Lie algebras, where an analogous story needs to be developed.]]> 0 <![CDATA[Connection matrices for graph parameters with values in a field have been introduced by M. Freedman, L. Lovász and A. Schrijver (2007). Graph parameters with connection matrices of finite rank can be computed in polynomial time on graph classes of bounded tree-width. We introduce join matrices, a generalization of connection matrices, and allow graph parameters to take values in the tropical rings (max-plus algebras) over the real numbers. We show that rank-finiteness of join matrices implies that these graph parameters can be computed in polynomial time on graph classes of bounded clique-width. In the case of graph parameters with values in arbitrary commutative semirings, this remains true for graph classes of bounded linear clique-width. B. Godlin, T. Kotek and J.A. Makowsky (2008) showed that definability of a graph parameter in Monadic Second Order Logic implies rank finiteness. We also show that there are uncountably many integer valued graph parameters with connection matrices or join matricesof fixed finite rank. This shows that rank finiteness is a much weaker assumption than any definability assumption.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2406 https://doi.org/10.46298/dmtcs.2406 Labai, Nadia Makowsky, Johann, Labai, Nadia Makowsky, Johann, <![CDATA[Connection matrices for graph parameters with values in a field have been introduced by M. Freedman, L. Lovász and A. Schrijver (2007). Graph parameters with connection matrices of finite rank can be computed in polynomial time on graph classes of bounded tree-width. We introduce join matrices, a generalization of connection matrices, and allow graph parameters to take values in the tropical rings (max-plus algebras) over the real numbers. We show that rank-finiteness of join matrices implies that these graph parameters can be computed in polynomial time on graph classes of bounded clique-width. In the case of graph parameters with values in arbitrary commutative semirings, this remains true for graph classes of bounded linear clique-width. B. Godlin, T. Kotek and J.A. Makowsky (2008) showed that definability of a graph parameter in Monadic Second Order Logic implies rank finiteness. We also show that there are uncountably many integer valued graph parameters with connection matrices or join matricesof fixed finite rank. This shows that rank finiteness is a much weaker assumption than any definability assumption.]]> 0 <![CDATA[A classical result of MacMahon states that inversion number and major index have the same distribution over permutations of a given multiset. In this work we prove a strengthening of this theorem originally conjectured by Haglund. Our result can be seen as an equidistribution theorem over the ordered partitions of a multiset into sets, which we call ordered multiset partitions. Our proof is bijective and involves a new generalization of Carlitz's insertion method. As an application, we develop refined Macdonald polynomials for hook shapes. We show that these polynomials are symmetric and give their Schur expansion.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2405 https://doi.org/10.46298/dmtcs.2405 Wilson, Andrew Timothy Wilson, Andrew Timothy <![CDATA[A classical result of MacMahon states that inversion number and major index have the same distribution over permutations of a given multiset. In this work we prove a strengthening of this theorem originally conjectured by Haglund. Our result can be seen as an equidistribution theorem over the ordered partitions of a multiset into sets, which we call ordered multiset partitions. Our proof is bijective and involves a new generalization of Carlitz's insertion method. As an application, we develop refined Macdonald polynomials for hook shapes. We show that these polynomials are symmetric and give their Schur expansion.]]> 0 <![CDATA[We prove the conjecture by M. Yip stating that counting genus one partitions by the number of their elements and parts yields, up to a shift of indices, the same array of numbers as counting genus one rooted hypermonopoles. Our proof involves representing each genus one permutation by a four-colored noncrossing partition. This representation may be selected in a unique way for permutations containing no trivial cycles. The conclusion follows from a general generating function formula that holds for any class of permutations that is closed under the removal and reinsertion of trivial cycles. Our method also provides another way to count rooted hypermonopoles of genus one, and puts the spotlight on a class of genus one permutations that is invariant under an obvious extension of the Kreweras duality map to genus one permutations.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2404 https://doi.org/10.46298/dmtcs.2404 Cori, Robert Hetyei, Gábor Cori, Robert Hetyei, Gábor <![CDATA[We prove the conjecture by M. Yip stating that counting genus one partitions by the number of their elements and parts yields, up to a shift of indices, the same array of numbers as counting genus one rooted hypermonopoles. Our proof involves representing each genus one permutation by a four-colored noncrossing partition. This representation may be selected in a unique way for permutations containing no trivial cycles. The conclusion follows from a general generating function formula that holds for any class of permutations that is closed under the removal and reinsertion of trivial cycles. Our method also provides another way to count rooted hypermonopoles of genus one, and puts the spotlight on a class of genus one permutations that is invariant under an obvious extension of the Kreweras duality map to genus one permutations.]]> 0 <![CDATA[Studying the problem of quasicommuting quantum minors, Leclerc and Zelevinsky introduced in 1998 the notion of weakly separated sets in $[n]:=\{1,\ldots, n\}$. Moreover, they raised several conjectures on the purity for this symmetric relation, in particular, on the Boolean cube $2^{[n]}$. In 0909.1423[math.CO] we proved these purity conjectures for the Boolean cube $2^{[n]}$, the discrete Grassmanian $\binom{[n]}{r}$, and some other set-systems. Oh, Postnikov, and Speyer in arxiv:1109.4434 proved the purity for weakly separated collections inside a positroid which contain a Grassmann necklace $\mathcal {N}$ defining the positroid. We denote such set-systems as $\mathcal{Int}(\mathcal {N} )$. In this paper we give an alternative (and shorter) proof of the purity of $\mathcal{Int}(\mathcal {N} )$ and present a stronger result. More precisely, we introduce a set-system $\mathcal{Out}(\mathcal {N} )$ complementary to $\mathcal{Int}(\mathcal {N })$, in a sense, and establish its purity. Moreover, we prove (Theorem~3) that these two set-systems are weakly separated from each other. As a consequence of Theorem~3, we obtain the purity of set-systems related to pairs of weakly separated necklaces (Proposition 4 and Corollaries 1 and 2). Finally, we raise a conjecture on the purity of both the interior and exterior of a generalized necklace.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2392 https://doi.org/10.46298/dmtcs.2392 Danilov, Vladimir, Karzanov, Alexander, Koshevoy, Gleb, Danilov, Vladimir, Karzanov, Alexander, Koshevoy, Gleb, <![CDATA[Studying the problem of quasicommuting quantum minors, Leclerc and Zelevinsky introduced in 1998 the notion of weakly separated sets in $[n]:=\{1,\ldots, n\}$. Moreover, they raised several conjectures on the purity for this symmetric relation, in particular, on the Boolean cube $2^{[n]}$. In 0909.1423[math.CO] we proved these purity conjectures for the Boolean cube $2^{[n]}$, the discrete Grassmanian $\binom{[n]}{r}$, and some other set-systems. Oh, Postnikov, and Speyer in arxiv:1109.4434 proved the purity for weakly separated collections inside a positroid which contain a Grassmann necklace $\mathcal {N}$ defining the positroid. We denote such set-systems as $\mathcal{Int}(\mathcal {N} )$. In this paper we give an alternative (and shorter) proof of the purity of $\mathcal{Int}(\mathcal {N} )$ and present a stronger result. More precisely, we introduce a set-system $\mathcal{Out}(\mathcal {N} )$ complementary to $\mathcal{Int}(\mathcal {N })$, in a sense, and establish its purity. Moreover, we prove (Theorem~3) that these two set-systems are weakly separated from each other. As a consequence of Theorem~3, we obtain the purity of set-systems related to pairs of weakly separated necklaces (Proposition 4 and Corollaries 1 and 2). Finally, we raise a conjecture on the purity of both the interior and exterior of a generalized necklace.]]> 0 <![CDATA[An associahedron is a polytope whose vertices correspond to the triangulations of a convex polygon and whose edges correspond to flips between them. J.-L. Loday gave a particularly elegant realization of the associahedron, which was then generalized in two directions: on the one hand to obtain realizations of graph associahedra, and on the other hand to obtain multiple realizations of the associahedron parametrized by a sequence of signs. The goal of this paper is to unify and extend these two constructions to signed tree associahedra.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2402 https://doi.org/10.46298/dmtcs.2402 Pilaud, Vincent Pilaud, Vincent <![CDATA[An associahedron is a polytope whose vertices correspond to the triangulations of a convex polygon and whose edges correspond to flips between them. J.-L. Loday gave a particularly elegant realization of the associahedron, which was then generalized in two directions: on the one hand to obtain realizations of graph associahedra, and on the other hand to obtain multiple realizations of the associahedron parametrized by a sequence of signs. The goal of this paper is to unify and extend these two constructions to signed tree associahedra.]]> 0 <![CDATA[The number of shortest factorizations into reflections for a Singer cycle in $GL_n(\mathbb{F}_q)$ is shown to be $(q^n-1)^{n-1}$. Formulas counting factorizations of any length, and counting those with reflections of fixed conjugacy classes are also given.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2401 https://doi.org/10.46298/dmtcs.2401 Lewis, J.B. Reiner, V. Stanton, D. Lewis, J.B. Reiner, V. Stanton, D. <![CDATA[The number of shortest factorizations into reflections for a Singer cycle in $GL_n(\mathbb{F}_q)$ is shown to be $(q^n-1)^{n-1}$. Formulas counting factorizations of any length, and counting those with reflections of fixed conjugacy classes are also given.]]> 0 <![CDATA[We revisit a classic partition theorem due to MacMahon that relates partitions with all parts repeated at least once and partitions with parts congruent to $2,3,4,6 \pmod{6}$, together with a generalization by Andrews and two others by Subbarao. Then we develop a unified bijective proof for all four theorems involved, and obtain a natural further generalization as a result.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2400 https://doi.org/10.46298/dmtcs.2400 Fu, Shishuo Sellers, James, Fu, Shishuo Sellers, James, <![CDATA[We revisit a classic partition theorem due to MacMahon that relates partitions with all parts repeated at least once and partitions with parts congruent to $2,3,4,6 \pmod{6}$, together with a generalization by Andrews and two others by Subbarao. Then we develop a unified bijective proof for all four theorems involved, and obtain a natural further generalization as a result.]]> 0 <![CDATA[The face numbers of simplicial polytopes that approximate $C^1$-convex bodies in the Hausdorff metric is studied. Several structural results about the skeleta of such polytopes are studied and used to derive a lower bound theorem for this class of polytopes. This partially resolves a conjecture made by Kalai in 1994: if a sequence $\{P_n\}_{n=0}^{\infty}$ of simplicial polytopes converges to a $C^1$-convex body in the Hausdorff distance, then the entries of the $g$-vector of $P_n$ converge to infinity.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2399 https://doi.org/10.46298/dmtcs.2399 Adiprasito, Karim Samper, José Alejandro Adiprasito, Karim Samper, José Alejandro <![CDATA[The face numbers of simplicial polytopes that approximate $C^1$-convex bodies in the Hausdorff metric is studied. Several structural results about the skeleta of such polytopes are studied and used to derive a lower bound theorem for this class of polytopes. This partially resolves a conjecture made by Kalai in 1994: if a sequence $\{P_n\}_{n=0}^{\infty}$ of simplicial polytopes converges to a $C^1$-convex body in the Hausdorff distance, then the entries of the $g$-vector of $P_n$ converge to infinity.]]> 0 <![CDATA[We define the bigraphical arrangement of a graph and show that the Pak-Stanley labels of its regions are the parking functions of a closely related graph, thus proving conjectures of Duval, Klivans, and Martin and of Hopkins and Perkinson. A consequence is a new proof of a bijection between labeled graphs and regions of the Shi arrangement first given by Stanley. We also give bounds on the number of regions of a bigraphical arrangement. The full version of this paper is forthcoming in the $\textit{Transactions of the American Mathematical Society}$]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2398 https://doi.org/10.46298/dmtcs.2398 Hopkins, Sam Perkinson, David Hopkins, Sam Perkinson, David <![CDATA[We define the bigraphical arrangement of a graph and show that the Pak-Stanley labels of its regions are the parking functions of a closely related graph, thus proving conjectures of Duval, Klivans, and Martin and of Hopkins and Perkinson. A consequence is a new proof of a bijection between labeled graphs and regions of the Shi arrangement first given by Stanley. We also give bounds on the number of regions of a bigraphical arrangement. The full version of this paper is forthcoming in the $\textit{Transactions of the American Mathematical Society}$]]> 0 <![CDATA[The set of all permutations, ordered by pattern containment, forms a poset. This extended abstract presents the first explicit major results on the topology of intervals in this poset. We show that almost all (open) intervals in this poset have a disconnected subinterval and are thus not shellable. Nevertheless, there seem to be large classes of intervals that are shellable and thus have the homotopy type of a wedge of spheres. We prove this to be the case for all intervals of layered permutations that have no disconnected subintervals of rank 3 or more. We also characterize in a simple way those intervals of layered permutations that are disconnected. These results carry over to the poset of generalized subword order when the ordering on the underlying alphabet is a rooted forest. We conjecture that the same applies to intervals of separable permutations, that is, that such an interval is shellable if and only if it has no disconnected subinterval of rank 3 or more. We also present a simplified version of the recursive formula for the Möbius function of decomposable permutations given by Burstein et al.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2397 https://doi.org/10.46298/dmtcs.2397 McNamara, Peter, Steingrımsson, Einar McNamara, Peter, Steingrımsson, Einar <![CDATA[The set of all permutations, ordered by pattern containment, forms a poset. This extended abstract presents the first explicit major results on the topology of intervals in this poset. We show that almost all (open) intervals in this poset have a disconnected subinterval and are thus not shellable. Nevertheless, there seem to be large classes of intervals that are shellable and thus have the homotopy type of a wedge of spheres. We prove this to be the case for all intervals of layered permutations that have no disconnected subintervals of rank 3 or more. We also characterize in a simple way those intervals of layered permutations that are disconnected. These results carry over to the poset of generalized subword order when the ordering on the underlying alphabet is a rooted forest. We conjecture that the same applies to intervals of separable permutations, that is, that such an interval is shellable if and only if it has no disconnected subinterval of rank 3 or more. We also present a simplified version of the recursive formula for the Möbius function of decomposable permutations given by Burstein et al.]]> 0 <![CDATA[We use a recently introduced combinatorial object, the $\textit{interval-poset}$, to describe two bijections on intervals of the Tamari lattice. Both bijections give a combinatorial proof of some previously known results. The first one is an inner bijection between Tamari intervals that exchanges the $\textit{initial rise}$ and $\textit{lower contacts}$ statistics. Those were introduced by Bousquet-Mélou, Fusy, and Préville-Ratelle who proved they were symmetrically distributed but had no combinatorial explanation. The second bijection sends a Tamari interval to a closed flow of an ordered forest. These combinatorial objects were studied by Chapoton in the context of the Pre-Lie operad and the connection with the Tamari order was still unclear.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2396 https://doi.org/10.46298/dmtcs.2396 Chapoton, Frédéric Chatel, Gregory Pons, Viviane Chapoton, Frédéric Chatel, Gregory Pons, Viviane <![CDATA[We use a recently introduced combinatorial object, the $\textit{interval-poset}$, to describe two bijections on intervals of the Tamari lattice. Both bijections give a combinatorial proof of some previously known results. The first one is an inner bijection between Tamari intervals that exchanges the $\textit{initial rise}$ and $\textit{lower contacts}$ statistics. Those were introduced by Bousquet-Mélou, Fusy, and Préville-Ratelle who proved they were symmetrically distributed but had no combinatorial explanation. The second bijection sends a Tamari interval to a closed flow of an ordered forest. These combinatorial objects were studied by Chapoton in the context of the Pre-Lie operad and the connection with the Tamari order was still unclear.]]> 0 <![CDATA[We introduce a quasisymmetric generalization of Berele and Regev's hook Schur functions and prove that these new quasisymmetric hook Schur functions decompose the hook Schur functions in a natural way. In this paper we examine the combinatorics of the quasisymmetric hook Schur functions, providing analogues of the Robinson-Schensted-Knuth algorithm and a generalized Cauchy Identity.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2395 https://doi.org/10.46298/dmtcs.2395 Mason, Sarah, Niese, Elizabeth Mason, Sarah, Niese, Elizabeth <![CDATA[We introduce a quasisymmetric generalization of Berele and Regev's hook Schur functions and prove that these new quasisymmetric hook Schur functions decompose the hook Schur functions in a natural way. In this paper we examine the combinatorics of the quasisymmetric hook Schur functions, providing analogues of the Robinson-Schensted-Knuth algorithm and a generalized Cauchy Identity.]]> 0 <![CDATA[The Rearrangement Conjecture states that if two words over $\mathbb{P}$ are Wilf-equivalent in the factor order on $\mathbb{P}^{\ast}$ then they are rearrangements of each other. We introduce the notion of strong Wilf-equivalence and prove that if two words over $\mathbb{P}$ are strongly Wilf-equivalent then they are rearrangements of each other. We further conjecture that Wilf-equivalence implies strong Wilf-equivalence.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2394 https://doi.org/10.46298/dmtcs.2394 Pantone, Jay Vatter, Vincent Pantone, Jay Vatter, Vincent <![CDATA[The Rearrangement Conjecture states that if two words over $\mathbb{P}$ are Wilf-equivalent in the factor order on $\mathbb{P}^{\ast}$ then they are rearrangements of each other. We introduce the notion of strong Wilf-equivalence and prove that if two words over $\mathbb{P}$ are strongly Wilf-equivalent then they are rearrangements of each other. We further conjecture that Wilf-equivalence implies strong Wilf-equivalence.]]> 0 <![CDATA[The number of standard Young tableaux of a fixed shape is famously given by the hook-length formula due to Frame, Robinson and Thrall. A bijective proof of Novelli, Pak and Stoyanovskii relies on a sorting algorithm akin to jeu-de-taquin which transforms an arbitrary filling of a partition into a standard Young tableau by exchanging adjacent entries. Recently, Krattenthaler and Müller defined the complexity of this algorithm as the average number of performed exchanges, and Neumann and the author proved it fulfils some nice symmetry properties. In this paper we recall and extend the previous results and provide new bijective proofs.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2393 https://doi.org/10.46298/dmtcs.2393 Sulzgruber, Robin Sulzgruber, Robin <![CDATA[The number of standard Young tableaux of a fixed shape is famously given by the hook-length formula due to Frame, Robinson and Thrall. A bijective proof of Novelli, Pak and Stoyanovskii relies on a sorting algorithm akin to jeu-de-taquin which transforms an arbitrary filling of a partition into a standard Young tableau by exchanging adjacent entries. Recently, Krattenthaler and Müller defined the complexity of this algorithm as the average number of performed exchanges, and Neumann and the author proved it fulfils some nice symmetry properties. In this paper we recall and extend the previous results and provide new bijective proofs.]]> 0 <![CDATA[We present a beautiful interplay between combinatorial topology and homological algebra for a class of monoids that arise naturally in algebraic combinatorics. We explore several applications of this interplay. For instance, we provide a new interpretation of the Leray number of a clique complex in terms of non-commutative algebra.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2381 https://doi.org/10.46298/dmtcs.2381 Margolis, Stuart Saliola, Franco Steinberg, Benjamin Margolis, Stuart Saliola, Franco Steinberg, Benjamin <![CDATA[We present a beautiful interplay between combinatorial topology and homological algebra for a class of monoids that arise naturally in algebraic combinatorics. We explore several applications of this interplay. For instance, we provide a new interpretation of the Leray number of a clique complex in terms of non-commutative algebra.]]> 0 <![CDATA[Recently, Kenyon and Wilson introduced Dyck tilings, which are certain tilings of the region between two Dyck paths. The enumeration of Dyck tilings is related with hook formulas for forests and the combinatorics of Hermite polynomials. The first goal of this work is to give an alternative point of view on Dyck tilings by making use of the weak order and the Bruhat order on permutations. Then we introduce two natural generalizations: $k$-Dyck tilings and symmetric Dyck tilings. We are led to consider Stirling permutations, and define an analogue of the Bruhat order on them. We show that certain families of $k$-Dyck tilings are in bijection with intervals in this order. We enumerate symmetric Dyck tilings and show that certain families of symmetric Dyck tilings are in bijection with intervals in the weak order on signed permutations.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2391 https://doi.org/10.46298/dmtcs.2391 Josuat-Vergès, Matthieu Kim, Jang Soo Josuat-Vergès, Matthieu Kim, Jang Soo <![CDATA[Recently, Kenyon and Wilson introduced Dyck tilings, which are certain tilings of the region between two Dyck paths. The enumeration of Dyck tilings is related with hook formulas for forests and the combinatorics of Hermite polynomials. The first goal of this work is to give an alternative point of view on Dyck tilings by making use of the weak order and the Bruhat order on permutations. Then we introduce two natural generalizations: $k$-Dyck tilings and symmetric Dyck tilings. We are led to consider Stirling permutations, and define an analogue of the Bruhat order on them. We show that certain families of $k$-Dyck tilings are in bijection with intervals in this order. We enumerate symmetric Dyck tilings and show that certain families of symmetric Dyck tilings are in bijection with intervals in the weak order on signed permutations.]]> 0 <![CDATA[We consider the multivariate generating series $F_P$ of $P-$partitions in infinitely many variables $x_1, x_2, \ldots$ . For some family of ranked posets $P$, it is natural to consider an analog $N_P$ with two infinite alphabets. When we collapse these two alphabets, we trivially recover $F_P$. Our main result is the converse, that is, the explicit construction of a map sending back $F_P$ onto $N_P$. We also give a noncommutative analog of the latter. An application is the construction of a basis of $\mathbf{WQSym}$ with a non-negative multiplication table, which lifts a basis of $\textit{QSym}$ introduced by K. Luoto.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2390 https://doi.org/10.46298/dmtcs.2390 Aval, Jean-Christophe Féray, Valentin Novelli, Jean-Christophe Thibon, J.-Y. Aval, Jean-Christophe Féray, Valentin Novelli, Jean-Christophe Thibon, J.-Y. <![CDATA[We consider the multivariate generating series $F_P$ of $P-$partitions in infinitely many variables $x_1, x_2, \ldots$ . For some family of ranked posets $P$, it is natural to consider an analog $N_P$ with two infinite alphabets. When we collapse these two alphabets, we trivially recover $F_P$. Our main result is the converse, that is, the explicit construction of a map sending back $F_P$ onto $N_P$. We also give a noncommutative analog of the latter. An application is the construction of a basis of $\mathbf{WQSym}$ with a non-negative multiplication table, which lifts a basis of $\textit{QSym}$ introduced by K. Luoto.]]> 0 <![CDATA[We present a very simple explicit technique to generate a large family of point configurations with neighborly Delaunay triangulations. This proves that there are superexponentially many combinatorially distinct neighborly $d$-polytopes with $n$ vertices that admit realizations inscribed on the sphere. These are the first examples of inscribable neighborly polytopes that are not cyclic polytopes, and provide the current best lower bound for the number of combinatorial types of inscribable polytopes (and thus also of Delaunay triangulations). It coincides with the current best lower bound for the number of combinatorial types of polytopes.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2389 https://doi.org/10.46298/dmtcs.2389 Gonska, Bernd Padrol, Arnau Gonska, Bernd Padrol, Arnau <![CDATA[We present a very simple explicit technique to generate a large family of point configurations with neighborly Delaunay triangulations. This proves that there are superexponentially many combinatorially distinct neighborly $d$-polytopes with $n$ vertices that admit realizations inscribed on the sphere. These are the first examples of inscribable neighborly polytopes that are not cyclic polytopes, and provide the current best lower bound for the number of combinatorial types of inscribable polytopes (and thus also of Delaunay triangulations). It coincides with the current best lower bound for the number of combinatorial types of polytopes.]]> 0 <![CDATA[We consider two aspects of Kronecker coefficients in the directions of representation theory and combinatorics. We consider a conjecture of Jan Saxl stating that the tensor square of the $S_n$-irreducible representation indexed by the staircase partition contains every irreducible representation of $S_n$. We present a sufficient condition allowing to determine whether an irreducible representation is a constituent of a tensor square and using this result together with some analytic statements on partitions we prove Saxl conjecture for several partition classes. We also use Kronecker coefficients to give a new proof and a generalization of the unimodality of Gaussian ($q$-binomial) coefficients as polynomials in $q$, and extend this to strict unimodality.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2388 https://doi.org/10.46298/dmtcs.2388 Pak, Igor Panova, Greta Vallejo, Ernesto Pak, Igor Panova, Greta Vallejo, Ernesto <![CDATA[We consider two aspects of Kronecker coefficients in the directions of representation theory and combinatorics. We consider a conjecture of Jan Saxl stating that the tensor square of the $S_n$-irreducible representation indexed by the staircase partition contains every irreducible representation of $S_n$. We present a sufficient condition allowing to determine whether an irreducible representation is a constituent of a tensor square and using this result together with some analytic statements on partitions we prove Saxl conjecture for several partition classes. We also use Kronecker coefficients to give a new proof and a generalization of the unimodality of Gaussian ($q$-binomial) coefficients as polynomials in $q$, and extend this to strict unimodality.]]> 0 <![CDATA[We study Newton polytopes of cluster variables in type $A_n$ cluster algebras, whose cluster and coefficient variables are indexed by the diagonals and boundary segments of a polygon. Our main results include an explicit description of the affine hull and facets of the Newton polytope of the Laurent expansion of any cluster variable, with respect to any cluster. In particular, we show that every Laurent monomial in a Laurent expansion of a type $A$ cluster variable corresponds to a vertex of the Newton polytope. We also describe the face lattice of each Newton polytope via an isomorphism with the lattice of elementary subgraphs of the associated snake graph.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2387 https://doi.org/10.46298/dmtcs.2387 Kalman, Adam Kalman, Adam <![CDATA[We study Newton polytopes of cluster variables in type $A_n$ cluster algebras, whose cluster and coefficient variables are indexed by the diagonals and boundary segments of a polygon. Our main results include an explicit description of the affine hull and facets of the Newton polytope of the Laurent expansion of any cluster variable, with respect to any cluster. In particular, we show that every Laurent monomial in a Laurent expansion of a type $A$ cluster variable corresponds to a vertex of the Newton polytope. We also describe the face lattice of each Newton polytope via an isomorphism with the lattice of elementary subgraphs of the associated snake graph.]]> 0 <![CDATA[We introduce a new method for showing that the roots of the characteristic polynomial of a finite lattice are all nonnegative integers. Our method gives two simple conditions under which the characteristic polynomial factors. We will see that Stanley's Supersolvability Theorem is a corollary of this result. We can also use this method to demonstrate a new result in graph theory and give new proofs of some classic results concerning the Möbius function.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2386 https://doi.org/10.46298/dmtcs.2386 Hallam, Joshua Sagan, Bruce, Hallam, Joshua Sagan, Bruce, <![CDATA[We introduce a new method for showing that the roots of the characteristic polynomial of a finite lattice are all nonnegative integers. Our method gives two simple conditions under which the characteristic polynomial factors. We will see that Stanley's Supersolvability Theorem is a corollary of this result. We can also use this method to demonstrate a new result in graph theory and give new proofs of some classic results concerning the Möbius function.]]> 0 <![CDATA[We define a $0$-Hecke action on composition tableaux, and then use it to derive $0$-Hecke modules whose quasisymmetric characteristic is a quasisymmetric Schur function. We then relate the modules to the weak Bruhat order and use them to derive a new basis for quasisymmetric functions. We also classify those modules that are tableau-cyclic and likewise indecomposable. Finally, we develop a restriction rule that reflects the coproduct of quasisymmetric Schur functions.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2385 https://doi.org/10.46298/dmtcs.2385 Tewari, Vasu, van Willigenburg, Stephanie, Tewari, Vasu, van Willigenburg, Stephanie, <![CDATA[We define a $0$-Hecke action on composition tableaux, and then use it to derive $0$-Hecke modules whose quasisymmetric characteristic is a quasisymmetric Schur function. We then relate the modules to the weak Bruhat order and use them to derive a new basis for quasisymmetric functions. We also classify those modules that are tableau-cyclic and likewise indecomposable. Finally, we develop a restriction rule that reflects the coproduct of quasisymmetric Schur functions.]]> 0 <![CDATA[In the recent study of infinite root systems, fractal patterns of ball packings were observed while visualizing roots in affine space. In fact, the observed fractals are exactly the ball packings described by Boyd and Maxwell. This correspondence is a corollary of a more fundamental result: given a geometric representation of a Coxeter group in Lorentz space, the set of limit directions of weights equals the set of limit roots.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2384 https://doi.org/10.46298/dmtcs.2384 Chen, Hao Labbé, Jean-Philippe Chen, Hao Labbé, Jean-Philippe <![CDATA[In the recent study of infinite root systems, fractal patterns of ball packings were observed while visualizing roots in affine space. In fact, the observed fractals are exactly the ball packings described by Boyd and Maxwell. This correspondence is a corollary of a more fundamental result: given a geometric representation of a Coxeter group in Lorentz space, the set of limit directions of weights equals the set of limit roots.]]> 0 <![CDATA[\textbfAbstract. We construct supercharacter theories of finite unipotent groups in the orthogonal, symplectic and unitary types. Our method utilizes group actions in a manner analogous to that of Diaconis and Isaacs in their construction of supercharacters of algebra groups. The resulting supercharacter theories agree with those of André and Neto in the case of the unipotent orthogonal and symplectic matrices and generalize to a large collection of subgroups. In the unitary group case, we describe the supercharacters and superclasses in terms of labeled set partitions and calculate the supercharacter table. \bigbreak]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2383 https://doi.org/10.46298/dmtcs.2383 Andrews, Scott Andrews, Scott <![CDATA[\textbfAbstract. We construct supercharacter theories of finite unipotent groups in the orthogonal, symplectic and unitary types. Our method utilizes group actions in a manner analogous to that of Diaconis and Isaacs in their construction of supercharacters of algebra groups. The resulting supercharacter theories agree with those of André and Neto in the case of the unipotent orthogonal and symplectic matrices and generalize to a large collection of subgroups. In the unitary group case, we describe the supercharacters and superclasses in terms of labeled set partitions and calculate the supercharacter table. \bigbreak]]> 0 <![CDATA[With a crystallographic root system $\Phi$ , there are associated two Catalan objects, the set of nonnesting partitions $NN(\Phi)$, and the cluster complex $\Delta (\Phi)$. These possess a number of enumerative coincidences, many of which are captured in a surprising identity, first conjectured by Chapoton. We prove this conjecture, and indicate its generalisation for the Fuß-Catalan objects $NN^{(k)}(\Phi)$ and $\Delta^{(k)}(\Phi)$, conjectured by Armstrong.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2382 https://doi.org/10.46298/dmtcs.2382 Thiel, Marko Thiel, Marko <![CDATA[With a crystallographic root system $\Phi$ , there are associated two Catalan objects, the set of nonnesting partitions $NN(\Phi)$, and the cluster complex $\Delta (\Phi)$. These possess a number of enumerative coincidences, many of which are captured in a surprising identity, first conjectured by Chapoton. We prove this conjecture, and indicate its generalisation for the Fuß-Catalan objects $NN^{(k)}(\Phi)$ and $\Delta^{(k)}(\Phi)$, conjectured by Armstrong.]]> 0 <![CDATA[We consider a carries process which is a generalization of that by Holte in the sense that (i) we take various digit sets, and (ii) we also consider negative base. Our results are : (i) eigenvalues and eigenvectors of the transition probability matrices, and their connection to combinatorics and representation theory, (ii) an application to the computation of the distribution of the sum of i.i.d. uniform r.v.'s on [0,1], (iii) a relation to riffle shuffle.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2380 https://doi.org/10.46298/dmtcs.2380 Fujita, Takahiko Nakano, Fumihiko Sadahiro, Taizo Fujita, Takahiko Nakano, Fumihiko Sadahiro, Taizo <![CDATA[We consider a carries process which is a generalization of that by Holte in the sense that (i) we take various digit sets, and (ii) we also consider negative base. Our results are : (i) eigenvalues and eigenvectors of the transition probability matrices, and their connection to combinatorics and representation theory, (ii) an application to the computation of the distribution of the sum of i.i.d. uniform r.v.'s on [0,1], (iii) a relation to riffle shuffle.]]> 0 <![CDATA[Let $X$ be a $(d \times N)$-matrix. We consider the variable polytope $\Pi_X(u) = \left\{ w \geq 0 : Xw = u \right\}$. It is known that the function $T_X$ that assigns to a parameter $u \in \mathbb{R}^N$ the volume of the polytope $\Pi_X(u)$ is piecewise polynomial. Formulas of Khovanskii-Pukhlikov and Brion-Vergne imply that the number of lattice points in $\Pi_X(u)$ can be obtained by applying a certain differential operator to the function $T_X$. In this extended abstract we slightly improve the formulas of Khovanskii-Pukhlikov and Brion-Vergne and we study the space of differential operators that are relevant for $T_X$ (ıe operators that do not annihilate $T_X$) and the space of nice differential operators (ıe operators that leave $T_X$ continuous). These two spaces are finite-dimensional homogeneous vector spaces and their Hilbert series are evaluations of the Tutte polynomial of the (arithmetic) matroid defined by $X$.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2379 https://doi.org/10.46298/dmtcs.2379 Lenz, Matthias Lenz, Matthias <![CDATA[Let $X$ be a $(d \times N)$-matrix. We consider the variable polytope $\Pi_X(u) = \left\{ w \geq 0 : Xw = u \right\}$. It is known that the function $T_X$ that assigns to a parameter $u \in \mathbb{R}^N$ the volume of the polytope $\Pi_X(u)$ is piecewise polynomial. Formulas of Khovanskii-Pukhlikov and Brion-Vergne imply that the number of lattice points in $\Pi_X(u)$ can be obtained by applying a certain differential operator to the function $T_X$. In this extended abstract we slightly improve the formulas of Khovanskii-Pukhlikov and Brion-Vergne and we study the space of differential operators that are relevant for $T_X$ (ıe operators that do not annihilate $T_X$) and the space of nice differential operators (ıe operators that leave $T_X$ continuous). These two spaces are finite-dimensional homogeneous vector spaces and their Hilbert series are evaluations of the Tutte polynomial of the (arithmetic) matroid defined by $X$.]]> 0 <![CDATA[The graph of overlapping permutations is defined in a way analogous to the De Bruijn graph on strings of symbols. However, instead of requiring the tail of one permutation to equal the head of another for them to be connected by an edge, we require that the head and tail in question have their letters appear in the same order of size. We give a formula for the number of cycles of length $d$ in the subgraph of overlapping $312$-avoiding permutations. Using this we also give a refinement of the enumeration of $312$-avoiding affine permutations.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2378 https://doi.org/10.46298/dmtcs.2378 Ehrenborg, Richard Kitaev, Sergey Steingrımsson, Einar Ehrenborg, Richard Kitaev, Sergey Steingrımsson, Einar <![CDATA[The graph of overlapping permutations is defined in a way analogous to the De Bruijn graph on strings of symbols. However, instead of requiring the tail of one permutation to equal the head of another for them to be connected by an edge, we require that the head and tail in question have their letters appear in the same order of size. We give a formula for the number of cycles of length $d$ in the subgraph of overlapping $312$-avoiding permutations. Using this we also give a refinement of the enumeration of $312$-avoiding affine permutations.]]> 0 <![CDATA[A driving question in (quantum) cohomology of flag varieties is to find non-recursive, positive combinatorial formulas for expressing the quantum product in a particularly nice basis, called the Schubert basis. Bertram, Ciocan-Fontanine and Fulton provide a way to compute quantum products of Schubert classes in the Grassmannian of $k$-planes in complex $n$-space by doing classical multiplication and then applying a combinatorial rimhook rule which yields the quantum parameter. In this paper, we provide a generalization of this rim hook rule to the setting in which there is also an action of the complex torus. Combining this result with Knutson and Tao's puzzle rule provides an effective algorithm for computing the equivariant quantum Littlewood-Richardson coefficients. Interestingly, this rule requires a specialization of torus weights that is tantalizingly similar to maps in affine Schubert calculus.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2377 https://doi.org/10.46298/dmtcs.2377 Beazley, Elizabeth Bertiger, Anna Taipale, Kaisa Beazley, Elizabeth Bertiger, Anna Taipale, Kaisa <![CDATA[A driving question in (quantum) cohomology of flag varieties is to find non-recursive, positive combinatorial formulas for expressing the quantum product in a particularly nice basis, called the Schubert basis. Bertram, Ciocan-Fontanine and Fulton provide a way to compute quantum products of Schubert classes in the Grassmannian of $k$-planes in complex $n$-space by doing classical multiplication and then applying a combinatorial rimhook rule which yields the quantum parameter. In this paper, we provide a generalization of this rim hook rule to the setting in which there is also an action of the complex torus. Combining this result with Knutson and Tao's puzzle rule provides an effective algorithm for computing the equivariant quantum Littlewood-Richardson coefficients. Interestingly, this rule requires a specialization of torus weights that is tantalizingly similar to maps in affine Schubert calculus.]]> 0 <![CDATA[We define an action of the $0$-Hecke algebra of type A on the Stanley-Reisner ring of the Boolean algebra. By studying this action we obtain a family of multivariate noncommutative symmetric functions, which specialize to the noncommutative Hall-Littlewood symmetric functions and their $(q,t)$-analogues introduced by Bergeron and Zabrocki. We also obtain multivariate quasisymmetric function identities, which specialize to a result of Garsia and Gessel on the generating function of the joint distribution of five permutation statistics.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2376 https://doi.org/10.46298/dmtcs.2376 Huang, Jia Huang, Jia <![CDATA[We define an action of the $0$-Hecke algebra of type A on the Stanley-Reisner ring of the Boolean algebra. By studying this action we obtain a family of multivariate noncommutative symmetric functions, which specialize to the noncommutative Hall-Littlewood symmetric functions and their $(q,t)$-analogues introduced by Bergeron and Zabrocki. We also obtain multivariate quasisymmetric function identities, which specialize to a result of Garsia and Gessel on the generating function of the joint distribution of five permutation statistics.]]> 0 <![CDATA[Kuniba, Nakanishi, and Suzuki (1994) have formulated a general conjecture expressing the positive solution of an $\ell$-restricted $Q$-system in terms of quantum dimensions of Kirillov-Reshetikhin modules. After presenting this conjecture, we sketch a proof for the exceptional type $E_6$ following our preprint (2013). In types $E_7$ and $E_8$, we prove positivity for a subset of the nodes of the Dynkin diagram, and we reduce the positivity for the remaining nodes to the conjectural iterated log-concavity of certain explicit sequences of real algebraic numbers.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2375 https://doi.org/10.46298/dmtcs.2375 Gleitz, Anne-Sophie Gleitz, Anne-Sophie <![CDATA[Kuniba, Nakanishi, and Suzuki (1994) have formulated a general conjecture expressing the positive solution of an $\ell$-restricted $Q$-system in terms of quantum dimensions of Kirillov-Reshetikhin modules. After presenting this conjecture, we sketch a proof for the exceptional type $E_6$ following our preprint (2013). In types $E_7$ and $E_8$, we prove positivity for a subset of the nodes of the Dynkin diagram, and we reduce the positivity for the remaining nodes to the conjectural iterated log-concavity of certain explicit sequences of real algebraic numbers.]]> 0 <![CDATA[At the end of the 1960s, Knuth characterised in terms of forbidden patterns the permutations that can be sorted using a stack. He also showed that they are in bijection with Dyck paths and thus counted by the Catalan numbers. Subsequently, Pratt and Tarjan asked about permutations that can be sorted using two stacks in parallel. This question is significantly harder, and the associated counting question has remained open for 40 years. We solve it by giving a pair of equations that characterise the generating function of such permutations. The first component of this system describes the generating function $Q(a,u)$ of square lattice loops confined to the positive quadrant, counted by the length and the number of North-West and East-South factors. Our analysis of the asymptotic number of sortable permutations relies at the moment on two intriguing conjectures dealing with this series. Given the recent activity on walks confined to cones, we believe them to be attractive $\textit{per se}$. We prove these conjectures for closed walks confined to the upper half plane, or not confined at all.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2425 https://doi.org/10.46298/dmtcs.2425 Albert, Michael Bousquet-Mélou, Mireille Albert, Michael Bousquet-Mélou, Mireille <![CDATA[At the end of the 1960s, Knuth characterised in terms of forbidden patterns the permutations that can be sorted using a stack. He also showed that they are in bijection with Dyck paths and thus counted by the Catalan numbers. Subsequently, Pratt and Tarjan asked about permutations that can be sorted using two stacks in parallel. This question is significantly harder, and the associated counting question has remained open for 40 years. We solve it by giving a pair of equations that characterise the generating function of such permutations. The first component of this system describes the generating function $Q(a,u)$ of square lattice loops confined to the positive quadrant, counted by the length and the number of North-West and East-South factors. Our analysis of the asymptotic number of sortable permutations relies at the moment on two intriguing conjectures dealing with this series. Given the recent activity on walks confined to cones, we believe them to be attractive $\textit{per se}$. We prove these conjectures for closed walks confined to the upper half plane, or not confined at all.]]> 0 <![CDATA[We show that an element $\mathcal{w}$ of a finite Weyl group W is rationally smooth if and only if the hyperplane arrangement $\mathcal{I} (\mathcal{w})$ associated to the inversion set of \mathcal{w} is inductively free, and the product $(d_1+1) ...(d_l+1)$ of the coexponents $d_1,\ldots,d_l$ is equal to the size of the Bruhat interval [e,w]. We also use Peterson translation of coconvex sets to give a Shapiro-Steinberg-Kostant rule for the exponents of $\mathcal{w}$.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2436 https://doi.org/10.46298/dmtcs.2436 Slofstra, William Slofstra, William <![CDATA[We show that an element $\mathcal{w}$ of a finite Weyl group W is rationally smooth if and only if the hyperplane arrangement $\mathcal{I} (\mathcal{w})$ associated to the inversion set of \mathcal{w} is inductively free, and the product $(d_1+1) ...(d_l+1)$ of the coexponents $d_1,\ldots,d_l$ is equal to the size of the Bruhat interval [e,w]. We also use Peterson translation of coconvex sets to give a Shapiro-Steinberg-Kostant rule for the exponents of $\mathcal{w}$.]]> 0 <![CDATA[There are numerous combinatorial objects associated to a Grassmannian permutation $w_λ$ that index cells of the totally nonnegative Grassmannian. We study some of these objects (rook placements, acyclic orientations, various restricted fillings) and their q-analogues in the case of permutations $\mathcal{w}$ that are not necessarily Grassmannian.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2435 https://doi.org/10.46298/dmtcs.2435 Lewis, Joel Brewster Morales, Alejandro, Lewis, Joel Brewster Morales, Alejandro, <![CDATA[There are numerous combinatorial objects associated to a Grassmannian permutation $w_λ$ that index cells of the totally nonnegative Grassmannian. We study some of these objects (rook placements, acyclic orientations, various restricted fillings) and their q-analogues in the case of permutations $\mathcal{w}$ that are not necessarily Grassmannian.]]> 0 <![CDATA[We present a generalization of the chromatic polynomial, and chromatic symmetric function, arising in the study of combinatorial species. These invariants are defined for modules over lattice rings in species. The primary examples are graphs and set partitions. For these new invariants, we present analogues of results regarding stable partitions, the bond lattice, the deletion-contraction recurrence, and the subset expansion formula. We also present two detailed examples, one related to enumerating subgraphs by their blocks, and a second example related to enumerating subgraphs of a directed graph by their strongly connected components.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2434 https://doi.org/10.46298/dmtcs.2434 White, Jacob, White, Jacob, <![CDATA[We present a generalization of the chromatic polynomial, and chromatic symmetric function, arising in the study of combinatorial species. These invariants are defined for modules over lattice rings in species. The primary examples are graphs and set partitions. For these new invariants, we present analogues of results regarding stable partitions, the bond lattice, the deletion-contraction recurrence, and the subset expansion formula. We also present two detailed examples, one related to enumerating subgraphs by their blocks, and a second example related to enumerating subgraphs of a directed graph by their strongly connected components.]]> 0 <![CDATA[We describe a new uniform generation tree for permutations with the specific property that, for most permutations, all of their descendants in the generation tree have the same number of fixed points. Our tree is optimal for the number of permutations having this property. We then use this tree to describe a new random generation algorithm for derangements, using an expected n+O(1) calls to a random number generator. Another application is a combinatorial algorithm for exact sampling from the Poisson distribution with parameter 1.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2433 https://doi.org/10.46298/dmtcs.2433 Duchon, Philippe Duvignau, Romaric Duchon, Philippe Duvignau, Romaric <![CDATA[We describe a new uniform generation tree for permutations with the specific property that, for most permutations, all of their descendants in the generation tree have the same number of fixed points. Our tree is optimal for the number of permutations having this property. We then use this tree to describe a new random generation algorithm for derangements, using an expected n+O(1) calls to a random number generator. Another application is a combinatorial algorithm for exact sampling from the Poisson distribution with parameter 1.]]> 0 <![CDATA[Sweep maps are a family of maps on words that, while simple to define, are not yet known to be injective in general. This family subsumes many of the "zeta maps" that have arisen in the study of q,t-Catalan numbers in the course of relating the three statistics of area, bounce and dinv. A sweep map can be defined for words over arbitrary alphabets with arbitrary weights. The latter property makes them particularly suitable for the study of rational Catalan combinatorics.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2432 https://doi.org/10.46298/dmtcs.2432 Loehr, Nicholas, Warrington, Gregory, Loehr, Nicholas, Warrington, Gregory, <![CDATA[Sweep maps are a family of maps on words that, while simple to define, are not yet known to be injective in general. This family subsumes many of the "zeta maps" that have arisen in the study of q,t-Catalan numbers in the course of relating the three statistics of area, bounce and dinv. A sweep map can be defined for words over arbitrary alphabets with arbitrary weights. The latter property makes them particularly suitable for the study of rational Catalan combinatorics.]]> 0 <![CDATA[We investigate the role that non-crossing partitions play in the study of positroids, a class of matroids introduced by Postnikov. We prove that every positroid can be constructed uniquely by choosing a non-crossing partition on the ground set, and then freely placing the structure of a connected positroid on each of the blocks of the partition. We use this to enumerate connected positroids, and we prove that the probability that a positroid on [n] is connected equals $1/e^2$ asymptotically. We also prove da Silva's 1987 conjecture that any positively oriented matroid is a positroid; that is, it can be realized by a set of vectors in a real vector space. It follows from this result that the positive matroid Grassmannian (or positive MacPhersonian) is homeomorphic to a closed ball.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2431 https://doi.org/10.46298/dmtcs.2431 Ardila, Federico Rincón, Felipe Williams, Lauren Ardila, Federico Rincón, Felipe Williams, Lauren <![CDATA[We investigate the role that non-crossing partitions play in the study of positroids, a class of matroids introduced by Postnikov. We prove that every positroid can be constructed uniquely by choosing a non-crossing partition on the ground set, and then freely placing the structure of a connected positroid on each of the blocks of the partition. We use this to enumerate connected positroids, and we prove that the probability that a positroid on [n] is connected equals $1/e^2$ asymptotically. We also prove da Silva's 1987 conjecture that any positively oriented matroid is a positroid; that is, it can be realized by a set of vectors in a real vector space. It follows from this result that the positive matroid Grassmannian (or positive MacPhersonian) is homeomorphic to a closed ball.]]> 0 <![CDATA[Partition the rows of a board into sets of $m$ rows called levels. An $m$-level rook placement is a subset of squares of the board with no two in the same column or the same level. We construct explicit bijections to prove three theorems about such placements. We start with two bijections between Ferrers boards having the same number of $m$-level rook placements. The first generalizes a map by Foata and Schützenberger and our proof applies to any Ferrers board. The second generalizes work of Loehr and Remmel. This construction only works for a special class of Ferrers boards but also yields a formula for calculating the rook numbers of these boards in terms of elementary symmetric functions. Finally we generalize another result of Loehr and Remmel giving a bijection between boards with the same hit numbers. The second and third bijections involve the Involution Principle of Garsia and Milne.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2430 https://doi.org/10.46298/dmtcs.2430 Barrese, Kenneth Sagan, Bruce, Barrese, Kenneth Sagan, Bruce, <![CDATA[Partition the rows of a board into sets of $m$ rows called levels. An $m$-level rook placement is a subset of squares of the board with no two in the same column or the same level. We construct explicit bijections to prove three theorems about such placements. We start with two bijections between Ferrers boards having the same number of $m$-level rook placements. The first generalizes a map by Foata and Schützenberger and our proof applies to any Ferrers board. The second generalizes work of Loehr and Remmel. This construction only works for a special class of Ferrers boards but also yields a formula for calculating the rook numbers of these boards in terms of elementary symmetric functions. Finally we generalize another result of Loehr and Remmel giving a bijection between boards with the same hit numbers. The second and third bijections involve the Involution Principle of Garsia and Milne.]]> 0 <![CDATA[For a random permutation sampled from the stationary distribution of the TASEP on a ring, we show that, conditioned on the event that the first entries are strictly larger than the last entries, the $\textit{order}$ of the first entries is independent of the $\textit{order}$ of the last entries. The proof uses multi-line queues as defined by Ferrari and Martin, and the theorem has an enumerative combinatorial interpretation in that setting. Finally, we present a conjecture for the case where the small and large entries are not separated.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2429 https://doi.org/10.46298/dmtcs.2429 Aas, Erik Sjöstrand, Jonas Aas, Erik Sjöstrand, Jonas <![CDATA[For a random permutation sampled from the stationary distribution of the TASEP on a ring, we show that, conditioned on the event that the first entries are strictly larger than the last entries, the $\textit{order}$ of the first entries is independent of the $\textit{order}$ of the last entries. The proof uses multi-line queues as defined by Ferrari and Martin, and the theorem has an enumerative combinatorial interpretation in that setting. Finally, we present a conjecture for the case where the small and large entries are not separated.]]> 0 <![CDATA[A general lattice theoretic construction of Reading constructs Hopf subalgebras of the Malvenuto-Reutenauer Hopf algebra (MR) of permutations. The products and coproducts of these Hopf subalgebras are defined extrinsically in terms of the embedding in MR. The goal of this paper is to find an intrinsic combinatorial description of a particular one of these Hopf subalgebras. This Hopf algebra has a natural basis given by permutations that we call Pell permutations. The Pell permutations are in bijection with combinatorial objects that we call sashes, that is, tilings of a 1 by n rectangle with three types of tiles: black 1 by 1 squares, white 1 by 1 squares, and white 1 by 2 rectangles. The bijection induces a Hopf algebra structure on sashes. We describe the product and coproduct in terms of sashes, and the natural partial order on sashes. We also describe the dual coproduct and dual product of the dual Hopf algebra of sashes.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2428 https://doi.org/10.46298/dmtcs.2428 Law, Shirley Law, Shirley <![CDATA[A general lattice theoretic construction of Reading constructs Hopf subalgebras of the Malvenuto-Reutenauer Hopf algebra (MR) of permutations. The products and coproducts of these Hopf subalgebras are defined extrinsically in terms of the embedding in MR. The goal of this paper is to find an intrinsic combinatorial description of a particular one of these Hopf subalgebras. This Hopf algebra has a natural basis given by permutations that we call Pell permutations. The Pell permutations are in bijection with combinatorial objects that we call sashes, that is, tilings of a 1 by n rectangle with three types of tiles: black 1 by 1 squares, white 1 by 1 squares, and white 1 by 2 rectangles. The bijection induces a Hopf algebra structure on sashes. We describe the product and coproduct in terms of sashes, and the natural partial order on sashes. We also describe the dual coproduct and dual product of the dual Hopf algebra of sashes.]]> 0 <![CDATA[We study a natural generalization of the noncrossing relation between pairs of elements in $[n]$ to $k$-tuples in $[n]$. We show that the flag simplicial complex on $\binom{[n]}{k}$ induced by this relation is a regular, unimodular and flag triangulation of the order polytope of the poset given by the product $[k] \times [n-k]$ of two chains, and it is the join of a simplex and a sphere (that is, it is a Gorenstein triangulation). This shows the existence of a flag simplicial polytope whose Stanley-Reisner ideal is an initial ideal of the Graßmann-Plücker ideal, while previous constructions of such a polytope did not guaranteed flagness. The simplicial complex and the polytope derived from it naturally reflect the relations between Graßmannians with different parameters, in particular the isomorphism $G_{k,n} \cong G_{n-k,n}$. This simplicial complex is closely related to the weak separability complex introduced by Zelevinsky and Leclerc.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2427 https://doi.org/10.46298/dmtcs.2427 Santos, Francisco Stump, Christian Welker, Volkmar Santos, Francisco Stump, Christian Welker, Volkmar <![CDATA[We study a natural generalization of the noncrossing relation between pairs of elements in $[n]$ to $k$-tuples in $[n]$. We show that the flag simplicial complex on $\binom{[n]}{k}$ induced by this relation is a regular, unimodular and flag triangulation of the order polytope of the poset given by the product $[k] \times [n-k]$ of two chains, and it is the join of a simplex and a sphere (that is, it is a Gorenstein triangulation). This shows the existence of a flag simplicial polytope whose Stanley-Reisner ideal is an initial ideal of the Graßmann-Plücker ideal, while previous constructions of such a polytope did not guaranteed flagness. The simplicial complex and the polytope derived from it naturally reflect the relations between Graßmannians with different parameters, in particular the isomorphism $G_{k,n} \cong G_{n-k,n}$. This simplicial complex is closely related to the weak separability complex introduced by Zelevinsky and Leclerc.]]> 0 <![CDATA[In this abstract, I will survey the story of two enumerative miracles that relate certain Coxeter-theoretic objects and other poset-theoretic objects. The first miracle relates reduced words and linear extensions, while the second may be thought of as relating group elements and order ideals. The purpose of this abstract is to use a conjecture from my thesis to present both miracles in the same light.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2426 https://doi.org/10.46298/dmtcs.2426 Williams, Nathan Williams, Nathan <![CDATA[In this abstract, I will survey the story of two enumerative miracles that relate certain Coxeter-theoretic objects and other poset-theoretic objects. The first miracle relates reduced words and linear extensions, while the second may be thought of as relating group elements and order ideals. The purpose of this abstract is to use a conjecture from my thesis to present both miracles in the same light.]]> 0 <![CDATA[We review and introduce several approaches to the study of centralizer algebras of the infinite symmetric group $S_{\infty}$. Our work is led by the double commutant relationship between finite symmetric groups and partition algebras; in the case of $S_{\infty}$, we obtain centralizer algebras that are contained in partition algebras. In view of the theory of symmetric functions in non-commuting variables, we consider representations of $S_{\infty}$ that are faithful and that contain invariant elements; namely, non-unitary representations on sequence spaces.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2414 https://doi.org/10.46298/dmtcs.2414 Daugherty, Zajj Herbrich, Peter Daugherty, Zajj Herbrich, Peter <![CDATA[We review and introduce several approaches to the study of centralizer algebras of the infinite symmetric group $S_{\infty}$. Our work is led by the double commutant relationship between finite symmetric groups and partition algebras; in the case of $S_{\infty}$, we obtain centralizer algebras that are contained in partition algebras. In view of the theory of symmetric functions in non-commuting variables, we consider representations of $S_{\infty}$ that are faithful and that contain invariant elements; namely, non-unitary representations on sequence spaces.]]> 0 <![CDATA[We establish a simple recurrence formula for the number $Q_g^n$ of rooted orientable maps counted by edges and genus. The formula is a consequence of the KP equation for the generating function of bipartite maps, coupled with a Tutte equation, and it was apparently unnoticed before. It gives by far the fastest known way of computing these numbers, or the fixed-genus generating functions, especially for large $g$. The formula is similar in look to the one discovered by Goulden and Jackson for triangulations (although the latter does not rely on an additional Tutte equation). Both of them have a very combinatorial flavour, but finding a bijective interpretation is currently unsolved - should such an interpretation exist, the history of bijective methods for maps would tend to show that the case treated here is easier to start with than the one of triangulations.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2424 https://doi.org/10.46298/dmtcs.2424 Carrell, Sean Chapuy, Guillaume Carrell, Sean Chapuy, Guillaume <![CDATA[We establish a simple recurrence formula for the number $Q_g^n$ of rooted orientable maps counted by edges and genus. The formula is a consequence of the KP equation for the generating function of bipartite maps, coupled with a Tutte equation, and it was apparently unnoticed before. It gives by far the fastest known way of computing these numbers, or the fixed-genus generating functions, especially for large $g$. The formula is similar in look to the one discovered by Goulden and Jackson for triangulations (although the latter does not rely on an additional Tutte equation). Both of them have a very combinatorial flavour, but finding a bijective interpretation is currently unsolved - should such an interpretation exist, the history of bijective methods for maps would tend to show that the case treated here is easier to start with than the one of triangulations.]]> 0 <![CDATA[We obtain a nonrecursive combinatorial formula for the Kazhdan-Lusztig polynomials which holds in complete generality and which is simpler and more explicit than any existing one, and which cannot be linearly simplified. Our proof uses a new basis of the peak subalgebra of the algebra of quasisymmetric functions.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2423 https://doi.org/10.46298/dmtcs.2423 Brenti, Francesco Caselli, Fabrizio Brenti, Francesco Caselli, Fabrizio <![CDATA[We obtain a nonrecursive combinatorial formula for the Kazhdan-Lusztig polynomials which holds in complete generality and which is simpler and more explicit than any existing one, and which cannot be linearly simplified. Our proof uses a new basis of the peak subalgebra of the algebra of quasisymmetric functions.]]> 0 <![CDATA[In our previous works "Pfaffian decomposition and a Pfaffian analogue of $q$-Catalan Hankel determinants'' (by M.Ishikawa, H. Tagawa and J. Zeng, J. Combin. Theory Ser. A, 120, 2013, 1263-1284) we have proposed several ways to evaluate certain Catalan-Hankel Pffafians and also formulated several conjectures. In this work we propose a new approach to compute these Catalan-Hankel Pffafians using Selberg's integral as well as their $q$-analogues. In particular, this approach permits us to settle most of the conjectures in our previous paper.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2422 https://doi.org/10.46298/dmtcs.2422 Ishikawa, Masao Zeng, Jiang Ishikawa, Masao Zeng, Jiang <![CDATA[In our previous works "Pfaffian decomposition and a Pfaffian analogue of $q$-Catalan Hankel determinants'' (by M.Ishikawa, H. Tagawa and J. Zeng, J. Combin. Theory Ser. A, 120, 2013, 1263-1284) we have proposed several ways to evaluate certain Catalan-Hankel Pffafians and also formulated several conjectures. In this work we propose a new approach to compute these Catalan-Hankel Pffafians using Selberg's integral as well as their $q$-analogues. In particular, this approach permits us to settle most of the conjectures in our previous paper.]]> 0 <![CDATA[The $\textit{parallel chip-firing game}$ is an automaton on graphs in which vertices "fire'' chips to their neighbors. This simple model, analogous to sandpiles forming and collapsing, contains much emergent complexity and has connections to different areas of mathematics including self-organized criticality and the study of the sandpile group. In this work, we study $\textit{firing sequences}$, which describe each vertex's interaction with its neighbors in this game. Our main contribution is a complete characterization of the periodic firing sequences that can occur in a game, which have a surprisingly simple combinatorial description. We also obtain other results about local behavior of the game after introducing the concept of $\textit{motors}$.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2421 https://doi.org/10.46298/dmtcs.2421 Scully, Ziv Jiang, Tian-Yi Zhang, Yan, Scully, Ziv Jiang, Tian-Yi Zhang, Yan, <![CDATA[The $\textit{parallel chip-firing game}$ is an automaton on graphs in which vertices "fire'' chips to their neighbors. This simple model, analogous to sandpiles forming and collapsing, contains much emergent complexity and has connections to different areas of mathematics including self-organized criticality and the study of the sandpile group. In this work, we study $\textit{firing sequences}$, which describe each vertex's interaction with its neighbors in this game. Our main contribution is a complete characterization of the periodic firing sequences that can occur in a game, which have a surprisingly simple combinatorial description. We also obtain other results about local behavior of the game after introducing the concept of $\textit{motors}$.]]> 0 <![CDATA[We expose a rule for multiplying a general Schubert polynomial with a power sum polynomial in $k$ variables. A signed sum over cyclic permutations replaces the signed sum over rim hooks in the classical Murgnahan-Nakayama rule. In the intersection theory of flag manifolds this computes all intersections of Schubert cycles with tautological classes coming from the Chern character. We also discuss extensions of this rule to small quantum cohomology.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2420 https://doi.org/10.46298/dmtcs.2420 Morrison, Andrew Morrison, Andrew <![CDATA[We expose a rule for multiplying a general Schubert polynomial with a power sum polynomial in $k$ variables. A signed sum over cyclic permutations replaces the signed sum over rim hooks in the classical Murgnahan-Nakayama rule. In the intersection theory of flag manifolds this computes all intersections of Schubert cycles with tautological classes coming from the Chern character. We also discuss extensions of this rule to small quantum cohomology.]]> 0 <![CDATA[We define piecewise-linear and birational analogues of toggle-involutions, rowmotion, and promotion on order ideals of a poset $P$ as studied by Striker and Williams. Piecewise-linear rowmotion relates to Stanley's transfer map for order polytopes; piecewise-linear promotion relates to Schützenberger promotion for semistandard Young tableaux. When $P = [a] \times [b]$, a reciprocal symmetry property recently proved by Grinberg and Roby implies that birational rowmotion (and consequently piecewise-linear rowmotion) is of order $a+b$. We prove some homomesy results, showing that for certain functions $f$, the average of $f$ over each rowmotion/promotion orbit is independent of the orbit chosen.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2419 https://doi.org/10.46298/dmtcs.2419 Einstein, David Propp, James Einstein, David Propp, James <![CDATA[We define piecewise-linear and birational analogues of toggle-involutions, rowmotion, and promotion on order ideals of a poset $P$ as studied by Striker and Williams. Piecewise-linear rowmotion relates to Stanley's transfer map for order polytopes; piecewise-linear promotion relates to Schützenberger promotion for semistandard Young tableaux. When $P = [a] \times [b]$, a reciprocal symmetry property recently proved by Grinberg and Roby implies that birational rowmotion (and consequently piecewise-linear rowmotion) is of order $a+b$. We prove some homomesy results, showing that for certain functions $f$, the average of $f$ over each rowmotion/promotion orbit is independent of the orbit chosen.]]> 0 <![CDATA[A Weyl arrangement is the arrangement defined by the root system of a finite Weyl group. When a set of positive roots is an ideal in the root poset, we call the corresponding arrangement an ideal subarrangement. Our main theorem asserts that any ideal subarrangement is a free arrangement and that its exponents are given by the dual partition of the height distribution, which was conjectured by Sommers-Tymoczko. In particular, when an ideal subarrangement is equal to the entire Weyl arrangement, our main theorem yields the celebrated formula by Shapiro, Steinberg, Kostant, and Macdonald. The proof of the main theorem is classification-free. It heavily depends on the theory of free arrangements and thus greatly differs from the earlier proofs of the formula.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2418 https://doi.org/10.46298/dmtcs.2418 Abe, Takuro Barakat, Mohamed Cuntz, Michael Hoge, Torsten Terao, Hiroaki Abe, Takuro Barakat, Mohamed Cuntz, Michael Hoge, Torsten Terao, Hiroaki <![CDATA[A Weyl arrangement is the arrangement defined by the root system of a finite Weyl group. When a set of positive roots is an ideal in the root poset, we call the corresponding arrangement an ideal subarrangement. Our main theorem asserts that any ideal subarrangement is a free arrangement and that its exponents are given by the dual partition of the height distribution, which was conjectured by Sommers-Tymoczko. In particular, when an ideal subarrangement is equal to the entire Weyl arrangement, our main theorem yields the celebrated formula by Shapiro, Steinberg, Kostant, and Macdonald. The proof of the main theorem is classification-free. It heavily depends on the theory of free arrangements and thus greatly differs from the earlier proofs of the formula.]]> 0 <![CDATA[We apply ideas from crystal theory to affine Schubert calculus and flag Gromov-Witten invariants. By defining operators on certain decompositions of elements in the type-$A$ affine Weyl group, we produce a crystal reflecting the internal structure of Specht modules associated to permutation diagrams. We show how this crystal framework can be applied to study the product of a Schur function with a $k$-Schur function. Consequently, we prove that a subclass of 3-point Gromov-Witten invariants of complete flag varieties for $\mathbb{C}^n$ enumerate the highest weight elements under these operators.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2417 https://doi.org/10.46298/dmtcs.2417 Morse, Jennifer Schilling, Anne Morse, Jennifer Schilling, Anne <![CDATA[We apply ideas from crystal theory to affine Schubert calculus and flag Gromov-Witten invariants. By defining operators on certain decompositions of elements in the type-$A$ affine Weyl group, we produce a crystal reflecting the internal structure of Specht modules associated to permutation diagrams. We show how this crystal framework can be applied to study the product of a Schur function with a $k$-Schur function. Consequently, we prove that a subclass of 3-point Gromov-Witten invariants of complete flag varieties for $\mathbb{C}^n$ enumerate the highest weight elements under these operators.]]> 0 <![CDATA[In this paper we improve a method of Robinson and Taulbee for computing Kronecker coefficients and show that for any partition $\overline{ν}$ of $d$ there is a polynomial $k_{\overline{ν}}$ with rational coefficients in variables $x_C$, where $C$ runs over the set of isomorphism classes of connected skew diagrams of size at most $d$, such that for all partitions $\lambda$ of $n$, the Kronecker coefficient $\mathsf{g}(\lambda, \lambda, (n-d, \overline{ν}))$ is obtained from $k_{\overline{ν}}(x_C)$ substituting each $x_C$ by the number of $\lambda$-removable diagrams in $C$. We present two applications. First we show that for $\rho_{k} = (k, k-1,\ldots, 2, 1)$ and any partition $\overline{ν}$ of size $d$ there is a piecewise polynomial function $s_{\overline{ν}}$ such that $\mathsf{g}(\rho_k, \rho_k, (|\rho_k| - d, \overline{ν})) = s_{\overline{ν}} (k)$ for all $k$ and that there is an interval of the form $[c, \infty)$ in which $s_{\overline{ν}}$ is polynomial of degree $d$ with leading coefficient the number of standard Young tableaux of shape $\overline{ν}$. The second application is new stability property for Kronecker coefficients.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2416 https://doi.org/10.46298/dmtcs.2416 Vallejo, Ernesto Vallejo, Ernesto <![CDATA[In this paper we improve a method of Robinson and Taulbee for computing Kronecker coefficients and show that for any partition $\overline{ν}$ of $d$ there is a polynomial $k_{\overline{ν}}$ with rational coefficients in variables $x_C$, where $C$ runs over the set of isomorphism classes of connected skew diagrams of size at most $d$, such that for all partitions $\lambda$ of $n$, the Kronecker coefficient $\mathsf{g}(\lambda, \lambda, (n-d, \overline{ν}))$ is obtained from $k_{\overline{ν}}(x_C)$ substituting each $x_C$ by the number of $\lambda$-removable diagrams in $C$. We present two applications. First we show that for $\rho_{k} = (k, k-1,\ldots, 2, 1)$ and any partition $\overline{ν}$ of size $d$ there is a piecewise polynomial function $s_{\overline{ν}}$ such that $\mathsf{g}(\rho_k, \rho_k, (|\rho_k| - d, \overline{ν})) = s_{\overline{ν}} (k)$ for all $k$ and that there is an interval of the form $[c, \infty)$ in which $s_{\overline{ν}}$ is polynomial of degree $d$ with leading coefficient the number of standard Young tableaux of shape $\overline{ν}$. The second application is new stability property for Kronecker coefficients.]]> 0 <![CDATA[This paper constructs a bijection between irreducible $k$-shapes and surjective pistols of height $k-1$, which carries the "free $k$-sites" to the fixed points of surjective pistols. The bijection confirms a conjecture of Hivert and Mallet (FPSAC 2011) that the number of irreducible $k$-shape is counted by the Genocchi number $G_{2k}$.]]> Tue, 31 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2415 https://doi.org/10.46298/dmtcs.2415 Bigeni, Ange Bigeni, Ange <![CDATA[This paper constructs a bijection between irreducible $k$-shapes and surjective pistols of height $k-1$, which carries the "free $k$-sites" to the fixed points of surjective pistols. The bijection confirms a conjecture of Hivert and Mallet (FPSAC 2011) that the number of irreducible $k$-shape is counted by the Genocchi number $G_{2k}$.]]> 0 <![CDATA[We study a graph parameter related to resolving sets and metric dimension, namely the resolving number, introduced by Chartrand, Poisson and Zhang. First, we establish an important difference between the two parameters: while computing the metric dimension of an arbitrary graph is known to be NP-hard, we show that the resolving number can be computed in polynomial time. We then relate the resolving number to classical graph parameters: diameter, girth, clique number, order and maximum degree. With these relations in hand, we characterize the graphs with resolving number 3 extending other studies that provide characterizations for smaller resolving number.]]> Mon, 09 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.615 https://doi.org/10.46298/dmtcs.615 Garijo, Delia González, Antonio Márquez, Alberto Garijo, Delia González, Antonio Márquez, Alberto <![CDATA[We study a graph parameter related to resolving sets and metric dimension, namely the resolving number, introduced by Chartrand, Poisson and Zhang. First, we establish an important difference between the two parameters: while computing the metric dimension of an arbitrary graph is known to be NP-hard, we show that the resolving number can be computed in polynomial time. We then relate the resolving number to classical graph parameters: diameter, girth, clique number, order and maximum degree. With these relations in hand, we characterize the graphs with resolving number 3 extending other studies that provide characterizations for smaller resolving number.]]> 0 <![CDATA[Given an arrangement of lines in the plane, what is the minimum number c of colors required to color the lines so that no cell of the arrangement is monochromatic? In this paper we give bounds on the number c both for the above question, as well as some of its variations. We redefine these problems as geometric hypergraph coloring problems. If we define $\Hlinecell$ as the hypergraph where vertices are lines and edges represent cells of the arrangement, the answer to the above question is equal to the chromatic number of this hypergraph. We prove that this chromatic number is between Ω(logn/loglogn). and O(n√). Similarly, we give bounds on the minimum size of a subset S of the intersections of the lines in A such that every cell is bounded by at least one of the vertices in S. This may be seen as a problem on guarding cells with vertices when the lines act as obstacles. The problem can also be defined as the minimum vertex cover problem in the hypergraph $\Hvertexcell$, the vertices of which are the line intersections, and the hyperedges are vertices of a cell. Analogously, we consider the problem of touching the lines with a minimum subset of the cells of the arrangement, which we identify as the minimum vertex cover problem in the $\Hcellzone$ hypergraph.]]> Tue, 03 Dec 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2072 https://doi.org/10.46298/dmtcs.2072 Bose, Prosenjit Cardinal, Jean Collette, Sébastien Hurtado, Ferran Korman, Matias Langerman, Stefan Taslakian, Perouz Bose, Prosenjit Cardinal, Jean Collette, Sébastien Hurtado, Ferran Korman, Matias Langerman, Stefan Taslakian, Perouz <![CDATA[Given an arrangement of lines in the plane, what is the minimum number c of colors required to color the lines so that no cell of the arrangement is monochromatic? In this paper we give bounds on the number c both for the above question, as well as some of its variations. We redefine these problems as geometric hypergraph coloring problems. If we define $\Hlinecell$ as the hypergraph where vertices are lines and edges represent cells of the arrangement, the answer to the above question is equal to the chromatic number of this hypergraph. We prove that this chromatic number is between Ω(logn/loglogn). and O(n√). Similarly, we give bounds on the minimum size of a subset S of the intersections of the lines in A such that every cell is bounded by at least one of the vertices in S. This may be seen as a problem on guarding cells with vertices when the lines act as obstacles. The problem can also be defined as the minimum vertex cover problem in the hypergraph $\Hvertexcell$, the vertices of which are the line intersections, and the hyperedges are vertices of a cell. Analogously, we consider the problem of touching the lines with a minimum subset of the cells of the arrangement, which we identify as the minimum vertex cover problem in the $\Hcellzone$ hypergraph.]]> 0 <![CDATA[In the frequency allocation problem, we are given a cellular telephone network whose geographical coverage area is divided into cells, where phone calls are serviced by assigned frequencies, so that none of the pairs of calls emanating from the same or neighboring cells is assigned the same frequency. The problem is to use the frequencies efficiently, i.e. minimize the span of frequencies used. The frequency allocation problem can be regarded as a multicoloring problem on a weighted hexagonal graph, where each vertex knows its position in the graph. We present a 1-local 33/24-competitive distributed algorithm for multicoloring a hexagonal graph, thereby improving the previous 1-local 7/5-competitive algorithm.]]> Wed, 20 Nov 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.614 https://doi.org/10.46298/dmtcs.614 Witkowski, Rafal Žerovnik, Janez Witkowski, Rafal Žerovnik, Janez <![CDATA[In the frequency allocation problem, we are given a cellular telephone network whose geographical coverage area is divided into cells, where phone calls are serviced by assigned frequencies, so that none of the pairs of calls emanating from the same or neighboring cells is assigned the same frequency. The problem is to use the frequencies efficiently, i.e. minimize the span of frequencies used. The frequency allocation problem can be regarded as a multicoloring problem on a weighted hexagonal graph, where each vertex knows its position in the graph. We present a 1-local 33/24-competitive distributed algorithm for multicoloring a hexagonal graph, thereby improving the previous 1-local 7/5-competitive algorithm.]]> 0 <![CDATA[We study time and message complexity of the problem of building a BFS tree by a spontaneously awaken node in ad hoc network. Computation is in synchronous rounds, and messages are sent via point-to-point bi-directional links. Network topology is modeled by a graph. Each node knows only its own id and the id's of its neighbors in the network and no pre-processing is allowed; therefore the solutions to the problem of spanning a BFS tree in this setting must be distributed. We deliver a deterministic distributed solution that trades time for messages, mainly, with time complexity O(D . min(D; n=f(n)) . logD . log n) and with the number of point-to-point messages sent O(n. (min(D; n=f(n))+f(n)) . logD. log n), for any n-node network with diameter D and for any monotonically non-decreasing sub-linear integer function f. Function f in the above formulas come from the threshold value on node degrees used by our algorithms, in the sense that nodes with degree at most f(n) are treated differently that the other nodes. This yields the first BFS-finding deterministic distributed algorithm in ad hoc networks working in time o(n) and with o(n2) message complexity, for some suitable functions f(n) = o(n= log2 n), provided D = o(n= log4 n).]]> Mon, 11 Nov 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.622 https://doi.org/10.46298/dmtcs.622 Kowalski, Dariusz Krzywdziński, Krzysztof Kowalski, Dariusz Krzywdziński, Krzysztof <![CDATA[We study time and message complexity of the problem of building a BFS tree by a spontaneously awaken node in ad hoc network. Computation is in synchronous rounds, and messages are sent via point-to-point bi-directional links. Network topology is modeled by a graph. Each node knows only its own id and the id's of its neighbors in the network and no pre-processing is allowed; therefore the solutions to the problem of spanning a BFS tree in this setting must be distributed. We deliver a deterministic distributed solution that trades time for messages, mainly, with time complexity O(D . min(D; n=f(n)) . logD . log n) and with the number of point-to-point messages sent O(n. (min(D; n=f(n))+f(n)) . logD. log n), for any n-node network with diameter D and for any monotonically non-decreasing sub-linear integer function f. Function f in the above formulas come from the threshold value on node degrees used by our algorithms, in the sense that nodes with degree at most f(n) are treated differently that the other nodes. This yields the first BFS-finding deterministic distributed algorithm in ad hoc networks working in time o(n) and with o(n2) message complexity, for some suitable functions f(n) = o(n= log2 n), provided D = o(n= log4 n).]]> 0 <![CDATA[The Magnus-Derek game (also called the maximal variant of the vector game), introduced by Nedev and Muthukrishnan is the following: a token is moved around a table with n positions. In each round of the game Magnus chooses a number and then Derek chooses a direction (clockwise or counterclockwise), and the token moves that many positions into that direction. The goal of Magnus is to maximize the number of positions visited, the goal of Derek is the opposite. In the minimal variant of the game the goals of the two players are exchanged: Magnus wants to minimize the number of positions visited and Derek wants the opposite. Here we introduce a generalization of these games: the token is moved in a group, Magnus chooses an element of the group and Derek decides if the current position is multiplied or divided by that element.]]> Mon, 11 Nov 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.621 https://doi.org/10.46298/dmtcs.621 Gerbner, Dániel Gerbner, Dániel <![CDATA[The Magnus-Derek game (also called the maximal variant of the vector game), introduced by Nedev and Muthukrishnan is the following: a token is moved around a table with n positions. In each round of the game Magnus chooses a number and then Derek chooses a direction (clockwise or counterclockwise), and the token moves that many positions into that direction. The goal of Magnus is to maximize the number of positions visited, the goal of Derek is the opposite. In the minimal variant of the game the goals of the two players are exchanged: Magnus wants to minimize the number of positions visited and Derek wants the opposite. Here we introduce a generalization of these games: the token is moved in a group, Magnus chooses an element of the group and Derek decides if the current position is multiplied or divided by that element.]]> 0 <![CDATA[The classical Erd˝os-Szekeres theorem states that a convex k-gon exists in every sufficiently large point set. This problem has been well studied and finding tight asymptotic bounds is considered a challenging open problem. Several variants of the Erd˝os-Szekeres problem have been posed and studied in the last two decades. The well studied variants include the empty convex k-gon problem, convex k-gon with specified number of interior points and the chromatic variant. In this paper, we introduce the following two player game variant of the Erdös-Szekeres problem: Consider a two player game where each player playing in alternate turns, place points in the plane. The objective of the game is to avoid the formation of the convex k-gon among the placed points. The game ends when a convex k-gon is formed and the player who placed the last point loses the game. In our paper we show a winning strategy for the player who plays second in the convex 5-gon game and the empty convex 5-gon game by considering convex layer configurations at each step. We prove that the game always ends in the 9th step by showing that the game reaches a specific set of configurations]]> Tue, 05 Nov 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.620 https://doi.org/10.46298/dmtcs.620 Kolipaka, Parikshit Govindarajan, Sathish Kolipaka, Parikshit Govindarajan, Sathish <![CDATA[The classical Erd˝os-Szekeres theorem states that a convex k-gon exists in every sufficiently large point set. This problem has been well studied and finding tight asymptotic bounds is considered a challenging open problem. Several variants of the Erd˝os-Szekeres problem have been posed and studied in the last two decades. The well studied variants include the empty convex k-gon problem, convex k-gon with specified number of interior points and the chromatic variant. In this paper, we introduce the following two player game variant of the Erdös-Szekeres problem: Consider a two player game where each player playing in alternate turns, place points in the plane. The objective of the game is to avoid the formation of the convex k-gon among the placed points. The game ends when a convex k-gon is formed and the player who placed the last point loses the game. In our paper we show a winning strategy for the player who plays second in the convex 5-gon game and the empty convex 5-gon game by considering convex layer configurations at each step. We prove that the game always ends in the 9th step by showing that the game reaches a specific set of configurations]]> 0 <![CDATA[The Cerný's conjecture states that for every synchronizing automaton with n states there exists a reset word of length not exceeding (n - 1)2. We prove this conjecture for a class of automata preserving certain properties of intervals of a directed graph. Our result unifies and generalizes some earlier results obtained by other authors.]]> Sat, 02 Nov 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.619 https://doi.org/10.46298/dmtcs.619 Grech, Mariusz Kisielewicz, Andrzej Grech, Mariusz Kisielewicz, Andrzej <![CDATA[The Cerný's conjecture states that for every synchronizing automaton with n states there exists a reset word of length not exceeding (n - 1)2. We prove this conjecture for a class of automata preserving certain properties of intervals of a directed graph. Our result unifies and generalizes some earlier results obtained by other authors.]]> 0 <![CDATA[One of the first and most famous results of cellular automata theory, Moore's Garden-of-Eden theorem has been proven to hold if and only if the underlying group possesses the measure-theoretic properties suggested by von Neumann to be the obstacle to the Banach-Tarski paradox. We show that several other results from the literature, already known to characterize surjective cellular automata in dimension d, hold precisely when the Garden-of-Eden theorem does. We focus in particular on the balancedness theorem, which has been proven by Bartholdi to fail on amenable groups, and we measure the amount of such failure.]]> Thu, 17 Oct 2013 22:00:00 +0000 https://doi.org/10.46298/dmtcs.618 https://doi.org/10.46298/dmtcs.618 Capobianco, Silvio Guillon, Pierre Kari, Jarkko Capobianco, Silvio Guillon, Pierre Kari, Jarkko <![CDATA[One of the first and most famous results of cellular automata theory, Moore's Garden-of-Eden theorem has been proven to hold if and only if the underlying group possesses the measure-theoretic properties suggested by von Neumann to be the obstacle to the Banach-Tarski paradox. We show that several other results from the literature, already known to characterize surjective cellular automata in dimension d, hold precisely when the Garden-of-Eden theorem does. We focus in particular on the balancedness theorem, which has been proven by Bartholdi to fail on amenable groups, and we measure the amount of such failure.]]> 0 <![CDATA[Fibonacci and Lucas cubes are induced subgraphs of hypercubes obtained by excluding certain binary strings from the vertex set. They appear as models for interconnection networks, as well as in chemistry. We derive a characterization of Lucas cubes that is based on a peripheral expansion of a unique convex subgraph of an appropriate Fibonacci cube. This serves as the foundation for a recognition algorithm of Lucas cubes that runs in linear time.]]> Tue, 01 Oct 2013 22:00:00 +0000 https://doi.org/10.46298/dmtcs.617 https://doi.org/10.46298/dmtcs.617 Taranenko, Andrej Taranenko, Andrej <![CDATA[Fibonacci and Lucas cubes are induced subgraphs of hypercubes obtained by excluding certain binary strings from the vertex set. They appear as models for interconnection networks, as well as in chemistry. We derive a characterization of Lucas cubes that is based on a peripheral expansion of a unique convex subgraph of an appropriate Fibonacci cube. This serves as the foundation for a recognition algorithm of Lucas cubes that runs in linear time.]]> 0 <![CDATA[We conjecture that every signed graph of unbalanced girth 2g, whose underlying graph is bipartite and planar, admits a homomorphism to the signed projective cube of dimension 2g1. Our main result is to show that for a given g, this conjecture is equivalent to the corresponding case (k = 2g) of a conjecture of Seymour claiming that every planar k-regular multigraph with no odd edge-cut of less than k edges is k-edge-colorable. To this end, we exhibit several properties of signed projective cubes and establish a folding lemma for planar even signed graphs.]]> Wed, 25 Sep 2013 22:00:00 +0000 https://doi.org/10.46298/dmtcs.612 https://doi.org/10.46298/dmtcs.612 Naserasr, Reza Rollova, Edita Sopena, Eric Naserasr, Reza Rollova, Edita Sopena, Eric <![CDATA[We conjecture that every signed graph of unbalanced girth 2g, whose underlying graph is bipartite and planar, admits a homomorphism to the signed projective cube of dimension 2g1. Our main result is to show that for a given g, this conjecture is equivalent to the corresponding case (k = 2g) of a conjecture of Seymour claiming that every planar k-regular multigraph with no odd edge-cut of less than k edges is k-edge-colorable. To this end, we exhibit several properties of signed projective cubes and establish a folding lemma for planar even signed graphs.]]> 0 <![CDATA[Let P be a set of n points in general position in the plane. A subset I of P is called an island if there exists a convex set C such that I = P \C. In this paper we define the generalized island Johnson graph of P as the graph whose vertex consists of all islands of P of cardinality k, two of which are adjacent if their intersection consists of exactly l elements. We show that for large enough values of n, this graph is connected, and give upper and lower bounds on its diameter.]]> Wed, 25 Sep 2013 22:00:00 +0000 https://doi.org/10.46298/dmtcs.613 https://doi.org/10.46298/dmtcs.613 Bautista-Santiago, Crevel Cano, Javier Fabila-Monroy, Ruy Flores-Peñaloza, David González-Aguilar, Hernàn Lara, Dolores Sarmiento, Eliseo Urrutia, Jorge Bautista-Santiago, Crevel Cano, Javier Fabila-Monroy, Ruy Flores-Peñaloza, David González-Aguilar, Hernàn Lara, Dolores Sarmiento, Eliseo Urrutia, Jorge <![CDATA[Let P be a set of n points in general position in the plane. A subset I of P is called an island if there exists a convex set C such that I = P \C. In this paper we define the generalized island Johnson graph of P as the graph whose vertex consists of all islands of P of cardinality k, two of which are adjacent if their intersection consists of exactly l elements. We show that for large enough values of n, this graph is connected, and give upper and lower bounds on its diameter.]]> 0 <![CDATA[A graph is extended P4-laden if each of its induced subgraphs with at most six vertices that contains more than two induced P4's is 2K2,C4-free. A cycle transversal (or feedback vertex set) of a graph G is a subset T ⊆ V (G) such that T ∩ V (C) 6= ∅ for every cycle C of G; if, in addition, T is a clique, then T is a clique cycle transversal (cct). Finding a cct in a graph G is equivalent to partitioning V (G) into subsets C and F such that C induces a complete subgraph and F an acyclic subgraph. This work considers the problem of characterizing extended P4-laden graphs admitting a cct. We characterize such graphs by means of a finite family of forbidden induced subgraphs, and present a linear-time algorithm to recognize them.]]> Wed, 25 Sep 2013 22:00:00 +0000 https://doi.org/10.46298/dmtcs.616 https://doi.org/10.46298/dmtcs.616 Bravo, Raquel Klein, Sulamita Nogueira, Loana Tito Protti, Fábio Bravo, Raquel Klein, Sulamita Nogueira, Loana Tito Protti, Fábio <![CDATA[A graph is extended P4-laden if each of its induced subgraphs with at most six vertices that contains more than two induced P4's is 2K2,C4-free. A cycle transversal (or feedback vertex set) of a graph G is a subset T ⊆ V (G) such that T ∩ V (C) 6= ∅ for every cycle C of G; if, in addition, T is a clique, then T is a clique cycle transversal (cct). Finding a cct in a graph G is equivalent to partitioning V (G) into subsets C and F such that C induces a complete subgraph and F an acyclic subgraph. This work considers the problem of characterizing extended P4-laden graphs admitting a cct. We characterize such graphs by means of a finite family of forbidden induced subgraphs, and present a linear-time algorithm to recognize them.]]> 0 <![CDATA[A maximal independent set is an independent set that is not a proper subset of any other independent set. Liu [J.Q. Liu, Maximal independent sets of bipartite graphs, J. Graph Theory, 17 (4) (1993) 495-507] determined the largest number of maximal independent sets among all n-vertex bipartite graphs. The corresponding extremal graphs are forests. It is natural and interesting for us to consider this problem on bipartite graphs with cycles. Let \mathscrBₙ (resp. \mathscrBₙ') be the set of all n-vertex bipartite graphs with at least one cycle for even (resp. odd) n. In this paper, the largest number of maximal independent sets of graphs in \mathscrBₙ (resp. \mathscrBₙ') is considered. Among \mathscrBₙ the disconnected graphs with the first-, second-, \ldots, \fracn-22-th largest number of maximal independent sets are characterized, while the connected graphs in \mathscrBₙ having the largest, the second largest number of maximal independent sets are determined. Among \mathscrBₙ' graphs have the largest number of maximal independent sets are identified.]]> Sun, 08 Sep 2013 22:00:00 +0000 https://doi.org/10.46298/dmtcs.607 https://doi.org/10.46298/dmtcs.607 Li, Shuchao Zhang, Huihui Zhang, Xiaoyan Li, Shuchao Zhang, Huihui Zhang, Xiaoyan <![CDATA[A maximal independent set is an independent set that is not a proper subset of any other independent set. Liu [J.Q. Liu, Maximal independent sets of bipartite graphs, J. Graph Theory, 17 (4) (1993) 495-507] determined the largest number of maximal independent sets among all n-vertex bipartite graphs. The corresponding extremal graphs are forests. It is natural and interesting for us to consider this problem on bipartite graphs with cycles. Let \mathscrBₙ (resp. \mathscrBₙ') be the set of all n-vertex bipartite graphs with at least one cycle for even (resp. odd) n. In this paper, the largest number of maximal independent sets of graphs in \mathscrBₙ (resp. \mathscrBₙ') is considered. Among \mathscrBₙ the disconnected graphs with the first-, second-, \ldots, \fracn-22-th largest number of maximal independent sets are characterized, while the connected graphs in \mathscrBₙ having the largest, the second largest number of maximal independent sets are determined. Among \mathscrBₙ' graphs have the largest number of maximal independent sets are identified.]]> 0 <![CDATA[The classic Coupon-Collector Problem (CCP) is generalized. Only basic probability theory is used. Centerpiece rather is an algorithm that efficiently counts all k-element transversals of a set system.]]> Sun, 08 Sep 2013 22:00:00 +0000 https://doi.org/10.46298/dmtcs.608 https://doi.org/10.46298/dmtcs.608 Wild, Marcel Janson, Svante Wagner, Stephan Laurie, Dirk Wild, Marcel Janson, Svante Wagner, Stephan Laurie, Dirk <![CDATA[The classic Coupon-Collector Problem (CCP) is generalized. Only basic probability theory is used. Centerpiece rather is an algorithm that efficiently counts all k-element transversals of a set system.]]> 0 <![CDATA[Given a graph, finding the maximal matching of minimum size (MMM) and the induced matching of maximum size (MIM) have been very popular research topics during the last decades. In this paper, we give new complexity results, namely the NP-hardness of MMM and MIM in induced subgrids and we point out some promising research directions. We also sketch the general framework of a unified approach to show the NP-hardness of some problems in subgrids.]]> Tue, 03 Sep 2013 22:00:00 +0000 https://doi.org/10.46298/dmtcs.606 https://doi.org/10.46298/dmtcs.606 Demange, Marc Ekim, Tınaz Demange, Marc Ekim, Tınaz <![CDATA[Given a graph, finding the maximal matching of minimum size (MMM) and the induced matching of maximum size (MIM) have been very popular research topics during the last decades. In this paper, we give new complexity results, namely the NP-hardness of MMM and MIM in induced subgrids and we point out some promising research directions. We also sketch the general framework of a unified approach to show the NP-hardness of some problems in subgrids.]]> 0 <![CDATA[A \em cyclic q-partition of a hypergraph (V,E) is a partition of the edge set E of the form \F,F^θ,F^θ², \ldots, F^θ^q-1\ for some permutation θ of the vertex set V. Let Vₙ = \ 1,2,\ldots,n\. For a positive integer k, Vₙ\choose k denotes the set of all k-subsets of Vₙ. For a nonempty subset K of V_n-1, we let \mathcalKₙ^(K) denote the hypergraph ≤ft(Vₙ, \bigcup_k∈ K Vₙ\choose k\right). In this paper, we find a necessary and sufficient condition on n, q and k for the existence of a cyclic q-partition of \mathcalKₙ^(V_k). In particular, we prove that if p is prime then there is a cyclic p^α-partition of \mathcalK^(Vₖ)ₙ if and only if p^α + β divides n, where β = \lfloor \logₚ k\rfloor. As an application of this result, we obtain two sufficient conditions on n₁,n₂,\ldots,n_t, k, α and a prime p for the existence of a cyclic p^α-partition of the complete t-partite k-uniform hypergraph \mathcal K^(k)_n₁,n₂,\ldots,n_t.]]> Fri, 23 Aug 2013 22:00:00 +0000 https://doi.org/10.46298/dmtcs.604 https://doi.org/10.46298/dmtcs.604 Gosselin, Shonda Szymański, Andrzej Wojda, Adam Pawel Gosselin, Shonda Szymański, Andrzej Wojda, Adam Pawel <![CDATA[A \em cyclic q-partition of a hypergraph (V,E) is a partition of the edge set E of the form \F,F^θ,F^θ², \ldots, F^θ^q-1\ for some permutation θ of the vertex set V. Let Vₙ = \ 1,2,\ldots,n\. For a positive integer k, Vₙ\choose k denotes the set of all k-subsets of Vₙ. For a nonempty subset K of V_n-1, we let \mathcalKₙ^(K) denote the hypergraph ≤ft(Vₙ, \bigcup_k∈ K Vₙ\choose k\right). In this paper, we find a necessary and sufficient condition on n, q and k for the existence of a cyclic q-partition of \mathcalKₙ^(V_k). In particular, we prove that if p is prime then there is a cyclic p^α-partition of \mathcalK^(Vₖ)ₙ if and only if p^α + β divides n, where β = \lfloor \logₚ k\rfloor. As an application of this result, we obtain two sufficient conditions on n₁,n₂,\ldots,n_t, k, α and a prime p for the existence of a cyclic p^α-partition of the complete t-partite k-uniform hypergraph \mathcal K^(k)_n₁,n₂,\ldots,n_t.]]> 0 <![CDATA[A graph containment problem is to decide whether one graph called the host graph can be modified into some other graph called the target graph by using a number of specified graph operations. We consider edge deletions, edge contractions, vertex deletions and vertex dissolutions as possible graph operations permitted. By allowing any combination of these four operations we capture the following problems: testing on (induced) minors, (induced) topological minors, (induced) subgraphs, (induced) spanning subgraphs, dissolutions and contractions. We show that these problems stay NP-complete even when the host and target belong to the class of line graphs, which form a subclass of the class of claw-free graphs, i.e., graphs with no induced 4-vertex star. A natural question is to study the computational complexity of these problems if the target graph is assumed to be fixed. We show that these problems may become computationally easier when the host graphs are restricted to be claw-free. In particular we consider the problems that are to test whether a given host graph contains a fixed target graph as a contraction.]]> Fri, 23 Aug 2013 22:00:00 +0000 https://doi.org/10.46298/dmtcs.605 https://doi.org/10.46298/dmtcs.605 Fiala, Jiří Kamiński, Marcin Paulusma, Daniël Fiala, Jiří Kamiński, Marcin Paulusma, Daniël <![CDATA[A graph containment problem is to decide whether one graph called the host graph can be modified into some other graph called the target graph by using a number of specified graph operations. We consider edge deletions, edge contractions, vertex deletions and vertex dissolutions as possible graph operations permitted. By allowing any combination of these four operations we capture the following problems: testing on (induced) minors, (induced) topological minors, (induced) subgraphs, (induced) spanning subgraphs, dissolutions and contractions. We show that these problems stay NP-complete even when the host and target belong to the class of line graphs, which form a subclass of the class of claw-free graphs, i.e., graphs with no induced 4-vertex star. A natural question is to study the computational complexity of these problems if the target graph is assumed to be fixed. We show that these problems may become computationally easier when the host graphs are restricted to be claw-free. In particular we consider the problems that are to test whether a given host graph contains a fixed target graph as a contraction.]]> 0 <![CDATA[A vertex ranking of a graph G is an assignment of positive integers (colors) to the vertices of G such that each path connecting two vertices of the same color contains a vertex of a higher color. Our main goal is to find a vertex ranking using as few colors as possible. Considering on-line algorithms for vertex ranking of split graphs, we prove that the worst case ratio of the number of colors used by any on-line ranking algorithm and the number of colors used in an optimal off-line solution may be arbitrarily large. This negative result motivates us to investigate semi on-line algorithms, where a split graph is presented on-line but its clique number is given in advance. We prove that there does not exist a (2-ɛ)-competitive semi on-line algorithm of this type. Finally, a 2-competitive semi on-line algorithm is given.]]> Wed, 21 Aug 2013 22:00:00 +0000 https://doi.org/10.46298/dmtcs.603 https://doi.org/10.46298/dmtcs.603 Borowiecki, Piotr Dereniowski, Dariusz Borowiecki, Piotr Dereniowski, Dariusz <![CDATA[A vertex ranking of a graph G is an assignment of positive integers (colors) to the vertices of G such that each path connecting two vertices of the same color contains a vertex of a higher color. Our main goal is to find a vertex ranking using as few colors as possible. Considering on-line algorithms for vertex ranking of split graphs, we prove that the worst case ratio of the number of colors used by any on-line ranking algorithm and the number of colors used in an optimal off-line solution may be arbitrarily large. This negative result motivates us to investigate semi on-line algorithms, where a split graph is presented on-line but its clique number is given in advance. We prove that there does not exist a (2-ɛ)-competitive semi on-line algorithm of this type. Finally, a 2-competitive semi on-line algorithm is given.]]> 0 <![CDATA[A graph is probe (unit) interval if its vertices can be partitioned into two sets: a set of probe vertices and a set of nonprobe vertices, so that the set of nonprobe vertices is a stable set and it is possible to obtain a (unit) interval graph by adding edges with both endpoints in the set of nonprobe vertices. Probe (unit) interval graphs form a superclass of (unit) interval graphs. Probe interval graphs were introduced by Zhang for an application concerning the physical mapping of DNA in the human genome project. The main results of this article are minimal forbidden induced subgraphs characterizations of probe interval and probe unit interval graphs within two superclasses of cographs: P4-tidy graphs and tree-cographs. Furthermore, we introduce the concept of graphs class with a companion which allows to describe all the minimally non-(probe G) graphs with disconnected complement for every graph class G with a companion.]]> Tue, 20 Aug 2013 22:00:00 +0000 https://doi.org/10.46298/dmtcs.602 https://doi.org/10.46298/dmtcs.602 Bonomo, Flavia Durán, Guillermo Grippo, Luciano N. Safe, Martın D. Bonomo, Flavia Durán, Guillermo Grippo, Luciano N. Safe, Martın D. <![CDATA[A graph is probe (unit) interval if its vertices can be partitioned into two sets: a set of probe vertices and a set of nonprobe vertices, so that the set of nonprobe vertices is a stable set and it is possible to obtain a (unit) interval graph by adding edges with both endpoints in the set of nonprobe vertices. Probe (unit) interval graphs form a superclass of (unit) interval graphs. Probe interval graphs were introduced by Zhang for an application concerning the physical mapping of DNA in the human genome project. The main results of this article are minimal forbidden induced subgraphs characterizations of probe interval and probe unit interval graphs within two superclasses of cographs: P4-tidy graphs and tree-cographs. Furthermore, we introduce the concept of graphs class with a companion which allows to describe all the minimally non-(probe G) graphs with disconnected complement for every graph class G with a companion.]]> 0 <![CDATA[We discuss an Alexandroff topology on ℤ2 having the property that its quotient topologies include the Khalimsky and Marcus-Wyse topologies. We introduce a further quotient topology and prove a Jordan curve theorem for it.]]> Wed, 31 Jul 2013 22:00:00 +0000 https://doi.org/10.46298/dmtcs.601 https://doi.org/10.46298/dmtcs.601 Šlapal, Josef Šlapal, Josef <![CDATA[We discuss an Alexandroff topology on ℤ2 having the property that its quotient topologies include the Khalimsky and Marcus-Wyse topologies. We introduce a further quotient topology and prove a Jordan curve theorem for it.]]> 0 <![CDATA[For a brick apart from a few small graphs, Lovász (1987) proposed a conjecture on the existence of an edge whose deletion results in a graph with only one brick in its tight cut decomposition. Carvalho, Lucchesi, and Murty (2002) confirmed this conjecture by showing the existence of such two edges. This paper generalizes the result obtained by Carvalho et al. to the case of irreducible near-brick, where a graph is irreducible if it contains no induced odd path of length 3 or more. Meanwhile, a lower bound on the number of removable edges of matching-covered bipartite graphs is presented.]]> Sun, 28 Jul 2013 22:00:00 +0000 https://doi.org/10.46298/dmtcs.600 https://doi.org/10.46298/dmtcs.600 Wang, Xiumei He, Cheng Lin, Yixun Wang, Xiumei He, Cheng Lin, Yixun <![CDATA[For a brick apart from a few small graphs, Lovász (1987) proposed a conjecture on the existence of an edge whose deletion results in a graph with only one brick in its tight cut decomposition. Carvalho, Lucchesi, and Murty (2002) confirmed this conjecture by showing the existence of such two edges. This paper generalizes the result obtained by Carvalho et al. to the case of irreducible near-brick, where a graph is irreducible if it contains no induced odd path of length 3 or more. Meanwhile, a lower bound on the number of removable edges of matching-covered bipartite graphs is presented.]]> 0 <![CDATA[The paper presents a condition necessarily satisfied by (tiling system) recognizable two-dimensional languages. The new recognizability condition is compared with all the other ones known in the literature (namely three conditions), once they are put in a uniform setting: they are stated as bounds on the growth of some complexity functions defined for two-dimensional languages. The gaps between such functions are analyzed and examples are shown that asymptotically separate them. Finally the new recognizability condition results to be the strongest one, while the remaining ones are its particular cases. The problem of deciding whether a two-dimensional language is recognizable is here related to the one of estimating the minimal size of finite automata recognizing a sequence of (one-dimensional) string languages.]]> Thu, 11 Jul 2013 22:00:00 +0000 https://doi.org/10.46298/dmtcs.599 https://doi.org/10.46298/dmtcs.599 Anselmo, Marcella Madonia, Maria Anselmo, Marcella Madonia, Maria <![CDATA[The paper presents a condition necessarily satisfied by (tiling system) recognizable two-dimensional languages. The new recognizability condition is compared with all the other ones known in the literature (namely three conditions), once they are put in a uniform setting: they are stated as bounds on the growth of some complexity functions defined for two-dimensional languages. The gaps between such functions are analyzed and examples are shown that asymptotically separate them. Finally the new recognizability condition results to be the strongest one, while the remaining ones are its particular cases. The problem of deciding whether a two-dimensional language is recognizable is here related to the one of estimating the minimal size of finite automata recognizing a sequence of (one-dimensional) string languages.]]> 0 <![CDATA[Our aim is to construct a finite automaton recognizing the set of words that are at a bounded distance from some word of a given regular language. We define new regular operators, the similarity operators, based on a generalization of the notion of distance and we introduce the family of regular expressions extended to similarity operators, that we call AREs (Approximate Regular Expressions). We set formulae to compute the Brzozowski derivatives and the Antimirov derivatives of an ARE, which allows us to give a solution to the ARE membership problem and to provide the construction of two recognizers for the language denoted by an ARE. As far as we know, the family of approximative regular expressions is introduced for the first time in this paper. Classical approximate regular expression matching algorithms are approximate matching algorithms on regular expressions. Our approach is rather to process an exact matching on approximate regular expressions.]]> Thu, 11 Jul 2013 22:00:00 +0000 https://doi.org/10.46298/dmtcs.598 https://doi.org/10.46298/dmtcs.598 Champarnaud, Jean-Marc Jeanne, Hadrien Mignot, Ludovic Champarnaud, Jean-Marc Jeanne, Hadrien Mignot, Ludovic <![CDATA[Our aim is to construct a finite automaton recognizing the set of words that are at a bounded distance from some word of a given regular language. We define new regular operators, the similarity operators, based on a generalization of the notion of distance and we introduce the family of regular expressions extended to similarity operators, that we call AREs (Approximate Regular Expressions). We set formulae to compute the Brzozowski derivatives and the Antimirov derivatives of an ARE, which allows us to give a solution to the ARE membership problem and to provide the construction of two recognizers for the language denoted by an ARE. As far as we know, the family of approximative regular expressions is introduced for the first time in this paper. Classical approximate regular expression matching algorithms are approximate matching algorithms on regular expressions. Our approach is rather to process an exact matching on approximate regular expressions.]]> 0 <![CDATA[In this paper we consider the problem of deciding whether a given r-uniform hypergraph H with minimum vertex degree at least c\binom|V(H)|-1r-1, or minimum degree of a pair of vertices at least c\binom|V(H)|-2r-2, has a vertex 2-coloring. Motivated by an old result of Edwards for graphs, we obtain first optimal dichotomy results for 2-colorings of r-uniform hypergraphs. For each problem, for every r≥q 3 we determine a threshold value depending on r such that the problem is NP-complete for c below the threshold, while for c strictly above the threshold it is polynomial. We provide an algorithm constructing the coloring with time complexity O(n^\lfloor 4/ε\rfloor+2\log n) with some ε>0. This algorithm becomes more efficient in the case of r=3,4,5 due to known Turán numbers of the triangle and the Fano plane. In addition, we determine the computational complexity of strong k-coloring of 3-uniform hypergraphs H with minimum vertex degree at least c\binom|V(H)|-12, for some c, leaving a gap for k≥q 5 which vanishes as k→ ∞.]]> Thu, 11 Jul 2013 22:00:00 +0000 https://doi.org/10.46298/dmtcs.596 https://doi.org/10.46298/dmtcs.596 Szymańska, Edyta Szymańska, Edyta <![CDATA[In this paper we consider the problem of deciding whether a given r-uniform hypergraph H with minimum vertex degree at least c\binom|V(H)|-1r-1, or minimum degree of a pair of vertices at least c\binom|V(H)|-2r-2, has a vertex 2-coloring. Motivated by an old result of Edwards for graphs, we obtain first optimal dichotomy results for 2-colorings of r-uniform hypergraphs. For each problem, for every r≥q 3 we determine a threshold value depending on r such that the problem is NP-complete for c below the threshold, while for c strictly above the threshold it is polynomial. We provide an algorithm constructing the coloring with time complexity O(n^\lfloor 4/ε\rfloor+2\log n) with some ε>0. This algorithm becomes more efficient in the case of r=3,4,5 due to known Turán numbers of the triangle and the Fano plane. In addition, we determine the computational complexity of strong k-coloring of 3-uniform hypergraphs H with minimum vertex degree at least c\binom|V(H)|-12, for some c, leaving a gap for k≥q 5 which vanishes as k→ ∞.]]> 0 <![CDATA[We draw the r-dimensional butterfly network with 1 / 44r+O(r2r) crossings which improves the previous estimate given by Cimikowski (1996). We also give a lower bound which matches the upper bound obtained in this paper.]]> Fri, 17 May 2013 22:00:00 +0000 https://doi.org/10.46298/dmtcs.611 https://doi.org/10.46298/dmtcs.611 Manuel, Paul D. Rajan, Bharati Rajasingh, Indra Beulah, P. Vasanthi Manuel, Paul D. Rajan, Bharati Rajasingh, Indra Beulah, P. Vasanthi <![CDATA[We draw the r-dimensional butterfly network with 1 / 44r+O(r2r) crossings which improves the previous estimate given by Cimikowski (1996). We also give a lower bound which matches the upper bound obtained in this paper.]]> 0 <![CDATA[Let T be a d-dimensional toroidal grid of n^d points. For a given range parameter ω, and a positive integer k ≤q d, we say that two points in T are mutually visible if they differ in at most k coordinates and are a distance at most ω apart, where distance is measured using the \ellₚ norm. We obtain a random d-dimensional line-of-sight graph G by placing a node at each point in T independently with some fixed probability p^* and connecting all pairs of mutually visible nodes. We prove an asymptotically tight connectivity result for this random graph.]]> Tue, 14 May 2013 22:00:00 +0000 https://doi.org/10.46298/dmtcs.610 https://doi.org/10.46298/dmtcs.610 Devroye, Luc Farczadi, Linda Devroye, Luc Farczadi, Linda <![CDATA[Let T be a d-dimensional toroidal grid of n^d points. For a given range parameter ω, and a positive integer k ≤q d, we say that two points in T are mutually visible if they differ in at most k coordinates and are a distance at most ω apart, where distance is measured using the \ellₚ norm. We obtain a random d-dimensional line-of-sight graph G by placing a node at each point in T independently with some fixed probability p^* and connecting all pairs of mutually visible nodes. We prove an asymptotically tight connectivity result for this random graph.]]> 0 <![CDATA[Let k be an integer and k ≥3. A graph G is k-chordal if G does not have an induced cycle of length greater than k. From the definition it is clear that 3-chordal graphs are precisely the class of chordal graphs. Duchet proved that, for every positive integer m, if Gm is chordal then so is Gm+2. Brandstädt et al. in [Andreas Brandstädt, Van Bang Le, and Thomas Szymczak. Duchet-type theorems for powers of HHD-free graphs. Discrete Mathematics, 177(1-3):9-16, 1997.] showed that if Gm is k-chordal, then so is Gm+2. Powering a bipartite graph does not preserve its bipartitedness. In order to preserve the bipartitedness of a bipartite graph while powering Chandran et al. introduced the notion of bipartite powering. This notion was introduced to aid their study of boxicity of chordal bipartite graphs. The m-th bipartite power G[m] of a bipartite graph G is the bipartite graph obtained from G by adding edges (u,v) where dG(u,v) is odd and less than or equal to m. Note that G[m] = G[m+1] for each odd m. In this paper we show that, given a bipartite graph G, if G is k-chordal then so is G[m], where k, m are positive integers with k≥4.]]> Sun, 05 May 2013 22:00:00 +0000 https://doi.org/10.46298/dmtcs.609 https://doi.org/10.46298/dmtcs.609 Chandran, Sunil, Mathew, Rogers Chandran, Sunil, Mathew, Rogers <![CDATA[Let k be an integer and k ≥3. A graph G is k-chordal if G does not have an induced cycle of length greater than k. From the definition it is clear that 3-chordal graphs are precisely the class of chordal graphs. Duchet proved that, for every positive integer m, if Gm is chordal then so is Gm+2. Brandstädt et al. in [Andreas Brandstädt, Van Bang Le, and Thomas Szymczak. Duchet-type theorems for powers of HHD-free graphs. Discrete Mathematics, 177(1-3):9-16, 1997.] showed that if Gm is k-chordal, then so is Gm+2. Powering a bipartite graph does not preserve its bipartitedness. In order to preserve the bipartitedness of a bipartite graph while powering Chandran et al. introduced the notion of bipartite powering. This notion was introduced to aid their study of boxicity of chordal bipartite graphs. The m-th bipartite power G[m] of a bipartite graph G is the bipartite graph obtained from G by adding edges (u,v) where dG(u,v) is odd and less than or equal to m. Note that G[m] = G[m+1] for each odd m. In this paper we show that, given a bipartite graph G, if G is k-chordal then so is G[m], where k, m are positive integers with k≥4.]]> 0 <![CDATA[We consider two repeat finding problems relative to sets of strings: (a) Find the largest substrings that occur in every string of a given set; (b) Find the maximal repeats in a given string that occur in no string of a given set. Our solutions are based on the suffix array construction, requiring O(m) memory, where m is the length of the longest input string, and O(n &log;m) time, where n is the the whole input size (the sum of the length of each string in the input). The most expensive part of our algorithms is the computation of several suffix arrays. We give an implementation and experimental results that evidence the efficiency of our algorithms in practice, even for very large inputs.]]> Sun, 05 May 2013 22:00:00 +0000 https://doi.org/10.46298/dmtcs.597 https://doi.org/10.46298/dmtcs.597 Barenbaum, Pablo Becher, Verónica Deymonnaz, Alejandro Halsband, Melisa Heiber, Pablo Ariel Barenbaum, Pablo Becher, Verónica Deymonnaz, Alejandro Halsband, Melisa Heiber, Pablo Ariel <![CDATA[We consider two repeat finding problems relative to sets of strings: (a) Find the largest substrings that occur in every string of a given set; (b) Find the maximal repeats in a given string that occur in no string of a given set. Our solutions are based on the suffix array construction, requiring O(m) memory, where m is the length of the longest input string, and O(n &log;m) time, where n is the the whole input size (the sum of the length of each string in the input). The most expensive part of our algorithms is the computation of several suffix arrays. We give an implementation and experimental results that evidence the efficiency of our algorithms in practice, even for very large inputs.]]> 0 <![CDATA[We consider the family of rational functions ψw= ∏( xwi - xwi+1 )-1 indexed by words with no repetition. We study the combinatorics of the sums ΨP of the functions ψw when w describes the linear extensions of a given poset P. In particular, we point out the connexions between some transformations on posets and elementary operations on the fraction ΨP. We prove that the denominator of ΨP has a closed expression in terms of the Hasse diagram of P, and we compute its numerator in some special cases. We show that the computation of ΨP can be reduced to the case of bipartite posets. Finally, we compute the numerators associated to some special bipartite graphs as Schubert polynomials.]]> Sat, 06 Apr 2013 22:00:00 +0000 https://doi.org/10.46298/dmtcs.595 https://doi.org/10.46298/dmtcs.595 Boussicault, Adrien Boussicault, Adrien <![CDATA[We consider the family of rational functions ψw= ∏( xwi - xwi+1 )-1 indexed by words with no repetition. We study the combinatorics of the sums ΨP of the functions ψw when w describes the linear extensions of a given poset P. In particular, we point out the connexions between some transformations on posets and elementary operations on the fraction ΨP. We prove that the denominator of ΨP has a closed expression in terms of the Hasse diagram of P, and we compute its numerator in some special cases. We show that the computation of ΨP can be reduced to the case of bipartite posets. Finally, we compute the numerators associated to some special bipartite graphs as Schubert polynomials.]]> 0 <![CDATA[We suggest an approach for the enumeration of minimal permutations having d descents which uses skew Young tableaux. We succeed in finding a general expression for the number of such permutations in terms of (several) sums of determinants. We then generalize the class of skew Young tableaux under consideration; this allows in particular to discover some presumably new results concerning Eulerian numbers.]]> Sat, 06 Apr 2013 22:00:00 +0000 https://doi.org/10.46298/dmtcs.636 https://doi.org/10.46298/dmtcs.636 Bouvel, Mathilde Ferrari, Luca Bouvel, Mathilde Ferrari, Luca <![CDATA[We suggest an approach for the enumeration of minimal permutations having d descents which uses skew Young tableaux. We succeed in finding a general expression for the number of such permutations in terms of (several) sums of determinants. We then generalize the class of skew Young tableaux under consideration; this allows in particular to discover some presumably new results concerning Eulerian numbers.]]> 0 <![CDATA[In this paper, we consider random words ω1ω2ω3⋯ωn of length n, where the letters ωi ∈ℕ are independently generated with a geometric probability such that Pωi=k=pqk-1 where p+q=1 . We have a descent at position i whenever ωi+1 < ωi. The size of such a descent is ωi-ωi+1 and the descent variation is the sum of all the descent sizes for that word. We study various types of random words over the infinite alphabet ℕ, where the letters have geometric probabilities, and find the probability generating functions for descent variation of such words.]]> Thu, 04 Apr 2013 22:00:00 +0000 https://doi.org/10.46298/dmtcs.594 https://doi.org/10.46298/dmtcs.594 Brennan, Charlotte Knopfmacher, Arnold Brennan, Charlotte Knopfmacher, Arnold <![CDATA[In this paper, we consider random words ω1ω2ω3⋯ωn of length n, where the letters ωi ∈ℕ are independently generated with a geometric probability such that Pωi=k=pqk-1 where p+q=1 . We have a descent at position i whenever ωi+1 < ωi. The size of such a descent is ωi-ωi+1 and the descent variation is the sum of all the descent sizes for that word. We study various types of random words over the infinite alphabet ℕ, where the letters have geometric probabilities, and find the probability generating functions for descent variation of such words.]]> 0 <![CDATA[Following the track of polyominoes, in particular the column-by-column construction of Temperley and its interpretation in terms of functional equations due to Bousquet-Mélou, we introduce a generic method for the enumeration of classes of directed polycubes the strata of which satisfy some property P. This method is applied to the enumeration of two new families of polycubes, the s-directed polycubes and the vertically-convex s-directed polycubes, with respect to width and volume. The case of non-directed polycubes is also studied and it is shown that the generic method can be applied in this case too. Finally the general case of d-dimensional polycubes, with d≥4, is investigated, and the generic method is extended in order to handle the enumeration of classes of directed d-polycubes.]]> Thu, 28 Mar 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.637 https://doi.org/10.46298/dmtcs.637 Champarnaud, Jean-Marc Dubernard, Jean-Philippe Jeanne, Hadrien Champarnaud, Jean-Marc Dubernard, Jean-Philippe Jeanne, Hadrien <![CDATA[Following the track of polyominoes, in particular the column-by-column construction of Temperley and its interpretation in terms of functional equations due to Bousquet-Mélou, we introduce a generic method for the enumeration of classes of directed polycubes the strata of which satisfy some property P. This method is applied to the enumeration of two new families of polycubes, the s-directed polycubes and the vertically-convex s-directed polycubes, with respect to width and volume. The case of non-directed polycubes is also studied and it is shown that the generic method can be applied in this case too. Finally the general case of d-dimensional polycubes, with d≥4, is investigated, and the generic method is extended in order to handle the enumeration of classes of directed d-polycubes.]]> 0 <![CDATA[A b-coloring of a graph G by k colors is a proper vertex coloring such that each color class contains a color-dominating vertex, that is, a vertex having neighbors in all other k-1 color classes. The b-chromatic number χb(G) is the maximum integer k for which G has a b-coloring by k colors. Let Cnr be the rth power of a cycle of order n. In 2003, Effantin and Kheddouci established the b-chromatic number χb(Cnr) for all values of n and r, except for 2r+3≤n≤3r. For the missing cases they presented the lower bound L:= min n-r-1,r+1+⌊ n-r-1 / 3⌋ and conjectured that χb(Cnr)=L. In this paper, we determine the exact value on χb(Cnr) for the missing cases. It turns out that χb(Cnr)>L for 2r+3≤n≤2r+3+r-6 / 4.]]> Sat, 23 Mar 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.631 https://doi.org/10.46298/dmtcs.631 Kohl, Anja Kohl, Anja <![CDATA[A b-coloring of a graph G by k colors is a proper vertex coloring such that each color class contains a color-dominating vertex, that is, a vertex having neighbors in all other k-1 color classes. The b-chromatic number χb(G) is the maximum integer k for which G has a b-coloring by k colors. Let Cnr be the rth power of a cycle of order n. In 2003, Effantin and Kheddouci established the b-chromatic number χb(Cnr) for all values of n and r, except for 2r+3≤n≤3r. For the missing cases they presented the lower bound L:= min n-r-1,r+1+⌊ n-r-1 / 3⌋ and conjectured that χb(Cnr)=L. In this paper, we determine the exact value on χb(Cnr) for the missing cases. It turns out that χb(Cnr)>L for 2r+3≤n≤2r+3+r-6 / 4.]]> 0 <![CDATA[We give a linear-time algorithm that checks for isomorphism between two 0-1 matrices that obey the circular-ones property. Our algorithm is similar to the isomorphism algorithm for interval graphs of Lueker and Booth, but works on PC trees, which are unrooted and have a cyclic nature, rather than with PQ trees, which are rooted. This algorithm leads to linear-time isomorphism algorithms for related graph classes, including Helly circular-arc graphs, Γ circular-arc graphs, proper circular-arc graphs and convex-round graphs.]]> Sat, 23 Mar 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.625 https://doi.org/10.46298/dmtcs.625 Curtis, Andrew R. Lin, Min Chih Mcconnell, Ross M. Nussbaum, Yahav Soulignac, Francisco Juan Spinrad, Jeremy P. Szwarcfiter, Jayme Luiz Curtis, Andrew R. Lin, Min Chih Mcconnell, Ross M. Nussbaum, Yahav Soulignac, Francisco Juan Spinrad, Jeremy P. Szwarcfiter, Jayme Luiz <![CDATA[We give a linear-time algorithm that checks for isomorphism between two 0-1 matrices that obey the circular-ones property. Our algorithm is similar to the isomorphism algorithm for interval graphs of Lueker and Booth, but works on PC trees, which are unrooted and have a cyclic nature, rather than with PQ trees, which are rooted. This algorithm leads to linear-time isomorphism algorithms for related graph classes, including Helly circular-arc graphs, Γ circular-arc graphs, proper circular-arc graphs and convex-round graphs.]]> 0 <![CDATA[In this paper we describe all edge-colored graphs that are fully symmetric with respect to colors and transitive on every set of edges of the same color. They correspond to fully symmetric homogeneous factorizations of complete graphs. Our description completes the work done in our previous paper, where we have shown, in particular, that there are no such graphs with more than 5 colors. Using some recent results, with a help of computer, we settle all the cases that was left open in the previous paper.]]> Thu, 21 Mar 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.630 https://doi.org/10.46298/dmtcs.630 Grech, Mariusz Kisielewicz, Andrzej Grech, Mariusz Kisielewicz, Andrzej <![CDATA[In this paper we describe all edge-colored graphs that are fully symmetric with respect to colors and transitive on every set of edges of the same color. They correspond to fully symmetric homogeneous factorizations of complete graphs. Our description completes the work done in our previous paper, where we have shown, in particular, that there are no such graphs with more than 5 colors. Using some recent results, with a help of computer, we settle all the cases that was left open in the previous paper.]]> 0 <![CDATA[We introduce a variation of chip-firing games on connected graphs. These 'burn-off' games incorporate the loss of energy that may occur in the physical processes that classical chip-firing games have been used to model. For a graph G=(V,E), a configuration of 'chips' on its nodes is a mapping C:V→ℕ. We study the configurations that can arise in the course of iterating a burn-off game. After characterizing the 'relaxed legal' configurations for general graphs, we enumerate the 'legal' ones for complete graphs Kn. The number of relaxed legal configurations on Kn coincides with the number tn+1 of spanning trees of Kn+1. Since our algorithmic, bijective proof of this fact does not invoke Cayley's Formula for tn, our main results yield secondarily a new proof of this formula.]]> Sun, 17 Mar 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.624 https://doi.org/10.46298/dmtcs.624 Kayll, Peter Mark Perkins, Dave Kayll, Peter Mark Perkins, Dave <![CDATA[We introduce a variation of chip-firing games on connected graphs. These 'burn-off' games incorporate the loss of energy that may occur in the physical processes that classical chip-firing games have been used to model. For a graph G=(V,E), a configuration of 'chips' on its nodes is a mapping C:V→ℕ. We study the configurations that can arise in the course of iterating a burn-off game. After characterizing the 'relaxed legal' configurations for general graphs, we enumerate the 'legal' ones for complete graphs Kn. The number of relaxed legal configurations on Kn coincides with the number tn+1 of spanning trees of Kn+1. Since our algorithmic, bijective proof of this fact does not invoke Cayley's Formula for tn, our main results yield secondarily a new proof of this formula.]]> 0 <![CDATA[A Krausz (k,m)-partition of a graph G is a decomposition of G into cliques, such that any vertex belongs to at most k cliques and any two cliques have at most m vertices in common. The m-Krausz dimension kdimm(G) of the graph G is the minimum number k such that G has a Krausz (k,m)-partition. In particular, 1-Krausz dimension or simply Krausz dimension kdim(G) is a well-known graph-theoretical parameter. In this paper we prove that the problem "kdim(G)≤3" is polynomially solvable for chordal graphs, thus partially solving the open problem of P. Hlineny and J. Kratochvil. We solve another open problem of P. Hlineny and J. Kratochvil by proving that the problem of finding Krausz dimension is NP-hard for split graphs and complements of bipartite graphs. We show that the problem of finding m-Krausz dimension is NP-hard for every m≥1, but the problem "kdimm(G)≤k" is is fixed-parameter tractable when parameterized by k and m for (∞,1)-polar graphs. Moreover, the class of (∞,1)-polar graphs with kdimm(G)≤k is characterized by a finite list of forbidden induced subgraphs for every k,m≥1.]]> Fri, 15 Mar 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.623 https://doi.org/10.46298/dmtcs.623 Glebova, Olga Metelsky, Yury Skums, Pavel Glebova, Olga Metelsky, Yury Skums, Pavel <![CDATA[A Krausz (k,m)-partition of a graph G is a decomposition of G into cliques, such that any vertex belongs to at most k cliques and any two cliques have at most m vertices in common. The m-Krausz dimension kdimm(G) of the graph G is the minimum number k such that G has a Krausz (k,m)-partition. In particular, 1-Krausz dimension or simply Krausz dimension kdim(G) is a well-known graph-theoretical parameter. In this paper we prove that the problem "kdim(G)≤3" is polynomially solvable for chordal graphs, thus partially solving the open problem of P. Hlineny and J. Kratochvil. We solve another open problem of P. Hlineny and J. Kratochvil by proving that the problem of finding Krausz dimension is NP-hard for split graphs and complements of bipartite graphs. We show that the problem of finding m-Krausz dimension is NP-hard for every m≥1, but the problem "kdimm(G)≤k" is is fixed-parameter tractable when parameterized by k and m for (∞,1)-polar graphs. Moreover, the class of (∞,1)-polar graphs with kdimm(G)≤k is characterized by a finite list of forbidden induced subgraphs for every k,m≥1.]]> 0 <![CDATA[Giving a planar graph G, let χ'l(G) and χ''l(G) denote the list edge chromatic number and list total chromatic number of G respectively. It is proved that if G is a planar graph without non-induced 7-cycles, then χ'l(G)≤Δ(G)+1 and χ''l(G)≤Δ(G)+2 where Δ(G)≥7.]]> Tue, 12 Mar 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.629 https://doi.org/10.46298/dmtcs.629 Dong, Aijun Liu, Guizhen Li, Guojun Dong, Aijun Liu, Guizhen Li, Guojun <![CDATA[Giving a planar graph G, let χ'l(G) and χ''l(G) denote the list edge chromatic number and list total chromatic number of G respectively. It is proved that if G is a planar graph without non-induced 7-cycles, then χ'l(G)≤Δ(G)+1 and χ''l(G)≤Δ(G)+2 where Δ(G)≥7.]]> 0 <![CDATA[Let f(n,k) be the minimum number of edges that must be removed from some complete geometric graph G on n points, so that there exists a tree on k vertices that is no longer a planar subgraph of G. In this paper we show that ( 1 / 2 )n2 / k-1-n / 2≤f(n,k) ≤2 n(n-2) / k-2. For the case when k=n, we show that 2 ≤f(n,n) ≤3. For the case when k=n and G is a geometric graph on a set of points in convex position, we completely solve the problem and prove that at least three edges must be removed.]]> Wed, 27 Feb 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.628 https://doi.org/10.46298/dmtcs.628 Barba, Luis Fabila-Monroy, Ruy Lara, Dolores Leaños, Jesús Rodrıguez, Cynthia Salazar, Gelasio Zaragoza, Francisco Barba, Luis Fabila-Monroy, Ruy Lara, Dolores Leaños, Jesús Rodrıguez, Cynthia Salazar, Gelasio Zaragoza, Francisco <![CDATA[Let f(n,k) be the minimum number of edges that must be removed from some complete geometric graph G on n points, so that there exists a tree on k vertices that is no longer a planar subgraph of G. In this paper we show that ( 1 / 2 )n2 / k-1-n / 2≤f(n,k) ≤2 n(n-2) / k-2. For the case when k=n, we show that 2 ≤f(n,n) ≤3. For the case when k=n and G is a geometric graph on a set of points in convex position, we completely solve the problem and prove that at least three edges must be removed.]]> 0 <![CDATA[Let g(n) denote the minimum number of edges of a maximal nontraceable (MNT) graph of order n. In 2005 Frick and Singleton (Lower bound for the size of maximal nontraceable graphs, Electronic Journal of Combinatorics, 12(1) R32, 2005) proved that g(n) = ⌈3n-22 ⌉ for n ≥54 as well as for n ∈I, where I= 12,13,22,23,30,31,38,39, 40,41,42,43,46,47,48,49,50,51 and they determined g(n) for n ≤9. We determine g(n) for 18 of the remaining 26 values of n, showing that g(n) = ⌈ 3n-22 ⌉ for n ≥54 as well as for n ∈I ∪18,19,20,21,24,25,26,27,28, 29,32,33 and g(n) = ⌈ 3n2 ⌉ for n ∈ 10, 11, 14, 15, 16, 17. We give results based on ''analytic'' proofs as well as computer searches.]]> Wed, 27 Feb 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.627 https://doi.org/10.46298/dmtcs.627 Burger, Alewyn Petrus Singleton, Joy Elizabeth Burger, Alewyn Petrus Singleton, Joy Elizabeth <![CDATA[Let g(n) denote the minimum number of edges of a maximal nontraceable (MNT) graph of order n. In 2005 Frick and Singleton (Lower bound for the size of maximal nontraceable graphs, Electronic Journal of Combinatorics, 12(1) R32, 2005) proved that g(n) = ⌈3n-22 ⌉ for n ≥54 as well as for n ∈I, where I= 12,13,22,23,30,31,38,39, 40,41,42,43,46,47,48,49,50,51 and they determined g(n) for n ≤9. We determine g(n) for 18 of the remaining 26 values of n, showing that g(n) = ⌈ 3n-22 ⌉ for n ≥54 as well as for n ∈I ∪18,19,20,21,24,25,26,27,28, 29,32,33 and g(n) = ⌈ 3n2 ⌉ for n ∈ 10, 11, 14, 15, 16, 17. We give results based on ''analytic'' proofs as well as computer searches.]]> 0 <![CDATA[A biclique is a set of vertices that induce a complete bipartite graph. A graph G is biclique-Helly when its family of maximal bicliques satisfies the Helly property. If every induced subgraph of G is also biclique-Helly, then G is hereditary biclique-Helly. A graph is C4-dominated when every cycle of length 4 contains a vertex that is dominated by the vertex of the cycle that is not adjacent to it. In this paper we show that the class of hereditary biclique-Helly graphs is formed precisely by those C4-dominated graphs that contain no triangles and no induced cycles of length either 5 or 6. Using this characterization, we develop an algorithm for recognizing hereditary biclique-Helly graphs in O(n2+αm) time and O(n+m) space. (Here n, m, and α= O(m1/2) are the number of vertices and edges, and the arboricity of the graph, respectively.) As a subprocedure, we show how to recognize those C4-dominated graphs that contain no triangles in O(αm) time and O(n+m) space. Finally, we show how to enumerate all the maximal bicliques of a C4-dominated graph with no triangles in O(n2 + αm) time and O(αm) space.]]> Sat, 09 Feb 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.626 https://doi.org/10.46298/dmtcs.626 Eguia, Martiniano Soulignac, Francisco Juan Eguia, Martiniano Soulignac, Francisco Juan <![CDATA[A biclique is a set of vertices that induce a complete bipartite graph. A graph G is biclique-Helly when its family of maximal bicliques satisfies the Helly property. If every induced subgraph of G is also biclique-Helly, then G is hereditary biclique-Helly. A graph is C4-dominated when every cycle of length 4 contains a vertex that is dominated by the vertex of the cycle that is not adjacent to it. In this paper we show that the class of hereditary biclique-Helly graphs is formed precisely by those C4-dominated graphs that contain no triangles and no induced cycles of length either 5 or 6. Using this characterization, we develop an algorithm for recognizing hereditary biclique-Helly graphs in O(n2+αm) time and O(n+m) space. (Here n, m, and α= O(m1/2) are the number of vertices and edges, and the arboricity of the graph, respectively.) As a subprocedure, we show how to recognize those C4-dominated graphs that contain no triangles in O(αm) time and O(n+m) space. Finally, we show how to enumerate all the maximal bicliques of a C4-dominated graph with no triangles in O(n2 + αm) time and O(αm) space.]]> 0 <![CDATA[If L is a language, the automaticity function A_L(n) (resp. N_L(n)) of L counts the number of states of a smallest deterministic (resp. non-deterministic) finite automaton that accepts a language that agrees with L on all inputs of length at most n. We provide bounds for the automaticity of the language of primitive words and the language of unbordered words over a k-letter alphabet. We also give a bound for the automaticity of the language of base-b representations of the irreducible polynomials over a finite field. This latter result is analogous to a result of Shallit concerning the base-k representations of the set of prime numbers.]]> Sun, 27 Jan 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.632 https://doi.org/10.46298/dmtcs.632 Lacroix, Anne Rampersad, Narad Lacroix, Anne Rampersad, Narad <![CDATA[If L is a language, the automaticity function A_L(n) (resp. N_L(n)) of L counts the number of states of a smallest deterministic (resp. non-deterministic) finite automaton that accepts a language that agrees with L on all inputs of length at most n. We provide bounds for the automaticity of the language of primitive words and the language of unbordered words over a k-letter alphabet. We also give a bound for the automaticity of the language of base-b representations of the irreducible polynomials over a finite field. This latter result is analogous to a result of Shallit concerning the base-k representations of the set of prime numbers.]]> 0 <![CDATA[The famous conjecture of Lovász and Plummer, very recently proven by Esperet et al. (2011), asserts that every cubic bridgeless graph has exponentially many perfect matchings. In this paper we improve the bound of Esperet et al. for a specific subclass of cubic bridgeless graphs called the Klee-graphs. We show that every Klee-graph with n ≥8 vertices has at least 3 *2(n+12)/60 perfect matchings.]]> Sun, 27 Jan 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.633 https://doi.org/10.46298/dmtcs.633 Cygan, Marek Pilipczuk, Marcin Škrekovski, Riste Cygan, Marek Pilipczuk, Marcin Škrekovski, Riste <![CDATA[The famous conjecture of Lovász and Plummer, very recently proven by Esperet et al. (2011), asserts that every cubic bridgeless graph has exponentially many perfect matchings. In this paper we improve the bound of Esperet et al. for a specific subclass of cubic bridgeless graphs called the Klee-graphs. We show that every Klee-graph with n ≥8 vertices has at least 3 *2(n+12)/60 perfect matchings.]]> 0 <![CDATA[Karonski, Luczak, and Thomason (2004) conjecture that, for any connected graph G on at least three vertices, there exists an edge weighting from 1, 2, 3 such that adjacent vertices receive different sums of incident edge weights. Bartnicki, Grytczuk, and Niwcyk (2009) make a stronger conjecture, that each edge's weight may be chosen from an arbitrary list of size 3 rather than 1, 2, 3. We examine a variation of these conjectures, where each vertex is coloured with a sequence of edge weights. Such a colouring relies on an ordering of E(G), and so two variations arise - one where we may choose any ordering of E(G) and one where the ordering is fixed. In the former case, we bound the list size required for any graph. In the latter, we obtain a bound on list sizes for graphs with sufficiently large minimum degree. We also extend our methods to a list variation of irregularity strength, where each vertex receives a distinct sequence of edge weights.]]> Sun, 27 Jan 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.635 https://doi.org/10.46298/dmtcs.635 Seamone, Ben Stevens, Brett Seamone, Ben Stevens, Brett <![CDATA[Karonski, Luczak, and Thomason (2004) conjecture that, for any connected graph G on at least three vertices, there exists an edge weighting from 1, 2, 3 such that adjacent vertices receive different sums of incident edge weights. Bartnicki, Grytczuk, and Niwcyk (2009) make a stronger conjecture, that each edge's weight may be chosen from an arbitrary list of size 3 rather than 1, 2, 3. We examine a variation of these conjectures, where each vertex is coloured with a sequence of edge weights. Such a colouring relies on an ordering of E(G), and so two variations arise - one where we may choose any ordering of E(G) and one where the ordering is fixed. In the former case, we bound the list size required for any graph. In the latter, we obtain a bound on list sizes for graphs with sufficiently large minimum degree. We also extend our methods to a list variation of irregularity strength, where each vertex receives a distinct sequence of edge weights.]]> 0 <![CDATA[A set of vertices S is a determining set of a graph G if every automorphism of G is uniquely determined by its action on S. The determining number of G is the minimum cardinality of a determining set of G. This paper studies the determining number of Kneser graphs. First, we compute the determining number of a wide range of Kneser graphs, concretely Kn:k with n≥k(k+1) / 2+1. In the language of group theory, these computations provide exact values for the base size of the symmetric group Sn acting on the k-subsets of 1,..., n. Then, we establish for which Kneser graphs Kn:k the determining number is equal to n-k, answering a question posed by Boutin. Finally, we find all Kneser graphs with fixed determining number 5, extending the study developed by Boutin for determining number 2, 3 or 4.]]> Sun, 27 Jan 2013 23:00:00 +0000 https://doi.org/10.46298/dmtcs.634 https://doi.org/10.46298/dmtcs.634 Cáceres, José Garijo, Delia González, Antonio Márquez, Alberto Puertas, Marıa Luz Cáceres, José Garijo, Delia González, Antonio Márquez, Alberto Puertas, Marıa Luz <![CDATA[A set of vertices S is a determining set of a graph G if every automorphism of G is uniquely determined by its action on S. The determining number of G is the minimum cardinality of a determining set of G. This paper studies the determining number of Kneser graphs. First, we compute the determining number of a wide range of Kneser graphs, concretely Kn:k with n≥k(k+1) / 2+1. In the language of group theory, these computations provide exact values for the base size of the symmetric group Sn acting on the k-subsets of 1,..., n. Then, we establish for which Kneser graphs Kn:k the determining number is equal to n-k, answering a question posed by Boutin. Finally, we find all Kneser graphs with fixed determining number 5, extending the study developed by Boutin for determining number 2, 3 or 4.]]> 0 <![CDATA[We consider the set $\mathcal{L}_n<$ of n-letters long Lyndon words on the alphabet $\mathcal{A}=\{0,1\}$. For a random uniform element ${L_n}$ of the set $\mathcal{L}_n$, the binary tree $\mathfrak{L} (L_n)$ obtained by successive standard factorization of $L_n$ and of the factors produced by these factorization is the $\textit{Lyndon tree}$ of $L_n$. We prove that the height $H_n$ of $\mathfrak{L} (L_n)$ satisfies $\lim \limits_n \frac{H_n}{\mathsf{ln}n}=\Delta$, in which the constant $\Delta$ is solution of an equation involving large deviation rate functions related to the asymptotics of Eulerian numbers ($\Delta ≃5.092\dots $). The convergence is the convergence in probability of random variables.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2357 https://doi.org/10.46298/dmtcs.2357 Mercier, Lucas Chassaing, Philippe Mercier, Lucas Chassaing, Philippe <![CDATA[We consider the set $\mathcal{L}_n<$ of n-letters long Lyndon words on the alphabet $\mathcal{A}=\{0,1\}$. For a random uniform element ${L_n}$ of the set $\mathcal{L}_n$, the binary tree $\mathfrak{L} (L_n)$ obtained by successive standard factorization of $L_n$ and of the factors produced by these factorization is the $\textit{Lyndon tree}$ of $L_n$. We prove that the height $H_n$ of $\mathfrak{L} (L_n)$ satisfies $\lim \limits_n \frac{H_n}{\mathsf{ln}n}=\Delta$, in which the constant $\Delta$ is solution of an equation involving large deviation rate functions related to the asymptotics of Eulerian numbers ($\Delta ≃5.092\dots $). The convergence is the convergence in probability of random variables.]]> 0 <![CDATA[We develop a method for counting words subject to various restrictions by finding a combinatorial interpretation for a product of formal sums of Laguerre polynomials. We use this method to find the generating function for $k$-ary words avoiding any vincular pattern that has only ones. We also give generating functions for $k$-ary words cyclically avoiding vincular patterns with only ones whose runs of ones between dashes are all of equal length, as well as the analogous results for compositions.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2369 https://doi.org/10.46298/dmtcs.2369 Taylor, Jair Taylor, Jair <![CDATA[We develop a method for counting words subject to various restrictions by finding a combinatorial interpretation for a product of formal sums of Laguerre polynomials. We use this method to find the generating function for $k$-ary words avoiding any vincular pattern that has only ones. We also give generating functions for $k$-ary words cyclically avoiding vincular patterns with only ones whose runs of ones between dashes are all of equal length, as well as the analogous results for compositions.]]> 0 <![CDATA[For irreducible characters $\{ \chi_q^{\lambda} | \lambda \vdash n\}$ and induced sign characters $\{\epsilon_q^{\lambda} | \lambda \vdash n\}$ of the Hecke algebra $H_n(q)$, and Kazhdan-Lusztig basis elements $C'_w(q)$ with $w$ avoiding the pattern 312, we combinatorially interpret the polynomials $\chi_q^{\lambda}(q^{\frac{\ell(w)}{2}} C'_w(q))$ and $\epsilon_q^{\lambda}(q^{\frac{\ell(w)}{2}} C'_w(q))$. This gives a new algebraic interpretation of $q$-chromatic symmetric functions of Shareshian and Wachs. We conjecture similar interpretations and generating functions corresponding to other $H_n(q)$-traces.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2368 https://doi.org/10.46298/dmtcs.2368 Clearman, Sam Hyatt, Matthew Shelton, Brittany Skandera, Mark Clearman, Sam Hyatt, Matthew Shelton, Brittany Skandera, Mark <![CDATA[For irreducible characters $\{ \chi_q^{\lambda} | \lambda \vdash n\}$ and induced sign characters $\{\epsilon_q^{\lambda} | \lambda \vdash n\}$ of the Hecke algebra $H_n(q)$, and Kazhdan-Lusztig basis elements $C'_w(q)$ with $w$ avoiding the pattern 312, we combinatorially interpret the polynomials $\chi_q^{\lambda}(q^{\frac{\ell(w)}{2}} C'_w(q))$ and $\epsilon_q^{\lambda}(q^{\frac{\ell(w)}{2}} C'_w(q))$. This gives a new algebraic interpretation of $q$-chromatic symmetric functions of Shareshian and Wachs. We conjecture similar interpretations and generating functions corresponding to other $H_n(q)$-traces.]]> 0 <![CDATA[This paper is devoted to the computation of the number of ordered factorizations of a long cycle in the symmetric group where the number of factors is arbitrary and the cycle structure of the factors is given. Jackson (1988) derived the first closed form expression for the generating series of these numbers using the theory of the irreducible characters of the symmetric group. Thanks to a direct bijection we compute a similar formula and provide the first purely combinatorial evaluation of these generating series.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2367 https://doi.org/10.46298/dmtcs.2367 Vassilieva, Ekaterina A. Vassilieva, Ekaterina A. <![CDATA[This paper is devoted to the computation of the number of ordered factorizations of a long cycle in the symmetric group where the number of factors is arbitrary and the cycle structure of the factors is given. Jackson (1988) derived the first closed form expression for the generating series of these numbers using the theory of the irreducible characters of the symmetric group. Thanks to a direct bijection we compute a similar formula and provide the first purely combinatorial evaluation of these generating series.]]> 0 <![CDATA[This paper studies the coefficients of algebraic functions. First, we recall the too-little-known fact that these coefficients $f_n$ have a closed form. Then, we study their asymptotics, known to be of the type $f_n \sim C A^n n^{\alpha}$. When the function is a power series associated to a context-free grammar, we solve a folklore conjecture: the appearing critical exponents $\alpha$ can not be $^1/_3$ or $^{-5}/_2$, they in fact belong to a subset of dyadic numbers. We extend what Philippe Flajolet called the Drmota-Lalley-Woods theorem (which is assuring $\alpha=^{-3}/_2$ as soon as a "dependency graph" associated to the algebraic system defining the function is strongly connected): We fully characterize the possible critical exponents in the non-strongly connected case. As a corollary, it shows that certain lattice paths and planar maps can not be generated by a context-free grammar (i.e., their generating function is not $\mathbb{N}-algebraic). We end by discussing some extensions of this work (limit laws, systems involving non-polynomial entire functions, algorithmic aspects).]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2366 https://doi.org/10.46298/dmtcs.2366 Banderier, Cyril Drmota, Michael Banderier, Cyril Drmota, Michael <![CDATA[This paper studies the coefficients of algebraic functions. First, we recall the too-little-known fact that these coefficients $f_n$ have a closed form. Then, we study their asymptotics, known to be of the type $f_n \sim C A^n n^{\alpha}$. When the function is a power series associated to a context-free grammar, we solve a folklore conjecture: the appearing critical exponents $\alpha$ can not be $^1/_3$ or $^{-5}/_2$, they in fact belong to a subset of dyadic numbers. We extend what Philippe Flajolet called the Drmota-Lalley-Woods theorem (which is assuring $\alpha=^{-3}/_2$ as soon as a "dependency graph" associated to the algebraic system defining the function is strongly connected): We fully characterize the possible critical exponents in the non-strongly connected case. As a corollary, it shows that certain lattice paths and planar maps can not be generated by a context-free grammar (i.e., their generating function is not $\mathbb{N}-algebraic). We end by discussing some extensions of this work (limit laws, systems involving non-polynomial entire functions, algorithmic aspects).]]> 0 <![CDATA[We study convolution powers $\mathtt{id}^{\ast n}$ of the identity of graded connected Hopf algebras $H$. (The antipode corresponds to $n=-1$.) The chief result is a complete description of the characteristic polynomial - both eigenvalues and multiplicity - for the action of the operator $\mathtt{id}^{\ast n}$ on each homogeneous component $H_m$. The multiplicities are independent of $n$. This follows from considering the action of the (higher) Eulerian idempotents on a certain Lie algebra $\mathfrak{g}$ associated to $H$. In case $H$ is cofree, we give an alternative (explicit and combinatorial) description in terms of palindromic words in free generators of $\mathfrak{g}$. We obtain identities involving partitions and compositions by specializing $H$ to some familiar combinatorial Hopf algebras.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2365 https://doi.org/10.46298/dmtcs.2365 Aguiar, Marcelo Lauve, Aaron Aguiar, Marcelo Lauve, Aaron <![CDATA[We study convolution powers $\mathtt{id}^{\ast n}$ of the identity of graded connected Hopf algebras $H$. (The antipode corresponds to $n=-1$.) The chief result is a complete description of the characteristic polynomial - both eigenvalues and multiplicity - for the action of the operator $\mathtt{id}^{\ast n}$ on each homogeneous component $H_m$. The multiplicities are independent of $n$. This follows from considering the action of the (higher) Eulerian idempotents on a certain Lie algebra $\mathfrak{g}$ associated to $H$. In case $H$ is cofree, we give an alternative (explicit and combinatorial) description in terms of palindromic words in free generators of $\mathfrak{g}$. We obtain identities involving partitions and compositions by specializing $H$ to some familiar combinatorial Hopf algebras.]]> 0 <![CDATA[We study height fluctuations around the limit shape of a measure on strict plane partitions. It was shown in our earlier work that this measure is a Pfaffian process. We show that the height fluctuations converge to a pullback of the Green's function for the Laplace operator with Dirichlet boundary conditions on the first quadrant. We use a Pfaffian formula for higher moments to show that the height fluctuations are governed by the Gaussian free field. The results follow from the correlation kernel asymptotics which is obtained by the steepest descent method.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2364 https://doi.org/10.46298/dmtcs.2364 Vuletić, Mirjana Vuletić, Mirjana <![CDATA[We study height fluctuations around the limit shape of a measure on strict plane partitions. It was shown in our earlier work that this measure is a Pfaffian process. We show that the height fluctuations converge to a pullback of the Green's function for the Laplace operator with Dirichlet boundary conditions on the first quadrant. We use a Pfaffian formula for higher moments to show that the height fluctuations are governed by the Gaussian free field. The results follow from the correlation kernel asymptotics which is obtained by the steepest descent method.]]> 0 <![CDATA[In this extended abstract we consider the poset of weighted partitions Π _n^w, introduced by Dotsenko and Khoroshkin in their study of a certain pair of dual operads. The maximal intervals of Π _n^w provide a generalization of the lattice Π _n of partitions, which we show possesses many of the well-known properties of Π _n. In particular, we prove these intervals are EL-shellable, we compute the Möbius invariant in terms of rooted trees, we find combinatorial bases for homology and cohomology, and we give an explicit sign twisted S_n-module isomorphism from cohomology to the multilinear component of the free Lie algebra with two compatible brackets. We also show that the characteristic polynomial of Π _n^w has a nice factorization analogous to that of Π _n.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2363 https://doi.org/10.46298/dmtcs.2363 D'León, Rafael González S. Wachs, Michelle L. D'León, Rafael González S. Wachs, Michelle L. <![CDATA[In this extended abstract we consider the poset of weighted partitions Π _n^w, introduced by Dotsenko and Khoroshkin in their study of a certain pair of dual operads. The maximal intervals of Π _n^w provide a generalization of the lattice Π _n of partitions, which we show possesses many of the well-known properties of Π _n. In particular, we prove these intervals are EL-shellable, we compute the Möbius invariant in terms of rooted trees, we find combinatorial bases for homology and cohomology, and we give an explicit sign twisted S_n-module isomorphism from cohomology to the multilinear component of the free Lie algebra with two compatible brackets. We also show that the characteristic polynomial of Π _n^w has a nice factorization analogous to that of Π _n.]]> 0 <![CDATA[The set of $n$ by $n$ upper-triangular nilpotent matrices with entries in a finite field $F_q$ has Jordan canonical forms indexed by partitions $λ \vdash n$. We study a connection between these matrices and non-attacking q-rook placements, which leads to a combinatorial formula for the number$ F_λ (q)$ of matrices of fixed Jordan type as a weighted sum over rook placements.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2362 https://doi.org/10.46298/dmtcs.2362 Yip, Martha Yip, Martha <![CDATA[The set of $n$ by $n$ upper-triangular nilpotent matrices with entries in a finite field $F_q$ has Jordan canonical forms indexed by partitions $λ \vdash n$. We study a connection between these matrices and non-attacking q-rook placements, which leads to a combinatorial formula for the number$ F_λ (q)$ of matrices of fixed Jordan type as a weighted sum over rook placements.]]> 0 <![CDATA[In this article, we show how the compositional refinement of the ``Shuffle Conjecture'' due to Jim Haglund, Jennifer Morse, and Mike Zabrocki can be used to express the image of a Schur function under the Bergeron-Garsia Nabla operator as a weighted sum of a suitable collection of ``Parking Functions.'' The validity of these expressions is, of course, going to be conjectural until the compositional refinement of the Shuffle Conjecture is established.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2361 https://doi.org/10.46298/dmtcs.2361 Kim, Yeonkyung Kim, Yeonkyung <![CDATA[In this article, we show how the compositional refinement of the ``Shuffle Conjecture'' due to Jim Haglund, Jennifer Morse, and Mike Zabrocki can be used to express the image of a Schur function under the Bergeron-Garsia Nabla operator as a weighted sum of a suitable collection of ``Parking Functions.'' The validity of these expressions is, of course, going to be conjectural until the compositional refinement of the Shuffle Conjecture is established.]]> 0 <![CDATA[We construct a divided difference operator using GKM theory. This generalizes the classical divided difference operator for the cohomology of the complete flag variety. This construction proves a special case of a recent conjecture of Shareshian and Wachs. Our methods are entirely combinatorial and algebraic, and rely heavily on the combinatorics of root systems and Bruhat order.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2360 https://doi.org/10.46298/dmtcs.2360 Teff, Nicholas Teff, Nicholas <![CDATA[We construct a divided difference operator using GKM theory. This generalizes the classical divided difference operator for the cohomology of the complete flag variety. This construction proves a special case of a recent conjecture of Shareshian and Wachs. Our methods are entirely combinatorial and algebraic, and rely heavily on the combinatorics of root systems and Bruhat order.]]> 0 <![CDATA[Using an analogue of the Robinson-Schensted-Knuth (RSK) algorithm for semi-skyline augmented fillings, due to Sarah Mason, we exhibit expansions of non-symmetric Cauchy kernels $∏_(i,j)∈\eta (1-x_iy_j)^-1$, where the product is over all cell-coordinates $(i,j)$ of the stair-type partition shape $\eta$ , consisting of the cells in a NW-SE diagonal of a rectangle diagram and below it, containing the biggest stair shape. In the spirit of the classical Cauchy kernel expansion for rectangle shapes, this RSK variation provides an interpretation of the kernel for stair-type shapes as a family of pairs of semi-skyline augmented fillings whose key tableaux, determined by their shapes, lead to expansions as a sum of products of two families of key polynomials, the basis of Demazure characters of type A, and the Demazure atoms. A previous expansion of the Cauchy kernel in type A, for the stair shape was given by Alain Lascoux, based on the structure of double crystal graphs, and by Amy M. Fu and Alain Lascoux, relying on Demazure operators, which was also used to recover expansions for Ferrers shapes.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2359 https://doi.org/10.46298/dmtcs.2359 Azenhas, Olga Emami, Aram Azenhas, Olga Emami, Aram <![CDATA[Using an analogue of the Robinson-Schensted-Knuth (RSK) algorithm for semi-skyline augmented fillings, due to Sarah Mason, we exhibit expansions of non-symmetric Cauchy kernels $∏_(i,j)∈\eta (1-x_iy_j)^-1$, where the product is over all cell-coordinates $(i,j)$ of the stair-type partition shape $\eta$ , consisting of the cells in a NW-SE diagonal of a rectangle diagram and below it, containing the biggest stair shape. In the spirit of the classical Cauchy kernel expansion for rectangle shapes, this RSK variation provides an interpretation of the kernel for stair-type shapes as a family of pairs of semi-skyline augmented fillings whose key tableaux, determined by their shapes, lead to expansions as a sum of products of two families of key polynomials, the basis of Demazure characters of type A, and the Demazure atoms. A previous expansion of the Cauchy kernel in type A, for the stair shape was given by Alain Lascoux, based on the structure of double crystal graphs, and by Amy M. Fu and Alain Lascoux, relying on Demazure operators, which was also used to recover expansions for Ferrers shapes.]]> 0 <![CDATA[We prove an analogue of the Murnaghan-Nakayama rule to express the product of a power symmetric function and a generalized Demazure atom in terms of generalized Demazure atoms.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2358 https://doi.org/10.46298/dmtcs.2358 LoBue, Jamine Remmel, Jeffrey B. LoBue, Jamine Remmel, Jeffrey B. <![CDATA[We prove an analogue of the Murnaghan-Nakayama rule to express the product of a power symmetric function and a generalized Demazure atom in terms of generalized Demazure atoms.]]> 0 <![CDATA[We define a subclass of totally symmetric self-complementary plane partitions (TSSCPPs) which we show is in direct bijection with permutation matrices. This bijection maps the inversion number of the permutation, the position of the 1 in the last column, and the position of the 1 in the last row to natural statistics on these TSSCPPs. We also discuss the possible extension of this approach to finding a bijection between alternating sign matrices and all TSSCPPs. Finally, we remark on a new poset structure on TSSCPPs arising from this perspective which is a distributive lattice when restricted to permutation TSSCPPs.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2344 https://doi.org/10.46298/dmtcs.2344 Striker, Jessica Striker, Jessica <![CDATA[We define a subclass of totally symmetric self-complementary plane partitions (TSSCPPs) which we show is in direct bijection with permutation matrices. This bijection maps the inversion number of the permutation, the position of the 1 in the last column, and the position of the 1 in the last row to natural statistics on these TSSCPPs. We also discuss the possible extension of this approach to finding a bijection between alternating sign matrices and all TSSCPPs. Finally, we remark on a new poset structure on TSSCPPs arising from this perspective which is a distributive lattice when restricted to permutation TSSCPPs.]]> 0 <![CDATA[Many cyclic actions $τ$ on a finite set $\mathcal{S}$ ; of combinatorial objects, along with a natural statistic $f$ on $\mathcal{S}$, exhibit ``homomesy'': the average of $f$ over each $τ$-orbit in $\mathcal{S} $ is the same as the average of $f$ over the whole set $\mathcal{S} $. This phenomenon was first noticed by Panyushev in 2007 in the context of antichains in root posets; Armstrong, Stump, and Thomas proved Panyushev's conjecture in 2011. We describe a theoretical framework for results of this kind and discuss old and new results for the actions of promotion and rowmotion on the poset that is the product of two chains.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2356 https://doi.org/10.46298/dmtcs.2356 Propp, James Roby, Tom Propp, James Roby, Tom <![CDATA[Many cyclic actions $τ$ on a finite set $\mathcal{S}$ ; of combinatorial objects, along with a natural statistic $f$ on $\mathcal{S}$, exhibit ``homomesy'': the average of $f$ over each $τ$-orbit in $\mathcal{S} $ is the same as the average of $f$ over the whole set $\mathcal{S} $. This phenomenon was first noticed by Panyushev in 2007 in the context of antichains in root posets; Armstrong, Stump, and Thomas proved Panyushev's conjecture in 2011. We describe a theoretical framework for results of this kind and discuss old and new results for the actions of promotion and rowmotion on the poset that is the product of two chains.]]> 0 <![CDATA[Each positive rational number $x>0$ can be written $\textbf{uniquely}$ as $x=a/(b-a)$ for coprime positive integers 0<$a$<$b$. We will identify $x$ with the pair $(a,b)$. In this extended abstract we use $\textit{rational Dyck paths}$ to define for each positive rational $x>0$ a simplicial complex $\mathsf{Ass} (x)=\mathsf{Ass} (a,b)$ called the $\textit{rational associahedron}$. It is a pure simplicial complex of dimension $a-2$, and its maximal faces are counted by the $\textit{rational Catalan number}$ $\mathsf{Cat} (x)=\mathsf{Cat}(a,b):=\frac{(a+b-1)! }{ a! b!}.$ The cases $(a,b)=(n,n+1)$ and $(a,b)=(n,kn+1)$ recover the classical associahedron and its Fuss-Catalan generalization studied by Athanasiadis-Tzanaki and Fomin-Reading. We prove that $\mathsf{Ass} (a,b)$ is shellable and give nice product formulas for its $h$-vector (the $\textit{rational Narayana numbers}$) and $f$-vector (the $\textit{rational Kirkman numbers}$). We define $\mathsf{Ass} (a,b)$ .]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2355 https://doi.org/10.46298/dmtcs.2355 Armstrong, Drew Rhoades, Brendon Williams, Nathan Armstrong, Drew Rhoades, Brendon Williams, Nathan <![CDATA[Each positive rational number $x>0$ can be written $\textbf{uniquely}$ as $x=a/(b-a)$ for coprime positive integers 0<$a$<$b$. We will identify $x$ with the pair $(a,b)$. In this extended abstract we use $\textit{rational Dyck paths}$ to define for each positive rational $x>0$ a simplicial complex $\mathsf{Ass} (x)=\mathsf{Ass} (a,b)$ called the $\textit{rational associahedron}$. It is a pure simplicial complex of dimension $a-2$, and its maximal faces are counted by the $\textit{rational Catalan number}$ $\mathsf{Cat} (x)=\mathsf{Cat}(a,b):=\frac{(a+b-1)! }{ a! b!}.$ The cases $(a,b)=(n,n+1)$ and $(a,b)=(n,kn+1)$ recover the classical associahedron and its Fuss-Catalan generalization studied by Athanasiadis-Tzanaki and Fomin-Reading. We prove that $\mathsf{Ass} (a,b)$ is shellable and give nice product formulas for its $h$-vector (the $\textit{rational Narayana numbers}$) and $f$-vector (the $\textit{rational Kirkman numbers}$). We define $\mathsf{Ass} (a,b)$ .]]> 0 <![CDATA[In 2007 Sami Assaf introduced dual equivalence graphs as a method for demonstrating that a quasisymmetric function is Schur positive. The method involves the creation of a graph whose vertices are weighted by Ira Gessel's fundamental quasisymmetric functions so that the sum of the weights of a connected component is a single Schur function. In this paper, we improve on Assaf's axiomatization of such graphs, giving locally testable criteria that are more easily verified by computers. We then demonstrate the utility of this result by giving explicit Schur expansions for a family of Lascoux-Leclerc-Thibon polynomials. This family properly contains the previously known case of polynomials indexed by two skew shapes, as was described in a 1995 paper by Christophe Carré and Bernard Leclerc. As an immediate corollary, we gain an explicit Schur expansion for a family of modified Macdonald polynomials in terms of Yamanouchi words. This family includes all polynomials indexed by shapes with less than four cells in the first row and strictly less than three cells in the second row, a slight improvement over the known two column case described in 2005 by James Haglund, Mark Haiman, and Nick Loehr.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2354 https://doi.org/10.46298/dmtcs.2354 Roberts, Austin Roberts, Austin <![CDATA[In 2007 Sami Assaf introduced dual equivalence graphs as a method for demonstrating that a quasisymmetric function is Schur positive. The method involves the creation of a graph whose vertices are weighted by Ira Gessel's fundamental quasisymmetric functions so that the sum of the weights of a connected component is a single Schur function. In this paper, we improve on Assaf's axiomatization of such graphs, giving locally testable criteria that are more easily verified by computers. We then demonstrate the utility of this result by giving explicit Schur expansions for a family of Lascoux-Leclerc-Thibon polynomials. This family properly contains the previously known case of polynomials indexed by two skew shapes, as was described in a 1995 paper by Christophe Carré and Bernard Leclerc. As an immediate corollary, we gain an explicit Schur expansion for a family of modified Macdonald polynomials in terms of Yamanouchi words. This family includes all polynomials indexed by shapes with less than four cells in the first row and strictly less than three cells in the second row, a slight improvement over the known two column case described in 2005 by James Haglund, Mark Haiman, and Nick Loehr.]]> 0 <![CDATA[Extending the notion of pattern avoidance in permutations, we study matchings and set partitions whose arc diagram representation avoids a given configuration of three arcs. These configurations, which generalize 3-crossings and 3-nestings, have an interpretation, in the case of matchings, in terms of patterns in full rook placements on Ferrers boards. We enumerate 312-avoiding matchings and partitions, obtaining algebraic generating functions, unlike in the 321-avoiding (i.e., 3-noncrossing) case. Our approach also provides a more direct proof of a formula of Bóna for the number of 1342-avoiding permutations. Additionally, we give a bijection proving the shape-Wilf-equivalence of the patterns 321 and 213 which simplifies existing proofs by Backelin–West–Xin and Jelínek.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2353 https://doi.org/10.46298/dmtcs.2353 Bloom, Jonathan Elizalde, Sergi Bloom, Jonathan Elizalde, Sergi <![CDATA[Extending the notion of pattern avoidance in permutations, we study matchings and set partitions whose arc diagram representation avoids a given configuration of three arcs. These configurations, which generalize 3-crossings and 3-nestings, have an interpretation, in the case of matchings, in terms of patterns in full rook placements on Ferrers boards. We enumerate 312-avoiding matchings and partitions, obtaining algebraic generating functions, unlike in the 321-avoiding (i.e., 3-noncrossing) case. Our approach also provides a more direct proof of a formula of Bóna for the number of 1342-avoiding permutations. Additionally, we give a bijection proving the shape-Wilf-equivalence of the patterns 321 and 213 which simplifies existing proofs by Backelin–West–Xin and Jelínek.]]> 0 <![CDATA[Let $\alpha$ be a string over $\mathbb{Z}_q$, where $q = 2^d$. The $j$-th elementary symmetric function evaluated at $\alpha$ is denoted $e_j(\alpha)$ . We study the cardinalities $S_q(m;\mathcal{T} _1,\mathcal{T} _2,\ldots,\mathcal{T} _t)$ of the set of length $m$ strings for which $e_j(\alpha) = \tau _i$. The $\textit{profile}$ k$(\alpha) = ⟨k_1,k_2,\ldots,k_(q-1) ⟩$ of a string $\alpha$ is the sequence of frequencies with which each letter occurs. The profile of $\alpha$ determines $e_j(\alpha)$ , and hence $S_q$. Let $h_n$ : $\mathbb{Z}_{2^{n+d-1}}^{(q-1)}$ $\mapsto \mathbb{Z}_{2^d} [z] $ mod $ z^{2^n}$ be the map that takes k$(\alpha)$ mod $2^{n+d-1}$ to the polynomial $1+ e_1(\alpha) z + e_2(\alpha) z^2 + ⋯+ e_{2^n-1}(\alpha)$ $z^{2^{n-1}}$. We show that $h_n$ is a group homomorphism and establish necessary conditions for membership in the kernel for fixed $d$. The kernel is determined for $d$ = 2,3. The range of $h_n$ is described for $d$ = 2. These results are used to efficiently compute $S_4(m;\mathcal{T} _1,\mathcal{T} _2,\ldots,\mathcal{T} _t)$ for $d$ = 2 and the number of complete factorizations of certain polynomials.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2352 https://doi.org/10.46298/dmtcs.2352 Miers, Charles Robert Ruskey, Franck Miers, Charles Robert Ruskey, Franck <![CDATA[Let $\alpha$ be a string over $\mathbb{Z}_q$, where $q = 2^d$. The $j$-th elementary symmetric function evaluated at $\alpha$ is denoted $e_j(\alpha)$ . We study the cardinalities $S_q(m;\mathcal{T} _1,\mathcal{T} _2,\ldots,\mathcal{T} _t)$ of the set of length $m$ strings for which $e_j(\alpha) = \tau _i$. The $\textit{profile}$ k$(\alpha) = ⟨k_1,k_2,\ldots,k_(q-1) ⟩$ of a string $\alpha$ is the sequence of frequencies with which each letter occurs. The profile of $\alpha$ determines $e_j(\alpha)$ , and hence $S_q$. Let $h_n$ : $\mathbb{Z}_{2^{n+d-1}}^{(q-1)}$ $\mapsto \mathbb{Z}_{2^d} [z] $ mod $ z^{2^n}$ be the map that takes k$(\alpha)$ mod $2^{n+d-1}$ to the polynomial $1+ e_1(\alpha) z + e_2(\alpha) z^2 + ⋯+ e_{2^n-1}(\alpha)$ $z^{2^{n-1}}$. We show that $h_n$ is a group homomorphism and establish necessary conditions for membership in the kernel for fixed $d$. The kernel is determined for $d$ = 2,3. The range of $h_n$ is described for $d$ = 2. These results are used to efficiently compute $S_4(m;\mathcal{T} _1,\mathcal{T} _2,\ldots,\mathcal{T} _t)$ for $d$ = 2 and the number of complete factorizations of certain polynomials.]]> 0 <![CDATA[We study the binomial and monomial ideals arising from linear equivalence of divisors on graphs from the point of view of Gröbner theory. We give an explicit description of a minimal Gröbner basis for each higher syzygy module. In each case the given minimal Gröbner basis is also a minimal generating set. The Betti numbers of $I_G$ and its initial ideal (with respect to a natural term order) coincide and they correspond to the number of ``connected flags'' in $G$. Moreover, the Betti numbers are independent of the characteristic of the base field.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2351 https://doi.org/10.46298/dmtcs.2351 Mohammadi, Fatemeh Shokrieh, Farbod Mohammadi, Fatemeh Shokrieh, Farbod <![CDATA[We study the binomial and monomial ideals arising from linear equivalence of divisors on graphs from the point of view of Gröbner theory. We give an explicit description of a minimal Gröbner basis for each higher syzygy module. In each case the given minimal Gröbner basis is also a minimal generating set. The Betti numbers of $I_G$ and its initial ideal (with respect to a natural term order) coincide and they correspond to the number of ``connected flags'' in $G$. Moreover, the Betti numbers are independent of the characteristic of the base field.]]> 0 <![CDATA[The periodic patterns of a map are the permutations realized by the relative order of the points in its periodic orbits. We give a combinatorial description of the periodic patterns of an arbitrary signed shift, in terms of the structure of the descent set of a certain transformation of the pattern. Signed shifts are an important family of one-dimensional dynamical systems. For particular types of signed shifts, namely shift maps, reverse shift maps, and the tent map, we give exact enumeration formulas for their periodic patterns. As a byproduct of our work, we recover some results of Gessel and Reutenauer and obtain new results on the enumeration of pattern-avoiding cycles.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2350 https://doi.org/10.46298/dmtcs.2350 Archer, Kassie Elizalde, Sergi Archer, Kassie Elizalde, Sergi <![CDATA[The periodic patterns of a map are the permutations realized by the relative order of the points in its periodic orbits. We give a combinatorial description of the periodic patterns of an arbitrary signed shift, in terms of the structure of the descent set of a certain transformation of the pattern. Signed shifts are an important family of one-dimensional dynamical systems. For particular types of signed shifts, namely shift maps, reverse shift maps, and the tent map, we give exact enumeration formulas for their periodic patterns. As a byproduct of our work, we recover some results of Gessel and Reutenauer and obtain new results on the enumeration of pattern-avoiding cycles.]]> 0 <![CDATA[The $A_2$-spider category encodes the representation theory of the $sl_3$ quantum group. Kuperberg (1996) introduced a combinatorial version of this category, wherein morphisms are represented by planar graphs called $\textit{webs}$ and the subset of $\textit{reduced webs}$ forms bases for morphism spaces. A great deal of recent interest has focused on the combinatorics of invariant webs for tensors powers of $V^+$, the standard representation of the quantum group. In particular, the invariant webs for the 3$n$th tensor power of $V^+$ correspond bijectively to $[n,n,n]$ standard Young tableaux. Kuperberg originally defined this map in terms of a graphical algorithm, and subsequent papers of Khovanov–Kuperberg (1999) and Tymoczko (2012) introduce algorithms for computing the inverse. The main result of this paper is a redefinition of Kuperberg's map through the representation theory of the symmetric group. In the classical limit, the space of invariant webs carries a symmetric group action. We use this structure in conjunction with Vogan's generalized tau-invariant and Kazhdan–Lusztig theory to show that Kuperberg's map is a direct analogue of the Robinson–Schensted correspondence.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2349 https://doi.org/10.46298/dmtcs.2349 Housley, Matthew Russell, Heather M. Tymoczko, Julianna Housley, Matthew Russell, Heather M. Tymoczko, Julianna <![CDATA[The $A_2$-spider category encodes the representation theory of the $sl_3$ quantum group. Kuperberg (1996) introduced a combinatorial version of this category, wherein morphisms are represented by planar graphs called $\textit{webs}$ and the subset of $\textit{reduced webs}$ forms bases for morphism spaces. A great deal of recent interest has focused on the combinatorics of invariant webs for tensors powers of $V^+$, the standard representation of the quantum group. In particular, the invariant webs for the 3$n$th tensor power of $V^+$ correspond bijectively to $[n,n,n]$ standard Young tableaux. Kuperberg originally defined this map in terms of a graphical algorithm, and subsequent papers of Khovanov–Kuperberg (1999) and Tymoczko (2012) introduce algorithms for computing the inverse. The main result of this paper is a redefinition of Kuperberg's map through the representation theory of the symmetric group. In the classical limit, the space of invariant webs carries a symmetric group action. We use this structure in conjunction with Vogan's generalized tau-invariant and Kazhdan–Lusztig theory to show that Kuperberg's map is a direct analogue of the Robinson–Schensted correspondence.]]> 0 <![CDATA[Given a reconfigurable system $X$, such as a robot moving on a grid or a set of particles traversing a graph without colliding, the possible positions of $X$ naturally form a cubical complex $\mathcal{S}(X)$. When $\mathcal{S}(X)$ is a CAT(0) space, we can explicitly construct the shortest path between any two points, for any of the four most natural metrics: distance, time, number of moves, and number of steps of simultaneous moves. CAT(0) cubical complexes are in correspondence with posets with inconsistent pairs (PIPs), so we can prove that a state complex $\mathcal{S}(X)$ is CAT(0) by identifying the corresponding PIP. We illustrate this very general strategy with one known and one new example: Abrams and Ghrist's ``positive robotic arm" on a square grid, and the robotic arm in a strip. We then use the PIP as a combinatorial ``remote control" to move these robots efficiently from one position to another.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2348 https://doi.org/10.46298/dmtcs.2348 Ardila, Federico Baker, Tia Yatchak, Rika Ardila, Federico Baker, Tia Yatchak, Rika <![CDATA[Given a reconfigurable system $X$, such as a robot moving on a grid or a set of particles traversing a graph without colliding, the possible positions of $X$ naturally form a cubical complex $\mathcal{S}(X)$. When $\mathcal{S}(X)$ is a CAT(0) space, we can explicitly construct the shortest path between any two points, for any of the four most natural metrics: distance, time, number of moves, and number of steps of simultaneous moves. CAT(0) cubical complexes are in correspondence with posets with inconsistent pairs (PIPs), so we can prove that a state complex $\mathcal{S}(X)$ is CAT(0) by identifying the corresponding PIP. We illustrate this very general strategy with one known and one new example: Abrams and Ghrist's ``positive robotic arm" on a square grid, and the robotic arm in a strip. We then use the PIP as a combinatorial ``remote control" to move these robots efficiently from one position to another.]]> 0 <![CDATA[A Gelfand model for a semisimple algebra $\mathsf{A}$ over $\mathbb{C}$ is a complex linear representation that contains each irreducible representation of $\mathsf{A}$ with multiplicity exactly one. We give a method of constructing these models that works uniformly for a large class of combinatorial diagram algebras including: the partition, Brauer, rook monoid, rook-Brauer, Temperley-Lieb, Motzkin, and planar rook monoid algebras. In each case, the model representation is given by diagrams acting via ``signed conjugation" on the linear span of their vertically symmetric diagrams. This representation is a generalization of the Saxl model for the symmetric group, and, in fact, our method is to use the Jones basic construction to lift the Saxl model from the symmetric group to each diagram algebra. In the case of the planar diagram algebras, our construction exactly produces the irreducible representations of the algebra.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2347 https://doi.org/10.46298/dmtcs.2347 Halverson, Tom Halverson, Tom <![CDATA[A Gelfand model for a semisimple algebra $\mathsf{A}$ over $\mathbb{C}$ is a complex linear representation that contains each irreducible representation of $\mathsf{A}$ with multiplicity exactly one. We give a method of constructing these models that works uniformly for a large class of combinatorial diagram algebras including: the partition, Brauer, rook monoid, rook-Brauer, Temperley-Lieb, Motzkin, and planar rook monoid algebras. In each case, the model representation is given by diagrams acting via ``signed conjugation" on the linear span of their vertically symmetric diagrams. This representation is a generalization of the Saxl model for the symmetric group, and, in fact, our method is to use the Jones basic construction to lift the Saxl model from the symmetric group to each diagram algebra. In the case of the planar diagram algebras, our construction exactly produces the irreducible representations of the algebra.]]> 0 <![CDATA[After extending classical results on simple varieties of trees to trees counted by their number of leaves, we describe a filtration of the set of permutations based on their strong interval trees. For each subclass we provide asymptotic formulas for number of trees (by leaves), average number of nodes of fixed arity, average subtree size sum, and average number of internal nodes. The filtration is motivated by genome comparison of related species.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2346 https://doi.org/10.46298/dmtcs.2346 Bouvel, Mathilde Mishna, Marni Nicaud, Cyril Bouvel, Mathilde Mishna, Marni Nicaud, Cyril <![CDATA[After extending classical results on simple varieties of trees to trees counted by their number of leaves, we describe a filtration of the set of permutations based on their strong interval trees. For each subclass we provide asymptotic formulas for number of trees (by leaves), average number of nodes of fixed arity, average subtree size sum, and average number of internal nodes. The filtration is motivated by genome comparison of related species.]]> 0 <![CDATA[In this article, we study some quotient sets on permutations built from peaks, valleys, double rises and double descents. One part is dedicated to the enumeration of the cosets using the bijection of Francon-Viennot which is a bijection between permutations and the so-called Laguerre histories. Then we study the algebraic properties of these quotient sets. After having shown that some of them give rise to quotient algebras of $\mathbf{FQSym}$, we prove that they are also free.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2345 https://doi.org/10.46298/dmtcs.2345 Vong, Vincent Vong, Vincent <![CDATA[In this article, we study some quotient sets on permutations built from peaks, valleys, double rises and double descents. One part is dedicated to the enumeration of the cosets using the bijection of Francon-Viennot which is a bijection between permutations and the so-called Laguerre histories. Then we study the algebraic properties of these quotient sets. After having shown that some of them give rise to quotient algebras of $\mathbf{FQSym}$, we prove that they are also free.]]> 0 <![CDATA[We show that given a poset $P$ and and a subposet $Q$, the integer points obtained by restricting linear extensions of $P$ to $Q$ can be explained via integer lattice points of a generalized permutohedron.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2319 https://doi.org/10.46298/dmtcs.2319 Croitoru, Dorian Oh, Suho Postnikov, Alexander Croitoru, Dorian Oh, Suho Postnikov, Alexander <![CDATA[We show that given a poset $P$ and and a subposet $Q$, the integer points obtained by restricting linear extensions of $P$ to $Q$ can be explained via integer lattice points of a generalized permutohedron.]]> 0 <![CDATA[Root-theoretic Young diagrams are a conceptual framework to discuss existence of a root-system uniform and manifestly non-negative combinatorial rule for Schubert calculus. Our main results use them to obtain formulas for (co)adjoint varieties of classical Lie type. This case is the simplest after the previously solved (co)minuscule family. Yet our formulas possess both uniform and non-uniform features.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2318 https://doi.org/10.46298/dmtcs.2318 Searles, Dominic Yong, Alexander Searles, Dominic Yong, Alexander <![CDATA[Root-theoretic Young diagrams are a conceptual framework to discuss existence of a root-system uniform and manifestly non-negative combinatorial rule for Schubert calculus. Our main results use them to obtain formulas for (co)adjoint varieties of classical Lie type. This case is the simplest after the previously solved (co)minuscule family. Yet our formulas possess both uniform and non-uniform features.]]> 0 <![CDATA[We provide exact and asymptotic counting formulas for five singular lattice path models in the quarter plane. Furthermore, we prove that these models have a non D-finite generating function.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2317 https://doi.org/10.46298/dmtcs.2317 Melczer, Stephen Mishna, Marni Melczer, Stephen Mishna, Marni <![CDATA[We provide exact and asymptotic counting formulas for five singular lattice path models in the quarter plane. Furthermore, we prove that these models have a non D-finite generating function.]]> 0 <![CDATA[In a recent paper, Diaconis, Ram and I constructed Markov chains using the coproduct-then-product map of a combinatorial Hopf algebra. We presented an algorithm for diagonalising a large class of these "Hopf-power chains", including the Gilbert-Shannon-Reeds model of riffle-shuffling of a deck of cards and a rock-breaking model. A very restrictive condition from that paper is removed in my thesis, and this extended abstract focuses on one application of the improved theory. Here, I use a new technique of lumping Hopf-power chains to show that the Hopf-power chain on the algebra of quasisymmetric functions is the induced chain on descent sets under riffle-shuffling. Moreover, I relate its right and left eigenfunctions to Garsia-Reutenauer idempotents and ribbon characters respectively, from which I recover an analogous result of Diaconis and Fulman (2012) concerning the number of descents under riffle-shuffling.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2316 https://doi.org/10.46298/dmtcs.2316 Pang, C.Y. Amy Pang, C.Y. Amy <![CDATA[In a recent paper, Diaconis, Ram and I constructed Markov chains using the coproduct-then-product map of a combinatorial Hopf algebra. We presented an algorithm for diagonalising a large class of these "Hopf-power chains", including the Gilbert-Shannon-Reeds model of riffle-shuffling of a deck of cards and a rock-breaking model. A very restrictive condition from that paper is removed in my thesis, and this extended abstract focuses on one application of the improved theory. Here, I use a new technique of lumping Hopf-power chains to show that the Hopf-power chain on the algebra of quasisymmetric functions is the induced chain on descent sets under riffle-shuffling. Moreover, I relate its right and left eigenfunctions to Garsia-Reutenauer idempotents and ribbon characters respectively, from which I recover an analogous result of Diaconis and Fulman (2012) concerning the number of descents under riffle-shuffling.]]> 0 <![CDATA[We consider the parameter rank introduced for graph configurations by M. Baker and S. Norine. We focus on complete graphs and obtain an efficient algorithm to determine the rank for these graphs. The analysis of this algorithm leads to the definition of a parameter on Dyck words, which we call prerank. We prove that the distribution of area and prerank on Dyck words of given length $2n$ leads to a polynomial with variables $q,t$ which is symmetric in these variables. This polynomial is different from the $q,t-$Catalan polynomial studied by A. Garsia, J. Haglund and M. Haiman.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2332 https://doi.org/10.46298/dmtcs.2332 Cori, Robert Borgne, Yvan Le Cori, Robert Borgne, Yvan Le <![CDATA[We consider the parameter rank introduced for graph configurations by M. Baker and S. Norine. We focus on complete graphs and obtain an efficient algorithm to determine the rank for these graphs. The analysis of this algorithm leads to the definition of a parameter on Dyck words, which we call prerank. We prove that the distribution of area and prerank on Dyck words of given length $2n$ leads to a polynomial with variables $q,t$ which is symmetric in these variables. This polynomial is different from the $q,t-$Catalan polynomial studied by A. Garsia, J. Haglund and M. Haiman.]]> 0 <![CDATA[Erdős and Rényi conjectured in 1960 that the limiting probability $p$ that a random graph with $n$ vertices and $M=n/2$ edges is planar exists. It has been shown that indeed p exists and is a constant strictly between 0 and 1. In this paper we answer completely this long standing question by finding an exact expression for this probability, whose approximate value turns out to be $p ≈0.99780$. More generally, we compute the probability of planarity at the critical window of width $n^{2/3}$ around the critical point $M=n/2$. We extend these results to some classes of graphs closed under taking minors. As an example, we show that the probability of being series-parallel converges to 0.98003. Our proofs rely on exploiting the structure of random graphs in the critical window, obtained previously by Janson, Łuczak and Wierman, by means of generating functions and analytic methods. This is a striking example of how analytic combinatorics can be applied to classical problems on random graphs.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2343 https://doi.org/10.46298/dmtcs.2343 Noy, Marc Ravelomanana, Vlady Rué, Juanjo Noy, Marc Ravelomanana, Vlady Rué, Juanjo <![CDATA[Erdős and Rényi conjectured in 1960 that the limiting probability $p$ that a random graph with $n$ vertices and $M=n/2$ edges is planar exists. It has been shown that indeed p exists and is a constant strictly between 0 and 1. In this paper we answer completely this long standing question by finding an exact expression for this probability, whose approximate value turns out to be $p ≈0.99780$. More generally, we compute the probability of planarity at the critical window of width $n^{2/3}$ around the critical point $M=n/2$. We extend these results to some classes of graphs closed under taking minors. As an example, we show that the probability of being series-parallel converges to 0.98003. Our proofs rely on exploiting the structure of random graphs in the critical window, obtained previously by Janson, Łuczak and Wierman, by means of generating functions and analytic methods. This is a striking example of how analytic combinatorics can be applied to classical problems on random graphs.]]> 0 <![CDATA[We study the $\textit{diagrams}$ of affine permutations and their $\textit{balanced}$ labellings. As in the finite case, which was investigated by Fomin, Greene, Reiner, and Shimozono, the balanced labellings give a natural encoding of reduced decompositions of affine permutations. In fact, we show that the sum of weight monomials of the $\textit{column strict}$ balanced labellings is the affine Stanley symmetric function defined by Lam and we give a simple algorithm to recover reduced words from balanced labellings. Applying this theory, we give a necessary and sufficient condition for a diagram to be an affine permutation diagram. Finally, we conjecture that if two affine permutations are $\textit{diagram equivalent}$ then their affine Stanley symmetric functions coincide.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2342 https://doi.org/10.46298/dmtcs.2342 Yoo, Hwanchul Yun, Taedong Yoo, Hwanchul Yun, Taedong <![CDATA[We study the $\textit{diagrams}$ of affine permutations and their $\textit{balanced}$ labellings. As in the finite case, which was investigated by Fomin, Greene, Reiner, and Shimozono, the balanced labellings give a natural encoding of reduced decompositions of affine permutations. In fact, we show that the sum of weight monomials of the $\textit{column strict}$ balanced labellings is the affine Stanley symmetric function defined by Lam and we give a simple algorithm to recover reduced words from balanced labellings. Applying this theory, we give a necessary and sufficient condition for a diagram to be an affine permutation diagram. Finally, we conjecture that if two affine permutations are $\textit{diagram equivalent}$ then their affine Stanley symmetric functions coincide.]]> 0 <![CDATA[Originally motivated by algebraic invariant theory, we present an algorithm to enumerate integer vectors modulo the action of a permutation group. This problem generalizes the generation of unlabeled graph up to an isomorphism. In this paper, we present the full development of a generation engine by describing the related theory, establishing a mathematical and practical complexity, and exposing some benchmarks. We next show two applications to effective invariant theory and effective Galois theory.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2341 https://doi.org/10.46298/dmtcs.2341 Borie, Nicolas Borie, Nicolas <![CDATA[Originally motivated by algebraic invariant theory, we present an algorithm to enumerate integer vectors modulo the action of a permutation group. This problem generalizes the generation of unlabeled graph up to an isomorphism. In this paper, we present the full development of a generation engine by describing the related theory, establishing a mathematical and practical complexity, and exposing some benchmarks. We next show two applications to effective invariant theory and effective Galois theory.]]> 0 <![CDATA[Many results involving Schur functions have analogues involving $k-$Schur functions. Standard strong marked tableaux play a role for $k-$Schur functions similar to the role standard Young tableaux play for Schur functions. We discuss results and conjectures toward an analogue of the hook length formula.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2340 https://doi.org/10.46298/dmtcs.2340 Fishel, Susanna Konvalinka, Matjaž Fishel, Susanna Konvalinka, Matjaž <![CDATA[Many results involving Schur functions have analogues involving $k-$Schur functions. Standard strong marked tableaux play a role for $k-$Schur functions similar to the role standard Young tableaux play for Schur functions. We discuss results and conjectures toward an analogue of the hook length formula.]]> 0 <![CDATA[The equidistribution of many crossing and nesting statistics exists in several combinatorial objects like matchings, set partitions, permutations, and embedded labelled graphs. The involutions switching nesting and crossing numbers for set partitions given by Krattenthaler, also by Chen, Deng, Du, Stanley, and Yan, and for permutations given by Burrill, Mishna, and Post involved passing through tableau-like objects. Recently, Chen and Guo for matchings, and Marberg for set partitions extended the result to coloured arc annotated diagrams. We prove that symmetric joint distribution continues to hold for arc-coloured permutations. As in Marberg's recent work, but through a different interpretation, we also conclude that the ordinary generating functions for all j-noncrossing, k-nonnesting, r-coloured permutations according to size n are rational functions. We use the interpretation to automate the generation of these rational series for both noncrossing and nonnesting coloured set partitions and permutations. otherlanguage*french L'équidistribution de plusieurs statistiques décrites en termes d'emboitements et de chevauchements d'arcs s'observes dans plusieurs familles d'objects combinatoires, tels que les couplages, partitions d'ensembles, permutations et graphes étiquetés. L'involution échangeant le nombre d'emboitements et de chevauchements dans les partitions d'ensemble due à Krattenthaler, et aussi Chen, Deng, Du, Stanley et Yan, et l'involution similaire dans les permutations due à Burrill, Mishna et Post, requièrent d'utiliser des objets de type tableaux. Récemment, Chen et Guo pour les couplages, et Marberg pour les partitions d'ensembles, ont étendu ces résultats au cas de diagrammes arc-annotés coloriés. Nous démontrons que la propriété d'équidistribution s'observe est aussi vraie dans le cas de permutations aux arcs coloriés. Tout comme dans le travail résent de Marberg, mais via un autre chemin, nous montrons que les séries génératrices ordinaires des permutations r-coloriées ayant au plus j chevauchements et k emboitements, comptées selon la taille n, sont des fonctions rationnelles. Nous décrivons aussi des algorithmes permettant de calculer ces fonctions rationnelles pour les partitions d'ensembles et les permutations coloriées sans emboitement ou sans chevauchement. otherlanguage*]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2339 https://doi.org/10.46298/dmtcs.2339 Yen, Lily Yen, Lily <![CDATA[The equidistribution of many crossing and nesting statistics exists in several combinatorial objects like matchings, set partitions, permutations, and embedded labelled graphs. The involutions switching nesting and crossing numbers for set partitions given by Krattenthaler, also by Chen, Deng, Du, Stanley, and Yan, and for permutations given by Burrill, Mishna, and Post involved passing through tableau-like objects. Recently, Chen and Guo for matchings, and Marberg for set partitions extended the result to coloured arc annotated diagrams. We prove that symmetric joint distribution continues to hold for arc-coloured permutations. As in Marberg's recent work, but through a different interpretation, we also conclude that the ordinary generating functions for all j-noncrossing, k-nonnesting, r-coloured permutations according to size n are rational functions. We use the interpretation to automate the generation of these rational series for both noncrossing and nonnesting coloured set partitions and permutations. otherlanguage*french L'équidistribution de plusieurs statistiques décrites en termes d'emboitements et de chevauchements d'arcs s'observes dans plusieurs familles d'objects combinatoires, tels que les couplages, partitions d'ensembles, permutations et graphes étiquetés. L'involution échangeant le nombre d'emboitements et de chevauchements dans les partitions d'ensemble due à Krattenthaler, et aussi Chen, Deng, Du, Stanley et Yan, et l'involution similaire dans les permutations due à Burrill, Mishna et Post, requièrent d'utiliser des objets de type tableaux. Récemment, Chen et Guo pour les couplages, et Marberg pour les partitions d'ensembles, ont étendu ces résultats au cas de diagrammes arc-annotés coloriés. Nous démontrons que la propriété d'équidistribution s'observe est aussi vraie dans le cas de permutations aux arcs coloriés. Tout comme dans le travail résent de Marberg, mais via un autre chemin, nous montrons que les séries génératrices ordinaires des permutations r-coloriées ayant au plus j chevauchements et k emboitements, comptées selon la taille n, sont des fonctions rationnelles. Nous décrivons aussi des algorithmes permettant de calculer ces fonctions rationnelles pour les partitions d'ensembles et les permutations coloriées sans emboitement ou sans chevauchement. otherlanguage*]]> 0 <![CDATA[We prove that the Mahonian-Stirling pairs of permutation statistics $(sor, cyc)$ and $(∈v , \mathrm{rlmin})$ are equidistributed on the set of permutations that correspond to arrangements of $n$ non-atacking rooks on a fixed Ferrers board with $n$ rows and $n$ columns. The proofs are combinatorial and use bijections between matchings and Dyck paths and a new statistic, sorting index for matchings, that we define. We also prove a refinement of this equidistribution result which describes the minimal elements in the permutation cycles and the right-to-left minimum letters.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2338 https://doi.org/10.46298/dmtcs.2338 Poznanović, Svetlana Poznanović, Svetlana <![CDATA[We prove that the Mahonian-Stirling pairs of permutation statistics $(sor, cyc)$ and $(∈v , \mathrm{rlmin})$ are equidistributed on the set of permutations that correspond to arrangements of $n$ non-atacking rooks on a fixed Ferrers board with $n$ rows and $n$ columns. The proofs are combinatorial and use bijections between matchings and Dyck paths and a new statistic, sorting index for matchings, that we define. We also prove a refinement of this equidistribution result which describes the minimal elements in the permutation cycles and the right-to-left minimum letters.]]> 0 <![CDATA[Motivated by the Gaussian symplectic ensemble, Mehta and Wang evaluated the $n×n$ determinant $\det ((a+j-i)Γ (b+j+i))$ in 2000. When $a=0$, Ciucu and Krattenthaler computed the associated Pfaffian $\mathrm{Pf}((j-i)Γ (b+j+i))$ with an application to the two dimensional dimer system in 2011. Recently we have generalized the latter Pfaffian formula with a $q$-analogue by replacing the Gamma function by the moment sequence of the little $q$-Jacobi polynomials. On the other hand, Nishizawa has found a q-analogue of the Mehta–Wang formula. Our purpose is to generalize both the Mehta-Wang and Nishizawa formulae by using the moment sequence of the little $q$-Jacobi polynomials. It turns out that the corresponding determinant can be evaluated explicitly in terms of the Askey-Wilson polynomials.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2337 https://doi.org/10.46298/dmtcs.2337 Guo, Victor J. W. Ishikawa, Masao Tagawa, Hiroyuki Zeng, Jiang Guo, Victor J. W. Ishikawa, Masao Tagawa, Hiroyuki Zeng, Jiang <![CDATA[Motivated by the Gaussian symplectic ensemble, Mehta and Wang evaluated the $n×n$ determinant $\det ((a+j-i)Γ (b+j+i))$ in 2000. When $a=0$, Ciucu and Krattenthaler computed the associated Pfaffian $\mathrm{Pf}((j-i)Γ (b+j+i))$ with an application to the two dimensional dimer system in 2011. Recently we have generalized the latter Pfaffian formula with a $q$-analogue by replacing the Gamma function by the moment sequence of the little $q$-Jacobi polynomials. On the other hand, Nishizawa has found a q-analogue of the Mehta–Wang formula. Our purpose is to generalize both the Mehta-Wang and Nishizawa formulae by using the moment sequence of the little $q$-Jacobi polynomials. It turns out that the corresponding determinant can be evaluated explicitly in terms of the Askey-Wilson polynomials.]]> 0 <![CDATA[We study variants of Gale-Robinson sequences, as motivated by cluster algebras with principal coefficients. For such cases, we give combinatorial interpretations of cluster variables using brane tilings, as from the physics literature.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2336 https://doi.org/10.46298/dmtcs.2336 Jeong, In-Jee Musiker, Gregg Zhang, Sicong Jeong, In-Jee Musiker, Gregg Zhang, Sicong <![CDATA[We study variants of Gale-Robinson sequences, as motivated by cluster algebras with principal coefficients. For such cases, we give combinatorial interpretations of cluster variables using brane tilings, as from the physics literature.]]> 0 <![CDATA[A function $g$, with domain the natural numbers, is a quasi-polynomial if there exists a period $m$ and polynomials $p_0,p_1,\ldots,p_m-1$ such that $g(t)=p_i(t)$ for $t\equiv i\bmod m$. Quasi-polynomials classically – and ``reasonably'' – appear in Ehrhart theory and in other contexts where one examines a family of polyhedra, parametrized by a variable t, and defined by linear inequalities of the form $a_1x_1+⋯+a_dx_d≤ b(t)$. Recent results of Chen, Li, Sam; Calegari, Walker; and Roune, Woods show a quasi-polynomial structure in several problems where the $a_i$ are also allowed to vary with $t$. We discuss these ``unreasonable'' results and conjecture a general class of sets that exhibit various (eventual) quasi-polynomial behaviors: sets $S_t$ that are defined with quantifiers $(\forall , ∃)$, boolean operations (and, or, not), and statements of the form $a_1(t)x_1+⋯+a_d(t)x_d ≤ b(t)$, where $a_i(t)$ and $b(t)$ are polynomials in $t$. These sets are a generalization of sets defined in the Presburger arithmetic. We prove several relationships between our conjectures, and we prove several special cases of the conjectures.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2335 https://doi.org/10.46298/dmtcs.2335 Woods, Kevin Woods, Kevin <![CDATA[A function $g$, with domain the natural numbers, is a quasi-polynomial if there exists a period $m$ and polynomials $p_0,p_1,\ldots,p_m-1$ such that $g(t)=p_i(t)$ for $t\equiv i\bmod m$. Quasi-polynomials classically – and ``reasonably'' – appear in Ehrhart theory and in other contexts where one examines a family of polyhedra, parametrized by a variable t, and defined by linear inequalities of the form $a_1x_1+⋯+a_dx_d≤ b(t)$. Recent results of Chen, Li, Sam; Calegari, Walker; and Roune, Woods show a quasi-polynomial structure in several problems where the $a_i$ are also allowed to vary with $t$. We discuss these ``unreasonable'' results and conjecture a general class of sets that exhibit various (eventual) quasi-polynomial behaviors: sets $S_t$ that are defined with quantifiers $(\forall , ∃)$, boolean operations (and, or, not), and statements of the form $a_1(t)x_1+⋯+a_d(t)x_d ≤ b(t)$, where $a_i(t)$ and $b(t)$ are polynomials in $t$. These sets are a generalization of sets defined in the Presburger arithmetic. We prove several relationships between our conjectures, and we prove several special cases of the conjectures.]]> 0 <![CDATA[We introduce the notion of $\textit{pattern}$ in the context of lattice paths, and investigate it in the specific case of Dyck paths. Similarly to the case of permutations, the pattern-containment relation defines a poset structure on the set of all Dyck paths, which we call the $\textit{Dyck pattern poset}$. Given a Dyck path $P$, we determine a formula for the number of Dyck paths covered by $P$, as well as for the number of Dyck paths covering $P$. We then address some typical pattern-avoidance issues, enumerating some classes of pattern-avoiding Dyck paths. Finally, we offer a conjecture concerning the asymptotic behavior of the sequence counting Dyck paths avoiding a generic pattern and we pose a series of open problems regarding the structure of the Dyck pattern poset.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2334 https://doi.org/10.46298/dmtcs.2334 Bernini, Antonio Ferrari, Luca Pinzani, Renzo West, Julian Bernini, Antonio Ferrari, Luca Pinzani, Renzo West, Julian <![CDATA[We introduce the notion of $\textit{pattern}$ in the context of lattice paths, and investigate it in the specific case of Dyck paths. Similarly to the case of permutations, the pattern-containment relation defines a poset structure on the set of all Dyck paths, which we call the $\textit{Dyck pattern poset}$. Given a Dyck path $P$, we determine a formula for the number of Dyck paths covered by $P$, as well as for the number of Dyck paths covering $P$. We then address some typical pattern-avoidance issues, enumerating some classes of pattern-avoiding Dyck paths. Finally, we offer a conjecture concerning the asymptotic behavior of the sequence counting Dyck paths avoiding a generic pattern and we pose a series of open problems regarding the structure of the Dyck pattern poset.]]> 0 <![CDATA[We study sorting operators $\textrm{A}$ on permutations that are obtained composing Knuth's stack sorting operator \textrmS and the reverse operator $\textrm{R}$, as many times as desired. For any such operator $\textrm{A}$, we provide a bijection between the set of permutations sorted by $\textrm{S} \circ \textrm{A}$ and the set of those sorted by $\textrm{S} \circ \textrm{R} \circ \textrm{A}$, proving that these sets are enumerated by the same sequence, but also that many classical permutation statistics are equidistributed across these two sets. The description of this family of bijections is based on an apparently novel bijection between the set of permutations avoiding the pattern $231$ and the set of those avoiding $132$ which preserves many permutation statistics. We also present other properties of this bijection, in particular for finding families of Wilf-equivalent permutation classes.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2333 https://doi.org/10.46298/dmtcs.2333 Albert, Michael Bouvel, Mathilde Albert, Michael Bouvel, Mathilde <![CDATA[We study sorting operators $\textrm{A}$ on permutations that are obtained composing Knuth's stack sorting operator \textrmS and the reverse operator $\textrm{R}$, as many times as desired. For any such operator $\textrm{A}$, we provide a bijection between the set of permutations sorted by $\textrm{S} \circ \textrm{A}$ and the set of those sorted by $\textrm{S} \circ \textrm{R} \circ \textrm{A}$, proving that these sets are enumerated by the same sequence, but also that many classical permutation statistics are equidistributed across these two sets. The description of this family of bijections is based on an apparently novel bijection between the set of permutations avoiding the pattern $231$ and the set of those avoiding $132$ which preserves many permutation statistics. We also present other properties of this bijection, in particular for finding families of Wilf-equivalent permutation classes.]]> 0 <![CDATA[For an arbitrary Coxeter group $W$, David Speyer and Nathan Reading defined Cambrian semilattices $C_{\gamma}$ as certain sub-semilattices of the weak order on $W$. In this article, we define an edge-labelling using the realization of Cambrian semilattices in terms of $\gamma$-sortable elements, and show that this is an EL-labelling for every closed interval of $C_{\gamma}$. In addition, we use our labelling to show that every finite open interval in a Cambrian semilattice is either contractible or spherical, and we characterize the spherical intervals, generalizing a result by Nathan Reading.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2320 https://doi.org/10.46298/dmtcs.2320 Kallipoliti, Myrto Mühle, Henri Kallipoliti, Myrto Mühle, Henri <![CDATA[For an arbitrary Coxeter group $W$, David Speyer and Nathan Reading defined Cambrian semilattices $C_{\gamma}$ as certain sub-semilattices of the weak order on $W$. In this article, we define an edge-labelling using the realization of Cambrian semilattices in terms of $\gamma$-sortable elements, and show that this is an EL-labelling for every closed interval of $C_{\gamma}$. In addition, we use our labelling to show that every finite open interval in a Cambrian semilattice is either contractible or spherical, and we characterize the spherical intervals, generalizing a result by Nathan Reading.]]> 0 <![CDATA[In a recent work, the combinatorial interpretation of the polynomial $\alpha (n; k_1,k_2,\ldots,k_n)$ counting the number of Monotone Triangles with bottom row $k_1 < k_2 < ⋯< k_n$ was extended to weakly decreasing sequences $k_1 ≥k_2 ≥⋯≥k_n$. In this case the evaluation of the polynomial is equal to a signed enumeration of objects called Decreasing Monotone Triangles. In this paper we define Generalized Monotone Triangles – a joint generalization of both ordinary Monotone Triangles and Decreasing Monotone Triangles. As main result of the paper we prove that the evaluation of $\alpha (n; k_1,k_2,\ldots,k_n)$ at arbitrary $(k_1,k_2,\ldots,k_n) ∈ \mathbb{Z}^n$ is a signed enumeration of Generalized Monotone Triangles with bottom row $(k_1,k_2,\ldots,k_n)$. Computational experiments indicate that certain evaluations of the polynomial at integral sequences yield well-known round numbers related to Alternating Sign Matrices. The main result provides a combinatorial interpretation of the conjectured identities and could turn out useful in giving bijective proofs.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2331 https://doi.org/10.46298/dmtcs.2331 Riegler, Lukas Riegler, Lukas <![CDATA[In a recent work, the combinatorial interpretation of the polynomial $\alpha (n; k_1,k_2,\ldots,k_n)$ counting the number of Monotone Triangles with bottom row $k_1 < k_2 < ⋯< k_n$ was extended to weakly decreasing sequences $k_1 ≥k_2 ≥⋯≥k_n$. In this case the evaluation of the polynomial is equal to a signed enumeration of objects called Decreasing Monotone Triangles. In this paper we define Generalized Monotone Triangles – a joint generalization of both ordinary Monotone Triangles and Decreasing Monotone Triangles. As main result of the paper we prove that the evaluation of $\alpha (n; k_1,k_2,\ldots,k_n)$ at arbitrary $(k_1,k_2,\ldots,k_n) ∈ \mathbb{Z}^n$ is a signed enumeration of Generalized Monotone Triangles with bottom row $(k_1,k_2,\ldots,k_n)$. Computational experiments indicate that certain evaluations of the polynomial at integral sequences yield well-known round numbers related to Alternating Sign Matrices. The main result provides a combinatorial interpretation of the conjectured identities and could turn out useful in giving bijective proofs.]]> 0 <![CDATA[In a recent paper with Bousquet-Mélou, de Gier, Duminil-Copin and Guttmann (2012), we proved that a model of self-avoiding walks on the honeycomb lattice, interacting with an impenetrable surface, undergoes an adsorption phase transition when the surface fugacity is 1+√2. Our proof used a generalisation of an identity obtained by Duminil-Copin and Smirnov (2012), and confirmed a conjecture of Batchelor and Yung (1995). Here we consider a similar model of self-avoiding walk adsorption on the honeycomb lattice, but with the impenetrable surface placed at a right angle to the previous orientation. For this model there also exists a conjecture for the critical surface fugacity, made by Batchelor, Bennett-Wood and Owczarek (1998). We adapt the methods of the earlier paper to this setting in order to prove the critical surface fugacity, but have to deal with several subtle complications which arise. This article is an abbreviated version of a paper of the same title, currently being prepared for submission.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2330 https://doi.org/10.46298/dmtcs.2330 Beaton, Nicholas R. Beaton, Nicholas R. <![CDATA[In a recent paper with Bousquet-Mélou, de Gier, Duminil-Copin and Guttmann (2012), we proved that a model of self-avoiding walks on the honeycomb lattice, interacting with an impenetrable surface, undergoes an adsorption phase transition when the surface fugacity is 1+√2. Our proof used a generalisation of an identity obtained by Duminil-Copin and Smirnov (2012), and confirmed a conjecture of Batchelor and Yung (1995). Here we consider a similar model of self-avoiding walk adsorption on the honeycomb lattice, but with the impenetrable surface placed at a right angle to the previous orientation. For this model there also exists a conjecture for the critical surface fugacity, made by Batchelor, Bennett-Wood and Owczarek (1998). We adapt the methods of the earlier paper to this setting in order to prove the critical surface fugacity, but have to deal with several subtle complications which arise. This article is an abbreviated version of a paper of the same title, currently being prepared for submission.]]> 0 <![CDATA[We study the statistics $\mathsf{area}$, $\mathsf{bounce}$ and $\mathsf{dinv}$ associated to polyominoes in a rectangular box $m$ times $n$. We show that the bi-statistics ($\mathsf{area}$,$\mathsf{bounce}$) and ($\mathsf{area}$,$\mathsf{dinv}$) give rise to the same $q,t-$analogue of Narayana numbers, which was introduced by two of these authors in a recent paper. We prove the main conjectures of that same work, i.e. the symmetries in $q$ and $t$, and in $m$ and $n$ of these polynomials, by providing a symmetric functions interpretation which relates them to the famous diagonal harmonics.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2329 https://doi.org/10.46298/dmtcs.2329 Aval, Jean-Christophe D'Adderio, Michele Dukes, Mark Hicks, Angela Le Borgne, Yvan Aval, Jean-Christophe D'Adderio, Michele Dukes, Mark Hicks, Angela Le Borgne, Yvan <![CDATA[We study the statistics $\mathsf{area}$, $\mathsf{bounce}$ and $\mathsf{dinv}$ associated to polyominoes in a rectangular box $m$ times $n$. We show that the bi-statistics ($\mathsf{area}$,$\mathsf{bounce}$) and ($\mathsf{area}$,$\mathsf{dinv}$) give rise to the same $q,t-$analogue of Narayana numbers, which was introduced by two of these authors in a recent paper. We prove the main conjectures of that same work, i.e. the symmetries in $q$ and $t$, and in $m$ and $n$ of these polynomials, by providing a symmetric functions interpretation which relates them to the famous diagonal harmonics.]]> 0 <![CDATA[We describe edge labelings of the increasing flip graph of a subword complex on a finite Coxeter group, and study applications thereof. On the one hand, we show that they provide canonical spanning trees of the facet-ridge graph of the subword complex, describe inductively these trees, and present their close relations to greedy facets. Searching these trees yields an efficient algorithm to generate all facets of the subword complex, which extends the greedy flip algorithm for pointed pseudotriangulations. On the other hand, when the increasing flip graph is a Hasse diagram, we show that the edge labeling is indeed an EL-labeling and derive further combinatorial properties of paths in the increasing flip graph. These results apply in particular to Cambrian lattices, in which case a similar EL-labeling was recently studied by M. Kallipoliti and H. Mühle.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2328 https://doi.org/10.46298/dmtcs.2328 Pilaud, Vincent Stump, Christian Pilaud, Vincent Stump, Christian <![CDATA[We describe edge labelings of the increasing flip graph of a subword complex on a finite Coxeter group, and study applications thereof. On the one hand, we show that they provide canonical spanning trees of the facet-ridge graph of the subword complex, describe inductively these trees, and present their close relations to greedy facets. Searching these trees yields an efficient algorithm to generate all facets of the subword complex, which extends the greedy flip algorithm for pointed pseudotriangulations. On the other hand, when the increasing flip graph is a Hasse diagram, we show that the edge labeling is indeed an EL-labeling and derive further combinatorial properties of paths in the increasing flip graph. These results apply in particular to Cambrian lattices, in which case a similar EL-labeling was recently studied by M. Kallipoliti and H. Mühle.]]> 0 <![CDATA[Let $\mathcal{A}$ be a minor-closed class of labelled graphs, and let $G_n$ be a random graph sampled uniformly from the set of n-vertex graphs of $\mathcal{A}$. When $n$ is large, what is the probability that $G_n$ is connected? How many components does it have? How large is its biggest component? Thanks to the work of McDiarmid and his collaborators, these questions are now solved when all excluded minors are 2-connected. Using exact enumeration, we study a collection of classes $\mathcal{A}$ excluding non-2-connected minors, and show that their asymptotic behaviour is sometimes rather different from the 2-connected case. This behaviour largely depends on the nature of the dominant singularity of the generating function $C(z)$ that counts connected graphs of $\mathcal{A}$. We classify our examples accordingly, thus taking a first step towards a classification of minor-closed classes of graphs. Furthermore, we investigate a parameter that has not received any attention in this context yet: the size of the root component. This follows non-gaussian limit laws (beta and gamma), and clearly deserves a systematic investigation.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2327 https://doi.org/10.46298/dmtcs.2327 Bousquet-Mélou, Mireille Weller, Kerstin Bousquet-Mélou, Mireille Weller, Kerstin <![CDATA[Let $\mathcal{A}$ be a minor-closed class of labelled graphs, and let $G_n$ be a random graph sampled uniformly from the set of n-vertex graphs of $\mathcal{A}$. When $n$ is large, what is the probability that $G_n$ is connected? How many components does it have? How large is its biggest component? Thanks to the work of McDiarmid and his collaborators, these questions are now solved when all excluded minors are 2-connected. Using exact enumeration, we study a collection of classes $\mathcal{A}$ excluding non-2-connected minors, and show that their asymptotic behaviour is sometimes rather different from the 2-connected case. This behaviour largely depends on the nature of the dominant singularity of the generating function $C(z)$ that counts connected graphs of $\mathcal{A}$. We classify our examples accordingly, thus taking a first step towards a classification of minor-closed classes of graphs. Furthermore, we investigate a parameter that has not received any attention in this context yet: the size of the root component. This follows non-gaussian limit laws (beta and gamma), and clearly deserves a systematic investigation.]]> 0 <![CDATA[We give closed combinatorial product formulas for Kazhdan–Lusztig poynomials and their parabolic analogue of type $q$ in the case of boolean elements, introduced in [M. Marietti, Boolean elements in Kazhdan–Lusztig theory, J. Algebra 295 (2006)], in Coxeter groups whose Coxeter graph is a tree. Such formulas involve Catalan numbers and use a combinatorial interpretation of the Coxeter graph of the group. In the case of classical Weyl groups, this combinatorial interpretation can be restated in terms of statistics of (signed) permutations. As an application of the formulas, we compute the intersection homology Poincaré polynomials of the Schubert varieties of boolean elements.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2326 https://doi.org/10.46298/dmtcs.2326 Mongelli, Pietro Mongelli, Pietro <![CDATA[We give closed combinatorial product formulas for Kazhdan–Lusztig poynomials and their parabolic analogue of type $q$ in the case of boolean elements, introduced in [M. Marietti, Boolean elements in Kazhdan–Lusztig theory, J. Algebra 295 (2006)], in Coxeter groups whose Coxeter graph is a tree. Such formulas involve Catalan numbers and use a combinatorial interpretation of the Coxeter graph of the group. In the case of classical Weyl groups, this combinatorial interpretation can be restated in terms of statistics of (signed) permutations. As an application of the formulas, we compute the intersection homology Poincaré polynomials of the Schubert varieties of boolean elements.]]> 0 <![CDATA[The action of the symmetric group $S_n$ on the set $\mathrm{Park}_n$ of parking functions of size $n$ has received a great deal of attention in algebraic combinatorics. We prove that the action of $S_n$ on $\mathrm{Park}_n$ extends to an action of $S_{n+1}$. More precisely, we construct a graded $S_{n+1}$-module $V_n$ such that the restriction of $V_n$ to $S_n$ is isomorphic to $\mathrm{Park}_n$. We describe the $S_n$-Frobenius characters of the module $V_n$ in all degrees and describe the $S_{n+1}$-Frobenius characters of $V_n$ in extreme degrees. We give a bivariate generalization $V_n^{(\ell, m)}$ of our module $V_n$ whose representation theory is governed by a bivariate generalization of Dyck paths. A Fuss generalization of our results is a special case of this bivariate generalization.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2325 https://doi.org/10.46298/dmtcs.2325 Berget, Andrew Rhoades, Brendon Berget, Andrew Rhoades, Brendon <![CDATA[The action of the symmetric group $S_n$ on the set $\mathrm{Park}_n$ of parking functions of size $n$ has received a great deal of attention in algebraic combinatorics. We prove that the action of $S_n$ on $\mathrm{Park}_n$ extends to an action of $S_{n+1}$. More precisely, we construct a graded $S_{n+1}$-module $V_n$ such that the restriction of $V_n$ to $S_n$ is isomorphic to $\mathrm{Park}_n$. We describe the $S_n$-Frobenius characters of the module $V_n$ in all degrees and describe the $S_{n+1}$-Frobenius characters of $V_n$ in extreme degrees. We give a bivariate generalization $V_n^{(\ell, m)}$ of our module $V_n$ whose representation theory is governed by a bivariate generalization of Dyck paths. A Fuss generalization of our results is a special case of this bivariate generalization.]]> 0 <![CDATA[We give noncommutative versions of the Redfield-Pólya theorem in $\mathrm{WSym}$, the algebra of word symmetric functions, and in other related combinatorial Hopf algebras.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2324 https://doi.org/10.46298/dmtcs.2324 Bultel, Jean-Paul Chouria, Ali Luque, Jean-Gabriel Mallet, Olivier Bultel, Jean-Paul Chouria, Ali Luque, Jean-Gabriel Mallet, Olivier <![CDATA[We give noncommutative versions of the Redfield-Pólya theorem in $\mathrm{WSym}$, the algebra of word symmetric functions, and in other related combinatorial Hopf algebras.]]> 0 <![CDATA[The Hecke algebra of the pair $(\mathcal{S}_{2n}, \mathcal{B}_n)$, where $\mathcal{B}_n$ is the hyperoctahedral subgroup of $\mathcal{S}_{2n}$, was introduced by James in 1961. It is a natural analogue of the center of the symmetric group algebra. In this paper, we give a polynomiality property of its structure coefficients. Our main tool is a combinatorial universal algebra which projects on the Hecke algebra of $(\mathcal{S}_{2n}, \mathcal{B}_n)$ for every $n$. To build it, we introduce new objects called partial bijections.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2323 https://doi.org/10.46298/dmtcs.2323 Tout, Omar Tout, Omar <![CDATA[The Hecke algebra of the pair $(\mathcal{S}_{2n}, \mathcal{B}_n)$, where $\mathcal{B}_n$ is the hyperoctahedral subgroup of $\mathcal{S}_{2n}$, was introduced by James in 1961. It is a natural analogue of the center of the symmetric group algebra. In this paper, we give a polynomiality property of its structure coefficients. Our main tool is a combinatorial universal algebra which projects on the Hecke algebra of $(\mathcal{S}_{2n}, \mathcal{B}_n)$ for every $n$. To build it, we introduce new objects called partial bijections.]]> 0 <![CDATA[We consider a deformation of Kerov character polynomials, linked to Jack symmetric functions. It has been introduced recently by M. Lassalle, who formulated several conjectures on these objects, suggesting some underlying combinatorics. We give a partial result in this direction, showing that some quantities are polynomials in the Jack parameter $\alpha$ with prescribed degree. Our result has several interesting consequences in various directions. Firstly, we give a new proof of the fact that the coefficients of Jack polynomials expanded in the monomial or power-sum basis depend polynomially in $\alpha$. Secondly, we describe asymptotically the shape of random Young diagrams under some deformation of Plancherel measure.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2322 https://doi.org/10.46298/dmtcs.2322 Féray, Valentin Dołęga, Maciej Féray, Valentin Dołęga, Maciej <![CDATA[We consider a deformation of Kerov character polynomials, linked to Jack symmetric functions. It has been introduced recently by M. Lassalle, who formulated several conjectures on these objects, suggesting some underlying combinatorics. We give a partial result in this direction, showing that some quantities are polynomials in the Jack parameter $\alpha$ with prescribed degree. Our result has several interesting consequences in various directions. Firstly, we give a new proof of the fact that the coefficients of Jack polynomials expanded in the monomial or power-sum basis depend polynomially in $\alpha$. Secondly, we describe asymptotically the shape of random Young diagrams under some deformation of Plancherel measure.]]> 0 <![CDATA[We give an intrinsic proof of a conjecture of Brenti that all the roots of the Eulerian polynomial of type $D$ are real and a proof of a conjecture of Dilks, Petersen, and Stembridge that all the roots of the affine Eulerian polynomial of type $B$ are real, as well.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2321 https://doi.org/10.46298/dmtcs.2321 Savage, Carla D. Visontai, Mirkó Savage, Carla D. Visontai, Mirkó <![CDATA[We give an intrinsic proof of a conjecture of Brenti that all the roots of the Eulerian polynomial of type $D$ are real and a proof of a conjecture of Dilks, Petersen, and Stembridge that all the roots of the affine Eulerian polynomial of type $B$ are real, as well.]]> 0 <![CDATA[A convex polyomino is $k$-$\textit{convex}$ if every pair of its cells can be connected by means of a $\textit{monotone path}$, internal to the polyomino, and having at most $k$ changes of direction. The number $k$-convex polyominoes of given semi-perimeter has been determined only for small values of $k$, precisely $k=1,2$. In this paper we consider the problem of enumerating a subclass of $k$-convex polyominoes, precisely the $k$-$\textit{convex parallelogram polyominoes}$ (briefly, $k$-$\textit{parallelogram polyominoes}$). For each $k \geq 1$, we give a recursive decomposition for the class of $k$-parallelogram polyominoes, and then use it to obtain the generating function of the class, which turns out to be a rational function. We are then able to express such a generating function in terms of the $\textit{Fibonacci polynomials}$.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2370 https://doi.org/10.46298/dmtcs.2370 Battaglino, Daniela Fédou, Jean-Marc Rinaldi, Simone Socci, Samanta Battaglino, Daniela Fédou, Jean-Marc Rinaldi, Simone Socci, Samanta <![CDATA[A convex polyomino is $k$-$\textit{convex}$ if every pair of its cells can be connected by means of a $\textit{monotone path}$, internal to the polyomino, and having at most $k$ changes of direction. The number $k$-convex polyominoes of given semi-perimeter has been determined only for small values of $k$, precisely $k=1,2$. In this paper we consider the problem of enumerating a subclass of $k$-convex polyominoes, precisely the $k$-$\textit{convex parallelogram polyominoes}$ (briefly, $k$-$\textit{parallelogram polyominoes}$). For each $k \geq 1$, we give a recursive decomposition for the class of $k$-parallelogram polyominoes, and then use it to obtain the generating function of the class, which turns out to be a rational function. We are then able to express such a generating function in terms of the $\textit{Fibonacci polynomials}$.]]> 0 <![CDATA[We prove that the complement of a complexified toric arrangement has the homotopy type of a minimal CW-complex, and thus its homology is torsion-free. To this end, we consider the toric Salvetti complex, a combinatorial model for the arrangement's complement. Using diagrams of acyclic categories we obtain a stratification of this combinatorial model that explicitly associates generators in homology to the "local no-broken-circuit sets'' defined in terms of the incidence relations of the arrangement. Then we apply a suitably generalized form of Discrete Morse Theory to describe a sequence of elementary collapses leading from the full model to a minimal complex.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2374 https://doi.org/10.46298/dmtcs.2374 d'Antonio, Giacomo Delucchi, Emanuele d'Antonio, Giacomo Delucchi, Emanuele <![CDATA[We prove that the complement of a complexified toric arrangement has the homotopy type of a minimal CW-complex, and thus its homology is torsion-free. To this end, we consider the toric Salvetti complex, a combinatorial model for the arrangement's complement. Using diagrams of acyclic categories we obtain a stratification of this combinatorial model that explicitly associates generators in homology to the "local no-broken-circuit sets'' defined in terms of the incidence relations of the arrangement. Then we apply a suitably generalized form of Discrete Morse Theory to describe a sequence of elementary collapses leading from the full model to a minimal complex.]]> 0 <![CDATA[For a given sequence $\alpha = [\alpha_1,\alpha_2,\ldots , \alpha_N, \alpha_{N+1}]$ of $N+1$ positive integers, we consider the combinatorial function $E(\alpha)(t)$ that counts the nonnegative integer solutions of the equation $\alpha_1x_1+\alpha_2 x_2+ \ldots+ \alpha_Nx_N+ \alpha_{N+1}x_{N+1}=t$, where the right-hand side $t$ is a varying nonnegative integer. It is well-known that $E(\alpha)(t)$ is a quasipolynomial function of $t$ of degree $N$. In combinatorial number theory this function is known as the $\textit{denumerant}$. Our main result is a new algorithm that, for every fixed number $k$, computes in polynomial time the highest $k+1$ coefficients of the quasi-polynomial $E(\alpha)(t)$ as step polynomials of $t$. Our algorithm is a consequence of a nice poset structure on the poles of the associated rational generating function for $E(\alpha)(t)$ and the geometric reinterpretation of some rational generating functions in terms of lattice points in polyhedral cones. Experiments using a $\texttt{MAPLE}$ implementation will be posted separately.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2373 https://doi.org/10.46298/dmtcs.2373 Baldoni, Velleda Berline, Nicole Dutra, Brandon Köppe, Matthias Vergne, Michele De Loera, Jesus Baldoni, Velleda Berline, Nicole Dutra, Brandon Köppe, Matthias Vergne, Michele De Loera, Jesus <![CDATA[For a given sequence $\alpha = [\alpha_1,\alpha_2,\ldots , \alpha_N, \alpha_{N+1}]$ of $N+1$ positive integers, we consider the combinatorial function $E(\alpha)(t)$ that counts the nonnegative integer solutions of the equation $\alpha_1x_1+\alpha_2 x_2+ \ldots+ \alpha_Nx_N+ \alpha_{N+1}x_{N+1}=t$, where the right-hand side $t$ is a varying nonnegative integer. It is well-known that $E(\alpha)(t)$ is a quasipolynomial function of $t$ of degree $N$. In combinatorial number theory this function is known as the $\textit{denumerant}$. Our main result is a new algorithm that, for every fixed number $k$, computes in polynomial time the highest $k+1$ coefficients of the quasi-polynomial $E(\alpha)(t)$ as step polynomials of $t$. Our algorithm is a consequence of a nice poset structure on the poles of the associated rational generating function for $E(\alpha)(t)$ and the geometric reinterpretation of some rational generating functions in terms of lattice points in polyhedral cones. Experiments using a $\texttt{MAPLE}$ implementation will be posted separately.]]> 0 <![CDATA[In a first part, we formalize the construction of combinatorial Hopf algebras from plactic-like monoids using polynomial realizations. Thank to this construction we reveal a lattice structure on those combinatorial Hopf algebras. As an application, we construct a new combinatorial Hopf algebra on binary trees with multiplicities and use it to prove a hook length formula for those trees.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2372 https://doi.org/10.46298/dmtcs.2372 Priez, Jean-Baptiste Priez, Jean-Baptiste <![CDATA[In a first part, we formalize the construction of combinatorial Hopf algebras from plactic-like monoids using polynomial realizations. Thank to this construction we reveal a lattice structure on those combinatorial Hopf algebras. As an application, we construct a new combinatorial Hopf algebra on binary trees with multiplicities and use it to prove a hook length formula for those trees.]]> 0 <![CDATA[We introduce two families of symmetric functions with an extra parameter $t$ that specialize to Schubert representatives for cohomology and homology of the affine Grassmannian when $t=1$. The families are defined by a statistic on combinatorial objects associated to the type-$A$ affine Weyl group and their transition matrix with Hall-Littlewood polynomials is $t$-positive. We conjecture that one family is the set of $k$-atoms.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2371 https://doi.org/10.46298/dmtcs.2371 Dalal, Avinash J. Morse, Jennifer Dalal, Avinash J. Morse, Jennifer <![CDATA[We introduce two families of symmetric functions with an extra parameter $t$ that specialize to Schubert representatives for cohomology and homology of the affine Grassmannian when $t=1$. The families are defined by a statistic on combinatorial objects associated to the type-$A$ affine Weyl group and their transition matrix with Hall-Littlewood polynomials is $t$-positive. We conjecture that one family is the set of $k$-atoms.]]> 0 <![CDATA[Consider the triangulations of a convex polygon with $n$ vertices. In 1988, Daniel Sleator, Robert Tarjan, and William Thurston have shown that the flip distance of two such triangulations is at most $2n-10$ when $n$ is greater than 12 and that this bound is sharp when $n$ is large enough. They also conjecture that `"large enough'' means greater than 12. A proof of this conjecture was recently announced by the author. A sketch of this proof is given here, with emphasis on the intuitions underlying the construction of lower bounds on the flip distance of two triangulations.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12788 https://doi.org/10.46298/dmtcs.12788 Pournin, Lionel Pournin, Lionel <![CDATA[Consider the triangulations of a convex polygon with $n$ vertices. In 1988, Daniel Sleator, Robert Tarjan, and William Thurston have shown that the flip distance of two such triangulations is at most $2n-10$ when $n$ is greater than 12 and that this bound is sharp when $n$ is large enough. They also conjecture that `"large enough'' means greater than 12. A proof of this conjecture was recently announced by the author. A sketch of this proof is given here, with emphasis on the intuitions underlying the construction of lower bounds on the flip distance of two triangulations.]]> 0 <![CDATA[Constellations and hypermaps generalize combinatorial maps, $\textit{i.e.}$ embedding of graphs in a surface, in terms of factorization of permutations. In this paper, we extend a result of Jackson and Visentin (1990) on an enumerative relation between quadrangulations and bipartite quadrangulations. We show a similar relation between hypermaps and constellations by generalizing a result in the original paper on factorization of characters. Using this enumerative relation, we recover a result on the asymptotic behavior of hypermaps of Chapuy (2009).]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12789 https://doi.org/10.46298/dmtcs.12789 Fang, Wenjie Fang, Wenjie <![CDATA[Constellations and hypermaps generalize combinatorial maps, $\textit{i.e.}$ embedding of graphs in a surface, in terms of factorization of permutations. In this paper, we extend a result of Jackson and Visentin (1990) on an enumerative relation between quadrangulations and bipartite quadrangulations. We show a similar relation between hypermaps and constellations by generalizing a result in the original paper on factorization of characters. Using this enumerative relation, we recover a result on the asymptotic behavior of hypermaps of Chapuy (2009).]]> 0 <![CDATA[We present a uniform construction of tensor products of one-column Kirillov–Reshetikhin (KR) crystals in all untwisted affine types, which uses a generalization of the Lakshmibai–Seshadri paths (in the theory of the Littelmann path model). This generalization is based on the graph on parabolic cosets of a Weyl group known as the parabolic quantum Bruhat graph. A related model is the so-called quantum alcove model. The proof is based on two lifts of the parabolic quantum Bruhat graph: to the Bruhat order on the affine Weyl group and to Littelmann's poset on level-zero weights. Our construction leads to a simple calculation of the energy function. It also implies the equality between a Macdonald polynomial specialized at $t=0$ and the graded character of a tensor product of KR modules.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12790 https://doi.org/10.46298/dmtcs.12790 Lenart, Cristian Naito, Satoshi Sagaki, Daisuke Schilling, Anne Shimozono, Mark Lenart, Cristian Naito, Satoshi Sagaki, Daisuke Schilling, Anne Shimozono, Mark <![CDATA[We present a uniform construction of tensor products of one-column Kirillov–Reshetikhin (KR) crystals in all untwisted affine types, which uses a generalization of the Lakshmibai–Seshadri paths (in the theory of the Littelmann path model). This generalization is based on the graph on parabolic cosets of a Weyl group known as the parabolic quantum Bruhat graph. A related model is the so-called quantum alcove model. The proof is based on two lifts of the parabolic quantum Bruhat graph: to the Bruhat order on the affine Weyl group and to Littelmann's poset on level-zero weights. Our construction leads to a simple calculation of the energy function. It also implies the equality between a Macdonald polynomial specialized at $t=0$ and the graded character of a tensor product of KR modules.]]> 0 <![CDATA[We develop a new method for studying the asymptotics of symmetric polynomials of representation–theoretic origin as the number of variables tends to infinity. Several applications of our method are presented: We prove a number of theorems concerning characters of infinite–dimensional unitary group and their $q$–deformations. We study the behavior of uniformly random lozenge tilings of large polygonal domains and find the GUE–eigenvalues distribution in the limit. We also investigate similar behavior for Alternating Sign Matrices (equivalently, six–vertex model with domain wall boundary conditions). Finally, we compute the asymptotic expansion of certain observables in the $O(n=1)$ dense loop model.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12791 https://doi.org/10.46298/dmtcs.12791 Gorin, Vadim Panova, Greta Gorin, Vadim Panova, Greta <![CDATA[We develop a new method for studying the asymptotics of symmetric polynomials of representation–theoretic origin as the number of variables tends to infinity. Several applications of our method are presented: We prove a number of theorems concerning characters of infinite–dimensional unitary group and their $q$–deformations. We study the behavior of uniformly random lozenge tilings of large polygonal domains and find the GUE–eigenvalues distribution in the limit. We also investigate similar behavior for Alternating Sign Matrices (equivalently, six–vertex model with domain wall boundary conditions). Finally, we compute the asymptotic expansion of certain observables in the $O(n=1)$ dense loop model.]]> 0 <![CDATA[This article investigates combinatorial properties of non-ambiguous trees. These objects we define may be seen either as binary trees drawn on a grid with some constraints, or as a subset of the tree-like tableaux previously defined by Aval, Boussicault and Nadeau. The enumeration of non-ambiguous trees satisfying some additional constraints allows us to give elegant combinatorial proofs of identities due to Carlitz, and to Ehrenborg and Steingrímsson. We also provide a hook formula to count the number of non-ambiguous trees with a given underlying tree. Finally, we use non-ambiguous trees to describe a very natural bijection between parallelogram polyominoes and binary trees.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12792 https://doi.org/10.46298/dmtcs.12792 Aval, Jean-Christophe Boussicault, Adrien Bouvel, Mathilde Silimbani, Matteo Aval, Jean-Christophe Boussicault, Adrien Bouvel, Mathilde Silimbani, Matteo <![CDATA[This article investigates combinatorial properties of non-ambiguous trees. These objects we define may be seen either as binary trees drawn on a grid with some constraints, or as a subset of the tree-like tableaux previously defined by Aval, Boussicault and Nadeau. The enumeration of non-ambiguous trees satisfying some additional constraints allows us to give elegant combinatorial proofs of identities due to Carlitz, and to Ehrenborg and Steingrímsson. We also provide a hook formula to count the number of non-ambiguous trees with a given underlying tree. Finally, we use non-ambiguous trees to describe a very natural bijection between parallelogram polyominoes and binary trees.]]> 0 <![CDATA[Deodhar introduced his decomposition of partial flag varieties as a tool for understanding Kazhdan-Lusztig polynomials. The Deodhar decomposition of the Grassmannian is also useful in the context of soliton solutions to the KP equation, as shown by Kodama and the second author. Deodhar components $S_D$ of the Grassmannian are in bijection with certain tableaux $D$ called $\textit{Go-diagrams}$, and each component is isomorphic to $(\mathbb{K}^*)^a ×\mathbb{K})^b$ for some non-negative integers $a$ and $b$. Our main result is an explicit parameterization of each Deodhar component in the Grassmannian in terms of networks. More specifically, from a Go-diagram $D$ we construct a weighted network $N_D$ and its $\textit{weight matrix}$ $W_D$, whose entries enumerate directed paths in $N_D$. By letting the weights in the network vary over $\mathbb{K}$ or $\mathbb{K} ^*$ as appropriate, one gets a parametrization of the Deodhar component $S_D$. One application of such a parametrization is that one may immediately determine which Plücker coordinates are vanishing and nonvanishing, by using the Lindstrom-Gessel-Viennot Lemma. We also give a (minimal) characterization of each Deodhar component in terms of Plücker coordinates.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12793 https://doi.org/10.46298/dmtcs.12793 Talaska, Kelli Williams, Lauren Talaska, Kelli Williams, Lauren <![CDATA[Deodhar introduced his decomposition of partial flag varieties as a tool for understanding Kazhdan-Lusztig polynomials. The Deodhar decomposition of the Grassmannian is also useful in the context of soliton solutions to the KP equation, as shown by Kodama and the second author. Deodhar components $S_D$ of the Grassmannian are in bijection with certain tableaux $D$ called $\textit{Go-diagrams}$, and each component is isomorphic to $(\mathbb{K}^*)^a ×\mathbb{K})^b$ for some non-negative integers $a$ and $b$. Our main result is an explicit parameterization of each Deodhar component in the Grassmannian in terms of networks. More specifically, from a Go-diagram $D$ we construct a weighted network $N_D$ and its $\textit{weight matrix}$ $W_D$, whose entries enumerate directed paths in $N_D$. By letting the weights in the network vary over $\mathbb{K}$ or $\mathbb{K} ^*$ as appropriate, one gets a parametrization of the Deodhar component $S_D$. One application of such a parametrization is that one may immediately determine which Plücker coordinates are vanishing and nonvanishing, by using the Lindstrom-Gessel-Viennot Lemma. We also give a (minimal) characterization of each Deodhar component in terms of Plücker coordinates.]]> 0 <![CDATA[We study the vector spaces and integer lattices of cuts and flows of an arbitrary finite CW complex, and their relationships to its critical group and related invariants. Our results extend the theory of cuts and flows in graphs, in particular the work of Bacher, de la Harpe and Nagnibeda. We construct explicit bases for the cut and flow spaces, interpret their coefficients topologically, and describe sufficient conditions for them to be integral bases of the cut and flow lattices. Second, we determine the precise relationships between the discriminant groups of the cut and flow lattices and the higher critical and cocritical groups; these are expressed as short exact sequences with error terms corresponding to torsion (co)homology. As an application, we generalize a result of Kotani and Sunada to give bounds for the complexity, girth, and connectivity of a complex in terms of Hermite's constant.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12794 https://doi.org/10.46298/dmtcs.12794 Duval, Art M. Klivans, Caroline J. Martin, Jeremy L. Duval, Art M. Klivans, Caroline J. Martin, Jeremy L. <![CDATA[We study the vector spaces and integer lattices of cuts and flows of an arbitrary finite CW complex, and their relationships to its critical group and related invariants. Our results extend the theory of cuts and flows in graphs, in particular the work of Bacher, de la Harpe and Nagnibeda. We construct explicit bases for the cut and flow spaces, interpret their coefficients topologically, and describe sufficient conditions for them to be integral bases of the cut and flow lattices. Second, we determine the precise relationships between the discriminant groups of the cut and flow lattices and the higher critical and cocritical groups; these are expressed as short exact sequences with error terms corresponding to torsion (co)homology. As an application, we generalize a result of Kotani and Sunada to give bounds for the complexity, girth, and connectivity of a complex in terms of Hermite's constant.]]> 0 <![CDATA[We present two simple descriptions of the denominator vectors of the cluster variables of a cluster algebra of finite type, with respect to any initial cluster seed: one in terms of the compatibility degrees between almost positive roots defined by S. Fomin and A. Zelevinsky, and the other in terms of the root function of a certain subword complex. These descriptions only rely on linear algebra, and provide simple proofs of the known fact that the $d$-vector of any non-initial cluster variable with respect to any initial cluster seed has non-negative entries and is different from zero.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12795 https://doi.org/10.46298/dmtcs.12795 Ceballos, Cesar Pilaud, Vincent Ceballos, Cesar Pilaud, Vincent <![CDATA[We present two simple descriptions of the denominator vectors of the cluster variables of a cluster algebra of finite type, with respect to any initial cluster seed: one in terms of the compatibility degrees between almost positive roots defined by S. Fomin and A. Zelevinsky, and the other in terms of the root function of a certain subword complex. These descriptions only rely on linear algebra, and provide simple proofs of the known fact that the $d$-vector of any non-initial cluster variable with respect to any initial cluster seed has non-negative entries and is different from zero.]]> 0 <![CDATA[The descent set of an oscillating (or up-down) tableau is introduced. This descent set plays the same role in the representation theory of the symplectic groups as the descent set of a standard tableau plays in the representation theory of the general linear groups. In particular, we show that the descent set is preserved by Sundaram's correspondence. This gives a direct combinatorial interpretation of the branching rules for the defining representations of the symplectic groups; equivalently, for the Frobenius character of the action of a symmetric group on an isotypic subspace in a tensor power of the defining representation of a symplectic group.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12796 https://doi.org/10.46298/dmtcs.12796 Rubey, Martin Sagan, Bruce E. Westbury, Bruce W. Rubey, Martin Sagan, Bruce E. Westbury, Bruce W. <![CDATA[The descent set of an oscillating (or up-down) tableau is introduced. This descent set plays the same role in the representation theory of the symplectic groups as the descent set of a standard tableau plays in the representation theory of the general linear groups. In particular, we show that the descent set is preserved by Sundaram's correspondence. This gives a direct combinatorial interpretation of the branching rules for the defining representations of the symplectic groups; equivalently, for the Frobenius character of the action of a symmetric group on an isotypic subspace in a tensor power of the defining representation of a symplectic group.]]> 0 <![CDATA[In this paper we give a new proof of the universality of the Tutte polynomial for matroids. This proof uses appropriate characters of Hopf algebra of matroids, algebra introduced by Schmitt (1994). We show that these Hopf algebra characters are solutions of some differential equations which are of the same type as the differential equations used to describe the renormalization group flow in quantum field theory. This approach allows us to also prove, in a different way, a matroid Tutte polynomial convolution formula published by Kook, Reiner and Stanton (1999). This FPSAC contribution is an extended abstract.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12821 https://doi.org/10.46298/dmtcs.12821 Duchamp, G. Hoang-Nghia, N. Krajewski, Thomas Tanasa, A. Duchamp, G. Hoang-Nghia, N. Krajewski, Thomas Tanasa, A. <![CDATA[In this paper we give a new proof of the universality of the Tutte polynomial for matroids. This proof uses appropriate characters of Hopf algebra of matroids, algebra introduced by Schmitt (1994). We show that these Hopf algebra characters are solutions of some differential equations which are of the same type as the differential equations used to describe the renormalization group flow in quantum field theory. This approach allows us to also prove, in a different way, a matroid Tutte polynomial convolution formula published by Kook, Reiner and Stanton (1999). This FPSAC contribution is an extended abstract.]]> 0 <![CDATA[$\textit{Adinkras}$ are graphical tools created for the study of supersymmetry representations. Besides having inherent interest for physicists, the study of adinkras has already shown connections with coding theory and Clifford algebras. Furthermore, adinkras offer many natural and accessible mathematical problems of combinatorial nature. We present the foundations for a mathematical audience, make new connections to other fields (homological algebra, poset theory, and polytopes), and solve some of these problems. Original results include the enumeration of all hypercube adinkras through dimension 5, the enumeration of odd dashings of adinkras for any dimension, and a connection between rankings and the chromatic polynomial for certain graphs.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12826 https://doi.org/10.46298/dmtcs.12826 Zhang, Yan X. Zhang, Yan X. <![CDATA[$\textit{Adinkras}$ are graphical tools created for the study of supersymmetry representations. Besides having inherent interest for physicists, the study of adinkras has already shown connections with coding theory and Clifford algebras. Furthermore, adinkras offer many natural and accessible mathematical problems of combinatorial nature. We present the foundations for a mathematical audience, make new connections to other fields (homological algebra, poset theory, and polytopes), and solve some of these problems. Original results include the enumeration of all hypercube adinkras through dimension 5, the enumeration of odd dashings of adinkras for any dimension, and a connection between rankings and the chromatic polynomial for certain graphs.]]> 0 <![CDATA[Weighted hypertrees have been used by C. Jensen, J. McCammond, and J. Meier to compute some Euler characteristics in group theory. We link them to decorated hypertrees and 2-coloured rooted trees. After the enumeration of pointed and non-pointed types of decorated hypertrees, we compute the character for the action of the symmetric group on these hypertrees.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12825 https://doi.org/10.46298/dmtcs.12825 Oger, Bérénice Oger, Bérénice <![CDATA[Weighted hypertrees have been used by C. Jensen, J. McCammond, and J. Meier to compute some Euler characteristics in group theory. We link them to decorated hypertrees and 2-coloured rooted trees. After the enumeration of pointed and non-pointed types of decorated hypertrees, we compute the character for the action of the symmetric group on these hypertrees.]]> 0 <![CDATA[We introduce new combinatorial objects, the interval-posets, that encode intervals of the Tamari lattice. We then find a combinatorial interpretation of the bilinear form that appears in the functional equation of Tamari intervals described by Chapoton. Thus, we retrieve this functional equation and prove that the polynomial recursively computed from the bilinear form on each tree $T$ counts the number of trees smaller than $T$ in the Tamari order.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12824 https://doi.org/10.46298/dmtcs.12824 Chatel, Grégory Pons, Viviane Chatel, Grégory Pons, Viviane <![CDATA[We introduce new combinatorial objects, the interval-posets, that encode intervals of the Tamari lattice. We then find a combinatorial interpretation of the bilinear form that appears in the functional equation of Tamari intervals described by Chapoton. Thus, we retrieve this functional equation and prove that the polynomial recursively computed from the bilinear form on each tree $T$ counts the number of trees smaller than $T$ in the Tamari order.]]> 0 <![CDATA[The $d$-dimensional simplicial, terminal, and reflexive polytopes with at least $3d-2$ vertices are classified. In particular, it turns out that all of them are smooth Fano polytopes. This improves previous results of Casagrande (2006) and Øbro (2008). Smooth Fano polytopes play a role in algebraic geometry and mathematical physics. This text is an extended abstract of Assarf et al. (2012).]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12823 https://doi.org/10.46298/dmtcs.12823 Assarf, Benjamin Joswig, Michael Paffenholz, Andreas Assarf, Benjamin Joswig, Michael Paffenholz, Andreas <![CDATA[The $d$-dimensional simplicial, terminal, and reflexive polytopes with at least $3d-2$ vertices are classified. In particular, it turns out that all of them are smooth Fano polytopes. This improves previous results of Casagrande (2006) and Øbro (2008). Smooth Fano polytopes play a role in algebraic geometry and mathematical physics. This text is an extended abstract of Assarf et al. (2012).]]> 0 <![CDATA[The $(q,r)$-Eulerian polynomials are the $(\mathrm{maj-exc, fix, exc})$ enumerative polynomials of permutations. Using Shareshian and Wachs' exponential generating function of these Eulerian polynomials, Chung and Graham proved two symmetrical $q$-Eulerian identities and asked for bijective proofs. We provide such proofs using Foata and Han's three-variable statistic $(\mathrm{inv-lec, pix, lec})$. We also prove a new recurrence formula for the $(q,r)$-Eulerian polynomials and study a $q$-analogue of Chung and Graham's restricted Eulerian polynomials. In particular, we obtain a symmetrical identity for these restricted $q$-Eulerian polynomials with a combinatorial proof.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12822 https://doi.org/10.46298/dmtcs.12822 Lin, Zhicong Lin, Zhicong <![CDATA[The $(q,r)$-Eulerian polynomials are the $(\mathrm{maj-exc, fix, exc})$ enumerative polynomials of permutations. Using Shareshian and Wachs' exponential generating function of these Eulerian polynomials, Chung and Graham proved two symmetrical $q$-Eulerian identities and asked for bijective proofs. We provide such proofs using Foata and Han's three-variable statistic $(\mathrm{inv-lec, pix, lec})$. We also prove a new recurrence formula for the $(q,r)$-Eulerian polynomials and study a $q$-analogue of Chung and Graham's restricted Eulerian polynomials. In particular, we obtain a symmetrical identity for these restricted $q$-Eulerian polynomials with a combinatorial proof.]]> 0 <![CDATA[A poset is $(3+1)$-free if it does not contain the disjoint union of chains of length 3 and 1 as an induced subposet. These posets are the subject of the $(3+1)$-free conjecture of Stanley and Stembridge. Recently, Lewis and Zhang have enumerated $\textit{graded}$ $(3+1)$-free posets, but until now the general enumeration problem has remained open. We enumerate all $(3+1)$-free posets by giving a decomposition into bipartite graphs, and obtain generating functions for $(3+1)$-free posets with labelled or unlabelled vertices.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12809 https://doi.org/10.46298/dmtcs.12809 Guay-Paquet, Mathieu Morales, Alejandro H. Rowland, Eric Guay-Paquet, Mathieu Morales, Alejandro H. Rowland, Eric <![CDATA[A poset is $(3+1)$-free if it does not contain the disjoint union of chains of length 3 and 1 as an induced subposet. These posets are the subject of the $(3+1)$-free conjecture of Stanley and Stembridge. Recently, Lewis and Zhang have enumerated $\textit{graded}$ $(3+1)$-free posets, but until now the general enumeration problem has remained open. We enumerate all $(3+1)$-free posets by giving a decomposition into bipartite graphs, and obtain generating functions for $(3+1)$-free posets with labelled or unlabelled vertices.]]> 0 <![CDATA[Given a finite list of vectors $X \subseteq \mathbb{R}^d$, one can define the box spline $B_X$. Box splines are piecewise polynomial functions that are used in approximation theory. They are also interesting from a combinatorial point of view and many of their properties solely depend on the structure of the matroid defined by the list $X$. The support of the box spline is the zonotope $Z(X)$. We show that if the list $X$ is totally unimodular, any real-valued function defined on the set of lattice points in the interior of $Z(X)$ can be extended to a function on $Z(X)$ of the form $p(D)B_X$ in a unique way, where $p(D)$ is a differential operator that is contained in the so-called internal $\mathcal{P}$-space. This was conjectured by Olga Holtz and Amos Ron. We also point out connections between this interpolation problem and matroid theory, including a deletion-contraction decomposition.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12820 https://doi.org/10.46298/dmtcs.12820 Lenz, Matthias Lenz, Matthias <![CDATA[Given a finite list of vectors $X \subseteq \mathbb{R}^d$, one can define the box spline $B_X$. Box splines are piecewise polynomial functions that are used in approximation theory. They are also interesting from a combinatorial point of view and many of their properties solely depend on the structure of the matroid defined by the list $X$. The support of the box spline is the zonotope $Z(X)$. We show that if the list $X$ is totally unimodular, any real-valued function defined on the set of lattice points in the interior of $Z(X)$ can be extended to a function on $Z(X)$ of the form $p(D)B_X$ in a unique way, where $p(D)$ is a differential operator that is contained in the so-called internal $\mathcal{P}$-space. This was conjectured by Olga Holtz and Amos Ron. We also point out connections between this interpolation problem and matroid theory, including a deletion-contraction decomposition.]]> 0 <![CDATA[The $\textit{simplicial rook graph}$ $SR(d,n)$ is the graph whose vertices are the lattice points in the $n$th dilate of the standard simplex in $\mathbb{R}^d$, with two vertices adjacent if they differ in exactly two coordinates. We prove that the adjacency and Laplacian matrices of $SR(3,n)$ have integral spectra for every $n$. We conjecture that $SR(d,n)$ is integral for all $d$ and $n$, and give a geometric construction of almost all eigenvectors in terms of characteristic vectors of lattice permutohedra. For $n \leq \binom{d}{2}$, we give an explicit construction of smallest-weight eigenvectors in terms of rook placements on Ferrers diagrams. The number of these eigenvectors appears to satisfy a Mahonian distribution.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12819 https://doi.org/10.46298/dmtcs.12819 Martin, Jeremy L. Wagner, Jennifer D. Martin, Jeremy L. Wagner, Jennifer D. <![CDATA[The $\textit{simplicial rook graph}$ $SR(d,n)$ is the graph whose vertices are the lattice points in the $n$th dilate of the standard simplex in $\mathbb{R}^d$, with two vertices adjacent if they differ in exactly two coordinates. We prove that the adjacency and Laplacian matrices of $SR(3,n)$ have integral spectra for every $n$. We conjecture that $SR(d,n)$ is integral for all $d$ and $n$, and give a geometric construction of almost all eigenvectors in terms of characteristic vectors of lattice permutohedra. For $n \leq \binom{d}{2}$, we give an explicit construction of smallest-weight eigenvectors in terms of rook placements on Ferrers diagrams. The number of these eigenvectors appears to satisfy a Mahonian distribution.]]> 0 <![CDATA[Just as the power series of $\log (1+X)$ is the analytical substitutional inverse of the series of $\exp (X)-1$, the (virtual) combinatorial species, $\mathrm{Lg} (1+X)$, is the combinatorial substitutional inverse of the combinatorial species, $E(X)-1$, of non-empty finite sets. This $\textit{combinatorial logarithm}$, $\mathrm{Lg} (1+X)$, has been introduced by A. Joyal in 1986 by making use of an iterative scheme. Given a species $F(X)$ (with $F(0)=1$), one of its main applications is to express the species, $F^{\mathrm{c}}(X)$, of $\textit{connected}$ $F$-structures through the formula $F{\mathrm{c}} = \mathrm{Lg} (F) = \mathrm{Lg} (1+F_+)$ where $F_+$ denotes the species of non-empty $F$-structures. Since its creation, equivalent descriptions of the combinatorial logarithm have been given by other combinatorialists (G. L., I. Gessel, J. Li), but its exact decomposition into irreducible components (molecular expansion) remained unclear. The main goal of the present work is to fill this gap by computing explicitly the molecular expansion of the combinatorial logarithm and of $-\mathrm{Lg}(1-X)$, a "cousin'' of the tensorial species, $\mathrm{Lie}(X)$, of free Lie algebras.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12818 https://doi.org/10.46298/dmtcs.12818 Labelle, Gilbert Labelle, Gilbert <![CDATA[Just as the power series of $\log (1+X)$ is the analytical substitutional inverse of the series of $\exp (X)-1$, the (virtual) combinatorial species, $\mathrm{Lg} (1+X)$, is the combinatorial substitutional inverse of the combinatorial species, $E(X)-1$, of non-empty finite sets. This $\textit{combinatorial logarithm}$, $\mathrm{Lg} (1+X)$, has been introduced by A. Joyal in 1986 by making use of an iterative scheme. Given a species $F(X)$ (with $F(0)=1$), one of its main applications is to express the species, $F^{\mathrm{c}}(X)$, of $\textit{connected}$ $F$-structures through the formula $F{\mathrm{c}} = \mathrm{Lg} (F) = \mathrm{Lg} (1+F_+)$ where $F_+$ denotes the species of non-empty $F$-structures. Since its creation, equivalent descriptions of the combinatorial logarithm have been given by other combinatorialists (G. L., I. Gessel, J. Li), but its exact decomposition into irreducible components (molecular expansion) remained unclear. The main goal of the present work is to fill this gap by computing explicitly the molecular expansion of the combinatorial logarithm and of $-\mathrm{Lg}(1-X)$, a "cousin'' of the tensorial species, $\mathrm{Lie}(X)$, of free Lie algebras.]]> 0 <![CDATA[We are interested in finding an explicit bijection between two families of combinatorial objects: Gog and Magog triangles. These two families are particular classes of Gelfand-Tsetlin triangles and are respectively in bijection with alternating sign matrices (ASM) and totally symmetric self complementary plane partitions (TSSCPP). For this purpose, we introduce left Gog and GOGAm trapezoids. We conjecture that these two families of trapezoids are equienumerated and we give an explicit bijection between the trapezoids with one or two diagonals.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12817 https://doi.org/10.46298/dmtcs.12817 Biane, Philippe Cheballah, Hayat Biane, Philippe Cheballah, Hayat <![CDATA[We are interested in finding an explicit bijection between two families of combinatorial objects: Gog and Magog triangles. These two families are particular classes of Gelfand-Tsetlin triangles and are respectively in bijection with alternating sign matrices (ASM) and totally symmetric self complementary plane partitions (TSSCPP). For this purpose, we introduce left Gog and GOGAm trapezoids. We conjecture that these two families of trapezoids are equienumerated and we give an explicit bijection between the trapezoids with one or two diagonals.]]> 0 <![CDATA[The main purpose of this paper is to show that the multiplication of a Schubert polynomial of finite type $A$ by a Schur function can be understood from the multiplication in the space of dual $k$-Schur functions. Using earlier work by the second author, we encode both problems by means of quasisymmetric functions. On the Schubert vs. Schur side, we study the $r$-Bruhat order given by Bergeron-Sottile, along with certain operators associated to this order. On the other side, we connect this poset with a graph on dual $k$-Schur functions given by studying the affine grassmannian order of Lam-Lapointe-Morse-Shimozono. Also, we define operators associated to the graph on dual $k$-Schur functions which are analogous to the ones given for the Schubert vs. Schur problem.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12816 https://doi.org/10.46298/dmtcs.12816 Benedetti, Carolina Bergeron, Nantel Benedetti, Carolina Bergeron, Nantel <![CDATA[The main purpose of this paper is to show that the multiplication of a Schubert polynomial of finite type $A$ by a Schur function can be understood from the multiplication in the space of dual $k$-Schur functions. Using earlier work by the second author, we encode both problems by means of quasisymmetric functions. On the Schubert vs. Schur side, we study the $r$-Bruhat order given by Bergeron-Sottile, along with certain operators associated to this order. On the other side, we connect this poset with a graph on dual $k$-Schur functions given by studying the affine grassmannian order of Lam-Lapointe-Morse-Shimozono. Also, we define operators associated to the graph on dual $k$-Schur functions which are analogous to the ones given for the Schubert vs. Schur problem.]]> 0 <![CDATA[An $\textit{increasing tableau}$ is a semistandard tableau with strictly increasing rows and columns. It is well known that the Catalan numbers enumerate both rectangular standard Young tableaux of two rows and also Dyck paths. We generalize this to a bijection between rectangular 2-row increasing tableaux and small Schröder paths. Using the jeu de taquin for increasing tableaux of [Thomas–Yong '09], we then present a new instance of the cyclic sieving phenomenon of [Reiner–Stanton–White '04].]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12815 https://doi.org/10.46298/dmtcs.12815 Pechenik, Oliver Pechenik, Oliver <![CDATA[An $\textit{increasing tableau}$ is a semistandard tableau with strictly increasing rows and columns. It is well known that the Catalan numbers enumerate both rectangular standard Young tableaux of two rows and also Dyck paths. We generalize this to a bijection between rectangular 2-row increasing tableaux and small Schröder paths. Using the jeu de taquin for increasing tableaux of [Thomas–Yong '09], we then present a new instance of the cyclic sieving phenomenon of [Reiner–Stanton–White '04].]]> 0 <![CDATA[Let $(W, S)$ be a Coxeter system. A $W$-graph is an encoding of a representation of the corresponding Iwahori-Hecke algebra. Especially important examples include the $W$-graph corresponding to the action of the Iwahori-Hecke algebra on the Kazhdan-Lusztig basis as well as this graph's strongly connected components (cells). In 2008, Stembridge identified some common features of the Kazhdan-Lusztig graphs ("admissibility'') and gave combinatorial rules for detecting admissible $W$-graphs. He conjectured, and checked up to $n=9$, that all admissible $A_n$-cells are Kazhdan-Lusztig cells. The current paper provides a possible first step toward a proof of the conjecture. More concretely, we prove that the connected subgraphs of $A_n$-cells consisting of simple (i.e. directed both ways) edges do fit into the Kazhdan-Lusztig cells.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12814 https://doi.org/10.46298/dmtcs.12814 Chmutov, Michael Chmutov, Michael <![CDATA[Let $(W, S)$ be a Coxeter system. A $W$-graph is an encoding of a representation of the corresponding Iwahori-Hecke algebra. Especially important examples include the $W$-graph corresponding to the action of the Iwahori-Hecke algebra on the Kazhdan-Lusztig basis as well as this graph's strongly connected components (cells). In 2008, Stembridge identified some common features of the Kazhdan-Lusztig graphs ("admissibility'') and gave combinatorial rules for detecting admissible $W$-graphs. He conjectured, and checked up to $n=9$, that all admissible $A_n$-cells are Kazhdan-Lusztig cells. The current paper provides a possible first step toward a proof of the conjecture. More concretely, we prove that the connected subgraphs of $A_n$-cells consisting of simple (i.e. directed both ways) edges do fit into the Kazhdan-Lusztig cells.]]> 0 <![CDATA[We introduce explicit combinatorial interpretations for the coefficients in some of the transition matrices relating to skew Hall-Littlewood polynomials $P_{\lambda / \mu}(x;t)$ and Hivert's quasisymmetric Hall-Littlewood polynomials $G_{\gamma}(x;t)$. More specifically, we provide the following: 1. $G_{\gamma}$-expansions of the $P_{\lambda}$, the monomial quasisymmetric functions, and Gessel's fundamental quasisymmetric functions $F_{\alpha}$, and 2. an expansion of the $P_{\lambda / \mu}$ in terms of the $F_{\alpha}$. The $F_{\alpha}$ expansion of the $P_{\lambda / \mu}$ is facilitated by introducing the set of $\textit{starred tableaux}$. In the full version of the article we also provide $G_{\gamma}$-expansions of the quasisymmetric Schur functions and the peak quasisymmetric functions of Stembridge.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12813 https://doi.org/10.46298/dmtcs.12813 Loehr, Nicolas Serrano, Luis Warrington, Gregory Loehr, Nicolas Serrano, Luis Warrington, Gregory <![CDATA[We introduce explicit combinatorial interpretations for the coefficients in some of the transition matrices relating to skew Hall-Littlewood polynomials $P_{\lambda / \mu}(x;t)$ and Hivert's quasisymmetric Hall-Littlewood polynomials $G_{\gamma}(x;t)$. More specifically, we provide the following: 1. $G_{\gamma}$-expansions of the $P_{\lambda}$, the monomial quasisymmetric functions, and Gessel's fundamental quasisymmetric functions $F_{\alpha}$, and 2. an expansion of the $P_{\lambda / \mu}$ in terms of the $F_{\alpha}$. The $F_{\alpha}$ expansion of the $P_{\lambda / \mu}$ is facilitated by introducing the set of $\textit{starred tableaux}$. In the full version of the article we also provide $G_{\gamma}$-expansions of the quasisymmetric Schur functions and the peak quasisymmetric functions of Stembridge.]]> 0 <![CDATA[We propose a new approach to study the Kronecker coefficients by using the Schur–Weyl duality between the symmetric group and the partition algebra.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12812 https://doi.org/10.46298/dmtcs.12812 Bowman, Christopher Visscher, Maud Orellana, Rosa Bowman, Christopher Visscher, Maud Orellana, Rosa <![CDATA[We propose a new approach to study the Kronecker coefficients by using the Schur–Weyl duality between the symmetric group and the partition algebra.]]> 0 <![CDATA[The goal of this paper is to introduce an algebraic structure on the space spanned by affine descent classes of a Weyl group, by analogy and in relation to the structure carried by ordinary descent classes. The latter classes span a subalgebra of the group algebra, Solomon's descent algebra. We show that the former span a left module over this algebra. The structure is obtained from geometric considerations involving hyperplane arrangements. We provide a combinatorial model for the case of the symmetric group.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12811 https://doi.org/10.46298/dmtcs.12811 Aguiar, Marcelo Petersen, Kile T. Aguiar, Marcelo Petersen, Kile T. <![CDATA[The goal of this paper is to introduce an algebraic structure on the space spanned by affine descent classes of a Weyl group, by analogy and in relation to the structure carried by ordinary descent classes. The latter classes span a subalgebra of the group algebra, Solomon's descent algebra. We show that the former span a left module over this algebra. The structure is obtained from geometric considerations involving hyperplane arrangements. We provide a combinatorial model for the case of the symmetric group.]]> 0 <![CDATA[We introduce a new basis of the non-commutative symmetric functions whose elements have Schur functions as their commutative images. Dually, we build a basis of the quasi-symmetric functions which expand positively in the fundamental quasi-symmetric functions and decompose Schur functions according to a signed combinatorial formula.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12810 https://doi.org/10.46298/dmtcs.12810 Berg, Chris Bergeron, Nantel Saliola, Franco Serrano, Luis Zabrocki, Mike Berg, Chris Bergeron, Nantel Saliola, Franco Serrano, Luis Zabrocki, Mike <![CDATA[We introduce a new basis of the non-commutative symmetric functions whose elements have Schur functions as their commutative images. Dually, we build a basis of the quasi-symmetric functions which expand positively in the fundamental quasi-symmetric functions and decompose Schur functions according to a signed combinatorial formula.]]> 0 <![CDATA[We define and study a recurrence relation in $\mathbb{Z}^3$, called the hexahedron recurrence, which is similar to the octahedron recurrence (Hirota bilinear difference equation) and cube recurrence (Miwa equation). Like these examples, solutions to the hexahedron recurrence are partition functions for configurations on a certain graph, and have a natural interpretation in terms of cluster algebras. We give an explicit correspondence between monomials in the Laurent expansions arising in the recurrence with certain double-dimer configurations of a graph. We compute limit shapes for the corresponding double-dimer configurations. The Kashaev difference equation arising in the Ising model star-triangle relation is a special case of the hexahedron recurrence. In particular this reveals the cluster nature underlying the Ising model. The above relation allows us to prove a Laurent phenomenon for the Kashaev difference equation.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12797 https://doi.org/10.46298/dmtcs.12797 Kenyon, Richard Pemantle, Robin Kenyon, Richard Pemantle, Robin <![CDATA[We define and study a recurrence relation in $\mathbb{Z}^3$, called the hexahedron recurrence, which is similar to the octahedron recurrence (Hirota bilinear difference equation) and cube recurrence (Miwa equation). Like these examples, solutions to the hexahedron recurrence are partition functions for configurations on a certain graph, and have a natural interpretation in terms of cluster algebras. We give an explicit correspondence between monomials in the Laurent expansions arising in the recurrence with certain double-dimer configurations of a graph. We compute limit shapes for the corresponding double-dimer configurations. The Kashaev difference equation arising in the Ising model star-triangle relation is a special case of the hexahedron recurrence. In particular this reveals the cluster nature underlying the Ising model. The above relation allows us to prove a Laurent phenomenon for the Kashaev difference equation.]]> 0 <![CDATA[We address the enumeration of $p$-valent planar maps equipped with a spanning forest, with a weight $z$ per face and a weight $u$ per component of the forest. Equivalently, we count regular maps equipped with a spanning tree, with a weight $z$ per face and a weight $\mu:=u+1$ per internally active edge, in the sense of Tutte. This enumeration problem corresponds to the limit $q \rightarrow 0$ of the $q$-state Potts model on the (dual) $p$-angulations. Our approach is purely combinatorial. The generating function, denoted by $F(z,u)$, is expressed in terms of a pair of series defined by an implicit system involving doubly hypergeometric functions. We derive from this system that $F(z,u)$ is $\textit{differentially algebraic}$, that is, satisfies a differential equation (in $z$) with polynomial coefficients in $z$ and $u$. This has recently been proved for the more general Potts model on 3-valent maps, but via a much more involved and less combinatorial proof. For $u \geq -1$, we study the singularities of $F(z,u)$ and the corresponding asymptotic behaviour of its $n^{\mathrm{th}}$ coefficient. For $u > 0$, we find the standard asymptotic behaviour of planar maps, with a subexponential factor $n^{-5/2}$. At $u=0$ we witness a phase transition with a factor $n^{-3}$. When $u \in[-1,0)$, we obtain an extremely unusual behaviour in $n^{-3}/(\log n)^2$. To our knowledge, this is a new ''universality class'' of planar maps.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12808 https://doi.org/10.46298/dmtcs.12808 Bousquet-Mélou, Mireille Courtiel, Julien Bousquet-Mélou, Mireille Courtiel, Julien <![CDATA[We address the enumeration of $p$-valent planar maps equipped with a spanning forest, with a weight $z$ per face and a weight $u$ per component of the forest. Equivalently, we count regular maps equipped with a spanning tree, with a weight $z$ per face and a weight $\mu:=u+1$ per internally active edge, in the sense of Tutte. This enumeration problem corresponds to the limit $q \rightarrow 0$ of the $q$-state Potts model on the (dual) $p$-angulations. Our approach is purely combinatorial. The generating function, denoted by $F(z,u)$, is expressed in terms of a pair of series defined by an implicit system involving doubly hypergeometric functions. We derive from this system that $F(z,u)$ is $\textit{differentially algebraic}$, that is, satisfies a differential equation (in $z$) with polynomial coefficients in $z$ and $u$. This has recently been proved for the more general Potts model on 3-valent maps, but via a much more involved and less combinatorial proof. For $u \geq -1$, we study the singularities of $F(z,u)$ and the corresponding asymptotic behaviour of its $n^{\mathrm{th}}$ coefficient. For $u > 0$, we find the standard asymptotic behaviour of planar maps, with a subexponential factor $n^{-5/2}$. At $u=0$ we witness a phase transition with a factor $n^{-3}$. When $u \in[-1,0)$, we obtain an extremely unusual behaviour in $n^{-3}/(\log n)^2$. To our knowledge, this is a new ''universality class'' of planar maps.]]> 0 <![CDATA[The Little map and the Edelman-Greene insertion algorithm, a generalization of the Robinson-Schensted correspondence, are both used for enumerating the reduced decompositions of an element of the symmetric group. We show the Little map factors through Edelman-Greene insertion and establish new results about each map as a consequence. In particular, we resolve some conjectures of Lam and Little.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12807 https://doi.org/10.46298/dmtcs.12807 Hamaker, Zachary Young, Benjamin Hamaker, Zachary Young, Benjamin <![CDATA[The Little map and the Edelman-Greene insertion algorithm, a generalization of the Robinson-Schensted correspondence, are both used for enumerating the reduced decompositions of an element of the symmetric group. We show the Little map factors through Edelman-Greene insertion and establish new results about each map as a consequence. In particular, we resolve some conjectures of Lam and Little.]]> 0 <![CDATA[This paper introduces a variation on the binomial coefficient that depends on a poset and interpolates between $q$-binomials and 1-binomials: a total order gives the usual $q$-binomial, and a poset with no relations gives the usual binomial coefficient. These coefficients arise naturally in the study of supercharacters of the finite groups of unipotent upper-triangular matrices, whose representation theory is dictated by the combinatorics of set partitions. In particular, we find a natural set of modules for these groups, whose characters have degrees given by $q$-binomials, and whose decomposition in terms of supercharacters are given by poset binomial coefficients. This results in a non-trivial family of formulas relating poset binomials to the usual $q$-binomials.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12806 https://doi.org/10.46298/dmtcs.12806 Bragg, Daniel Thiem, Nathaniel Bragg, Daniel Thiem, Nathaniel <![CDATA[This paper introduces a variation on the binomial coefficient that depends on a poset and interpolates between $q$-binomials and 1-binomials: a total order gives the usual $q$-binomial, and a poset with no relations gives the usual binomial coefficient. These coefficients arise naturally in the study of supercharacters of the finite groups of unipotent upper-triangular matrices, whose representation theory is dictated by the combinatorics of set partitions. In particular, we find a natural set of modules for these groups, whose characters have degrees given by $q$-binomials, and whose decomposition in terms of supercharacters are given by poset binomial coefficients. This results in a non-trivial family of formulas relating poset binomials to the usual $q$-binomials.]]> 0 <![CDATA[Generalizing the notion of a vexillary permutation, we introduce a filtration of $S_{\infty}$ by the number of Edelman-Greene tableaux of a permutation, and show that each filtration level is characterized by avoiding a finite set of patterns. In doing so, we show that if $w$ is a permutation containing $v$ as a pattern, then there is an injection from the set of Edelman-Greene tableaux of $v$ to the set of Edelman-Greene tableaux of $w$ which respects inclusion of shapes. We also consider the set of permutations whose Edelman-Greene tableaux have distinct shapes, and show that it is closed under taking patterns.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12805 https://doi.org/10.46298/dmtcs.12805 Billey, Sara Pawlowski, Brendan Billey, Sara Pawlowski, Brendan <![CDATA[Generalizing the notion of a vexillary permutation, we introduce a filtration of $S_{\infty}$ by the number of Edelman-Greene tableaux of a permutation, and show that each filtration level is characterized by avoiding a finite set of patterns. In doing so, we show that if $w$ is a permutation containing $v$ as a pattern, then there is an injection from the set of Edelman-Greene tableaux of $v$ to the set of Edelman-Greene tableaux of $w$ which respects inclusion of shapes. We also consider the set of permutations whose Edelman-Greene tableaux have distinct shapes, and show that it is closed under taking patterns.]]> 0 <![CDATA[An action on order ideals of posets considered by Fon-Der-Flaass is analyzed in the case of posets arising from minuscule representations of complex simple Lie algebras. For these minuscule posets, it is shown that the Fon-Der-Flaass action exhibits the cyclic sieving phenomenon, as defined by Reiner, Stanton, and White. A uniform proof is given by investigation of a bijection due to Stembridge between order ideals of minuscule posets and fully commutative Weyl group elements. This bijection is proven to be equivariant with respect to a conjugate of the Fon-Der-Flaass action and an arbitrary Coxeter element. If $P$ is a minuscule poset, it is shown that the Fon-Der-Flaass action on order ideals of the Cartesian product $P \times [2]$ also exhibits the cyclic sieving phenomenon, only the proof is by appeal to the classification of minuscule posets and is not uniform.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12804 https://doi.org/10.46298/dmtcs.12804 Rush, David B. Shi, Xiaolin Rush, David B. Shi, Xiaolin <![CDATA[An action on order ideals of posets considered by Fon-Der-Flaass is analyzed in the case of posets arising from minuscule representations of complex simple Lie algebras. For these minuscule posets, it is shown that the Fon-Der-Flaass action exhibits the cyclic sieving phenomenon, as defined by Reiner, Stanton, and White. A uniform proof is given by investigation of a bijection due to Stembridge between order ideals of minuscule posets and fully commutative Weyl group elements. This bijection is proven to be equivariant with respect to a conjugate of the Fon-Der-Flaass action and an arbitrary Coxeter element. If $P$ is a minuscule poset, it is shown that the Fon-Der-Flaass action on order ideals of the Cartesian product $P \times [2]$ also exhibits the cyclic sieving phenomenon, only the proof is by appeal to the classification of minuscule posets and is not uniform.]]> 0 <![CDATA[The notion of $r$-stackedness for simplicial polytopes was introduced by McMullen and Walkup in 1971 as a generalization of stacked polytopes. In this paper, we define the $r$-stackedness for triangulated homology manifolds and study their basic properties. In addition, we find a new necessary condition for face vectors of triangulated manifolds when all the vertex links are polytopal.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12803 https://doi.org/10.46298/dmtcs.12803 Murai, Satoshi Nevo, Eran Murai, Satoshi Nevo, Eran <![CDATA[The notion of $r$-stackedness for simplicial polytopes was introduced by McMullen and Walkup in 1971 as a generalization of stacked polytopes. In this paper, we define the $r$-stackedness for triangulated homology manifolds and study their basic properties. In addition, we find a new necessary condition for face vectors of triangulated manifolds when all the vertex links are polytopal.]]> 0 <![CDATA[New formulas for the $n^{\mathrm{th}}$ moment $\mu_n(a,b,c,d;q)$ of the Askey-Wilson polynomials are given. These are derived using analytic techniques, and by considering three combinatorial models for the moments: Motzkin paths, matchings, and staircase tableaux. A related positivity theorem is given and another one is conjectured.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12802 https://doi.org/10.46298/dmtcs.12802 Kim, Jang Soo Stanton, Dennis Kim, Jang Soo Stanton, Dennis <![CDATA[New formulas for the $n^{\mathrm{th}}$ moment $\mu_n(a,b,c,d;q)$ of the Askey-Wilson polynomials are given. These are derived using analytic techniques, and by considering three combinatorial models for the moments: Motzkin paths, matchings, and staircase tableaux. A related positivity theorem is given and another one is conjectured.]]> 0 <![CDATA[We introduce the notion of a matroid $M$ over a commutative ring $R$, assigning to every subset of the ground set an $R$-module according to some axioms. When $R$ is a field, we recover matroids. When $R=\mathbb{Z}$, and when $R$ is a DVR, we get (structures which contain all the data of) quasi-arithmetic matroids, and valuated matroids, respectively. More generally, whenever $R$ is a Dedekind domain, we extend the usual properties and operations holding for matroids (e.g., duality), and we compute the Tutte-Grothendieck group of matroids over $R$.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12801 https://doi.org/10.46298/dmtcs.12801 Fink, Alex Moci, Luca Fink, Alex Moci, Luca <![CDATA[We introduce the notion of a matroid $M$ over a commutative ring $R$, assigning to every subset of the ground set an $R$-module according to some axioms. When $R$ is a field, we recover matroids. When $R=\mathbb{Z}$, and when $R$ is a DVR, we get (structures which contain all the data of) quasi-arithmetic matroids, and valuated matroids, respectively. More generally, whenever $R$ is a Dedekind domain, we extend the usual properties and operations holding for matroids (e.g., duality), and we compute the Tutte-Grothendieck group of matroids over $R$.]]> 0 <![CDATA[An element of a Coxeter group $W$ is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. These elements were extensively studied by Stembridge in the finite case. In this work we deal with any finite or affine Coxeter group $W$, and we enumerate fully commutative elements according to their Coxeter length. Our approach consists in encoding these elements by various classes of lattice walks, and we then use recursive decompositions of these walks in order to obtain the desired generating functions. In type $A$, this reproves a theorem of Barcucci et al.; in type $\tilde{A}$, it simplifies and refines results of Hanusa and Jones. For all other finite and affine groups, our results are new.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12800 https://doi.org/10.46298/dmtcs.12800 Biagioli, Riccardo Jouhet, Frédéric Nadeau, Philippe Biagioli, Riccardo Jouhet, Frédéric Nadeau, Philippe <![CDATA[An element of a Coxeter group $W$ is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. These elements were extensively studied by Stembridge in the finite case. In this work we deal with any finite or affine Coxeter group $W$, and we enumerate fully commutative elements according to their Coxeter length. Our approach consists in encoding these elements by various classes of lattice walks, and we then use recursive decompositions of these walks in order to obtain the desired generating functions. In type $A$, this reproves a theorem of Barcucci et al.; in type $\tilde{A}$, it simplifies and refines results of Hanusa and Jones. For all other finite and affine groups, our results are new.]]> 0 <![CDATA[We show the $\mathrm{cd}$-index exists for Whitney stratified manifolds by extending the notion of a graded poset to that of a quasi-graded poset. This is a poset endowed with an order-preserving rank function and a weighted zeta function. This allows us to generalize the classical notion of Eulerianness, and obtain a $\mathrm{cd}$-index in the quasi-graded poset arena. We also extend the semi-suspension operation to that of embedding a complex in the boundary of a higher dimensional ball and study the shelling components of the simplex.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12799 https://doi.org/10.46298/dmtcs.12799 Ehrenborg, Richard Goresky, Mark Readdy, Margaret Ehrenborg, Richard Goresky, Mark Readdy, Margaret <![CDATA[We show the $\mathrm{cd}$-index exists for Whitney stratified manifolds by extending the notion of a graded poset to that of a quasi-graded poset. This is a poset endowed with an order-preserving rank function and a weighted zeta function. This allows us to generalize the classical notion of Eulerianness, and obtain a $\mathrm{cd}$-index in the quasi-graded poset arena. We also extend the semi-suspension operation to that of embedding a complex in the boundary of a higher dimensional ball and study the shelling components of the simplex.]]> 0 <![CDATA[We consider the Ehrhart $h^*$-vector for the hypersimplex. It is well-known that the sum of the $h_i^*$ is the normalized volume which equals an Eulerian number. The main result is a proof of a conjecture by R. Stanley which gives an interpretation of the $h^*_i$ coefficients in terms of descents and excedances. Our proof is geometric using a careful book-keeping of a shelling of a unimodular triangulation. We generalize this result to other closely related polytopes.]]> Mon, 31 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.12798 https://doi.org/10.46298/dmtcs.12798 Li, Nan Li, Nan <![CDATA[We consider the Ehrhart $h^*$-vector for the hypersimplex. It is well-known that the sum of the $h_i^*$ is the normalized volume which equals an Eulerian number. The main result is a proof of a conjecture by R. Stanley which gives an interpretation of the $h^*_i$ coefficients in terms of descents and excedances. Our proof is geometric using a careful book-keeping of a shelling of a unimodular triangulation. We generalize this result to other closely related polytopes.]]> 0 <![CDATA[For a positive integer k, a k-tuple dominating set of a graph G is a subset S of V (G) such that |N [v] ∩ S| ≥ k for every vertex v, where N [v] = {v} ∪ {u ∈ V (G) : uv ∈ E(G)}. The upper k-tuple domination number of G, denoted by Γ×k (G), is the maximum cardinality of a minimal k-tuple dominating set of G. In this paper we present an upper bound on Γ×k (G) for r-regular graphs G with r ≥ k, and characterize extremal graphs achieving the upper bound. We also establish an upper bound on Γ×2 (G) for claw-free r-regular graphs. For the algorithmic aspect, we show that the upper k-tuple domination problem is NP-complete for bipartite graphs and for chordal graphs.]]> Tue, 11 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.593 https://doi.org/10.46298/dmtcs.593 Chang, Gerard Jennhwa Dorbec, Paul Kim, Hye Kyung Raspaud, André Wang, Haichao Zhao, Weiliang Chang, Gerard Jennhwa Dorbec, Paul Kim, Hye Kyung Raspaud, André Wang, Haichao Zhao, Weiliang <![CDATA[For a positive integer k, a k-tuple dominating set of a graph G is a subset S of V (G) such that |N [v] ∩ S| ≥ k for every vertex v, where N [v] = {v} ∪ {u ∈ V (G) : uv ∈ E(G)}. The upper k-tuple domination number of G, denoted by Γ×k (G), is the maximum cardinality of a minimal k-tuple dominating set of G. In this paper we present an upper bound on Γ×k (G) for r-regular graphs G with r ≥ k, and characterize extremal graphs achieving the upper bound. We also establish an upper bound on Γ×2 (G) for claw-free r-regular graphs. For the algorithmic aspect, we show that the upper k-tuple domination problem is NP-complete for bipartite graphs and for chordal graphs.]]> 0 <![CDATA[For a binary code Γ of length v, a v-word w produces by a set of codewords {w1,...,wr}⊆Γ if for all i=1,...,v, we have wi∈{w1i,...,wri} . We call a code r-secure frameproof of size t if |Γ|=t and for any v-word that is produced by two sets C1 and C2 of size at most r then the intersection of these sets is nonempty. A d-biclique cover of size v of a graph G is a collection of v-complete bipartite subgraphs of G such that each edge of G belongs to at least d of these complete bipartite subgraphs. In this paper, we show that for t≥2r, an r-secure frameproof code of size t and length v exists if and only if there exists a 1-biclique cover of size v for the Kneser graph KG(t,r) whose vertices are all r-subsets of a t-element set and two r-subsets are adjacent if their intersection is empty. Then we investigate some connection between the minimum size of d-biclique covers of Kneser graphs and cover-free families, where an (r,w;d) cover-free family is a family of subsets of a finite set such that the intersection of any r members of the family contains at least d elements that are not in the union of any other w members. Also, we present an upper bound for 1-biclique covering number of Kneser graphs.]]> Mon, 03 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.585 https://doi.org/10.46298/dmtcs.585 Hajiabolhassan, Hossein Moazami, Farokhlagha Hajiabolhassan, Hossein Moazami, Farokhlagha <![CDATA[For a binary code Γ of length v, a v-word w produces by a set of codewords {w1,...,wr}⊆Γ if for all i=1,...,v, we have wi∈{w1i,...,wri} . We call a code r-secure frameproof of size t if |Γ|=t and for any v-word that is produced by two sets C1 and C2 of size at most r then the intersection of these sets is nonempty. A d-biclique cover of size v of a graph G is a collection of v-complete bipartite subgraphs of G such that each edge of G belongs to at least d of these complete bipartite subgraphs. In this paper, we show that for t≥2r, an r-secure frameproof code of size t and length v exists if and only if there exists a 1-biclique cover of size v for the Kneser graph KG(t,r) whose vertices are all r-subsets of a t-element set and two r-subsets are adjacent if their intersection is empty. Then we investigate some connection between the minimum size of d-biclique covers of Kneser graphs and cover-free families, where an (r,w;d) cover-free family is a family of subsets of a finite set such that the intersection of any r members of the family contains at least d elements that are not in the union of any other w members. Also, we present an upper bound for 1-biclique covering number of Kneser graphs.]]> 0 <![CDATA[This article deals with some stochastic population protocols, motivated by theoretical aspects of distributed computing. We modelize the problem by a large urn of black and white balls from which at every time unit a fixed number of balls are drawn and their colors are changed according to the number of black balls among them. When the time and the number of balls both tend to infinity the proportion of black balls converges to an algebraic number. We prove that, surprisingly enough, not every algebraic number can be ''computed'' this way.]]> Mon, 03 Dec 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.565 https://doi.org/10.46298/dmtcs.565 Albenque, Marie Gerin, Lucas Albenque, Marie Gerin, Lucas <![CDATA[This article deals with some stochastic population protocols, motivated by theoretical aspects of distributed computing. We modelize the problem by a large urn of black and white balls from which at every time unit a fixed number of balls are drawn and their colors are changed according to the number of black balls among them. When the time and the number of balls both tend to infinity the proportion of black balls converges to an algebraic number. We prove that, surprisingly enough, not every algebraic number can be ''computed'' this way.]]> 0 <![CDATA[For any fixed integer R≥2 we characterise the typical structure of undirected graphs with vertices 1,...,n and maximum degree R, as n tends to infinity. The information is used to prove that such graphs satisfy a labelled limit law for first-order logic. If R≥5 then also an unlabelled limit law holds.]]> Thu, 22 Nov 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.592 https://doi.org/10.46298/dmtcs.592 Koponen, Vera Koponen, Vera <![CDATA[For any fixed integer R≥2 we characterise the typical structure of undirected graphs with vertices 1,...,n and maximum degree R, as n tends to infinity. The information is used to prove that such graphs satisfy a labelled limit law for first-order logic. If R≥5 then also an unlabelled limit law holds.]]> 0 <![CDATA[For two given graphs G and H , the planar Ramsey number P R ( G; H ) is the smallest integer n such that every planar graph F on n vertices either contains a copy of G , or its complement contains a copy of H . In this paper, we determine all planar Ramsey numbers for a triangle versus wheels.]]> Thu, 22 Nov 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.584 https://doi.org/10.46298/dmtcs.584 Zhou, Guofei Chen, Yaojun Miao, Zhengke Pirzada, Shariefuddin Zhou, Guofei Chen, Yaojun Miao, Zhengke Pirzada, Shariefuddin <![CDATA[For two given graphs G and H , the planar Ramsey number P R ( G; H ) is the smallest integer n such that every planar graph F on n vertices either contains a copy of G , or its complement contains a copy of H . In this paper, we determine all planar Ramsey numbers for a triangle versus wheels.]]> 0 <![CDATA[A predicate P: {-1, 1}k →{0, 1} can be associated with a constraint satisfaction problem Max CSP(P). P is called ''approximation resistant'' if Max CSP(P) cannot be approximated better than the approximation obtained by choosing a random assignment, and ''approximable'' otherwise. This classification of predicates has proved to be an important and challenging open problem. Motivated by a recent result of Austrin and Mossel (Computational Complexity, 2009), we consider a natural subclass of predicates defined by signs of quadratic polynomials, including the special case of predicates defined by signs of linear forms, and supply algorithms to approximate them as follows. In the quadratic case we prove that every symmetric predicate is approximable. We introduce a new rounding algorithm for the standard semidefinite programming relaxation of Max CSP(P) for any predicate P: {-1, 1}k →{0, 1} and analyze its approximation ratio. Our rounding scheme operates by first manipulating the optimal SDP solution so that all the vectors are nearly perpendicular and then applying a form of hyperplane rounding to obtain an integral solution. The advantage of this method is that we are able to analyze the behaviour of a set of k rounded variables together as opposed to just a pair of rounded variables in most previous methods. In the linear case we prove that a predicate called ''Monarchy'' is approximable. This predicate is not amenable to our algorithm for the quadratic case, nor to other LP/SDP-based approaches we are aware of.]]> Thu, 22 Nov 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.591 https://doi.org/10.46298/dmtcs.591 Austrin, Per Benabbas, Siavosh Magen, Avner Austrin, Per Benabbas, Siavosh Magen, Avner <![CDATA[A predicate P: {-1, 1}k →{0, 1} can be associated with a constraint satisfaction problem Max CSP(P). P is called ''approximation resistant'' if Max CSP(P) cannot be approximated better than the approximation obtained by choosing a random assignment, and ''approximable'' otherwise. This classification of predicates has proved to be an important and challenging open problem. Motivated by a recent result of Austrin and Mossel (Computational Complexity, 2009), we consider a natural subclass of predicates defined by signs of quadratic polynomials, including the special case of predicates defined by signs of linear forms, and supply algorithms to approximate them as follows. In the quadratic case we prove that every symmetric predicate is approximable. We introduce a new rounding algorithm for the standard semidefinite programming relaxation of Max CSP(P) for any predicate P: {-1, 1}k →{0, 1} and analyze its approximation ratio. Our rounding scheme operates by first manipulating the optimal SDP solution so that all the vectors are nearly perpendicular and then applying a form of hyperplane rounding to obtain an integral solution. The advantage of this method is that we are able to analyze the behaviour of a set of k rounded variables together as opposed to just a pair of rounded variables in most previous methods. In the linear case we prove that a predicate called ''Monarchy'' is approximable. This predicate is not amenable to our algorithm for the quadratic case, nor to other LP/SDP-based approaches we are aware of.]]> 0 <![CDATA[We investigate structural complexity measures on digraphs, in particular the cycle rank. This concept is intimately related to a classical topic in formal language theory, namely the star height of regular languages. We explore this connection, and obtain several new algorithmic insights regarding both cycle rank and star height. Among other results, we show that computing the cycle rank is NP-complete, even for sparse digraphs of maximum outdegree 2. Notwithstanding, we provide both a polynomial-time approximation algorithm and an exponential-time exact algorithm for this problem. The former algorithm yields an O((log n)^(3/2))- approximation in polynomial time, whereas the latter yields the optimum solution, and runs in time and space O*(1.9129^n) on digraphs of maximum outdegree at most two. Regarding the star height problem, we identify a subclass of the regular languages for which we can precisely determine the computational complexity of the star height problem. Namely, the star height problem for bideterministic languages is NP-complete, and this holds already for binary alphabets. Then we translate the algorithmic results concerning cycle rank to the bideterministic star height problem, thus giving a polynomial-time approximation as well as a reasonably fast exact exponential algorithm for bideterministic star height.]]> Sun, 11 Nov 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.583 https://doi.org/10.46298/dmtcs.583 Gruber, Hermann Gruber, Hermann <![CDATA[We investigate structural complexity measures on digraphs, in particular the cycle rank. This concept is intimately related to a classical topic in formal language theory, namely the star height of regular languages. We explore this connection, and obtain several new algorithmic insights regarding both cycle rank and star height. Among other results, we show that computing the cycle rank is NP-complete, even for sparse digraphs of maximum outdegree 2. Notwithstanding, we provide both a polynomial-time approximation algorithm and an exponential-time exact algorithm for this problem. The former algorithm yields an O((log n)^(3/2))- approximation in polynomial time, whereas the latter yields the optimum solution, and runs in time and space O*(1.9129^n) on digraphs of maximum outdegree at most two. Regarding the star height problem, we identify a subclass of the regular languages for which we can precisely determine the computational complexity of the star height problem. Namely, the star height problem for bideterministic languages is NP-complete, and this holds already for binary alphabets. Then we translate the algorithmic results concerning cycle rank to the bideterministic star height problem, thus giving a polynomial-time approximation as well as a reasonably fast exact exponential algorithm for bideterministic star height.]]> 0 <![CDATA[A 4-valent first-kind Frobenius circulant graph is a connected Cayley graph DLn(1, h) = Cay(Zn, H) on the additive group of integers modulo n, where each prime factor of n is congruent to 1 modulo 4 and H = {[1], [h], −[1], −[h]} with h a solution to the congruence equation x 2 + 1 ≡ 0 (mod n). In [A. Thomson and S. Zhou, Frobenius circulant graphs of valency four, J. Austral. Math. Soc. 85 (2008), 269-282] it was proved that such graphs admit 'perfect ' routing and gossiping schemes in some sense, making them attractive candidates for modelling interconnection networks. In the present paper we prove that DLn(1, h) has the smallest possible broadcasting time, namely its diameter plus two, and we explicitly give an optimal broadcasting in DLn(1, h). Using number theory we prove that it is possible to recursively construct larger 4-valent first-kind Frobenius circulants from smaller ones, and we give a methodology for such a construction. These and existing results suggest that, among all 4-valent circulant graphs, 4-valent first-kind Frobenius circulants are extremely efficient in terms of routing, gossiping, broadcasting and recursive construction.]]> Tue, 30 Oct 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.582 https://doi.org/10.46298/dmtcs.582 Zhou, Sanming Zhou, Sanming <![CDATA[A 4-valent first-kind Frobenius circulant graph is a connected Cayley graph DLn(1, h) = Cay(Zn, H) on the additive group of integers modulo n, where each prime factor of n is congruent to 1 modulo 4 and H = {[1], [h], −[1], −[h]} with h a solution to the congruence equation x 2 + 1 ≡ 0 (mod n). In [A. Thomson and S. Zhou, Frobenius circulant graphs of valency four, J. Austral. Math. Soc. 85 (2008), 269-282] it was proved that such graphs admit 'perfect ' routing and gossiping schemes in some sense, making them attractive candidates for modelling interconnection networks. In the present paper we prove that DLn(1, h) has the smallest possible broadcasting time, namely its diameter plus two, and we explicitly give an optimal broadcasting in DLn(1, h). Using number theory we prove that it is possible to recursively construct larger 4-valent first-kind Frobenius circulants from smaller ones, and we give a methodology for such a construction. These and existing results suggest that, among all 4-valent circulant graphs, 4-valent first-kind Frobenius circulants are extremely efficient in terms of routing, gossiping, broadcasting and recursive construction.]]> 0 <![CDATA[An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a'(G). A graph G is called fully subdivided if it is obtained from another graph H by replacing every edge by a path of length at least two. Fully subdivided graphs are known to be acyclically edge colorable using Δ+1 colors since they are properly contained in 2-degenerate graphs which are acyclically edge colorable using Δ+1 colors. Muthu, Narayanan and Subramanian gave a simple direct proof of this fact for the fully subdivided graphs. Fiamcik has shown that if we subdivide every edge in a cubic graph with at most two exceptions to get a graph G, then a'(G)=3. In this paper we generalise the bound to Δ for all fully subdivided graphs improving the result of Muthu et al. In particular, we prove that if G is a fully subdivided graph and Δ(G) ≥3, then a'(G)=Δ(G). Consider a graph G=(V,E), with E=E(T) ∪E(C) where T is a rooted tree on the vertex set V and C is a simple cycle on the leaves of T. Such a graph G is called a Halin graph if G has a planar embedding and T has no vertices of degree 2. Let Kn denote a complete graph on n vertices. Let G be a Halin graph with maximum degree Δ. We prove that, a'(G) = 5 if G is K4, 4 if Δ = 3 and G is not K4, and Δ otherwise.]]> Mon, 29 Oct 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.590 https://doi.org/10.46298/dmtcs.590 Basavaraju, Manu Basavaraju, Manu <![CDATA[An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a'(G). A graph G is called fully subdivided if it is obtained from another graph H by replacing every edge by a path of length at least two. Fully subdivided graphs are known to be acyclically edge colorable using Δ+1 colors since they are properly contained in 2-degenerate graphs which are acyclically edge colorable using Δ+1 colors. Muthu, Narayanan and Subramanian gave a simple direct proof of this fact for the fully subdivided graphs. Fiamcik has shown that if we subdivide every edge in a cubic graph with at most two exceptions to get a graph G, then a'(G)=3. In this paper we generalise the bound to Δ for all fully subdivided graphs improving the result of Muthu et al. In particular, we prove that if G is a fully subdivided graph and Δ(G) ≥3, then a'(G)=Δ(G). Consider a graph G=(V,E), with E=E(T) ∪E(C) where T is a rooted tree on the vertex set V and C is a simple cycle on the leaves of T. Such a graph G is called a Halin graph if G has a planar embedding and T has no vertices of degree 2. Let Kn denote a complete graph on n vertices. Let G be a Halin graph with maximum degree Δ. We prove that, a'(G) = 5 if G is K4, 4 if Δ = 3 and G is not K4, and Δ otherwise.]]> 0 <![CDATA[The relationship between graph coloring and the immersion order is considered. Vertex connectivity, edge connectivity and related issues are explored. It is shown that a t-chromatic graph G contains either an immersed Kt or an immersed t-chromatic subgraph that is both 4-vertex-connected and t-edge-connected. This gives supporting evidence of our conjecture that if G requires at least t colors, then Kt is immersed in G.]]> Mon, 29 Oct 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.589 https://doi.org/10.46298/dmtcs.589 Abu-Khzam, Faisal N. Langston, Michael A. Abu-Khzam, Faisal N. Langston, Michael A. <![CDATA[The relationship between graph coloring and the immersion order is considered. Vertex connectivity, edge connectivity and related issues are explored. It is shown that a t-chromatic graph G contains either an immersed Kt or an immersed t-chromatic subgraph that is both 4-vertex-connected and t-edge-connected. This gives supporting evidence of our conjecture that if G requires at least t colors, then Kt is immersed in G.]]> 0 <![CDATA[In the running time analysis of the algorithm Find and versions of it appear as limiting distributions solutions of stochastic fixed points equation of the form X D = Sigma(i) AiXi o Bi + C on the space D of cadlag functions. The distribution of the D-valued process X is invariant by some random linear affine transformation of space and random time change. We show the existence of solutions in some generality via the Weighted Branching Process. Finite exponential moments are connected to stochastic fixed point of supremum type X D = sup(i) (A(i)X(i) + C-i) on the positive reals. Specifically we present a running time analysis of m-median and adapted versions of Find. The finite dimensional distributions converge in L-1 and are continuous in the cylinder coordinates. We present the optimal adapted version in the sense of low asymptotic average number of comparisons. The limit distribution of the optimal adapted version of Find is a point measure on the function [0, 1] there exists t -> 1 + mint, 1 - t.]]> Sat, 22 Sep 2012 22:00:00 +0000 https://doi.org/10.46298/dmtcs.581 https://doi.org/10.46298/dmtcs.581 Knof, Diether Roesler, Uwe Knof, Diether Roesler, Uwe <![CDATA[In the running time analysis of the algorithm Find and versions of it appear as limiting distributions solutions of stochastic fixed points equation of the form X D = Sigma(i) AiXi o Bi + C on the space D of cadlag functions. The distribution of the D-valued process X is invariant by some random linear affine transformation of space and random time change. We show the existence of solutions in some generality via the Weighted Branching Process. Finite exponential moments are connected to stochastic fixed point of supremum type X D = sup(i) (A(i)X(i) + C-i) on the positive reals. Specifically we present a running time analysis of m-median and adapted versions of Find. The finite dimensional distributions converge in L-1 and are continuous in the cylinder coordinates. We present the optimal adapted version in the sense of low asymptotic average number of comparisons. The limit distribution of the optimal adapted version of Find is a point measure on the function [0, 1] there exists t -> 1 + mint, 1 - t.]]> 0 <![CDATA[We study a leader election protocol that we call the Swedish leader election protocol. This name comes from a protocol presented by L. Bondesson, T. Nilsson, and G. Wikstrand (2007). The goal is to select one among n > 0 players, by proceeding through a number of rounds. If there is only one player remaining, the protocol stops and the player is declared the leader. Otherwise, all remaining players flip a biased coin; with probability q the player survives to the next round, with probability p = 1 - q the player loses (is killed) and plays no further ... unless all players lose in a given round (null round), so all of them play again. In the classical leader election protocol, any number of null rounds may take place, and with probability 1 some player will ultimately be elected. In the Swedish leader election protocol there is a maximum number tau of consecutive null rounds, and if the threshold is attained the protocol fails without declaring a leader. In this paper, several parameters are asymptotically analyzed, among them: Success Probability, Number of rounds R-n, Number of null rounds T-n. This paper is a companion paper to Louchard, Martinez and Prodinger (2011) where De-Poissonization was used, together with the Mellin transform. While this works fine as far as it goes, there are limitations, in particular of a computational nature. The approach chosen here is similar to earlier efforts of the same authors - Louchard and Prodinger (2004,2005,2009). Identifying some underlying distributions as Gumbel (type) distributions, one can start with approximations at a very early stage and compute (at least in principle) all moments asymptotically. This is in contrast to the companion work, where only expected values were considered. In an appendix, it is shown that, whereever results are given in both papers, they coincide, although they are presented in different ways.]]> Sun, 16 Sep 2012 22:00:00 +0000 https://doi.org/10.46298/dmtcs.580 https://doi.org/10.46298/dmtcs.580 Louchard, Guy Prodinger, Helmut Louchard, Guy Prodinger, Helmut <![CDATA[We study a leader election protocol that we call the Swedish leader election protocol. This name comes from a protocol presented by L. Bondesson, T. Nilsson, and G. Wikstrand (2007). The goal is to select one among n > 0 players, by proceeding through a number of rounds. If there is only one player remaining, the protocol stops and the player is declared the leader. Otherwise, all remaining players flip a biased coin; with probability q the player survives to the next round, with probability p = 1 - q the player loses (is killed) and plays no further ... unless all players lose in a given round (null round), so all of them play again. In the classical leader election protocol, any number of null rounds may take place, and with probability 1 some player will ultimately be elected. In the Swedish leader election protocol there is a maximum number tau of consecutive null rounds, and if the threshold is attained the protocol fails without declaring a leader. In this paper, several parameters are asymptotically analyzed, among them: Success Probability, Number of rounds R-n, Number of null rounds T-n. This paper is a companion paper to Louchard, Martinez and Prodinger (2011) where De-Poissonization was used, together with the Mellin transform. While this works fine as far as it goes, there are limitations, in particular of a computational nature. The approach chosen here is similar to earlier efforts of the same authors - Louchard and Prodinger (2004,2005,2009). Identifying some underlying distributions as Gumbel (type) distributions, one can start with approximations at a very early stage and compute (at least in principle) all moments asymptotically. This is in contrast to the companion work, where only expected values were considered. In an appendix, it is shown that, whereever results are given in both papers, they coincide, although they are presented in different ways.]]> 0 <![CDATA[We consider random Cayley digraphs of order n with uniformly distributed generating sets of size k. Specifically, we are interested in the asymptotics of the probability that such a Cayley digraph has diameter two as n -> infinity and k = f(n), focusing on the functions f(n) = left perpendicularn(delta)right perpendicular and f(n) = left perpendicularcnright perpendicular. In both instances we show that this probability converges to 1 as n -> infinity for arbitrary fixed delta is an element of (1/2, 1) and c is an element of (0, 1/2), respectively, with a much larger convergence rate in the second case and with sharper results for Abelian groups.]]> Sun, 09 Sep 2012 22:00:00 +0000 https://doi.org/10.46298/dmtcs.588 https://doi.org/10.46298/dmtcs.588 Lladser, Manuel E. Potočnik, Primož Širáň, Jozef Wilson, Mark C. Lladser, Manuel E. Potočnik, Primož Širáň, Jozef Wilson, Mark C. <![CDATA[We consider random Cayley digraphs of order n with uniformly distributed generating sets of size k. Specifically, we are interested in the asymptotics of the probability that such a Cayley digraph has diameter two as n -> infinity and k = f(n), focusing on the functions f(n) = left perpendicularn(delta)right perpendicular and f(n) = left perpendicularcnright perpendicular. In both instances we show that this probability converges to 1 as n -> infinity for arbitrary fixed delta is an element of (1/2, 1) and c is an element of (0, 1/2), respectively, with a much larger convergence rate in the second case and with sharper results for Abelian groups.]]> 0 <![CDATA[We study graphs G in which the maximum number of vertex-disjoint cycles nu(G) is close to the cyclomatic number mu(G), which is a natural upper bound for nu(G). Our main result is the existence of a finite set P(k) of graphs for all k is an element of N-0 such that every 2-connected graph G with mu(G)-nu(G) = k arises by applying a simple extension rule to a graph in P(k). As an algorithmic consequence we describe algorithms calculating minmu(G)-nu(G), k + 1 in linear time for fixed k.]]> Sat, 08 Sep 2012 22:00:00 +0000 https://doi.org/10.46298/dmtcs.587 https://doi.org/10.46298/dmtcs.587 Rautenbach, Dieter Regen, Friedrich Rautenbach, Dieter Regen, Friedrich <![CDATA[We study graphs G in which the maximum number of vertex-disjoint cycles nu(G) is close to the cyclomatic number mu(G), which is a natural upper bound for nu(G). Our main result is the existence of a finite set P(k) of graphs for all k is an element of N-0 such that every 2-connected graph G with mu(G)-nu(G) = k arises by applying a simple extension rule to a graph in P(k). As an algorithmic consequence we describe algorithms calculating minmu(G)-nu(G), k + 1 in linear time for fixed k.]]> 0 <![CDATA[In this paper we deal from an algorithmic perspective with different questions regarding properly edge-colored (or PEC) paths, trails and closed trails. Given a c-edge-colored graph G(c), we show how to polynomially determine, if any, a PEC closed trail subgraph whose number of visits at each vertex is specified before hand. As a consequence, we solve a number of interesting related problems. For instance, given subset S of vertices in G(c), we show how to maximize in polynomial time the number of S-restricted vertex (resp., edge) disjoint PEC paths (resp., trails) in G(c) with endpoints in S. Further, if G(c) contains no PEC closed trails, we show that the problem of finding a PEC s-t trail visiting a given subset of vertices can be solved in polynomial time and prove that it becomes NP-complete if we are restricted to graphs with no PEC cycles. We also deal with graphs G(c) containing no (almost) PEC cycles or closed trails through s or t. We prove that finding 2 PEC s-t paths (resp., trails) with length at most L > 0 is NP-complete in the strong sense even for graphs with maximum degree equal to 3 and present an approximation algorithm for computing k vertex (resp., edge) disjoint PEC s-t paths (resp., trails) so that the maximum path (resp., trail) length is no more than k times the PEC path (resp., trail) length in an optimal solution. Further, we prove that finding 2 vertex disjoint s-t paths with exactly one PEC s-t path is NP-complete. This result is interesting since as proved in Abouelaoualim et. al.(2008), the determination of two or more vertex disjoint PEC s-t paths can be done in polynomial time. Finally, if G(c) is an arbitrary c-edge-colored graph with maximum vertex degree equal to four, we prove that finding two monochromatic vertex disjoint s-t paths with different colors is NP-complete. We also propose some related problems.]]> Sun, 02 Sep 2012 22:00:00 +0000 https://doi.org/10.46298/dmtcs.586 https://doi.org/10.46298/dmtcs.586 Gourvès, Laurent Lyra, Adria Martinhon, Carlos A. Monnot, Jérôme Gourvès, Laurent Lyra, Adria Martinhon, Carlos A. Monnot, Jérôme <![CDATA[In this paper we deal from an algorithmic perspective with different questions regarding properly edge-colored (or PEC) paths, trails and closed trails. Given a c-edge-colored graph G(c), we show how to polynomially determine, if any, a PEC closed trail subgraph whose number of visits at each vertex is specified before hand. As a consequence, we solve a number of interesting related problems. For instance, given subset S of vertices in G(c), we show how to maximize in polynomial time the number of S-restricted vertex (resp., edge) disjoint PEC paths (resp., trails) in G(c) with endpoints in S. Further, if G(c) contains no PEC closed trails, we show that the problem of finding a PEC s-t trail visiting a given subset of vertices can be solved in polynomial time and prove that it becomes NP-complete if we are restricted to graphs with no PEC cycles. We also deal with graphs G(c) containing no (almost) PEC cycles or closed trails through s or t. We prove that finding 2 PEC s-t paths (resp., trails) with length at most L > 0 is NP-complete in the strong sense even for graphs with maximum degree equal to 3 and present an approximation algorithm for computing k vertex (resp., edge) disjoint PEC s-t paths (resp., trails) so that the maximum path (resp., trail) length is no more than k times the PEC path (resp., trail) length in an optimal solution. Further, we prove that finding 2 vertex disjoint s-t paths with exactly one PEC s-t path is NP-complete. This result is interesting since as proved in Abouelaoualim et. al.(2008), the determination of two or more vertex disjoint PEC s-t paths can be done in polynomial time. Finally, if G(c) is an arbitrary c-edge-colored graph with maximum vertex degree equal to four, we prove that finding two monochromatic vertex disjoint s-t paths with different colors is NP-complete. We also propose some related problems.]]> 0 <![CDATA[This article deals with the Maximum Weight Stable Set (MWS) problem (and some other related NP-hard problems) and the class of P-6-free graphs. The complexity status of MWS is open for P-6-free graphs and is open even for P-5-free graphs (as a long standing open problem). Several results are known for MWS on subclasses of P-5-free: in particular, MWS can be solved for k-colorable P-5-free graphs in polynomial time for every k (depending on k) and more generally for (P-5, K-p)-free graphs (depending on p), which is a useful result since for every graph G one can easily compute a k-coloring of G, with k not necessarily minimum. This article studies the MWS problem for k-colorable P-6-free graphs and more generally for (P-6, K-p)-free graphs. Though we were not able to define a polynomial time algorithm for this problem for every k, this article introduces: (i) some structure properties of P-6-free graphs with respect to stable sets, (ii) two reductions for MWS on (P-6; K-p)-free graphs for every p, (iii) three polynomial time algorithms to solve MWS respectively for 3-colorable P-6-free, for 4-colorable P-6-free, and for (P-6, K-4)-free graphs (the latter allows one to state, together with other known results, that MWS can be solved for (P-6, F)-free graphs in polynomial time where F is any four vertex graph).]]> Mon, 27 Aug 2012 22:00:00 +0000 https://doi.org/10.46298/dmtcs.579 https://doi.org/10.46298/dmtcs.579 Mosca, Raffaele Mosca, Raffaele <![CDATA[This article deals with the Maximum Weight Stable Set (MWS) problem (and some other related NP-hard problems) and the class of P-6-free graphs. The complexity status of MWS is open for P-6-free graphs and is open even for P-5-free graphs (as a long standing open problem). Several results are known for MWS on subclasses of P-5-free: in particular, MWS can be solved for k-colorable P-5-free graphs in polynomial time for every k (depending on k) and more generally for (P-5, K-p)-free graphs (depending on p), which is a useful result since for every graph G one can easily compute a k-coloring of G, with k not necessarily minimum. This article studies the MWS problem for k-colorable P-6-free graphs and more generally for (P-6, K-p)-free graphs. Though we were not able to define a polynomial time algorithm for this problem for every k, this article introduces: (i) some structure properties of P-6-free graphs with respect to stable sets, (ii) two reductions for MWS on (P-6; K-p)-free graphs for every p, (iii) three polynomial time algorithms to solve MWS respectively for 3-colorable P-6-free, for 4-colorable P-6-free, and for (P-6, K-4)-free graphs (the latter allows one to state, together with other known results, that MWS can be solved for (P-6, F)-free graphs in polynomial time where F is any four vertex graph).]]> 0 <![CDATA[We consider the property that in a random definite Horn formula of size-3 clauses over n variables, where every such clause is included with probability p, there is a pair of variables for which forward chaining produces all other variables. We show that with high probability the property does not hold for p <= 1/(11n ln n), and does hold for p >= (5 1n ln n)/(n ln n).]]> Wed, 15 Aug 2012 22:00:00 +0000 https://doi.org/10.46298/dmtcs.578 https://doi.org/10.46298/dmtcs.578 Sloan, Robert H. Stasi, Despina Turán, György Sloan, Robert H. Stasi, Despina Turán, György <![CDATA[We consider the property that in a random definite Horn formula of size-3 clauses over n variables, where every such clause is included with probability p, there is a pair of variables for which forward chaining produces all other variables. We show that with high probability the property does not hold for p <= 1/(11n ln n), and does hold for p >= (5 1n ln n)/(n ln n).]]> 0 <![CDATA[An edge colouring of a graph is said to be neighbour-distinguishing if any two adjacent vertices have distinct sets of colours of their incident edges. In this paper the list version of the problem of determining the minimum number of colours in a neighbour-distinguishing colouring of a given graph is considered.]]> Wed, 15 Aug 2012 22:00:00 +0000 https://doi.org/10.46298/dmtcs.577 https://doi.org/10.46298/dmtcs.577 Horňák, Mirko WoźniaK, Mariusz Horňák, Mirko WoźniaK, Mariusz <![CDATA[An edge colouring of a graph is said to be neighbour-distinguishing if any two adjacent vertices have distinct sets of colours of their incident edges. In this paper the list version of the problem of determining the minimum number of colours in a neighbour-distinguishing colouring of a given graph is considered.]]> 0 <![CDATA[The notion of graph powers is a well-studied topic in graph theory and its applications. In this paper, we investigate a bipartite analogue of graph powers, which we call bipartite powers of bigraphs. We show that the classes of bipartite permutation graphs and interval bigraphs are closed under taking bipartite power. We also show that the problem of recognizing bipartite powers is NP-complete in general.]]> Sat, 04 Aug 2012 22:00:00 +0000 https://doi.org/10.46298/dmtcs.576 https://doi.org/10.46298/dmtcs.576 Okamoto, Yoshio Otachi, Yota Uehara, Ryuhei Okamoto, Yoshio Otachi, Yota Uehara, Ryuhei <![CDATA[The notion of graph powers is a well-studied topic in graph theory and its applications. In this paper, we investigate a bipartite analogue of graph powers, which we call bipartite powers of bigraphs. We show that the classes of bipartite permutation graphs and interval bigraphs are closed under taking bipartite power. We also show that the problem of recognizing bipartite powers is NP-complete in general.]]> 0 <![CDATA[In 1982, Opsut showed that the competition number of a line graph is at most two and gave a necessary and sufficient condition for the competition number of a line graph being one. In this paper, we generalize this result to the competition numbers of generalized line graphs, that is, we show that the competition number of a generalized line graph is at most two, and give necessary conditions and sufficient conditions for the competition number of a generalized line graph being one.]]> Sat, 04 Aug 2012 22:00:00 +0000 https://doi.org/10.46298/dmtcs.575 https://doi.org/10.46298/dmtcs.575 Park, Boram Sano, Yoshio Park, Boram Sano, Yoshio <![CDATA[In 1982, Opsut showed that the competition number of a line graph is at most two and gave a necessary and sufficient condition for the competition number of a line graph being one. In this paper, we generalize this result to the competition numbers of generalized line graphs, that is, we show that the competition number of a generalized line graph is at most two, and give necessary conditions and sufficient conditions for the competition number of a generalized line graph being one.]]> 0 <![CDATA[The join of two varieties is the smallest variety containing both. In finite semigroup theory, the varieties of R-trivial and L-trivial monoids are two of the most prominent classes of finite monoids. Their join is known to be decidable due to a result of Almeida and Azevedo. In this paper, we give a new proof for Almeida and Azevedo's effective characterization of the join of R-trivial and L-trivial monoids. This characterization is a single identity of omega-terms using three variables.]]> Sun, 24 Jun 2012 22:00:00 +0000 https://doi.org/10.46298/dmtcs.564 https://doi.org/10.46298/dmtcs.564 Kufleitner, Manfred Lauser, Alexander Kufleitner, Manfred Lauser, Alexander <![CDATA[The join of two varieties is the smallest variety containing both. In finite semigroup theory, the varieties of R-trivial and L-trivial monoids are two of the most prominent classes of finite monoids. Their join is known to be decidable due to a result of Almeida and Azevedo. In this paper, we give a new proof for Almeida and Azevedo's effective characterization of the join of R-trivial and L-trivial monoids. This characterization is a single identity of omega-terms using three variables.]]> 0 <![CDATA[We present theoretical and computational results on alpha-labelings of trees. The theorems proved in this paper were inspired by the results of a computer investigation of alpha-labelings of all trees with up to 26 vertices, all trees with maximum degree 3 and up to 36 vertices, all trees with maximum degree 4 and up to 32 vertices and all trees with maximum degree 5 and up to 31 vertices. We generalise a criterion for trees to have nonzero alpha-deficit, and prove an unexpected result on the alpha-deficit of trees with a vertex of large degree compared to the order of the tree.]]> Fri, 08 Jun 2012 22:00:00 +0000 https://doi.org/10.46298/dmtcs.569 https://doi.org/10.46298/dmtcs.569 Brinkmann, Gunnar Crevals, Simon Melot, Hadrien Rylands, Leanne Steffen, Eckhard Brinkmann, Gunnar Crevals, Simon Melot, Hadrien Rylands, Leanne Steffen, Eckhard <![CDATA[We present theoretical and computational results on alpha-labelings of trees. The theorems proved in this paper were inspired by the results of a computer investigation of alpha-labelings of all trees with up to 26 vertices, all trees with maximum degree 3 and up to 36 vertices, all trees with maximum degree 4 and up to 32 vertices and all trees with maximum degree 5 and up to 31 vertices. We generalise a criterion for trees to have nonzero alpha-deficit, and prove an unexpected result on the alpha-deficit of trees with a vertex of large degree compared to the order of the tree.]]> 0 <![CDATA[By an Euler walk in a 3-uniform hypergraph H we mean an alternating sequence v(0), epsilon(1), v(1), epsilon(2), v(2), ... , v(m-1), epsilon(m), v(m) of vertices and edges in H such that each edge of H appears in this sequence exactly once and v(i-1); v(i) is an element of epsilon(i), v(i-1) not equal v(i), for every i = 1, 2, ... , m. This concept is a natural extension of the graph theoretic notion of an Euler walk to the case of 3-uniform hypergraphs. We say that a 3-uniform hypergraph H is strongly connected if it has no isolated vertices and for each two edges e and f in H there is a sequence of edges starting with e and ending with f such that each two consecutive edges in this sequence have two vertices in common. In this paper we give an algorithm that constructs an Euler walk in a strongly connected 3-uniform hypergraph (it is known that such a walk in such a hypergraph always exists). The algorithm runs in time O(m), where m is the number of edges in the input hypergraph.]]> Mon, 04 Jun 2012 22:00:00 +0000 https://doi.org/10.46298/dmtcs.568 https://doi.org/10.46298/dmtcs.568 Lonc, Zbigniew Naroski, Pawel Lonc, Zbigniew Naroski, Pawel <![CDATA[By an Euler walk in a 3-uniform hypergraph H we mean an alternating sequence v(0), epsilon(1), v(1), epsilon(2), v(2), ... , v(m-1), epsilon(m), v(m) of vertices and edges in H such that each edge of H appears in this sequence exactly once and v(i-1); v(i) is an element of epsilon(i), v(i-1) not equal v(i), for every i = 1, 2, ... , m. This concept is a natural extension of the graph theoretic notion of an Euler walk to the case of 3-uniform hypergraphs. We say that a 3-uniform hypergraph H is strongly connected if it has no isolated vertices and for each two edges e and f in H there is a sequence of edges starting with e and ending with f such that each two consecutive edges in this sequence have two vertices in common. In this paper we give an algorithm that constructs an Euler walk in a strongly connected 3-uniform hypergraph (it is known that such a walk in such a hypergraph always exists). The algorithm runs in time O(m), where m is the number of edges in the input hypergraph.]]> 0 <![CDATA[We consider the problem of determining an orientation with minimum diameter MOD(G) of a connected and bridge-less graph G. In 2001 Fomin et al. discovered the relation MOD(G) <= 9 gamma(G) - 5 between the minimum oriented diameter and the size gamma(G) of a minimum dominating set. We improve their upper bound to MOD(G) <= 4 gamma(G).]]> Sat, 12 May 2012 22:00:00 +0000 https://doi.org/10.46298/dmtcs.567 https://doi.org/10.46298/dmtcs.567 Kurz, Sascha Laetsch, Martin Kurz, Sascha Laetsch, Martin <![CDATA[We consider the problem of determining an orientation with minimum diameter MOD(G) of a connected and bridge-less graph G. In 2001 Fomin et al. discovered the relation MOD(G) <= 9 gamma(G) - 5 between the minimum oriented diameter and the size gamma(G) of a minimum dominating set. We improve their upper bound to MOD(G) <= 4 gamma(G).]]> 0 <![CDATA[Let T be a monadic-second order class of finite trees, and let T(x) be its (ordinary) generating function, with radius of convergence rho. If rho >= 1 then T has an explicit specification (without using recursion) in terms of the operations of union, sum, stack, and the multiset operators n and (>= n). Using this, one has an explicit expression for T(x) in terms of the initial functions x and x . (1 - x(n))(-1), the operations of addition and multiplication, and the Polya exponentiation operators E-n, E-(>= n). Let F be a monadic-second order class of finite forests, and let F (x) = Sigma(n) integral(n)x(n) be its (ordinary) generating function. Suppose F is closed under extraction of component trees and sums of forests. Using the above-mentioned structure theory for the class T of trees in F, Compton's theory of 0-1 laws, and a significantly strengthened version of 2003 results of Bell and Burris on generating functions, we show that F has a monadic second-order 0-1 law iff the radius of convergence of F (x) is 1 iff the radius of convergence of T (x) is >= 1.]]> Mon, 07 May 2012 22:00:00 +0000 https://doi.org/10.46298/dmtcs.566 https://doi.org/10.46298/dmtcs.566 Bell, Jason P. Burris, Stanley N. Yeats, Karen A. Bell, Jason P. Burris, Stanley N. Yeats, Karen A. <![CDATA[Let T be a monadic-second order class of finite trees, and let T(x) be its (ordinary) generating function, with radius of convergence rho. If rho >= 1 then T has an explicit specification (without using recursion) in terms of the operations of union, sum, stack, and the multiset operators n and (>= n). Using this, one has an explicit expression for T(x) in terms of the initial functions x and x . (1 - x(n))(-1), the operations of addition and multiplication, and the Polya exponentiation operators E-n, E-(>= n). Let F be a monadic-second order class of finite forests, and let F (x) = Sigma(n) integral(n)x(n) be its (ordinary) generating function. Suppose F is closed under extraction of component trees and sums of forests. Using the above-mentioned structure theory for the class T of trees in F, Compton's theory of 0-1 laws, and a significantly strengthened version of 2003 results of Bell and Burris on generating functions, we show that F has a monadic second-order 0-1 law iff the radius of convergence of F (x) is 1 iff the radius of convergence of T (x) is >= 1.]]> 0 <![CDATA[In this paper, we consider a concept of adaptive identification of vertices and sets of vertices in different graphs, which was recently introduced by Ben-Haim, Gravier, Lobstein and Moncel (2008). The motivation for adaptive identification comes from applications such as sensor networks and fault detection in multiprocessor systems. We present an optimal adaptive algorithm for identifying vertices in cycles. We also give efficient adaptive algorithms for identifying sets of vertices in different graphs such as cycles, king lattices and square lattices. Adaptive identification is also considered in Hamming spaces, which is one of the most widely studied graphs in the field of identifying codes.]]> Mon, 16 Apr 2012 22:00:00 +0000 https://doi.org/10.46298/dmtcs.574 https://doi.org/10.46298/dmtcs.574 Junnila, Ville Junnila, Ville <![CDATA[In this paper, we consider a concept of adaptive identification of vertices and sets of vertices in different graphs, which was recently introduced by Ben-Haim, Gravier, Lobstein and Moncel (2008). The motivation for adaptive identification comes from applications such as sensor networks and fault detection in multiprocessor systems. We present an optimal adaptive algorithm for identifying vertices in cycles. We also give efficient adaptive algorithms for identifying sets of vertices in different graphs such as cycles, king lattices and square lattices. Adaptive identification is also considered in Hamming spaces, which is one of the most widely studied graphs in the field of identifying codes.]]> 0 <![CDATA[For a parallel computer system with m identical computers, we study optimal performance precaution for one possible computer crash. We want to calculate the cost of crash precaution in the case of no crash. We thus define a tolerance level r meaning that we only tolerate that the completion time of a parallel program after a crash is at most a factor r + 1 larger than if we use optimal allocation on m - 1 computers. This is an r-dependent restriction of the set of allocations of a program. Then, what is the worst-case ratio of the optimal r-dependent completion time in the case of no crash and the unrestricted optimal completion time of the same parallel program? We denote the maximal ratio of completion times f(r, m) - i.e., the ratio for worst-case programs. In the paper we establish upper and lower bounds of the worst-case cost function f (r, m) and characterize worst-case programs.]]> Thu, 22 Mar 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.573 https://doi.org/10.46298/dmtcs.573 Laksman, Efraim Lennerstad, Hakan Lundberg, Lars Laksman, Efraim Lennerstad, Hakan Lundberg, Lars <![CDATA[For a parallel computer system with m identical computers, we study optimal performance precaution for one possible computer crash. We want to calculate the cost of crash precaution in the case of no crash. We thus define a tolerance level r meaning that we only tolerate that the completion time of a parallel program after a crash is at most a factor r + 1 larger than if we use optimal allocation on m - 1 computers. This is an r-dependent restriction of the set of allocations of a program. Then, what is the worst-case ratio of the optimal r-dependent completion time in the case of no crash and the unrestricted optimal completion time of the same parallel program? We denote the maximal ratio of completion times f(r, m) - i.e., the ratio for worst-case programs. In the paper we establish upper and lower bounds of the worst-case cost function f (r, m) and characterize worst-case programs.]]> 0 <![CDATA[The generalized connectivity of a graph, which was introduced by Chartrand et al. in 1984, is a generalization of the concept of vertex connectivity. Let S be a nonempty set of vertices of G, a collection \T-1, T (2), ... , T-r\ of trees in G is said to be internally disjoint trees connecting S if E(T-i) boolean AND E(T-j) - empty set and V (T-i) boolean AND V(T-j) = S for any pair of distinct integers i, j, where 1 <= i, j <= r. For an integer k with 2 <= k <= n, the k-connectivity kappa(k) (G) of G is the greatest positive integer r for which G contains at least r internally disjoint trees connecting S for any set S of k vertices of G. Obviously, kappa(2)(G) = kappa(G) is the connectivity of G. Sabidussi's Theorem showed that kappa(G square H) >= kappa(G) + kappa(H) for any two connected graphs G and H. In this paper, we prove that for any two connected graphs G and H with kappa(3) (G) >= kappa(3) (H), if kappa(G) > kappa(3) (G), then kappa(3) (G square H) >= kappa(3) (G) + kappa(3) (H); if kappa(G) = kappa(3)(G), then kappa(3)(G square H) >= kappa(3)(G) + kappa(3) (H) - 1. Our result could be seen as an extension of Sabidussi's Theorem. Moreover, all the bounds are sharp.]]> Mon, 12 Mar 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.572 https://doi.org/10.46298/dmtcs.572 Li, Hengzhe Li, Xueliang Sun, Yuefang Li, Hengzhe Li, Xueliang Sun, Yuefang <![CDATA[The generalized connectivity of a graph, which was introduced by Chartrand et al. in 1984, is a generalization of the concept of vertex connectivity. Let S be a nonempty set of vertices of G, a collection \T-1, T (2), ... , T-r\ of trees in G is said to be internally disjoint trees connecting S if E(T-i) boolean AND E(T-j) - empty set and V (T-i) boolean AND V(T-j) = S for any pair of distinct integers i, j, where 1 <= i, j <= r. For an integer k with 2 <= k <= n, the k-connectivity kappa(k) (G) of G is the greatest positive integer r for which G contains at least r internally disjoint trees connecting S for any set S of k vertices of G. Obviously, kappa(2)(G) = kappa(G) is the connectivity of G. Sabidussi's Theorem showed that kappa(G square H) >= kappa(G) + kappa(H) for any two connected graphs G and H. In this paper, we prove that for any two connected graphs G and H with kappa(3) (G) >= kappa(3) (H), if kappa(G) > kappa(3) (G), then kappa(3) (G square H) >= kappa(3) (G) + kappa(3) (H); if kappa(G) = kappa(3)(G), then kappa(3)(G square H) >= kappa(3)(G) + kappa(3) (H) - 1. Our result could be seen as an extension of Sabidussi's Theorem. Moreover, all the bounds are sharp.]]> 0 <![CDATA[An independent dominating set D of a graph G = (V,E) is a subset of vertices such that every vertex in V \ D has at least one neighbor in D and D is an independent set, i.e. no two vertices of D are adjacent in G. Finding a minimum independent dominating set in a graph is an NP-hard problem. Whereas it is hard to cope with this problem using parameterized and approximation algorithms, there is a simple exact O(1.4423^n)-time algorithm solving the problem by enumerating all maximal independent sets. In this paper we improve the latter result, providing the first non trivial algorithm computing a minimum independent dominating set of a graph in time O(1.3569^n). Furthermore, we give a lower bound of \Omega(1.3247^n) on the worst-case running time of this algorithm, showing that the running time analysis is almost tight.]]> Mon, 06 Feb 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.563 https://doi.org/10.46298/dmtcs.563 Gaspers, Serge Liedloff, Mathieu Gaspers, Serge Liedloff, Mathieu <![CDATA[An independent dominating set D of a graph G = (V,E) is a subset of vertices such that every vertex in V \ D has at least one neighbor in D and D is an independent set, i.e. no two vertices of D are adjacent in G. Finding a minimum independent dominating set in a graph is an NP-hard problem. Whereas it is hard to cope with this problem using parameterized and approximation algorithms, there is a simple exact O(1.4423^n)-time algorithm solving the problem by enumerating all maximal independent sets. In this paper we improve the latter result, providing the first non trivial algorithm computing a minimum independent dominating set of a graph in time O(1.3569^n). Furthermore, we give a lower bound of \Omega(1.3247^n) on the worst-case running time of this algorithm, showing that the running time analysis is almost tight.]]> 0 <![CDATA[We say that a hypergraph H is hamiltonian chain saturated if H does not contain a hamiltonian chain but by adding any new edge we create a hamiltonian chain in H. In this paper we ask about the smallest size of a k-uniform hamiltonian chain saturated hypergraph. We present a construction of a family of k-uniform hamiltonian chain saturated hypergraphs with O(n(k-1/2)) edges.]]> Wed, 25 Jan 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.571 https://doi.org/10.46298/dmtcs.571 Dudek, Aneta Zak, Andrzej Dudek, Aneta Zak, Andrzej <![CDATA[We say that a hypergraph H is hamiltonian chain saturated if H does not contain a hamiltonian chain but by adding any new edge we create a hamiltonian chain in H. In this paper we ask about the smallest size of a k-uniform hamiltonian chain saturated hypergraph. We present a construction of a family of k-uniform hamiltonian chain saturated hypergraphs with O(n(k-1/2)) edges.]]> 0 <![CDATA[An edge-weighting vertex colouring of a graph is an edge-weight assignment such that the accumulated weights at the vertices yields a proper vertex colouring. If such an assignment from a set S exists, we say the graph is S-weight colourable. It is conjectured that every graph with no isolated edge is \1, 2, 3\-weight colourable. We explore the problem of classifying those graphs which are \1, 2\ -weight colourable. We establish that a number of classes of graphs are S -weight colourable for much more general sets S of size 2. In particular, we show that any graph having only cycles of length 0 mod 4 is S -weight colourable for most sets S of size 2. As a consequence, we classify the minimal graphs which are not \1, 2\-weight colourable with respect to subgraph containment. We also demonstrate techniques for constructing graphs which are not \1, 2\-weight colourable.]]> Mon, 16 Jan 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.570 https://doi.org/10.46298/dmtcs.570 Khatirinejad, Mahdad Naserasr, Reza Newman, Mike Seamone, Ben Stevens, Brett Khatirinejad, Mahdad Naserasr, Reza Newman, Mike Seamone, Ben Stevens, Brett <![CDATA[An edge-weighting vertex colouring of a graph is an edge-weight assignment such that the accumulated weights at the vertices yields a proper vertex colouring. If such an assignment from a set S exists, we say the graph is S-weight colourable. It is conjectured that every graph with no isolated edge is \1, 2, 3\-weight colourable. We explore the problem of classifying those graphs which are \1, 2\ -weight colourable. We establish that a number of classes of graphs are S -weight colourable for much more general sets S of size 2. In particular, we show that any graph having only cycles of length 0 mod 4 is S -weight colourable for most sets S of size 2. As a consequence, we classify the minimal graphs which are not \1, 2\-weight colourable with respect to subgraph containment. We also demonstrate techniques for constructing graphs which are not \1, 2\-weight colourable.]]> 0 <![CDATA[The metric dimension of a graph Gamma is the least number of vertices in a set with the property that the list of distances from any vertex to those in the set uniquely identifies that vertex. We consider the Grassmann graph G(q)(n, k) (whose vertices are the k-subspaces of F-q(n), and are adjacent if they intersect in a (k 1)-subspace) for k \textgreater= 2. We find an upper bound on its metric dimension, which is equal to the number of 1-dimensional subspaces of F-q(n). We also give a construction of a resolving set of this size in the case where k + 1 divides n, and a related construction in other cases.]]> Sat, 07 Jan 2012 23:00:00 +0000 https://doi.org/10.46298/dmtcs.532 https://doi.org/10.46298/dmtcs.532 Bailey, Robert F. Meagher, Karen Bailey, Robert F. Meagher, Karen <![CDATA[The metric dimension of a graph Gamma is the least number of vertices in a set with the property that the list of distances from any vertex to those in the set uniquely identifies that vertex. We consider the Grassmann graph G(q)(n, k) (whose vertices are the k-subspaces of F-q(n), and are adjacent if they intersect in a (k 1)-subspace) for k \textgreater= 2. We find an upper bound on its metric dimension, which is equal to the number of 1-dimensional subspaces of F-q(n). We also give a construction of a resolving set of this size in the case where k + 1 divides n, and a related construction in other cases.]]> 0 <![CDATA[We give a polyomino characterisation of recurrent configurations of the sandpile model on the complete bipartite graph $K_{m,n}$ in which one designated vertex is the sink. We present a bijection from these recurrent configurations to decorated parallelogram polyominoes whose bounding box is a $m×n$ rectangle. Other combinatorial structures appear in special cases of this correspondence: for example bicomposition matrices (a matrix analogue of set partitions), and (2+2)-free posets. A canonical toppling process for recurrent configurations gives rise to a path within the associated parallelogram polyominoes. We define a collection of polynomials that we call $q,t$-Narayana polynomials, the generating functions of the bistatistic $(\mathsf{area ,parabounce} )$ on the set of parallelogram polyominoes, akin to Haglund's $(\mathsf{area ,hagbounce} )$ bistatistic on Dyck paths. In doing so, we have extended a bistatistic of Egge et al. to the set of parallelogram polyominoes. This is one answer to their question concerning extensions to other combinatorial objects. We conjecture the $q,t$-Narayana polynomials to be symmetric and discuss the proofs for numerous special cases. We also show a relationship between the $q,t$-Catalan polynomials and our bistatistic $(\mathsf{area ,parabounce}) $on a subset of parallelogram polyominoes.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3044 https://doi.org/10.46298/dmtcs.3044 Dukes, Mark Le Borgne, Yvan Dukes, Mark Le Borgne, Yvan <![CDATA[We give a polyomino characterisation of recurrent configurations of the sandpile model on the complete bipartite graph $K_{m,n}$ in which one designated vertex is the sink. We present a bijection from these recurrent configurations to decorated parallelogram polyominoes whose bounding box is a $m×n$ rectangle. Other combinatorial structures appear in special cases of this correspondence: for example bicomposition matrices (a matrix analogue of set partitions), and (2+2)-free posets. A canonical toppling process for recurrent configurations gives rise to a path within the associated parallelogram polyominoes. We define a collection of polynomials that we call $q,t$-Narayana polynomials, the generating functions of the bistatistic $(\mathsf{area ,parabounce} )$ on the set of parallelogram polyominoes, akin to Haglund's $(\mathsf{area ,hagbounce} )$ bistatistic on Dyck paths. In doing so, we have extended a bistatistic of Egge et al. to the set of parallelogram polyominoes. This is one answer to their question concerning extensions to other combinatorial objects. We conjecture the $q,t$-Narayana polynomials to be symmetric and discuss the proofs for numerous special cases. We also show a relationship between the $q,t$-Catalan polynomials and our bistatistic $(\mathsf{area ,parabounce}) $on a subset of parallelogram polyominoes.]]> 0 <![CDATA[We produce skew Pieri Rules for Hall–Littlewood functions in the spirit of Assaf and McNamara (FPSAC, 2010). The first two were conjectured by the first author (FPSAC, 2011). The key ingredients in the proofs are a q-binomial identity for skew partitions that are horizontal strips and a Hopf algebraic identity that expands products of skew elements in terms of the coproduct and antipode.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3054 https://doi.org/10.46298/dmtcs.3054 Konvalinka, Matjaž Lauve, Aaron Konvalinka, Matjaž Lauve, Aaron <![CDATA[We produce skew Pieri Rules for Hall–Littlewood functions in the spirit of Assaf and McNamara (FPSAC, 2010). The first two were conjectured by the first author (FPSAC, 2011). The key ingredients in the proofs are a q-binomial identity for skew partitions that are horizontal strips and a Hopf algebraic identity that expands products of skew elements in terms of the coproduct and antipode.]]> 0 <![CDATA[The well-known Gilbert-Shannon-Reeds model for riffle shuffles assumes that the cards are initially cut `about in half' and then riffled together. We analyze a natural variant where the initial cut is biased. Extending results of Fulman (1998), we show a sharp cutoff in separation and L-infinity distances. This analysis is possible due to the close connection between shuffling and quasisymmetric functions along with some complex analysis of a generating function.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3053 https://doi.org/10.46298/dmtcs.3053 Assaf, Sami Diaconis, Persi Soundararajan, Kannan Assaf, Sami Diaconis, Persi Soundararajan, Kannan <![CDATA[The well-known Gilbert-Shannon-Reeds model for riffle shuffles assumes that the cards are initially cut `about in half' and then riffled together. We analyze a natural variant where the initial cut is biased. Extending results of Fulman (1998), we show a sharp cutoff in separation and L-infinity distances. This analysis is possible due to the close connection between shuffling and quasisymmetric functions along with some complex analysis of a generating function.]]> 0 <![CDATA[We prove that the Lam-Shimozono ``down operator'' on the affine Weyl group induces a derivation of the affine Fomin-Stanley subalgebra of the affine nilCoxeter algebra. We use this to verify a conjecture of Berg, Bergeron, Pon and Zabrocki describing the expansion of k-Schur functions of ``near rectangles'' in the affine nilCoxeter algebra. Consequently, we obtain a combinatorial interpretation of the corresponding k-Littlewood–Richardson coefficients.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3052 https://doi.org/10.46298/dmtcs.3052 Berg, Chris Saliola, Franco Serrano, Luis Berg, Chris Saliola, Franco Serrano, Luis <![CDATA[We prove that the Lam-Shimozono ``down operator'' on the affine Weyl group induces a derivation of the affine Fomin-Stanley subalgebra of the affine nilCoxeter algebra. We use this to verify a conjecture of Berg, Bergeron, Pon and Zabrocki describing the expansion of k-Schur functions of ``near rectangles'' in the affine nilCoxeter algebra. Consequently, we obtain a combinatorial interpretation of the corresponding k-Littlewood–Richardson coefficients.]]> 0 <![CDATA[We investigate the combinatorics and geometry of permutation polytopes associated to cyclic permutation groups, i.e., the convex hulls of cyclic groups of permutation matrices. In the situation that the generator of the group consists of at most two orbits, we can give a complete combinatorial description of the associated permutation polytope. In the case of three orbits the facet structure is already quite complex. For a large class of examples we show that there exist exponentially many facets.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3051 https://doi.org/10.46298/dmtcs.3051 Baumeister, Barbara Haase, Christian Nill, Benjamin Paffenholz, Andreas Baumeister, Barbara Haase, Christian Nill, Benjamin Paffenholz, Andreas <![CDATA[We investigate the combinatorics and geometry of permutation polytopes associated to cyclic permutation groups, i.e., the convex hulls of cyclic groups of permutation matrices. In the situation that the generator of the group consists of at most two orbits, we can give a complete combinatorial description of the associated permutation polytope. In the case of three orbits the facet structure is already quite complex. For a large class of examples we show that there exist exponentially many facets.]]> 0 <![CDATA[We describe a generating tree approach to the enumeration and exhaustive generation of k-nonnesting set partitions and permutations. Unlike previous work in the literature using the connections of these objects to Young tableaux and restricted lattice walks, our approach deals directly with partition and permutation diagrams. We provide explicit functional equations for the generating functions, with k as a parameter.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3050 https://doi.org/10.46298/dmtcs.3050 Burrill, Sophie Elizalde, Sergi Mishna, Marni Yen, Lily Burrill, Sophie Elizalde, Sergi Mishna, Marni Yen, Lily <![CDATA[We describe a generating tree approach to the enumeration and exhaustive generation of k-nonnesting set partitions and permutations. Unlike previous work in the literature using the connections of these objects to Young tableaux and restricted lattice walks, our approach deals directly with partition and permutation diagrams. We provide explicit functional equations for the generating functions, with k as a parameter.]]> 0 <![CDATA[The pentagram map, introduced by R. Schwartz, is a birational map on the configuration space of polygons in the projective plane. We study the singularities of the iterates of the pentagram map. We show that a ``typical'' singularity disappears after a finite number of iterations, a confinement phenomenon first discovered by Schwartz. We provide a method to bypass such a singular patch by directly constructing the first subsequent iterate that is well-defined on the singular locus under consideration. The key ingredient of this construction is the notion of a decorated (twisted) polygon, and the extension of the pentagram map to the corresponding decorated configuration space.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3049 https://doi.org/10.46298/dmtcs.3049 Glick, Max Glick, Max <![CDATA[The pentagram map, introduced by R. Schwartz, is a birational map on the configuration space of polygons in the projective plane. We study the singularities of the iterates of the pentagram map. We show that a ``typical'' singularity disappears after a finite number of iterations, a confinement phenomenon first discovered by Schwartz. We provide a method to bypass such a singular patch by directly constructing the first subsequent iterate that is well-defined on the singular locus under consideration. The key ingredient of this construction is the notion of a decorated (twisted) polygon, and the extension of the pentagram map to the corresponding decorated configuration space.]]> 0 <![CDATA[We study the fluctuations of models of random partitions $(\mathbb{P}_n,ω )_n ∈\mathbb{N}$ stemming from the representation theory of the infinite symmetric group. Using the theory of polynomial functions on Young diagrams, we establish a central limit theorem for the values of the irreducible characters $χ ^λ$ of the symmetric groups, with $λ$ taken randomly according to the laws $\mathbb{P}_n,ω$ . This implies a central limit theorem for the rows and columns of the random partitions, and these ``geometric'' fluctuations of our models can be recovered by relating central measures on partitions, generalized riffle shuffles, and Brownian motions conditioned to stay in a Weyl chamber.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3048 https://doi.org/10.46298/dmtcs.3048 Mèliot, Pierre-Loïc Mèliot, Pierre-Loïc <![CDATA[We study the fluctuations of models of random partitions $(\mathbb{P}_n,ω )_n ∈\mathbb{N}$ stemming from the representation theory of the infinite symmetric group. Using the theory of polynomial functions on Young diagrams, we establish a central limit theorem for the values of the irreducible characters $χ ^λ$ of the symmetric groups, with $λ$ taken randomly according to the laws $\mathbb{P}_n,ω$ . This implies a central limit theorem for the rows and columns of the random partitions, and these ``geometric'' fluctuations of our models can be recovered by relating central measures on partitions, generalized riffle shuffles, and Brownian motions conditioned to stay in a Weyl chamber.]]> 0 <![CDATA[It is well-known that the derivation modules of Coxeter arrangements are free. Holm began to study the freeness of modules of differential operators on hyperplane arrangements. In this paper, we study the cases of the Coxter arrangements of type A, B and D. In this case, we prove that the modules of differential operators of order 2 are free. We give examples of all the 3-dimensional classical Coxeter arrangements. Two keys for the proof are ``Cauchy–Sylvester's theorem on compound determinants'' and ``Saito–Holm's criterion''.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3047 https://doi.org/10.46298/dmtcs.3047 Nakashima, Norihiro Nakashima, Norihiro <![CDATA[It is well-known that the derivation modules of Coxeter arrangements are free. Holm began to study the freeness of modules of differential operators on hyperplane arrangements. In this paper, we study the cases of the Coxter arrangements of type A, B and D. In this case, we prove that the modules of differential operators of order 2 are free. We give examples of all the 3-dimensional classical Coxeter arrangements. Two keys for the proof are ``Cauchy–Sylvester's theorem on compound determinants'' and ``Saito–Holm's criterion''.]]> 0 <![CDATA[Recently, Kenyon and Wilson introduced a certain matrix M in order to compute pairing probabilities of what they call the double-dimer model. They showed that the absolute value of each entry of the inverse matrix $M^-1$ is equal to the number of certain Dyck tilings of a skew shape. They conjectured two formulas on the sum of the absolute values of the entries in a row or a column of $M^-1$. In this paper we prove the two conjectures. As a consequence we obtain that the sum of the absolute values of all entries of $M^-1$ is equal to the number of complete matchings. We also find a bijection between Dyck tilings and complete matchings.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3046 https://doi.org/10.46298/dmtcs.3046 Kim, Jang Soo Kim, Jang Soo <![CDATA[Recently, Kenyon and Wilson introduced a certain matrix M in order to compute pairing probabilities of what they call the double-dimer model. They showed that the absolute value of each entry of the inverse matrix $M^-1$ is equal to the number of certain Dyck tilings of a skew shape. They conjectured two formulas on the sum of the absolute values of the entries in a row or a column of $M^-1$. In this paper we prove the two conjectures. As a consequence we obtain that the sum of the absolute values of all entries of $M^-1$ is equal to the number of complete matchings. We also find a bijection between Dyck tilings and complete matchings.]]> 0 <![CDATA[An $m$-ballot path of size $n$ is a path on the square grid consisting of north and east unit steps, starting at (0,0), ending at $(mn,n)$, and never going below the line $\{x=my\}$. The set of these paths can be equipped with a lattice structure, called the $m$-Tamari lattice and denoted by $\mathcal{T}{_n}^{(m)}$, which generalizes the usual Tamari lattice $\mathcal{T}n$ obtained when $m=1$. This lattice was introduced by F. Bergeron in connection with the study of diagonally coinvariant spaces in three sets of $n$ variables. The representation of the symmetric group $\mathfrak{S}_n$ on these spaces is conjectured to be closely related to the natural representation of $\mathfrak{S}_n$ on (labelled) intervals of the $m$-Tamari lattice studied in this paper. An interval $[P,Q$] of $\mathcal{T}{_n}^{(m)}$ is labelled if the north steps of $Q$ are labelled from 1 to $n$ in such a way the labels increase along any sequence of consecutive north steps. The symmetric group $\mathfrak{S}_n$ acts on labelled intervals of $\mathcal{T}{_n}^{(m)}$by permutation of the labels. We prove an explicit formula, conjectured by F. Bergeron and the third author, for the character of the associated representation of $\mathfrak{S}_n$. In particular, the dimension of the representation, that is, the number of labelled $m$-Tamari intervals of size $n$, is found to be $(m+1)^n(mn+1)^{n-2}$. These results are new, even when $m=1$. The form of these numbers suggests a connection with parking functions, but our proof is not bijective. The starting point is a recursive description of $m$-Tamari intervals. It yields an equation for an associated generating function, which is a refined version of the Frobenius series of the representation. The form of this equation is highly non-standard: it involves two additional variables $x$ and $y$, a derivative with respect to $y$ and iterated divided differences with respect to $x$. The hardest part of the proof consists in solving it, and we develop original techniques to do so.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3045 https://doi.org/10.46298/dmtcs.3045 Bousquet-Mélou, Mireille Chapuy, Guillaume Préville-Ratelle, Louis-François Bousquet-Mélou, Mireille Chapuy, Guillaume Préville-Ratelle, Louis-François <![CDATA[An $m$-ballot path of size $n$ is a path on the square grid consisting of north and east unit steps, starting at (0,0), ending at $(mn,n)$, and never going below the line $\{x=my\}$. The set of these paths can be equipped with a lattice structure, called the $m$-Tamari lattice and denoted by $\mathcal{T}{_n}^{(m)}$, which generalizes the usual Tamari lattice $\mathcal{T}n$ obtained when $m=1$. This lattice was introduced by F. Bergeron in connection with the study of diagonally coinvariant spaces in three sets of $n$ variables. The representation of the symmetric group $\mathfrak{S}_n$ on these spaces is conjectured to be closely related to the natural representation of $\mathfrak{S}_n$ on (labelled) intervals of the $m$-Tamari lattice studied in this paper. An interval $[P,Q$] of $\mathcal{T}{_n}^{(m)}$ is labelled if the north steps of $Q$ are labelled from 1 to $n$ in such a way the labels increase along any sequence of consecutive north steps. The symmetric group $\mathfrak{S}_n$ acts on labelled intervals of $\mathcal{T}{_n}^{(m)}$by permutation of the labels. We prove an explicit formula, conjectured by F. Bergeron and the third author, for the character of the associated representation of $\mathfrak{S}_n$. In particular, the dimension of the representation, that is, the number of labelled $m$-Tamari intervals of size $n$, is found to be $(m+1)^n(mn+1)^{n-2}$. These results are new, even when $m=1$. The form of these numbers suggests a connection with parking functions, but our proof is not bijective. The starting point is a recursive description of $m$-Tamari intervals. It yields an equation for an associated generating function, which is a refined version of the Frobenius series of the representation. The form of this equation is highly non-standard: it involves two additional variables $x$ and $y$, a derivative with respect to $y$ and iterated divided differences with respect to $x$. The hardest part of the proof consists in solving it, and we develop original techniques to do so.]]> 0 <![CDATA[We consider unicellular maps, or polygon gluings, of fixed genus. In FPSAC '09 the first author gave a recursive bijection transforming unicellular maps into trees, explaining the presence of Catalan numbers in counting formulas for these objects. In this paper, we give another bijection that explicitly describes the ``recursive part'' of the first bijection. As a result we obtain a very simple description of unicellular maps as pairs made by a plane tree and a permutation-like structure. All the previously known formulas follow as an immediate corollary or easy exercise, thus giving a bijective proof for each of them, in a unified way. For some of these formulas, this is the first bijective proof, e.g. the Harer-Zagier recurrence formula, or the Lehman-Walsh/Goupil-Schaeffer formulas. Thanks to previous work of the second author this also leads us to a new expression for Stanley character polynomials, which evaluate irreducible characters of the symmetric group.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3033 https://doi.org/10.46298/dmtcs.3033 Chapuy, Guillaume Feray, Valentin Fusy, Eric Chapuy, Guillaume Feray, Valentin Fusy, Eric <![CDATA[We consider unicellular maps, or polygon gluings, of fixed genus. In FPSAC '09 the first author gave a recursive bijection transforming unicellular maps into trees, explaining the presence of Catalan numbers in counting formulas for these objects. In this paper, we give another bijection that explicitly describes the ``recursive part'' of the first bijection. As a result we obtain a very simple description of unicellular maps as pairs made by a plane tree and a permutation-like structure. All the previously known formulas follow as an immediate corollary or easy exercise, thus giving a bijective proof for each of them, in a unified way. For some of these formulas, this is the first bijective proof, e.g. the Harer-Zagier recurrence formula, or the Lehman-Walsh/Goupil-Schaeffer formulas. Thanks to previous work of the second author this also leads us to a new expression for Stanley character polynomials, which evaluate irreducible characters of the symmetric group.]]> 0 <![CDATA[We introduce the notion of arithmetic matroid, whose main example is provided by a list of elements in a finitely generated abelian group. We study the representability of its dual, and, guided by the geometry of toric arrangements, we give a combinatorial interpretation of the associated arithmetic Tutte polynomial, which can be seen as a generalization of Crapo's formula.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3043 https://doi.org/10.46298/dmtcs.3043 D'Adderio, Michele Moci, Luca D'Adderio, Michele Moci, Luca <![CDATA[We introduce the notion of arithmetic matroid, whose main example is provided by a list of elements in a finitely generated abelian group. We study the representability of its dual, and, guided by the geometry of toric arrangements, we give a combinatorial interpretation of the associated arithmetic Tutte polynomial, which can be seen as a generalization of Crapo's formula.]]> 0 <![CDATA[The number of Monotone Triangles with bottom row $k_1 < k_2 < ⋯< k_n$ is given by a polynomial $\alpha (n; k_1,\ldots,k_n)$ in $n$ variables. The evaluation of this polynomial at weakly decreasing sequences $k_1 ≥k_2 ≥⋯≥k_n $turns out to be interpretable as signed enumeration of new combinatorial objects called Decreasing Monotone Triangles. There exist surprising connections between the two classes of objects – in particular it is shown that $\alpha (n;1,2,\ldots,n) = \alpha (2n; n,n,n-1,n-1,\ldots,1,1)$. In perfect analogy to the correspondence between Monotone Triangles and Alternating Sign Matrices, the set of Decreasing Monotone Triangles with bottom row $(n,n,n-1,n-1,\ldots,1,1)$ is in one-to-one correspondence with a certain set of ASM-like matrices, which also play an important role in proving the claimed identity algebraically. Finding a bijective proof remains an open problem.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3042 https://doi.org/10.46298/dmtcs.3042 Fischer, Ilse Riegler, Lukas Fischer, Ilse Riegler, Lukas <![CDATA[The number of Monotone Triangles with bottom row $k_1 < k_2 < ⋯< k_n$ is given by a polynomial $\alpha (n; k_1,\ldots,k_n)$ in $n$ variables. The evaluation of this polynomial at weakly decreasing sequences $k_1 ≥k_2 ≥⋯≥k_n $turns out to be interpretable as signed enumeration of new combinatorial objects called Decreasing Monotone Triangles. There exist surprising connections between the two classes of objects – in particular it is shown that $\alpha (n;1,2,\ldots,n) = \alpha (2n; n,n,n-1,n-1,\ldots,1,1)$. In perfect analogy to the correspondence between Monotone Triangles and Alternating Sign Matrices, the set of Decreasing Monotone Triangles with bottom row $(n,n,n-1,n-1,\ldots,1,1)$ is in one-to-one correspondence with a certain set of ASM-like matrices, which also play an important role in proving the claimed identity algebraically. Finding a bijective proof remains an open problem.]]> 0 <![CDATA[The symmetric edge polytopes of odd cycles (del Pezzo polytopes) are known as smooth Fano polytopes. In this extended abstract, we show that if the length of the cycle is 127, then the Ehrhart polynomial has a root whose real part is greater than the dimension. As a result, we have a smooth Fano polytope that is a counterexample to the two conjectures on the roots of Ehrhart polynomials.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3041 https://doi.org/10.46298/dmtcs.3041 Ohsugi, Hidefumi Shibata, Kazuki Ohsugi, Hidefumi Shibata, Kazuki <![CDATA[The symmetric edge polytopes of odd cycles (del Pezzo polytopes) are known as smooth Fano polytopes. In this extended abstract, we show that if the length of the cycle is 127, then the Ehrhart polynomial has a root whose real part is greater than the dimension. As a result, we have a smooth Fano polytope that is a counterexample to the two conjectures on the roots of Ehrhart polynomials.]]> 0 <![CDATA[We give a multivariate analog of the type B Eulerian polynomial introduced by Brenti. We prove that this multivariate polynomial is stable generalizing Brenti's result that every root of the type B Eulerian polynomial is real. Our proof combines a refinement of the descent statistic for signed permutations with the notion of real stability—a generalization of real-rootedness to polynomials in multiple variables. The key is that our refined multivariate Eulerian polynomials satisfy a recurrence given by a stability-preserving linear operator.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3040 https://doi.org/10.46298/dmtcs.3040 Visontai, Mirkó Williams, Nathan Visontai, Mirkó Williams, Nathan <![CDATA[We give a multivariate analog of the type B Eulerian polynomial introduced by Brenti. We prove that this multivariate polynomial is stable generalizing Brenti's result that every root of the type B Eulerian polynomial is real. Our proof combines a refinement of the descent statistic for signed permutations with the notion of real stability—a generalization of real-rootedness to polynomials in multiple variables. The key is that our refined multivariate Eulerian polynomials satisfy a recurrence given by a stability-preserving linear operator.]]> 0 <![CDATA[We investigate the Zariski closure of the projective equivalence class of a matrix. New results are presented regarding the matrices in this variety and their matroids, and we give equations for the variety. We also discuss the K-polynomial of the closure of a projective equivalence class, and two other geometric invariants that can be obtained from this.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3039 https://doi.org/10.46298/dmtcs.3039 Berget, Andrew Fink, Alex Berget, Andrew Fink, Alex <![CDATA[We investigate the Zariski closure of the projective equivalence class of a matrix. New results are presented regarding the matrices in this variety and their matroids, and we give equations for the variety. We also discuss the K-polynomial of the closure of a projective equivalence class, and two other geometric invariants that can be obtained from this.]]> 0 <![CDATA[We present an equivariant bijection between two actions—promotion and rowmotion—on order ideals in certain posets. This bijection simultaneously generalizes a result of R. Stanley concerning promotion on the linear extensions of two disjoint chains and certain cases of recent work of D. Armstrong, C. Stump, and H. Thomas on noncrossing and nonnesting partitions. We apply this bijection to several classes of posets, obtaining equivariant bijections to various known objects under rotation. We extend the same idea to give an equivariant bijection between alternating sign matrices under rowmotion and under B. Wieland's gyration. Lastly, we define two actions with related orders on alternating sign matrices and totally symmetric self-complementary plane partitions.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3038 https://doi.org/10.46298/dmtcs.3038 Striker, Jessica Williams, Nathan Striker, Jessica Williams, Nathan <![CDATA[We present an equivariant bijection between two actions—promotion and rowmotion—on order ideals in certain posets. This bijection simultaneously generalizes a result of R. Stanley concerning promotion on the linear extensions of two disjoint chains and certain cases of recent work of D. Armstrong, C. Stump, and H. Thomas on noncrossing and nonnesting partitions. We apply this bijection to several classes of posets, obtaining equivariant bijections to various known objects under rotation. We extend the same idea to give an equivariant bijection between alternating sign matrices under rowmotion and under B. Wieland's gyration. Lastly, we define two actions with related orders on alternating sign matrices and totally symmetric self-complementary plane partitions.]]> 0 <![CDATA[Arc permutations and unimodal permutations were introduced in the study of triangulations and characters. In this paper we describe combinatorial properties of these permutations, including characterizations in terms of pattern avoidance, connections to Young tableaux, and an affine Weyl group action on them.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3037 https://doi.org/10.46298/dmtcs.3037 Elizalde, Sergi Roichman, Yuval Elizalde, Sergi Roichman, Yuval <![CDATA[Arc permutations and unimodal permutations were introduced in the study of triangulations and characters. In this paper we describe combinatorial properties of these permutations, including characterizations in terms of pattern avoidance, connections to Young tableaux, and an affine Weyl group action on them.]]> 0 <![CDATA[We use the cluster method in order to enumerate permutations avoiding consecutive patterns. We reprove and generalize in a unified way several known results and obtain new ones, including some patterns of length 4 and 5, as well as some infinite families of patterns of a given shape. Our main tool is the cluster method of Goulden and Jackson. We also prove some that, for a large class of patterns, the inverse of the exponential generating function counting occurrences is an entire function, but we conjecture that it is not D-finite in general.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3036 https://doi.org/10.46298/dmtcs.3036 Elizalde, Sergi Noy, Marc Elizalde, Sergi Noy, Marc <![CDATA[We use the cluster method in order to enumerate permutations avoiding consecutive patterns. We reprove and generalize in a unified way several known results and obtain new ones, including some patterns of length 4 and 5, as well as some infinite families of patterns of a given shape. Our main tool is the cluster method of Goulden and Jackson. We also prove some that, for a large class of patterns, the inverse of the exponential generating function counting occurrences is an entire function, but we conjecture that it is not D-finite in general.]]> 0 <![CDATA[We give in a particular case a combinatorial proof of a recent algebraicity result of Kontsevich; the proof uses generalized one-sided and two-sided Dyck words, or equivalently, excursions and bridges. We indicate a noncommutative version of these notions, which could lead to a full proof. We show also a relation with pointed planar maps.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3035 https://doi.org/10.46298/dmtcs.3035 Reutenauer, Christophe Robado, Marco Reutenauer, Christophe Robado, Marco <![CDATA[We give in a particular case a combinatorial proof of a recent algebraicity result of Kontsevich; the proof uses generalized one-sided and two-sided Dyck words, or equivalently, excursions and bridges. We indicate a noncommutative version of these notions, which could lead to a full proof. We show also a relation with pointed planar maps.]]> 0 <![CDATA[We introduce a functorial construction which, from a monoid, produces a set-operad. We obtain new (symmetric or not) operads as suboperads or quotients of the operad obtained from the additive monoid. These involve various familiar combinatorial objects: parking functions, packed words, planar rooted trees, generalized Dyck paths, Schröder trees, Motzkin paths, integer compositions, directed animals, etc. We also retrieve some known operads: the magmatic operad, the commutative associative operad, and the diassociative operad.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3034 https://doi.org/10.46298/dmtcs.3034 Giraudo, Samuele Giraudo, Samuele <![CDATA[We introduce a functorial construction which, from a monoid, produces a set-operad. We obtain new (symmetric or not) operads as suboperads or quotients of the operad obtained from the additive monoid. These involve various familiar combinatorial objects: parking functions, packed words, planar rooted trees, generalized Dyck paths, Schröder trees, Motzkin paths, integer compositions, directed animals, etc. We also retrieve some known operads: the magmatic operad, the commutative associative operad, and the diassociative operad.]]> 0 <![CDATA[We obtain a very simple formula for the generating function of bipartite (resp. quasi-bipartite) planar maps with boundaries (holes) of prescribed lengths, which generalizes certain expressions obtained by Eynard in a book to appear. The formula is derived from a bijection due to Bouttier, Di Francesco and Guitter combined with a process (reminiscent of a construction of Pitman) of aggregating connected components of a forest into a single tree.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3067 https://doi.org/10.46298/dmtcs.3067 Collet, Gwendal Fusy, Eric Collet, Gwendal Fusy, Eric <![CDATA[We obtain a very simple formula for the generating function of bipartite (resp. quasi-bipartite) planar maps with boundaries (holes) of prescribed lengths, which generalizes certain expressions obtained by Eynard in a book to appear. The formula is derived from a bijection due to Bouttier, Di Francesco and Guitter combined with a process (reminiscent of a construction of Pitman) of aggregating connected components of a forest into a single tree.]]> 0 <![CDATA[Using the expansion of the inverse of the Kostka matrix in terms of tabloids as presented by Eğecioğlu and Remmel, we show that the fusion coefficients can be expressed as an alternating sum over cylindric tableaux. Cylindric tableaux are skew tableaux with a certain cyclic symmetry. When the skew shape of the tableau has a cutting point, meaning that the cylindric skew shape is not connected, or if its weight has at most two parts, we give a positive combinatorial formula for the fusion coefficients. The proof uses a slight modification of a sign-reversing involution introduced by Remmel and Shimozono. We discuss how this approach may work in general.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3078 https://doi.org/10.46298/dmtcs.3078 Morse, Jennifer Schilling, Anne Morse, Jennifer Schilling, Anne <![CDATA[Using the expansion of the inverse of the Kostka matrix in terms of tabloids as presented by Eğecioğlu and Remmel, we show that the fusion coefficients can be expressed as an alternating sum over cylindric tableaux. Cylindric tableaux are skew tableaux with a certain cyclic symmetry. When the skew shape of the tableau has a cutting point, meaning that the cylindric skew shape is not connected, or if its weight has at most two parts, we give a positive combinatorial formula for the fusion coefficients. The proof uses a slight modification of a sign-reversing involution introduced by Remmel and Shimozono. We discuss how this approach may work in general.]]> 0 <![CDATA[Baxter numbers are known to count several families of combinatorial objects, all of which come equipped with natural involutions. In this paper, we add a combinatorial family to the list, and show that the known bijections between these objects respect these involutions. We also give a formula for the number of objects fixed under this involution, showing that it is an instance of Stembridge's "$q=-1$ phenomenon''.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3077 https://doi.org/10.46298/dmtcs.3077 Dilks, Kevin Dilks, Kevin <![CDATA[Baxter numbers are known to count several families of combinatorial objects, all of which come equipped with natural involutions. In this paper, we add a combinatorial family to the list, and show that the known bijections between these objects respect these involutions. We also give a formula for the number of objects fixed under this involution, showing that it is an instance of Stembridge's "$q=-1$ phenomenon''.]]> 0 <![CDATA[Given a fan $\Delta$ and a cone $\sigma \in \Delta$ let $star^1(\sigma )$ be the set of cones that contain $\sigma$ and are one dimension bigger than $\sigma$ . In this paper we study two cones of piecewise linear functions defined on $\delta$ : the cone of functions which are convex on $star^1(σ\sigma)$ for all cones, and the cone of functions which are convex on $star^1(σ\sigma)$ for all cones of codimension 1. We give nice combinatorial descriptions for these two cones given two different fan structures on the tropical linear space of complete graphs. For the complete graph $K_5$, we prove that with the finer fan subdivision the two cones are not equal, but with the coarser subdivision they are the same. This gives a negative answer to a question of Gibney-Maclagan that for the finer subdivision the two cones are the same.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3076 https://doi.org/10.46298/dmtcs.3076 Escobar, Laura Escobar, Laura <![CDATA[Given a fan $\Delta$ and a cone $\sigma \in \Delta$ let $star^1(\sigma )$ be the set of cones that contain $\sigma$ and are one dimension bigger than $\sigma$ . In this paper we study two cones of piecewise linear functions defined on $\delta$ : the cone of functions which are convex on $star^1(σ\sigma)$ for all cones, and the cone of functions which are convex on $star^1(σ\sigma)$ for all cones of codimension 1. We give nice combinatorial descriptions for these two cones given two different fan structures on the tropical linear space of complete graphs. For the complete graph $K_5$, we prove that with the finer fan subdivision the two cones are not equal, but with the coarser subdivision they are the same. This gives a negative answer to a question of Gibney-Maclagan that for the finer subdivision the two cones are the same.]]> 0 <![CDATA[The Hilbert series of the Garsia-Haiman module can be written as a generating function of standard fillings of Ferrers diagrams. It is conjectured by Haglund and Loehr that the Hilbert series of the diagonal harmonics can be written as a generating function of parking functions. In this paper we present a weight-preserving injection from standard fillings to parking functions for certain cases.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3075 https://doi.org/10.46298/dmtcs.3075 Niese, Elizabeth Niese, Elizabeth <![CDATA[The Hilbert series of the Garsia-Haiman module can be written as a generating function of standard fillings of Ferrers diagrams. It is conjectured by Haglund and Loehr that the Hilbert series of the diagonal harmonics can be written as a generating function of parking functions. In this paper we present a weight-preserving injection from standard fillings to parking functions for certain cases.]]> 0 <![CDATA[The q-semicircular law as introduced by Bożejko and Speicher interpolates between the Gaussian law and the semicircular law, and its moments have a combinatorial interpretation in terms of matchings and crossings. We prove that the cumulants of this law are, up to some factor, polynomials in q with nonnegative coefficients. This is done by showing that they are obtained by an enumeration of connected matchings, weighted by the evaluation at (1,q) of a Tutte polynomial. The two particular cases q=0 and q=2 have also alternative proofs, related with the fact that these particular evaluation of the Tutte polynomials count some orientations on graphs. Our methods also give a combinatorial model for the cumulants of the free Poisson law.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3074 https://doi.org/10.46298/dmtcs.3074 Josuat-Vergès, Matthieu Josuat-Vergès, Matthieu <![CDATA[The q-semicircular law as introduced by Bożejko and Speicher interpolates between the Gaussian law and the semicircular law, and its moments have a combinatorial interpretation in terms of matchings and crossings. We prove that the cumulants of this law are, up to some factor, polynomials in q with nonnegative coefficients. This is done by showing that they are obtained by an enumeration of connected matchings, weighted by the evaluation at (1,q) of a Tutte polynomial. The two particular cases q=0 and q=2 have also alternative proofs, related with the fact that these particular evaluation of the Tutte polynomials count some orientations on graphs. Our methods also give a combinatorial model for the cumulants of the free Poisson law.]]> 0 <![CDATA[We give a combinatorial proof of Goulden and Jackson's formula for the number of minimal transitive factorizations of a permutation when the permutation has two cycles. We use the recent result of Goulden, Nica, and Oancea on the number of maximal chains of annular noncrossing partitions of type B.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3073 https://doi.org/10.46298/dmtcs.3073 Kim, Jang Soo Seo, Seunghyun Shin, Heesung Kim, Jang Soo Seo, Seunghyun Shin, Heesung <![CDATA[We give a combinatorial proof of Goulden and Jackson's formula for the number of minimal transitive factorizations of a permutation when the permutation has two cycles. We use the recent result of Goulden, Nica, and Oancea on the number of maximal chains of annular noncrossing partitions of type B.]]> 0 <![CDATA[We introduce an arithmetic version of the multivariate Tutte polynomial recently studied by Sokal, and a quasi-polynomial that interpolates between the two. We provide a generalized Fortuin-Kasteleyn representation for representable arithmetic matroids, with applications to arithmetic colorings and flows. We give a new proof of the positivity of the coefficients of the arithmetic Tutte polynomial in the more general framework of pseudo-arithmetic matroids. In the case of a representable arithmetic matroid, we provide a geometric interpretation of the coefficients of the arithmetic Tutte polynomial.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3072 https://doi.org/10.46298/dmtcs.3072 Brändèn, Petter Moci, Luca Brändèn, Petter Moci, Luca <![CDATA[We introduce an arithmetic version of the multivariate Tutte polynomial recently studied by Sokal, and a quasi-polynomial that interpolates between the two. We provide a generalized Fortuin-Kasteleyn representation for representable arithmetic matroids, with applications to arithmetic colorings and flows. We give a new proof of the positivity of the coefficients of the arithmetic Tutte polynomial in the more general framework of pseudo-arithmetic matroids. In the case of a representable arithmetic matroid, we provide a geometric interpretation of the coefficients of the arithmetic Tutte polynomial.]]> 0 <![CDATA[\textbfAbstract. The purpose of this paper is to present an algorithm which generates linear extensions for a non-simply-laced d-complete poset with uniform probability. ≠wline]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3071 https://doi.org/10.46298/dmtcs.3071 Nakada, Kento Nakada, Kento <![CDATA[\textbfAbstract. The purpose of this paper is to present an algorithm which generates linear extensions for a non-simply-laced d-complete poset with uniform probability. ≠wline]]> 0 <![CDATA[We study random lozenge tilings of a certain shape in the plane called the Novak half-hexagon, and compute the correlation functions for this process. This model was introduced by Nordenstam and Young (2011) and has many intriguing similarities with a more well-studied model, domino tilings of the Aztec diamond. The most difficult step in the present paper is to compute the inverse of the matrix whose (i,j)-entry is the binomial coefficient $C(A, B_j-i)$ for indeterminate variables $A$ and $B_1, \dots , B_n.$]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3070 https://doi.org/10.46298/dmtcs.3070 Nordenstam, Eric Young, Benjamin Nordenstam, Eric Young, Benjamin <![CDATA[We study random lozenge tilings of a certain shape in the plane called the Novak half-hexagon, and compute the correlation functions for this process. This model was introduced by Nordenstam and Young (2011) and has many intriguing similarities with a more well-studied model, domino tilings of the Aztec diamond. The most difficult step in the present paper is to compute the inverse of the matrix whose (i,j)-entry is the binomial coefficient $C(A, B_j-i)$ for indeterminate variables $A$ and $B_1, \dots , B_n.$]]> 0 <![CDATA[We obtain several properties of extremal statistics in non-crossing configurations with n vertices. We prove that the maximum degree and the largest component are of logarithmic order, and the diameter is of order $\sqrt{n}$. The proofs are based on singularity analysis, an application of the first and second moment method and on the analysis of iterated functions.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3069 https://doi.org/10.46298/dmtcs.3069 Mier, Anna, Noy, Marc Mier, Anna, Noy, Marc <![CDATA[We obtain several properties of extremal statistics in non-crossing configurations with n vertices. We prove that the maximum degree and the largest component are of logarithmic order, and the diameter is of order $\sqrt{n}$. The proofs are based on singularity analysis, an application of the first and second moment method and on the analysis of iterated functions.]]> 0 <![CDATA[We give a new description of the combinatorics of triangulations of even-dimensional cyclic polytopes, and of their bistellar flips. We show that the tropical exchange relation governing the number of intersections between diagonals of a polygon and a lamination (which generalizes to arbitrary surfaces) can also be generalized in a different way, to the setting of higher dimensional cyclic polytopes.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3068 https://doi.org/10.46298/dmtcs.3068 Oppermann, Steffen Thomas, Hugh Oppermann, Steffen Thomas, Hugh <![CDATA[We give a new description of the combinatorics of triangulations of even-dimensional cyclic polytopes, and of their bistellar flips. We show that the tropical exchange relation governing the number of intersections between diagonals of a polygon and a lamination (which generalizes to arbitrary surfaces) can also be generalized in a different way, to the setting of higher dimensional cyclic polytopes.]]> 0 <![CDATA[Cayley polytopes were defined recently as convex hulls of Cayley compositions introduced by Cayley in 1857. In this paper we resolve Braun's conjecture, which expresses the volume of Cayley polytopes in terms of the number of connected graphs. We extend this result to a two-variable deformations, which we call Tutte polytopes. The volume of the latter is given via an evaluation of the Tutte polynomial of the complete graph. Our approach is based on an explicit triangulation of the Cayley and Tutte polytope. We prove that simplices in the triangulations correspond to labeled trees and forests. The heart of the proof is a direct bijection based on the neighbors-first search graph traversal algorithm.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3055 https://doi.org/10.46298/dmtcs.3055 Konvalinka, Matjaž Pak, Igor Konvalinka, Matjaž Pak, Igor <![CDATA[Cayley polytopes were defined recently as convex hulls of Cayley compositions introduced by Cayley in 1857. In this paper we resolve Braun's conjecture, which expresses the volume of Cayley polytopes in terms of the number of connected graphs. We extend this result to a two-variable deformations, which we call Tutte polytopes. The volume of the latter is given via an evaluation of the Tutte polynomial of the complete graph. Our approach is based on an explicit triangulation of the Cayley and Tutte polytope. We prove that simplices in the triangulations correspond to labeled trees and forests. The heart of the proof is a direct bijection based on the neighbors-first search graph traversal algorithm.]]> 0 <![CDATA[We introduce an algorithm to determine when a sorting operation, such as stack-sort or bubble-sort, outputs a given pattern. The algorithm provides a new proof of the description of West-2-stack-sortable permutations, that is permutations that are completely sorted when passed twice through a stack, in terms of patterns. We also solve the long-standing problem of describing West-3-stack-sortable permutations. This requires a new type of generalized permutation pattern we call a decorated pattern.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3066 https://doi.org/10.46298/dmtcs.3066 Claesson, Anders Úlfarsson, Henning Claesson, Anders Úlfarsson, Henning <![CDATA[We introduce an algorithm to determine when a sorting operation, such as stack-sort or bubble-sort, outputs a given pattern. The algorithm provides a new proof of the description of West-2-stack-sortable permutations, that is permutations that are completely sorted when passed twice through a stack, in terms of patterns. We also solve the long-standing problem of describing West-3-stack-sortable permutations. This requires a new type of generalized permutation pattern we call a decorated pattern.]]> 0 <![CDATA[Let $δ (\mathcal{P} )=(δ _0,δ _1,\ldots,δ _d)$ be the $δ$ -vector of an integral convex polytope $\mathcal{P}$ of dimension $d$. First, by using two well-known inequalities on $δ$ -vectors, we classify the possible $δ$ -vectors with $\sum_{i=0}^d δ _i ≤3$. Moreover, by means of Hermite normal forms of square matrices, we also classify the possible $δ$ -vectors with $\sum_{i=0}^d δ _i = 4$. In addition, for $\sum_{i=0}^d δ _i ≥5$, we characterize the $δ$ -vectors of integral simplices when $\sum_{i=0}^d δ _i$ is prime.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3065 https://doi.org/10.46298/dmtcs.3065 Higashitani, Akihiro Higashitani, Akihiro <![CDATA[Let $δ (\mathcal{P} )=(δ _0,δ _1,\ldots,δ _d)$ be the $δ$ -vector of an integral convex polytope $\mathcal{P}$ of dimension $d$. First, by using two well-known inequalities on $δ$ -vectors, we classify the possible $δ$ -vectors with $\sum_{i=0}^d δ _i ≤3$. Moreover, by means of Hermite normal forms of square matrices, we also classify the possible $δ$ -vectors with $\sum_{i=0}^d δ _i = 4$. In addition, for $\sum_{i=0}^d δ _i ≥5$, we characterize the $δ$ -vectors of integral simplices when $\sum_{i=0}^d δ _i$ is prime.]]> 0 <![CDATA[We study the problem of expanding the product of two Stanley symmetric functions $F_w·F_u$ into Stanley symmetric functions in some natural way. Our approach is to consider a Stanley symmetric function as a stabilized Schubert polynomial $F_w=\lim _n→∞\mathfrak{S}_{1^n×w}$, and study the behavior of the expansion of $\mathfrak{S} _{1^n×w}·\mathfrak{S} _{1^n×u}$ into Schubert polynomials, as $n$ increases. We prove that this expansion stabilizes and thus we get a natural expansion for the product of two Stanley symmetric functions. In the case when one permutation is Grassmannian, we have a better understanding of this stability.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3064 https://doi.org/10.46298/dmtcs.3064 Li, Nan Li, Nan <![CDATA[We study the problem of expanding the product of two Stanley symmetric functions $F_w·F_u$ into Stanley symmetric functions in some natural way. Our approach is to consider a Stanley symmetric function as a stabilized Schubert polynomial $F_w=\lim _n→∞\mathfrak{S}_{1^n×w}$, and study the behavior of the expansion of $\mathfrak{S} _{1^n×w}·\mathfrak{S} _{1^n×u}$ into Schubert polynomials, as $n$ increases. We prove that this expansion stabilizes and thus we get a natural expansion for the product of two Stanley symmetric functions. In the case when one permutation is Grassmannian, we have a better understanding of this stability.]]> 0 <![CDATA[We discuss some properties of a subposet of the Tamari lattice introduced by Pallo (1986), which we call the comb poset. We show that three binary functions that are not well-behaved in the Tamari lattice are remarkably well-behaved within an interval of the comb poset: rotation distance, meets and joins, and the common parse words function for a pair of trees. We relate this poset to a partial order on the symmetric group studied by Edelman (1989).]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3063 https://doi.org/10.46298/dmtcs.3063 Csar, Sebastian A. Sengupta, Rik Suksompong, Warut Csar, Sebastian A. Sengupta, Rik Suksompong, Warut <![CDATA[We discuss some properties of a subposet of the Tamari lattice introduced by Pallo (1986), which we call the comb poset. We show that three binary functions that are not well-behaved in the Tamari lattice are remarkably well-behaved within an interval of the comb poset: rotation distance, meets and joins, and the common parse words function for a pair of trees. We relate this poset to a partial order on the symmetric group studied by Edelman (1989).]]> 0 <![CDATA[We evaluate induced sign characters of $H_n(q)$ at certain elements of $H_n(q)$ and conjecture an interpretation for the resulting polynomials as generating functions for $P$-tableaux by a certain statistic. Our conjecture relates the quantum chromatic symmetric functions of Shareshian and Wachs to $H_n(q)$ characters.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3062 https://doi.org/10.46298/dmtcs.3062 Shelton, Brittany Skandera, Mark Shelton, Brittany Skandera, Mark <![CDATA[We evaluate induced sign characters of $H_n(q)$ at certain elements of $H_n(q)$ and conjecture an interpretation for the resulting polynomials as generating functions for $P$-tableaux by a certain statistic. Our conjecture relates the quantum chromatic symmetric functions of Shareshian and Wachs to $H_n(q)$ characters.]]> 0 <![CDATA[A chromatic root is a zero of the chromatic polynomial of a graph. At a Newton Institute workshop on Combinatorics and Statistical Mechanics in 2008, two conjectures were proposed on the subject of which algebraic integers can be chromatic roots, known as the ``$α +n$ conjecture'' and the ``$nα$ conjecture''. These say, respectively, that given any algebraic integer α there is a natural number $n$ such that $α +n$ is a chromatic root, and that any positive integer multiple of a chromatic root is also a chromatic root. By computing the chromatic polynomials of two large families of graphs, we prove the $α +n$ conjecture for quadratic and cubic integers, and show that the set of chromatic roots satisfying the nα conjecture is dense in the complex plane.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3061 https://doi.org/10.46298/dmtcs.3061 Bohn, Adam Bohn, Adam <![CDATA[A chromatic root is a zero of the chromatic polynomial of a graph. At a Newton Institute workshop on Combinatorics and Statistical Mechanics in 2008, two conjectures were proposed on the subject of which algebraic integers can be chromatic roots, known as the ``$α +n$ conjecture'' and the ``$nα$ conjecture''. These say, respectively, that given any algebraic integer α there is a natural number $n$ such that $α +n$ is a chromatic root, and that any positive integer multiple of a chromatic root is also a chromatic root. By computing the chromatic polynomials of two large families of graphs, we prove the $α +n$ conjecture for quadratic and cubic integers, and show that the set of chromatic roots satisfying the nα conjecture is dense in the complex plane.]]> 0 <![CDATA[In this article we prove that the poset of m-divisible noncrossing partitions is EL-shellable for every well-generated complex reflection group. This was an open problem for type G(d,d,n) and for the exceptional types, for which a proof is given case-by-case.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3060 https://doi.org/10.46298/dmtcs.3060 Mühle, Henri Mühle, Henri <![CDATA[In this article we prove that the poset of m-divisible noncrossing partitions is EL-shellable for every well-generated complex reflection group. This was an open problem for type G(d,d,n) and for the exceptional types, for which a proof is given case-by-case.]]> 0 <![CDATA[We define new families of noncommutative symmetric functions and quasi-symmetric functions depending on two matrices of parameters, and more generally on parameters associated with paths in a binary tree. Appropriate specializations of both matrices then give back the two-vector families of Hivert, Lascoux, and Thibon and the noncommutative Macdonald functions of Bergeron and Zabrocki.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3059 https://doi.org/10.46298/dmtcs.3059 Lascoux, Alain Novelli, Jean-Christophe Thibon, Jean-Yves Lascoux, Alain Novelli, Jean-Christophe Thibon, Jean-Yves <![CDATA[We define new families of noncommutative symmetric functions and quasi-symmetric functions depending on two matrices of parameters, and more generally on parameters associated with paths in a binary tree. Appropriate specializations of both matrices then give back the two-vector families of Hivert, Lascoux, and Thibon and the noncommutative Macdonald functions of Bergeron and Zabrocki.]]> 0 <![CDATA[We will discuss some recent theorems relating the space of weighted phylogenetic trees to the tropical varieties of each flag variety of type A. We will also discuss the tropicalizations of the functions corresponding to semi-standard tableaux, in particular we relate them to familiar functions from phylogenetics. We close with some remarks on the generalization of these results to the tropical geometry of arbitrary flag varieties. This involves the family of Bergman complexes derived from the hyperplane arrangements associated to simple Dynkin diagrams.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3058 https://doi.org/10.46298/dmtcs.3058 Manon, Christopher Manon, Christopher <![CDATA[We will discuss some recent theorems relating the space of weighted phylogenetic trees to the tropical varieties of each flag variety of type A. We will also discuss the tropicalizations of the functions corresponding to semi-standard tableaux, in particular we relate them to familiar functions from phylogenetics. We close with some remarks on the generalization of these results to the tropical geometry of arbitrary flag varieties. This involves the family of Bergman complexes derived from the hyperplane arrangements associated to simple Dynkin diagrams.]]> 0 <![CDATA[Motivated by a result of Fiebig (2007), we categorify some properties of Kazhdan-Lusztig polynomials via sheaves on Bruhat moment graphs. In order to do this, we develop new techniques and apply them to the combinatorial data encoded in these moment graphs.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3057 https://doi.org/10.46298/dmtcs.3057 Lanini, Martina Lanini, Martina <![CDATA[Motivated by a result of Fiebig (2007), we categorify some properties of Kazhdan-Lusztig polynomials via sheaves on Bruhat moment graphs. In order to do this, we develop new techniques and apply them to the combinatorial data encoded in these moment graphs.]]> 0 <![CDATA[Let $G=(V,E)$ be a finite acyclic directed graph. Being motivated by a study of certain aspects of cluster algebras, we are interested in a class of triangulations of the cone of non-negative flows in $G, \mathcal F_+(G)$. To construct a triangulation, we fix a raming at each inner vertex $v$ of $G$, which consists of two linear orders: one on the set of incoming edges, and the other on the set of outgoing edges of $v$. A digraph $G$ endowed with a framing at each inner vertex is called $framed$. Given a framing on $G$, we define a reflexive and symmetric binary relation on the set of extreme rays of $\mathcal F_+ (G)$. We prove that that the complex of cliques formed by this binary relation is a pure simplicial complex, and that the cones spanned by cliques constitute a unimodular simplicial regular fan $Σ (G)$ covering the entire $\mathcal F_+(G)$.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3056 https://doi.org/10.46298/dmtcs.3056 Danilov, Vladimir I. Karzanov, Alexander V. Koshevoy, Gleb A. Danilov, Vladimir I. Karzanov, Alexander V. Koshevoy, Gleb A. <![CDATA[Let $G=(V,E)$ be a finite acyclic directed graph. Being motivated by a study of certain aspects of cluster algebras, we are interested in a class of triangulations of the cone of non-negative flows in $G, \mathcal F_+(G)$. To construct a triangulation, we fix a raming at each inner vertex $v$ of $G$, which consists of two linear orders: one on the set of incoming edges, and the other on the set of outgoing edges of $v$. A digraph $G$ endowed with a framing at each inner vertex is called $framed$. Given a framing on $G$, we define a reflexive and symmetric binary relation on the set of extreme rays of $\mathcal F_+ (G)$. We prove that that the complex of cliques formed by this binary relation is a pure simplicial complex, and that the cones spanned by cliques constitute a unimodular simplicial regular fan $Σ (G)$ covering the entire $\mathcal F_+(G)$.]]> 0 <![CDATA[In this paper, we show that data streams can sometimes usefully be studied as random permutations. This simple observation allows a wealth of classical and recent results from combinatorics to be recycled, with minimal effort, as estimators for various statistics over data streams. We illustrate this by introducing RECORDINALITY, an algorithm which estimates the number of distinct elements in a stream by counting the number of $k$-records occurring in it. The algorithm has a score of interesting properties, such as providing a random sample of the set underlying the stream. To the best of our knowledge, a modified version of RECORDINALITY is the first cardinality estimation algorithm which, in the random-order model, uses neither sampling nor hashing.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3002 https://doi.org/10.46298/dmtcs.3002 Helmi, Ahmed Lumbroso, Jérémie Martínez, Conrado Viola, Alfredo Helmi, Ahmed Lumbroso, Jérémie Martínez, Conrado Viola, Alfredo <![CDATA[In this paper, we show that data streams can sometimes usefully be studied as random permutations. This simple observation allows a wealth of classical and recent results from combinatorics to be recycled, with minimal effort, as estimators for various statistics over data streams. We illustrate this by introducing RECORDINALITY, an algorithm which estimates the number of distinct elements in a stream by counting the number of $k$-records occurring in it. The algorithm has a score of interesting properties, such as providing a random sample of the set underlying the stream. To the best of our knowledge, a modified version of RECORDINALITY is the first cardinality estimation algorithm which, in the random-order model, uses neither sampling nor hashing.]]> 0 <![CDATA[Given a planar triangulation, a 3-orientation is an orientation of the internal edges so all internal vertices have out-degree three. Each 3-orientation gives rise to a unique edge coloring known as a $\textit{Schnyder wood}$ that has proven useful for various computing and combinatorics applications. We consider natural Markov chains for sampling uniformly from the set of 3-orientations. First, we study a "triangle-reversing'' chain on the space of 3-orientations of a fixed triangulation that reverses the orientation of the edges around a triangle in each move. We show that (i) when restricted to planar triangulations of maximum degree six, the Markov chain is rapidly mixing, and (ii) there exists a triangulation with high degree on which this Markov chain mixes slowly. Next, we consider an "edge-flipping'' chain on the larger state space consisting of 3-orientations of all planar triangulations on a fixed number of vertices. It was also shown previously that this chain connects the state space and we prove that the chain is always rapidly mixing.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3010 https://doi.org/10.46298/dmtcs.3010 Miracle, Sarah Randall, Dana Streib, Amanda Pascoe Tetali, Prasad Miracle, Sarah Randall, Dana Streib, Amanda Pascoe Tetali, Prasad <![CDATA[Given a planar triangulation, a 3-orientation is an orientation of the internal edges so all internal vertices have out-degree three. Each 3-orientation gives rise to a unique edge coloring known as a $\textit{Schnyder wood}$ that has proven useful for various computing and combinatorics applications. We consider natural Markov chains for sampling uniformly from the set of 3-orientations. First, we study a "triangle-reversing'' chain on the space of 3-orientations of a fixed triangulation that reverses the orientation of the edges around a triangle in each move. We show that (i) when restricted to planar triangulations of maximum degree six, the Markov chain is rapidly mixing, and (ii) there exists a triangulation with high degree on which this Markov chain mixes slowly. Next, we consider an "edge-flipping'' chain on the larger state space consisting of 3-orientations of all planar triangulations on a fixed number of vertices. It was also shown previously that this chain connects the state space and we prove that the chain is always rapidly mixing.]]> 0 <![CDATA[Extending an idea of Suppakitpaisarn, Edahiro and Imai, a dynamic programming approach for computing digital expansions of minimal weight is transformed into an asymptotic analysis of minimal weight expansions. The minimal weight of an optimal expansion of a random input of length $\ell$ is shown to be asymptotically normally distributed under suitable conditions. After discussing the general framework, we focus on expansions to the base of $\tau$, where $\tau$ is a root of the polynomial $X^2- \mu X + 2$ for $\mu \in \{ \pm 1\}$. As the Frobenius endomorphism on a binary Koblitz curve fulfils the same equation, digit expansions to the base of this $\tau$ can be used for scalar multiplication and linear combination in elliptic curve cryptosystems over these curves.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3009 https://doi.org/10.46298/dmtcs.3009 Heigl, Florian Heuberger, Clemens Heigl, Florian Heuberger, Clemens <![CDATA[Extending an idea of Suppakitpaisarn, Edahiro and Imai, a dynamic programming approach for computing digital expansions of minimal weight is transformed into an asymptotic analysis of minimal weight expansions. The minimal weight of an optimal expansion of a random input of length $\ell$ is shown to be asymptotically normally distributed under suitable conditions. After discussing the general framework, we focus on expansions to the base of $\tau$, where $\tau$ is a root of the polynomial $X^2- \mu X + 2$ for $\mu \in \{ \pm 1\}$. As the Frobenius endomorphism on a binary Koblitz curve fulfils the same equation, digit expansions to the base of this $\tau$ can be used for scalar multiplication and linear combination in elliptic curve cryptosystems over these curves.]]> 0 <![CDATA[We use probabilistic and combinatorial tools on strings to discover the average number of 2-protected nodes in tries and in suffix trees. Our analysis covers both the uniform and non-uniform cases. For instance, in a uniform trie with $n$ leaves, the number of 2-protected nodes is approximately 0.803$n$, plus small first-order fluctuations. The 2-protected nodes are an emerging way to distinguish the interior of a tree from the fringe.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3008 https://doi.org/10.46298/dmtcs.3008 Gaither, Jeffrey Homma, Yushi Sellke, Mark Ward, Mark Daniel Gaither, Jeffrey Homma, Yushi Sellke, Mark Ward, Mark Daniel <![CDATA[We use probabilistic and combinatorial tools on strings to discover the average number of 2-protected nodes in tries and in suffix trees. Our analysis covers both the uniform and non-uniform cases. For instance, in a uniform trie with $n$ leaves, the number of 2-protected nodes is approximately 0.803$n$, plus small first-order fluctuations. The 2-protected nodes are an emerging way to distinguish the interior of a tree from the fringe.]]> 0 <![CDATA[In the paper "How to select a looser'' Prodinger was analyzing an algorithm where $n$ participants are selecting a leader by flipping fair coins, where recursively, the 0-party (those who i.e. have tossed heads) continues until the leader is chosen. We give an answer to the question stated in the Prodinger's paper – what happens if not a 0-party is recursively looking for a leader but always a party with a smaller cardinality. We show the lower bound on the number of rounds of the greedy algorithm (for fair coin).]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3007 https://doi.org/10.46298/dmtcs.3007 Gołębiewski, Zbigniew Zagórski, Filip Gołębiewski, Zbigniew Zagórski, Filip <![CDATA[In the paper "How to select a looser'' Prodinger was analyzing an algorithm where $n$ participants are selecting a leader by flipping fair coins, where recursively, the 0-party (those who i.e. have tossed heads) continues until the leader is chosen. We give an answer to the question stated in the Prodinger's paper – what happens if not a 0-party is recursively looking for a leader but always a party with a smaller cardinality. We show the lower bound on the number of rounds of the greedy algorithm (for fair coin).]]> 0 <![CDATA[In the paper, bike sharing systems with stations having a finite capacity are studied as stochastic networks. The inhomogeneity is modeled by clusters. We use a mean field limit to compute the limiting stationary distribution of the number of bikes at the stations. This method is an alternative to analytical methods. It can be used even if a closed form expression for the stationary distribution is out of reach as illustrated on a variant. Both models are compared. A practical conclusion is that avoiding empty or full stations does not improve overall performance.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3006 https://doi.org/10.46298/dmtcs.3006 Fricker, Christine Gast, Nicolas Mohamed, Hanene Fricker, Christine Gast, Nicolas Mohamed, Hanene <![CDATA[In the paper, bike sharing systems with stations having a finite capacity are studied as stochastic networks. The inhomogeneity is modeled by clusters. We use a mean field limit to compute the limiting stationary distribution of the number of bikes at the stations. This method is an alternative to analytical methods. It can be used even if a closed form expression for the stationary distribution is out of reach as illustrated on a variant. Both models are compared. A practical conclusion is that avoiding empty or full stations does not improve overall performance.]]> 0 <![CDATA[Consider a pair of $\textit{interlacing regular convex polygons}$, each with $2(n + 2)$ vertices, which we will be referring to as $\textit{red}$ and $\textit{black}$ ones. One can place these vertices on the unit circle $|z | = 1$ in the complex plane; the vertices of the red polygon at $\epsilon^{2k}, k = 0, \ldots , 2n − 1$, of the black polygon at $\epsilon^{2k+1}, k = 0, \ldots , 2n − 1$; here $\epsilon = \exp(i \pi /(2n + 2))$. We assign to the vertices of each polygon alternating (within each polygon) signs. Note that all the pairwise intersections of red and black sides are oriented consistently. We declare the corresponding orientation positive.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3005 https://doi.org/10.46298/dmtcs.3005 Baryshnikov, Yu. Hickok, L. Orlow, N. Son, S. Baryshnikov, Yu. Hickok, L. Orlow, N. Son, S. <![CDATA[Consider a pair of $\textit{interlacing regular convex polygons}$, each with $2(n + 2)$ vertices, which we will be referring to as $\textit{red}$ and $\textit{black}$ ones. One can place these vertices on the unit circle $|z | = 1$ in the complex plane; the vertices of the red polygon at $\epsilon^{2k}, k = 0, \ldots , 2n − 1$, of the black polygon at $\epsilon^{2k+1}, k = 0, \ldots , 2n − 1$; here $\epsilon = \exp(i \pi /(2n + 2))$. We assign to the vertices of each polygon alternating (within each polygon) signs. Note that all the pairwise intersections of red and black sides are oriented consistently. We declare the corresponding orientation positive.]]> 0 <![CDATA[In a continuous-time setting, Fill (2012) proved, for a large class of probabilistic sources, that the number of symbol comparisons used by $\texttt{QuickSort}$, when centered by subtracting the mean and scaled by dividing by time, has a limiting distribution, but proved little about that limiting random variable $Y$—not even that it is nondegenerate. We establish the nondegeneracy of $Y$. The proof is perhaps surprisingly difficult.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3004 https://doi.org/10.46298/dmtcs.3004 Bindjeme, Patrick fill, james Allen Bindjeme, Patrick fill, james Allen <![CDATA[In a continuous-time setting, Fill (2012) proved, for a large class of probabilistic sources, that the number of symbol comparisons used by $\texttt{QuickSort}$, when centered by subtracting the mean and scaled by dividing by time, has a limiting distribution, but proved little about that limiting random variable $Y$—not even that it is nondegenerate. We establish the nondegeneracy of $Y$. The proof is perhaps surprisingly difficult.]]> 0 <![CDATA[Using a recursive approach, we obtain a simple exact expression for the $L^2$-distance from the limit in the classical limit theorem of Régnier (1989) for the number of key comparisons required by $\texttt{QuickSort}$. A previous study by Fill and Janson (2002) using a similar approach found that the $d_2$-distance is of order between $n^{-1} \log{n}$ and $n^{-1/2}$, and another by Neininger and Ruschendorf (2002) found that the Zolotarev $\zeta _3$-distance is of exact order $n^{-1} \log{n}$. Our expression reveals that the $L^2$-distance is asymptotically equivalent to $(2 n^{-1} \ln{n})^{1/2}$.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3003 https://doi.org/10.46298/dmtcs.3003 Bindjeme, Patrick fill, james Allen Bindjeme, Patrick fill, james Allen <![CDATA[Using a recursive approach, we obtain a simple exact expression for the $L^2$-distance from the limit in the classical limit theorem of Régnier (1989) for the number of key comparisons required by $\texttt{QuickSort}$. A previous study by Fill and Janson (2002) using a similar approach found that the $d_2$-distance is of order between $n^{-1} \log{n}$ and $n^{-1/2}$, and another by Neininger and Ruschendorf (2002) found that the Zolotarev $\zeta _3$-distance is of exact order $n^{-1} \log{n}$. Our expression reveals that the $L^2$-distance is asymptotically equivalent to $(2 n^{-1} \ln{n})^{1/2}$.]]> 0 <![CDATA[The space requirements of an $m$-ary search tree satisfies a well-known phase transition: when $m\leq 26$, the second order asymptotics is Gaussian. When $m\geq 27$, it is not Gaussian any longer and a limit $W$ of a complex-valued martingale arises. We show that the distribution of $W$ has a square integrable density on the complex plane, that its support is the whole complex plane, and that it has finite exponential moments. The proofs are based on the study of the distributional equation $ W \overset{\mathcal{L}}{=} \sum_{k=1}^mV_k^{\lambda}W_k$, where $V_1, ..., V_m$ are the spacings of $(m-1)$ independent random variables uniformly distributed on $[0,1]$, $W_1, ..., W_m$ are independent copies of W which are also independent of $(V_1, ..., V_m)$ and $\lambda$ is a complex number.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2994 https://doi.org/10.46298/dmtcs.2994 Chauvin, Brigitte Liu, Quansheng Pouyanne, Nicolas Chauvin, Brigitte Liu, Quansheng Pouyanne, Nicolas <![CDATA[The space requirements of an $m$-ary search tree satisfies a well-known phase transition: when $m\leq 26$, the second order asymptotics is Gaussian. When $m\geq 27$, it is not Gaussian any longer and a limit $W$ of a complex-valued martingale arises. We show that the distribution of $W$ has a square integrable density on the complex plane, that its support is the whole complex plane, and that it has finite exponential moments. The proofs are based on the study of the distributional equation $ W \overset{\mathcal{L}}{=} \sum_{k=1}^mV_k^{\lambda}W_k$, where $V_1, ..., V_m$ are the spacings of $(m-1)$ independent random variables uniformly distributed on $[0,1]$, $W_1, ..., W_m$ are independent copies of W which are also independent of $(V_1, ..., V_m)$ and $\lambda$ is a complex number.]]> 0 <![CDATA[String complexity is defined as the cardinality of a set of all distinct words (factors) of a given string. For two strings, we define $\textit{joint string complexity}$ as the set of words that are common to both strings. We also relax this definition and introduce $\textit{joint semi-complexity}$ restricted to the common words appearing at least twice in both strings. String complexity finds a number of applications from capturing the richness of a language to finding similarities between two genome sequences. In this paper we analyze joint complexity and joint semi-complexity when both strings are generated by a Markov source. The problem turns out to be quite challenging requiring subtle singularity analysis and saddle point method over infinity many saddle points leading to novel oscillatory phenomena with single and double periodicities.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3001 https://doi.org/10.46298/dmtcs.3001 Jacquet, Philippe Szpankowski, Wojciech Jacquet, Philippe Szpankowski, Wojciech <![CDATA[String complexity is defined as the cardinality of a set of all distinct words (factors) of a given string. For two strings, we define $\textit{joint string complexity}$ as the set of words that are common to both strings. We also relax this definition and introduce $\textit{joint semi-complexity}$ restricted to the common words appearing at least twice in both strings. String complexity finds a number of applications from capturing the richness of a language to finding similarities between two genome sequences. In this paper we analyze joint complexity and joint semi-complexity when both strings are generated by a Markov source. The problem turns out to be quite challenging requiring subtle singularity analysis and saddle point method over infinity many saddle points leading to novel oscillatory phenomena with single and double periodicities.]]> 0 <![CDATA[We consider Euclid’s gcd algorithm for two integers $(p, q)$ with $1 \leq p \leq q \leq N$, with the uniform distribution on input pairs. We study the distribution of the total cost of execution of the algorithm for an additive cost function $d$ on the set of possible digits, asymptotically for $N \to \infty$. For any additive cost of moderate growth $d$, Baladi and Vallée obtained a central limit theorem, and –in the case when the cost $d$ is lattice– a local limit theorem. In both cases, the optimal speed was attained. When the cost is non lattice, the problem was later considered by Baladi and Hachemi, who obtained a local limit theorem under an intertwined diophantine condition which involves the cost $d$ together with the “canonical” cost $c$ of the underlying dynamical system. The speed depends on the irrationality exponent that intervenes in the diophantine condition. We show here how to replace this diophantine condition by another diophantine condition, much more natural, which already intervenes in simpler problems of the same vein, and only involves the cost $d$. This “replacement” is made possible by using the additivity of cost $d$, together with a strong property satisfied by the Euclidean Dynamical System, which states that the cost $c$ is both “strongly” non additive and diophantine in a precise sense. We thus obtain a local limit theorem, whose speed is related to the irrationality exponent which intervenes in the new diophantine condition. We mainly use the previous proof of Baladi and Hachemi, and “just” explain how their diophantine condition may be replaced by our condition. Our result also provides a precise comparison between the rational trajectories of the Euclid dynamical system and the real trajectories.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3000 https://doi.org/10.46298/dmtcs.3000 Vallée, Brigitte Vallée, Brigitte <![CDATA[We consider Euclid’s gcd algorithm for two integers $(p, q)$ with $1 \leq p \leq q \leq N$, with the uniform distribution on input pairs. We study the distribution of the total cost of execution of the algorithm for an additive cost function $d$ on the set of possible digits, asymptotically for $N \to \infty$. For any additive cost of moderate growth $d$, Baladi and Vallée obtained a central limit theorem, and –in the case when the cost $d$ is lattice– a local limit theorem. In both cases, the optimal speed was attained. When the cost is non lattice, the problem was later considered by Baladi and Hachemi, who obtained a local limit theorem under an intertwined diophantine condition which involves the cost $d$ together with the “canonical” cost $c$ of the underlying dynamical system. The speed depends on the irrationality exponent that intervenes in the diophantine condition. We show here how to replace this diophantine condition by another diophantine condition, much more natural, which already intervenes in simpler problems of the same vein, and only involves the cost $d$. This “replacement” is made possible by using the additivity of cost $d$, together with a strong property satisfied by the Euclidean Dynamical System, which states that the cost $c$ is both “strongly” non additive and diophantine in a precise sense. We thus obtain a local limit theorem, whose speed is related to the irrationality exponent which intervenes in the new diophantine condition. We mainly use the previous proof of Baladi and Hachemi, and “just” explain how their diophantine condition may be replaced by our condition. Our result also provides a precise comparison between the rational trajectories of the Euclid dynamical system and the real trajectories.]]> 0 <![CDATA[We assign a uniform probability to the set consisting of partitions of a positive integer $n$ such that the multiplicity of each summand is less than a given number $d$ and we study the limiting distribution of the number of summands in a random partition. It is known from a result by Erdős and Lehner published in 1941 that the distributions of the length in random restricted $(d=2)$ and random unrestricted $(d \geq n+1)$ partitions behave very differently. In this paper we show that as the bound $d$ increases we observe a phase transition in which the distribution goes from the Gaussian distribution of the restricted case to the Gumbel distribution of the unrestricted case.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2999 https://doi.org/10.46298/dmtcs.2999 Ralaivaosaona, Dimbinaina Ralaivaosaona, Dimbinaina <![CDATA[We assign a uniform probability to the set consisting of partitions of a positive integer $n$ such that the multiplicity of each summand is less than a given number $d$ and we study the limiting distribution of the number of summands in a random partition. It is known from a result by Erdős and Lehner published in 1941 that the distributions of the length in random restricted $(d=2)$ and random unrestricted $(d \geq n+1)$ partitions behave very differently. In this paper we show that as the bound $d$ increases we observe a phase transition in which the distribution goes from the Gaussian distribution of the restricted case to the Gumbel distribution of the unrestricted case.]]> 0 <![CDATA[We consider the word collector problem, i.e. the expected number of calls to a random weighted generator before all the words of a given length in a language are generated. The originality of this instance of the non-uniform coupon collector lies in the, potentially large, multiplicity of the words/coupons of a given probability/composition. We obtain a general theorem that gives an asymptotic equivalent for the expected waiting time of a general version of the Coupon Collector. This theorem is especially well-suited for classes of coupons featuring high multiplicities. Its application to a given language essentially necessitates knowledge on the number of words of a given composition/probability. We illustrate the application of our theorem, in a step-by-step fashion, on four exemplary languages, whose analyses reveal a large diversity of asymptotic waiting times, generally expressible as $\kappa \cdot m^p \cdot (\log{m})^q \cdot (\log \log{m})^r$, for $m$ the number of words, and $p, q, r$ some positive real numbers.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2998 https://doi.org/10.46298/dmtcs.2998 Du Boisberranger, Jérémie Gardy, Danièle Ponty, Yann Du Boisberranger, Jérémie Gardy, Danièle Ponty, Yann <![CDATA[We consider the word collector problem, i.e. the expected number of calls to a random weighted generator before all the words of a given length in a language are generated. The originality of this instance of the non-uniform coupon collector lies in the, potentially large, multiplicity of the words/coupons of a given probability/composition. We obtain a general theorem that gives an asymptotic equivalent for the expected waiting time of a general version of the Coupon Collector. This theorem is especially well-suited for classes of coupons featuring high multiplicities. Its application to a given language essentially necessitates knowledge on the number of words of a given composition/probability. We illustrate the application of our theorem, in a step-by-step fashion, on four exemplary languages, whose analyses reveal a large diversity of asymptotic waiting times, generally expressible as $\kappa \cdot m^p \cdot (\log{m})^q \cdot (\log \log{m})^r$, for $m$ the number of words, and $p, q, r$ some positive real numbers.]]> 0 <![CDATA[We define the notion of $t$-free for locally restricted compositions, which means roughly that if such a composition contains a part $c_i$ and nearby parts are at least $t$ smaller, then $c_i$ can be replaced by any larger part. Two well-known examples are Carlitz and alternating compositions. We show that large parts have asymptotically geometric distributions. This leads to asymptotically independent Poisson variables for numbers of various large parts. Based on this we obtain asymptotic formulas for the probability of being gap free and for the expected values of the largest part and number distinct parts, all accurate to $o(1)$.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2997 https://doi.org/10.46298/dmtcs.2997 Bender, Edward, Canfield, Rodney, Gao, Zhicheng Bender, Edward, Canfield, Rodney, Gao, Zhicheng <![CDATA[We define the notion of $t$-free for locally restricted compositions, which means roughly that if such a composition contains a part $c_i$ and nearby parts are at least $t$ smaller, then $c_i$ can be replaced by any larger part. Two well-known examples are Carlitz and alternating compositions. We show that large parts have asymptotically geometric distributions. This leads to asymptotically independent Poisson variables for numbers of various large parts. Based on this we obtain asymptotic formulas for the probability of being gap free and for the expected values of the largest part and number distinct parts, all accurate to $o(1)$.]]> 0 <![CDATA[This paper develops an analytic theory for the study of some Pólya urns with random rules. The idea is to extend the isomorphism theorem in Flajolet et al. (2006), which connects deterministic balanced urns to a differential system for the generating function. The methodology is based upon adaptation of operators and use of a weighted probability generating function. Systems of differential equations are developed, and when they can be solved, they lead to characterization of the exact distributions underlying the urn evolution. We give a few illustrative examples.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2996 https://doi.org/10.46298/dmtcs.2996 Morcrette, Basile Mahmoud, Hosam M. Morcrette, Basile Mahmoud, Hosam M. <![CDATA[This paper develops an analytic theory for the study of some Pólya urns with random rules. The idea is to extend the isomorphism theorem in Flajolet et al. (2006), which connects deterministic balanced urns to a differential system for the generating function. The methodology is based upon adaptation of operators and use of a weighted probability generating function. Systems of differential equations are developed, and when they can be solved, they lead to characterization of the exact distributions underlying the urn evolution. We give a few illustrative examples.]]> 0 <![CDATA[This paper sheds light on universal coding with respect to classes of memoryless sources over a countable alphabet defined by an envelope function with finite and non-decreasing hazard rate. We prove that the auto-censuring (AC) code introduced by Bontemps (2011) is adaptive with respect to the collection of such classes. The analysis builds on the tight characterization of universal redundancy rate in terms of metric entropy by Haussler and Opper (1997) and on a careful analysis of the performance of the AC-coding algorithm. The latter relies on non-asymptotic bounds for maxima of samples from discrete distributions with finite and non-decreasing hazard rate.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2995 https://doi.org/10.46298/dmtcs.2995 Bontemps, Dominique Boucheron, Stephane Gassiat, Elisabeth Bontemps, Dominique Boucheron, Stephane Gassiat, Elisabeth <![CDATA[This paper sheds light on universal coding with respect to classes of memoryless sources over a countable alphabet defined by an envelope function with finite and non-decreasing hazard rate. We prove that the auto-censuring (AC) code introduced by Bontemps (2011) is adaptive with respect to the collection of such classes. The analysis builds on the tight characterization of universal redundancy rate in terms of metric entropy by Haussler and Opper (1997) and on a careful analysis of the performance of the AC-coding algorithm. The latter relies on non-asymptotic bounds for maxima of samples from discrete distributions with finite and non-decreasing hazard rate.]]> 0 <![CDATA[We study integral ratios of hook products of quotient partitions. This question is motivated by an analogous question in number theory concerning integral factorial ratios. We prove an analogue of a theorem of Landau that already applied in the factorial case. Under the additional condition that the ratio has one more factor on the denominator than the numerator, we provide a complete classification. Ultimately this relies on Kneser's theorem in additive combinatorics.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3022 https://doi.org/10.46298/dmtcs.3022 Dehaye, Paul-Olivier Dehaye, Paul-Olivier <![CDATA[We study integral ratios of hook products of quotient partitions. This question is motivated by an analogous question in number theory concerning integral factorial ratios. We prove an analogue of a theorem of Landau that already applied in the factorial case. Under the additional condition that the ratio has one more factor on the denominator than the numerator, we provide a complete classification. Ultimately this relies on Kneser's theorem in additive combinatorics.]]> 0 <![CDATA[A $d$-polytope $P$ is neighborly if every subset of $\lfloor\frac{d}{2}\rfloor $vertices is a face of $P$. In 1982, Shemer introduced a sewing construction that allows to add a vertex to a neighborly polytope in such a way as to obtain a new neighborly polytope. With this, he constructed superexponentially many different neighborly polytopes. The concept of neighborliness extends naturally to oriented matroids. Duals of neighborly oriented matroids also have a nice characterization: balanced oriented matroids. In this paper, we generalize Shemer's sewing construction to oriented matroids, providing a simpler proof. Moreover we provide a new technique that allows to construct balanced oriented matroids. In the dual setting, it constructs a neighborly oriented matroid whose contraction at a particular vertex is a prescribed neighborly oriented matroid. We compare the families of polytopes that can be constructed with both methods, and show that the new construction allows to construct many new polytopes.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3032 https://doi.org/10.46298/dmtcs.3032 Padrol, Arnau Padrol, Arnau <![CDATA[A $d$-polytope $P$ is neighborly if every subset of $\lfloor\frac{d}{2}\rfloor $vertices is a face of $P$. In 1982, Shemer introduced a sewing construction that allows to add a vertex to a neighborly polytope in such a way as to obtain a new neighborly polytope. With this, he constructed superexponentially many different neighborly polytopes. The concept of neighborliness extends naturally to oriented matroids. Duals of neighborly oriented matroids also have a nice characterization: balanced oriented matroids. In this paper, we generalize Shemer's sewing construction to oriented matroids, providing a simpler proof. Moreover we provide a new technique that allows to construct balanced oriented matroids. In the dual setting, it constructs a neighborly oriented matroid whose contraction at a particular vertex is a prescribed neighborly oriented matroid. We compare the families of polytopes that can be constructed with both methods, and show that the new construction allows to construct many new polytopes.]]> 0 <![CDATA[We investigate the combinatorial Hopf algebra based on uniform block permutations and we realize this algebra in terms of noncommutative polynomials in infinitely many bi-letters.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3031 https://doi.org/10.46298/dmtcs.3031 Maurice, Rémi Maurice, Rémi <![CDATA[We investigate the combinatorial Hopf algebra based on uniform block permutations and we realize this algebra in terms of noncommutative polynomials in infinitely many bi-letters.]]> 0 <![CDATA[Bergman complexes are polyhedral complexes associated to matroids. Faces of these complexes are certain matroids, called matroid types, too. In order to understand the structure of these faces we decompose matroid types into direct summands. Ardila/Klivans proved that the Bergman Complex of a matroid can be subdivided into the order complex of the proper part of its lattice of flats. Beyond that Feichtner/Sturmfels showed that the Bergman complex can even be subdivided to the even coarser nested set complex. We will give a much shorter and more general proof of this fact. Generalizing formulas proposed by Ardila/Klivans and Feichtner/Sturmfels for special cases, we present a decomposition into direct sums working for faces of any of these complexes. Additionally we show that it is the finest possible decomposition for faces of the Bergman complex.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3030 https://doi.org/10.46298/dmtcs.3030 Dlugosch, Martin Dlugosch, Martin <![CDATA[Bergman complexes are polyhedral complexes associated to matroids. Faces of these complexes are certain matroids, called matroid types, too. In order to understand the structure of these faces we decompose matroid types into direct summands. Ardila/Klivans proved that the Bergman Complex of a matroid can be subdivided into the order complex of the proper part of its lattice of flats. Beyond that Feichtner/Sturmfels showed that the Bergman complex can even be subdivided to the even coarser nested set complex. We will give a much shorter and more general proof of this fact. Generalizing formulas proposed by Ardila/Klivans and Feichtner/Sturmfels for special cases, we present a decomposition into direct sums working for faces of any of these complexes. Additionally we show that it is the finest possible decomposition for faces of the Bergman complex.]]> 0 <![CDATA[Based on the notion of colored and absolute excedances introduced by Bagno and Garber we give an analogue of the derangement polynomials. We obtain some basic properties of these polynomials. Moreover, we define an excedance statistic for the affine Weyl groups of type $\widetilde{B}_n, \widetilde {C}_n$ and $\widetilde {D}_n$ and determine the generating functions of their distributions. This paper is inspired by one of Clark and Ehrenborg (2011) in which the authors introduce the excedance statistic for the group of affine permutations and ask if this statistic can be defined for other affine groups.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3029 https://doi.org/10.46298/dmtcs.3029 Mongelli, Pietro Mongelli, Pietro <![CDATA[Based on the notion of colored and absolute excedances introduced by Bagno and Garber we give an analogue of the derangement polynomials. We obtain some basic properties of these polynomials. Moreover, we define an excedance statistic for the affine Weyl groups of type $\widetilde{B}_n, \widetilde {C}_n$ and $\widetilde {D}_n$ and determine the generating functions of their distributions. This paper is inspired by one of Clark and Ehrenborg (2011) in which the authors introduce the excedance statistic for the group of affine permutations and ask if this statistic can be defined for other affine groups.]]> 0 <![CDATA[We introduce a certain class of algebras associated to matroids. We prove the Lefschetz property of the algebras for some special cases. Our result implies the Sperner property for the Boolean lattice and the vector space lattice.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3028 https://doi.org/10.46298/dmtcs.3028 Maeno, Toshiaki Numata, Yasuhide Maeno, Toshiaki Numata, Yasuhide <![CDATA[We introduce a certain class of algebras associated to matroids. We prove the Lefschetz property of the algebras for some special cases. Our result implies the Sperner property for the Boolean lattice and the vector space lattice.]]> 0 <![CDATA[In a recent paper, Duane, Garsia, and Zabrocki introduced a new statistic, "ndinv'', on a family of parking functions. The definition was guided by a recursion satisfied by the polynomial $\langle\Delta_{h_m}C_p1C_p2...C_{pk}1,e_n\rangle$, for $\Delta_{h_m}$ a Macdonald eigenoperator, $C_{p_i}$ a modified Hall-Littlewood operator and $(p_1,p_2,\dots ,p_k)$ a composition of n. Using their new statistics, they are able to give a new interpretation for the polynomial $\langle\nabla e_n, h_j h_n-j\rangle$ as a q,t numerator of parking functions by area and ndinv. We recall that in the shuffle conjecture, parking functions are q,t enumerated by area and diagonal inversion number (dinv). Since their definition is recursive, they pose the problem of obtaining a non recursive definition. We solved this problem by giving an explicit formula for ndinv similar to the classical definition of dinv. In this paper, we describe the work we did to construct this formula and to prove that the resulting ndinv is the same as the one recursively defined by Duane, Garsia, and Zabrocki.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3027 https://doi.org/10.46298/dmtcs.3027 Hicks, Angela Kim, Yeonkyung Hicks, Angela Kim, Yeonkyung <![CDATA[In a recent paper, Duane, Garsia, and Zabrocki introduced a new statistic, "ndinv'', on a family of parking functions. The definition was guided by a recursion satisfied by the polynomial $\langle\Delta_{h_m}C_p1C_p2...C_{pk}1,e_n\rangle$, for $\Delta_{h_m}$ a Macdonald eigenoperator, $C_{p_i}$ a modified Hall-Littlewood operator and $(p_1,p_2,\dots ,p_k)$ a composition of n. Using their new statistics, they are able to give a new interpretation for the polynomial $\langle\nabla e_n, h_j h_n-j\rangle$ as a q,t numerator of parking functions by area and ndinv. We recall that in the shuffle conjecture, parking functions are q,t enumerated by area and diagonal inversion number (dinv). Since their definition is recursive, they pose the problem of obtaining a non recursive definition. We solved this problem by giving an explicit formula for ndinv similar to the classical definition of dinv. In this paper, we describe the work we did to construct this formula and to prove that the resulting ndinv is the same as the one recursively defined by Duane, Garsia, and Zabrocki.]]> 0 <![CDATA[Tropical oriented matroids were defined by Ardila and Develin in 2007. They are a tropical analogue of classical oriented matroids in the sense that they encode the properties of the types of points in an arrangement of tropical hyperplanes – in much the same way as the covectors of (classical) oriented matroids describe the types in arrangements of linear hyperplanes. Not every oriented matroid can be realised by an arrangement of linear hyperplanes though. The famous Topological Representation Theorem by Folkman and Lawrence, however, states that every oriented matroid can be represented as an arrangement of $\textit{pseudo}$hyperplanes. Ardila and Develin proved that tropical oriented matroids can be represented as mixed subdivisions of dilated simplices. In this paper I prove that this correspondence is a bijection. Moreover, I present a tropical analogue for the Topological Representation Theorem.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3026 https://doi.org/10.46298/dmtcs.3026 Horn, Silke Horn, Silke <![CDATA[Tropical oriented matroids were defined by Ardila and Develin in 2007. They are a tropical analogue of classical oriented matroids in the sense that they encode the properties of the types of points in an arrangement of tropical hyperplanes – in much the same way as the covectors of (classical) oriented matroids describe the types in arrangements of linear hyperplanes. Not every oriented matroid can be realised by an arrangement of linear hyperplanes though. The famous Topological Representation Theorem by Folkman and Lawrence, however, states that every oriented matroid can be represented as an arrangement of $\textit{pseudo}$hyperplanes. Ardila and Develin proved that tropical oriented matroids can be represented as mixed subdivisions of dilated simplices. In this paper I prove that this correspondence is a bijection. Moreover, I present a tropical analogue for the Topological Representation Theorem.]]> 0 <![CDATA[This paper is devoted to the explicit computation of generating series for the connection coefficients of two commutative subalgebras of the group algebra of the symmetric group, the class algebra and the double coset algebra. As shown by Hanlon, Stanley and Stembridge (1992), these series gives the spectral distribution of some random matrices that are of interest to statisticians. Morales and Vassilieva (2009, 2011) found explicit formulas for these generating series in terms of monomial symmetric functions by introducing a bijection between partitioned hypermaps on (locally) orientable surfaces and some decorated forests and trees. Thanks to purely algebraic means, we recover the formula for the class algebra and provide a new simpler formula for the double coset algebra. As a salient ingredient, we compute an explicit formulation for zonal polynomials indexed by partitions of type $[a,b,1^{n-a-b}]$.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3025 https://doi.org/10.46298/dmtcs.3025 Vassilieva, Ekaterina A. Vassilieva, Ekaterina A. <![CDATA[This paper is devoted to the explicit computation of generating series for the connection coefficients of two commutative subalgebras of the group algebra of the symmetric group, the class algebra and the double coset algebra. As shown by Hanlon, Stanley and Stembridge (1992), these series gives the spectral distribution of some random matrices that are of interest to statisticians. Morales and Vassilieva (2009, 2011) found explicit formulas for these generating series in terms of monomial symmetric functions by introducing a bijection between partitioned hypermaps on (locally) orientable surfaces and some decorated forests and trees. Thanks to purely algebraic means, we recover the formula for the class algebra and provide a new simpler formula for the double coset algebra. As a salient ingredient, we compute an explicit formulation for zonal polynomials indexed by partitions of type $[a,b,1^{n-a-b}]$.]]> 0 <![CDATA[In a 2010 paper Haglund, Morse, and Zabrocki studied the family of polynomials $\nabla C_{p1}\dots C_{pk}1$ , where $p=(p_1,\ldots,p_k)$ is a composition, $\nabla$ is the Bergeron-Garsia Macdonald operator and the $C_\alpha$ are certain slightly modified Hall-Littlewood vertex operators. They conjecture that these polynomials enumerate a composition indexed family of parking functions by area, dinv and an appropriate quasi-symmetric function. This refinement of the nearly decade old ``Shuffle Conjecture,'' when combined with properties of the Hall-Littlewood operators can be shown to imply the existence of certain bijections between these families of parking functions. In previous work to appear in her PhD thesis, the author has shown that the existence of these bijections follows from some relatively simple properties of a certain family of polynomials in one variable x with coefficients in $\mathbb{N}[q]$. In this paper we introduce those polynomials, explain their connection to the conjecture of Haglund, Morse, and Zabrocki, and explore some of their surprising properties, both proven and conjectured.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3024 https://doi.org/10.46298/dmtcs.3024 Hicks, Angela Hicks, Angela <![CDATA[In a 2010 paper Haglund, Morse, and Zabrocki studied the family of polynomials $\nabla C_{p1}\dots C_{pk}1$ , where $p=(p_1,\ldots,p_k)$ is a composition, $\nabla$ is the Bergeron-Garsia Macdonald operator and the $C_\alpha$ are certain slightly modified Hall-Littlewood vertex operators. They conjecture that these polynomials enumerate a composition indexed family of parking functions by area, dinv and an appropriate quasi-symmetric function. This refinement of the nearly decade old ``Shuffle Conjecture,'' when combined with properties of the Hall-Littlewood operators can be shown to imply the existence of certain bijections between these families of parking functions. In previous work to appear in her PhD thesis, the author has shown that the existence of these bijections follows from some relatively simple properties of a certain family of polynomials in one variable x with coefficients in $\mathbb{N}[q]$. In this paper we introduce those polynomials, explain their connection to the conjecture of Haglund, Morse, and Zabrocki, and explore some of their surprising properties, both proven and conjectured.]]> 0 <![CDATA[We generalize the symmetry on Young's lattice, found by Suter, to a symmetry on the $k$-bounded partition lattice of Lapointe, Lascoux and Morse.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3023 https://doi.org/10.46298/dmtcs.3023 Berg, Chris Zabrocki, Mike Berg, Chris Zabrocki, Mike <![CDATA[We generalize the symmetry on Young's lattice, found by Suter, to a symmetry on the $k$-bounded partition lattice of Lapointe, Lascoux and Morse.]]> 0 <![CDATA[Consider a countable alphabet $\mathcal{A}$. A multi-modular hidden pattern is an $r$-tuple $(w_1,\ldots , w_r)$, where each $w_i$ is a word over $\mathcal{A}$ called a module. The hidden pattern is said to occur in a text $t$ when the later admits the decomposition $t = v_0 w_1v_1 \cdots v_{r−1}w_r v_r$, for arbitrary words $v_i$ over $\mathcal{A}$. Flajolet, Szpankowski and Vallée (2006) proved via the method of moments that the number of matches (or occurrences) with a multi-modular hidden pattern in a random text $X_1\cdots X_n$ is asymptotically Normal, when $(X_n)_{n\geq1}$ are independent and identically distributed $\mathcal{A}$-valued random variables. Bourdon and Vallée (2002) had conjectured however that asymptotic Normality holds more generally when $(X_n)_{n\geq1}$ is produced by an expansive dynamical source. Whereas memoryless and Markovian sequences are instances of dynamical sources with finite memory length, general dynamical sources may be non-Markovian i.e. convey an infinite memory length. The technical difficulty to count hidden patterns under sources with memory is the context-free nature of these patterns as well as the lack of logarithm-and exponential-type transformations to rewrite the product of non-commuting transfer operators. In this paper, we address a case study in which we have successfully overpassed the aforementioned difficulties and which may illuminate how to address more general cases via auto-correlation operators. Our main result shows that the number of matches with a bi-modular pattern $(w_1, w_2)$ normalized by the number of matches with the pattern $w_1$, where $w_1$ and $w_2$ are different alphabet characters, is indeed asymptotically Normal when $(X_n)_{n\geq1}$ is produced by a holomorphic probabilistic dynamical source.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3011 https://doi.org/10.46298/dmtcs.3011 Lhote, Loïck Lladser, Manuel E. Lhote, Loïck Lladser, Manuel E. <![CDATA[Consider a countable alphabet $\mathcal{A}$. A multi-modular hidden pattern is an $r$-tuple $(w_1,\ldots , w_r)$, where each $w_i$ is a word over $\mathcal{A}$ called a module. The hidden pattern is said to occur in a text $t$ when the later admits the decomposition $t = v_0 w_1v_1 \cdots v_{r−1}w_r v_r$, for arbitrary words $v_i$ over $\mathcal{A}$. Flajolet, Szpankowski and Vallée (2006) proved via the method of moments that the number of matches (or occurrences) with a multi-modular hidden pattern in a random text $X_1\cdots X_n$ is asymptotically Normal, when $(X_n)_{n\geq1}$ are independent and identically distributed $\mathcal{A}$-valued random variables. Bourdon and Vallée (2002) had conjectured however that asymptotic Normality holds more generally when $(X_n)_{n\geq1}$ is produced by an expansive dynamical source. Whereas memoryless and Markovian sequences are instances of dynamical sources with finite memory length, general dynamical sources may be non-Markovian i.e. convey an infinite memory length. The technical difficulty to count hidden patterns under sources with memory is the context-free nature of these patterns as well as the lack of logarithm-and exponential-type transformations to rewrite the product of non-commuting transfer operators. In this paper, we address a case study in which we have successfully overpassed the aforementioned difficulties and which may illuminate how to address more general cases via auto-correlation operators. Our main result shows that the number of matches with a bi-modular pattern $(w_1, w_2)$ normalized by the number of matches with the pattern $w_1$, where $w_1$ and $w_2$ are different alphabet characters, is indeed asymptotically Normal when $(X_n)_{n\geq1}$ is produced by a holomorphic probabilistic dynamical source.]]> 0 <![CDATA[We generalize the brick polytope of V. Pilaud and F. Santos to spherical subword complexes for finite Coxeter groups. This construction provides polytopal realizations for a certain class of subword complexes containing all cluster complexes of finite types. For the latter, the brick polytopes turn out to coincide with the known realizations of generalized associahedra, thus opening new perspectives on these constructions. This new approach yields in particular the vertex description and a relevant Minkowski sum decomposition of generalized associahedra.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3021 https://doi.org/10.46298/dmtcs.3021 Pilaud, Vincent Stump, Christian Pilaud, Vincent Stump, Christian <![CDATA[We generalize the brick polytope of V. Pilaud and F. Santos to spherical subword complexes for finite Coxeter groups. This construction provides polytopal realizations for a certain class of subword complexes containing all cluster complexes of finite types. For the latter, the brick polytopes turn out to coincide with the known realizations of generalized associahedra, thus opening new perspectives on these constructions. This new approach yields in particular the vertex description and a relevant Minkowski sum decomposition of generalized associahedra.]]> 0 <![CDATA[The degree chromatic polynomial $P_m(G,k)$ of a graph $G$ counts the number of $k$ -colorings in which no vertex has m adjacent vertices of its same color. We prove Humpert and Martin's conjecture on the leading terms of the degree chromatic polynomial of a tree.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3020 https://doi.org/10.46298/dmtcs.3020 Cifuentes, Diego Cifuentes, Diego <![CDATA[The degree chromatic polynomial $P_m(G,k)$ of a graph $G$ counts the number of $k$ -colorings in which no vertex has m adjacent vertices of its same color. We prove Humpert and Martin's conjecture on the leading terms of the degree chromatic polynomial of a tree.]]> 0 <![CDATA[The notion of (3+1)-avoidance appears in many places in enumerative combinatorics, but the natural goal of enumerating all (3+1)-avoiding posets remains open. In this paper, we enumerate \emphgraded (3+1)-avoiding posets. Our proof consists of a number of structural theorems followed by some generating function magic.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3019 https://doi.org/10.46298/dmtcs.3019 Lewis Brewster, Joel Zhang, Yan X Lewis Brewster, Joel Zhang, Yan X <![CDATA[The notion of (3+1)-avoidance appears in many places in enumerative combinatorics, but the natural goal of enumerating all (3+1)-avoiding posets remains open. In this paper, we enumerate \emphgraded (3+1)-avoiding posets. Our proof consists of a number of structural theorems followed by some generating function magic.]]> 0 <![CDATA[In their paper on Wilf-equivalence for singleton classes, Backelin, West, and Xin introduced a transformation $\phi^*$, defined by an iterative process and operating on (all) full rook placements on Ferrers boards. Bousquet-Mélou and Steingrimsson proved the analogue of the main result of Backelin, West, and Xin in the context of involutions, and in so doing they needed to prove that $\phi^*$ commutes with the operation of taking inverses. The proof of this commutation result was long and difficult, and Bousquet-Mélou and Steingrimsson asked if $\phi^*$ might be reformulated in such a way as to make this result obvious. In the present paper we provide such a reformulation of $\phi^*$, by modifying the growth diagram algorithm of Fomin. This also answers a question of Krattenthaler, who noted that a bijection defined by the unmodified Fomin algorithm obviously commutes with inverses, and asked what the connection is between this bijection and $\phi^*$.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3018 https://doi.org/10.46298/dmtcs.3018 Bloom, Jonathan Saracino, Dan Bloom, Jonathan Saracino, Dan <![CDATA[In their paper on Wilf-equivalence for singleton classes, Backelin, West, and Xin introduced a transformation $\phi^*$, defined by an iterative process and operating on (all) full rook placements on Ferrers boards. Bousquet-Mélou and Steingrimsson proved the analogue of the main result of Backelin, West, and Xin in the context of involutions, and in so doing they needed to prove that $\phi^*$ commutes with the operation of taking inverses. The proof of this commutation result was long and difficult, and Bousquet-Mélou and Steingrimsson asked if $\phi^*$ might be reformulated in such a way as to make this result obvious. In the present paper we provide such a reformulation of $\phi^*$, by modifying the growth diagram algorithm of Fomin. This also answers a question of Krattenthaler, who noted that a bijection defined by the unmodified Fomin algorithm obviously commutes with inverses, and asked what the connection is between this bijection and $\phi^*$.]]> 0 <![CDATA[We provide a Hopf algebra structure on the supercharacter theory for the unipotent upper triangular group of type {D} over a finite field. Also, we make further comments with respect to types {B} and {C}. Type {A} was explored by M. Aguiar et. al (2010), thus this extended abstract is a contribution to understand combinatorially the supercharacter theory of the other classical Lie types.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3017 https://doi.org/10.46298/dmtcs.3017 Benedetti, Carolina Benedetti, Carolina <![CDATA[We provide a Hopf algebra structure on the supercharacter theory for the unipotent upper triangular group of type {D} over a finite field. Also, we make further comments with respect to types {B} and {C}. Type {A} was explored by M. Aguiar et. al (2010), thus this extended abstract is a contribution to understand combinatorially the supercharacter theory of the other classical Lie types.]]> 0 <![CDATA[Let $P$ be a poset and let $P^*$ be the set of all finite length words over $P$. Generalized subword order is the partial order on $P^*$ obtained by letting $u≤ w$ if and only if there is a subword $u'$ of $w$ having the same length as $u$ such that each element of $u$ is less than or equal to the corresponding element of $u'$ in the partial order on $P$. Classical subword order arises when $P$ is an antichain, while letting $P$ be a chain gives an order on compositions. For any finite poset $P$, we give a simple formula for the Möbius function of $P^*$ in terms of the Möbius function of $P$. This permits us to rederive in an easy and uniform manner previous results of Björner, Sagan and Vatter, and Tomie. We are also able to determine the homotopy type of all intervals in $P^*$ for any finite $P$ of rank at most 1.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3016 https://doi.org/10.46298/dmtcs.3016 McNamara, Peter R. W. Sagan, Bruce E. McNamara, Peter R. W. Sagan, Bruce E. <![CDATA[Let $P$ be a poset and let $P^*$ be the set of all finite length words over $P$. Generalized subword order is the partial order on $P^*$ obtained by letting $u≤ w$ if and only if there is a subword $u'$ of $w$ having the same length as $u$ such that each element of $u$ is less than or equal to the corresponding element of $u'$ in the partial order on $P$. Classical subword order arises when $P$ is an antichain, while letting $P$ be a chain gives an order on compositions. For any finite poset $P$, we give a simple formula for the Möbius function of $P^*$ in terms of the Möbius function of $P$. This permits us to rederive in an easy and uniform manner previous results of Björner, Sagan and Vatter, and Tomie. We are also able to determine the homotopy type of all intervals in $P^*$ for any finite $P$ of rank at most 1.]]> 0 <![CDATA[The Ram–Yip formula for Macdonald polynomials (at t=0) provides a statistic which we call charge. In types ${A}$ and ${C}$ it can be defined on tensor products of Kashiwara–Nakashima single column crystals. In this paper we show that the charge is equal to the (negative of the) energy function on affine crystals. The algorithm for computing charge is much simpler than the recursive definition of energy in terms of the combinatorial ${R}$-matrix.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3015 https://doi.org/10.46298/dmtcs.3015 Lenart, Cristian Schilling, Anne Lenart, Cristian Schilling, Anne <![CDATA[The Ram–Yip formula for Macdonald polynomials (at t=0) provides a statistic which we call charge. In types ${A}$ and ${C}$ it can be defined on tensor products of Kashiwara–Nakashima single column crystals. In this paper we show that the charge is equal to the (negative of the) energy function on affine crystals. The algorithm for computing charge is much simpler than the recursive definition of energy in terms of the combinatorial ${R}$-matrix.]]> 0 <![CDATA[We present a family of simplicial complexes called \emphmulti-cluster complexes. These complexes generalize the concept of cluster complexes, and extend the notion of multi-associahedra of types ${A}$ and ${B}$ to general finite Coxeter groups. We study combinatorial and geometric properties of these objects and, in particular, provide a simple combinatorial description of the compatibility relation among the set of almost positive roots in the cluster complex.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3014 https://doi.org/10.46298/dmtcs.3014 Ceballos, Cesar Labbé, Jean-Philippe Stump, Christian Ceballos, Cesar Labbé, Jean-Philippe Stump, Christian <![CDATA[We present a family of simplicial complexes called \emphmulti-cluster complexes. These complexes generalize the concept of cluster complexes, and extend the notion of multi-associahedra of types ${A}$ and ${B}$ to general finite Coxeter groups. We study combinatorial and geometric properties of these objects and, in particular, provide a simple combinatorial description of the compatibility relation among the set of almost positive roots in the cluster complex.]]> 0 <![CDATA[We give a unified treatment of the limit, as the size tends to infinity, of random simply generated trees, including both the well-known result in the standard case of critical Galton-Watson trees and similar but less well-known results in the other cases (i.e., when no equivalent critical Galton-Watson tree exists). There is a well-defined limit in the form of an infinite random tree in all cases; for critical Galton-Watson trees this tree is locally finite but for the other cases the random limit has exactly one node of infinite degree. The random infinite limit tree can in all cases be constructed by a modified Galton-Watson process. In the standard case of a critical Galton-Watson tree, the limit tree has an infinite "spine", where the offspring distribution is size-biased. In the other cases, the spine has finite length and ends with a vertex with infinite degree. A node of infinite degree in the limit corresponds to the existence of one node with very high degree in the finite random trees; in physics terminology, this is a type of condensation. In simple cases, there is one node with a degree that is roughly a constant times the number of nodes, while all other degrees are much smaller; however, more complicated behaviour is also possible. The proofs use a well-known connection to a random allocation model that we call balls-in-boxes, and we prove corresponding results for this model.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3013 https://doi.org/10.46298/dmtcs.3013 Janson, Svante Janson, Svante <![CDATA[We give a unified treatment of the limit, as the size tends to infinity, of random simply generated trees, including both the well-known result in the standard case of critical Galton-Watson trees and similar but less well-known results in the other cases (i.e., when no equivalent critical Galton-Watson tree exists). There is a well-defined limit in the form of an infinite random tree in all cases; for critical Galton-Watson trees this tree is locally finite but for the other cases the random limit has exactly one node of infinite degree. The random infinite limit tree can in all cases be constructed by a modified Galton-Watson process. In the standard case of a critical Galton-Watson tree, the limit tree has an infinite "spine", where the offspring distribution is size-biased. In the other cases, the spine has finite length and ends with a vertex with infinite degree. A node of infinite degree in the limit corresponds to the existence of one node with very high degree in the finite random trees; in physics terminology, this is a type of condensation. In simple cases, there is one node with a degree that is roughly a constant times the number of nodes, while all other degrees are much smaller; however, more complicated behaviour is also possible. The proofs use a well-known connection to a random allocation model that we call balls-in-boxes, and we prove corresponding results for this model.]]> 0 <![CDATA[In this paper infinite systems of functional equations in finitely or infinitely many random variables arising in combinatorial enumeration problems are studied. We prove sufficient conditions under which the combinatorial random variables encoded in the generating function of the system tend to a finite or infinite dimensional limiting distribution.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3012 https://doi.org/10.46298/dmtcs.3012 Drmota, Michael Gittenberger, Bernhard Morgenbesser, Johannes F. Drmota, Michael Gittenberger, Bernhard Morgenbesser, Johannes F. <![CDATA[In this paper infinite systems of functional equations in finitely or infinitely many random variables arising in combinatorial enumeration problems are studied. We prove sufficient conditions under which the combinatorial random variables encoded in the generating function of the system tend to a finite or infinite dimensional limiting distribution.]]> 0 <![CDATA[no abstract]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2985 https://doi.org/10.46298/dmtcs.2985 Eisner, Idan Vainshtein, Alek Eisner, Idan Vainshtein, Alek <![CDATA[no abstract]]> 0 <![CDATA[In the paper we discuss a technology based on Bernstein polynomials of asymptotic analysis of a class of binomial sums that arise in information theory. Our method gives a quick derivation of required sums and can be generalized to multinomial distributions. As an example we derive a formula for the entropy of multinomial distributions. Our method simplifies previous work of Jacquet, Szpankowski and Flajolet from 1999.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2993 https://doi.org/10.46298/dmtcs.2993 Cichoń, Jacek Gołębiewski, Zbigniew Cichoń, Jacek Gołębiewski, Zbigniew <![CDATA[In the paper we discuss a technology based on Bernstein polynomials of asymptotic analysis of a class of binomial sums that arise in information theory. Our method gives a quick derivation of required sums and can be generalized to multinomial distributions. As an example we derive a formula for the entropy of multinomial distributions. Our method simplifies previous work of Jacquet, Szpankowski and Flajolet from 1999.]]> 0 <![CDATA[We give simple probabilistic algorithms that approximately maximize the volume of overlap of two solid, i.e. full-dimensional, shapes under translations and rigid motions. The shapes are subsets of $ℝ^d$ where $d≥ 2$. The algorithms approximate with respect to an pre-specified additive error and succeed with high probability. Apart from measurability assumptions, we only require that points from the shapes can be generated uniformly at random. An important example are shapes given as finite unions of simplices that have pairwise disjoint interiors.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2992 https://doi.org/10.46298/dmtcs.2992 Schymura, Daria Schymura, Daria <![CDATA[We give simple probabilistic algorithms that approximately maximize the volume of overlap of two solid, i.e. full-dimensional, shapes under translations and rigid motions. The shapes are subsets of $ℝ^d$ where $d≥ 2$. The algorithms approximate with respect to an pre-specified additive error and succeed with high probability. Apart from measurability assumptions, we only require that points from the shapes can be generated uniformly at random. An important example are shapes given as finite unions of simplices that have pairwise disjoint interiors.]]> 0 <![CDATA[We consider a generalization of the uniform word-based distribution for finitely generated subgroups of a free group. In our setting, the number of generators is not fixed, the length of each generator is determined by a random variable with some simple constraints and the distribution of words of a fixed length is specified by a Markov process. We show by probabilistic arguments that under rather relaxed assumptions, the good properties of the uniform word-based distribution are preserved: generically (but maybe not exponentially generically), the tuple we pick is a basis of the subgroup it generates, this subgroup is malnormal and the group presentation defined by this tuple satisfies a small cancellation condition.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2991 https://doi.org/10.46298/dmtcs.2991 Bassino, Frédérique Nicaud, Cyril Weil, Pascal Bassino, Frédérique Nicaud, Cyril Weil, Pascal <![CDATA[We consider a generalization of the uniform word-based distribution for finitely generated subgroups of a free group. In our setting, the number of generators is not fixed, the length of each generator is determined by a random variable with some simple constraints and the distribution of words of a fixed length is specified by a Markov process. We show by probabilistic arguments that under rather relaxed assumptions, the good properties of the uniform word-based distribution are preserved: generically (but maybe not exponentially generically), the tuple we pick is a basis of the subgroup it generates, this subgroup is malnormal and the group presentation defined by this tuple satisfies a small cancellation condition.]]> 0 <![CDATA[We study transversals in random trees with n vertices asymptotically as n tends to infinity. Our investigation treats the average number of transversals of fixed size, the size of a random transversal as well as the probability that a random subset of the vertex set of a tree is a transversal for the class of simply generated trees and for Pólya trees. The last parameter was already studied by Devroye for simply generated trees. We offer an alternative proof based on generating functions and singularity analysis and extend the result to Pólya trees.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2990 https://doi.org/10.46298/dmtcs.2990 Gittenberger, Bernhard Kraus, Veronika Gittenberger, Bernhard Kraus, Veronika <![CDATA[We study transversals in random trees with n vertices asymptotically as n tends to infinity. Our investigation treats the average number of transversals of fixed size, the size of a random transversal as well as the probability that a random subset of the vertex set of a tree is a transversal for the class of simply generated trees and for Pólya trees. The last parameter was already studied by Devroye for simply generated trees. We offer an alternative proof based on generating functions and singularity analysis and extend the result to Pólya trees.]]> 0 <![CDATA[This paper is devoted to the construction of Boltzmann samplers according to various distributions, and uses stochastic bias on the parameter of a Boltzmann sampler, to produce a sampler with a different distribution for the size of the output. As a significant application, we produce Boltzmann samplers for words defined by regular specifications containing shuffle operators and linear recursions. This sampler has linear complexity in the size of the output, where the complexity is measured in terms of real-arithmetic operations and evaluations of generating functions.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2989 https://doi.org/10.46298/dmtcs.2989 Darrasse, Alexis Panagiotou, Konstantinos Roussel, Olivier Soria, Michele Darrasse, Alexis Panagiotou, Konstantinos Roussel, Olivier Soria, Michele <![CDATA[This paper is devoted to the construction of Boltzmann samplers according to various distributions, and uses stochastic bias on the parameter of a Boltzmann sampler, to produce a sampler with a different distribution for the size of the output. As a significant application, we produce Boltzmann samplers for words defined by regular specifications containing shuffle operators and linear recursions. This sampler has linear complexity in the size of the output, where the complexity is measured in terms of real-arithmetic operations and evaluations of generating functions.]]> 0 <![CDATA[Enumeration of planar lattice walks is a classical topic in combinatorics, at the cross-roads of several domains (e.g., probability, statistical physics, computer science). The aim of this paper is to propose a new approach to obtain some exact asymptotics for walks confined to the quarter plane.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2988 https://doi.org/10.46298/dmtcs.2988 Fayolle, Guy Raschel, Kilian Fayolle, Guy Raschel, Kilian <![CDATA[Enumeration of planar lattice walks is a classical topic in combinatorics, at the cross-roads of several domains (e.g., probability, statistical physics, computer science). The aim of this paper is to propose a new approach to obtain some exact asymptotics for walks confined to the quarter plane.]]> 0 <![CDATA[We prove a total variation approximation for the distribution of component vector of a weakly logarithmic random assembly. The proof demonstrates an analytic approach based on a comparative analysis of the coefficients of two power series.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2987 https://doi.org/10.46298/dmtcs.2987 Manstavičius, Eugenijus Manstavičius, Eugenijus <![CDATA[We prove a total variation approximation for the distribution of component vector of a weakly logarithmic random assembly. The proof demonstrates an analytic approach based on a comparative analysis of the coefficients of two power series.]]> 0 <![CDATA[In this paper, we study the shuffle operator on concurrent processes (represented as trees) using analytic combinatorics tools. As a first result, we show that the mean width of shuffle trees is exponentially smaller than the worst case upper-bound. We also study the expected size (in total number of nodes) of shuffle trees. We notice, rather unexpectedly, that only a small ratio of all nodes do not belong to the last two levels. We also provide a precise characterization of what ``exponential growth'' means in the case of the shuffle on trees. Two practical outcomes of our quantitative study are presented: (1) a linear-time algorithm to compute the probability of a concurrent run prefix, and (2) an efficient algorithm for uniform random generation of concurrent runs.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2986 https://doi.org/10.46298/dmtcs.2986 Bodini, Olivier Genitrini, Antoine Peschanski, Frédéric Bodini, Olivier Genitrini, Antoine Peschanski, Frédéric <![CDATA[In this paper, we study the shuffle operator on concurrent processes (represented as trees) using analytic combinatorics tools. As a first result, we show that the mean width of shuffle trees is exponentially smaller than the worst case upper-bound. We also study the expected size (in total number of nodes) of shuffle trees. We notice, rather unexpectedly, that only a small ratio of all nodes do not belong to the last two levels. We also provide a precise characterization of what ``exponential growth'' means in the case of the shuffle on trees. Two practical outcomes of our quantitative study are presented: (1) a linear-time algorithm to compute the probability of a concurrent run prefix, and (2) an efficient algorithm for uniform random generation of concurrent runs.]]> 0 <![CDATA[Many parameters of trees are additive in the sense that they can be computed recursively from the sum of the branches plus a certain toll function. For instance, such parameters occur very frequently in the analysis of divide-and-conquer algorithms. Here we are interested in the situation that the toll function is small (the average over all trees of a given size $n$ decreases exponentially with $n$). We prove a general central limit theorem for random labelled trees and apply it to a number of examples. The main motivation is the study of the number of subtrees in a random labelled tree, but it also applies to classical instances such as the number of leaves.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2984 https://doi.org/10.46298/dmtcs.2984 Wagner, Stephan Wagner, Stephan <![CDATA[Many parameters of trees are additive in the sense that they can be computed recursively from the sum of the branches plus a certain toll function. For instance, such parameters occur very frequently in the analysis of divide-and-conquer algorithms. Here we are interested in the situation that the toll function is small (the average over all trees of a given size $n$ decreases exponentially with $n$). We prove a general central limit theorem for random labelled trees and apply it to a number of examples. The main motivation is the study of the number of subtrees in a random labelled tree, but it also applies to classical instances such as the number of leaves.]]> 0 <![CDATA[Let $k≥2$ be a fixed integer. Given a bounded sequence of real numbers $(a_n:n≥k)$, then for any sequence $(f_n:n≥1)$ of real numbers satisfying the divide-and-conquer recurrence $f_n = (k-mod(n,k))f_⌊n/k⌋+mod(n,k)f_⌈n/k⌉ + a_n, n ≥k$, there is a unique continuous periodic function $f^*:\mathbb{R}→\mathbb{R}$ with period 1 such that $f_n = nf^*(\log _kn)+o(n)$. If $(a_n)$ is periodic with period $k, a_k=0$, and the initial conditions $(f_i:1 ≤i ≤k-1)$ are all zero, we obtain a specific iterated function system $S$, consisting of $k$ continuous functions from $[0,1]×\mathbb{R}$ into itself, such that the attractor of $S$ is $\{(x,f^*(x)): 0 ≤x ≤1\}$. Using the system $S$, an accurate plot of $f^*$ can be rapidly obtained.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2983 https://doi.org/10.46298/dmtcs.2983 Kieffer, John, Kieffer, John, <![CDATA[Let $k≥2$ be a fixed integer. Given a bounded sequence of real numbers $(a_n:n≥k)$, then for any sequence $(f_n:n≥1)$ of real numbers satisfying the divide-and-conquer recurrence $f_n = (k-mod(n,k))f_⌊n/k⌋+mod(n,k)f_⌈n/k⌉ + a_n, n ≥k$, there is a unique continuous periodic function $f^*:\mathbb{R}→\mathbb{R}$ with period 1 such that $f_n = nf^*(\log _kn)+o(n)$. If $(a_n)$ is periodic with period $k, a_k=0$, and the initial conditions $(f_i:1 ≤i ≤k-1)$ are all zero, we obtain a specific iterated function system $S$, consisting of $k$ continuous functions from $[0,1]×\mathbb{R}$ into itself, such that the attractor of $S$ is $\{(x,f^*(x)): 0 ≤x ≤1\}$. Using the system $S$, an accurate plot of $f^*$ can be rapidly obtained.]]> 0 <![CDATA[The purpose of this article is to present a general method to find limiting laws for some renormalized statistics on random permutations. The model considered here is Ewens sampling model, which generalizes uniform random permutations. We describe the asymptotic behavior of a large family of statistics, including the number of occurrences of any given dashed pattern. Our approach is based on the method of moments and relies on the following intuition: two events involving the images of different integers are almost independent.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2982 https://doi.org/10.46298/dmtcs.2982 Feray, Valentin Feray, Valentin <![CDATA[The purpose of this article is to present a general method to find limiting laws for some renormalized statistics on random permutations. The model considered here is Ewens sampling model, which generalizes uniform random permutations. We describe the asymptotic behavior of a large family of statistics, including the number of occurrences of any given dashed pattern. Our approach is based on the method of moments and relies on the following intuition: two events involving the images of different integers are almost independent.]]> 0 <![CDATA[We use death processes and embeddings into continuous time in order to analyze several urn models with a diminishing content. In particular we discuss generalizations of the pill's problem, originally introduced by Knuth and McCarthy, and generalizations of the well known sampling without replacement urn models, and OK Corral urn models.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2981 https://doi.org/10.46298/dmtcs.2981 Kuba, Markus Panholzer, Alois Kuba, Markus Panholzer, Alois <![CDATA[We use death processes and embeddings into continuous time in order to analyze several urn models with a diminishing content. In particular we discuss generalizations of the pill's problem, originally introduced by Knuth and McCarthy, and generalizations of the well known sampling without replacement urn models, and OK Corral urn models.]]> 0 <![CDATA[Approximate counting is an algorithm that provides a count of a huge number of objects within an error tolerance. The first detailed analysis of this algorithm was given by Flajolet. In this paper, we propose a new analysis via the Poisson-Laplace-Mellin approach, a method devised for analyzing shape parameters of digital search trees in a recent paper of Hwang et al. Our approach yields a different and more compact expression for the periodic function from the asymptotic expansion of the variance. We show directly that our expression coincides with the one obtained by Flajolet. Moreover, we apply our method to variations of approximate counting, too.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2980 https://doi.org/10.46298/dmtcs.2980 Fuchs, Michael Lee, Chung-Kuei Prodinger, Helmut Fuchs, Michael Lee, Chung-Kuei Prodinger, Helmut <![CDATA[Approximate counting is an algorithm that provides a count of a huge number of objects within an error tolerance. The first detailed analysis of this algorithm was given by Flajolet. In this paper, we propose a new analysis via the Poisson-Laplace-Mellin approach, a method devised for analyzing shape parameters of digital search trees in a recent paper of Hwang et al. Our approach yields a different and more compact expression for the periodic function from the asymptotic expansion of the variance. We show directly that our expression coincides with the one obtained by Flajolet. Moreover, we apply our method to variations of approximate counting, too.]]> 0 <![CDATA[In a recently proposed graphical compression algorithm by Choi and Szpankowski (2009), the following tree arose in the course of the analysis. The root contains n balls that are consequently distributed between two subtrees according to a simple rule: In each step, all balls independently move down to the left subtree (say with probability $p$) or the right subtree (with probability 1-$p$). A new node is created as long as there is at least one ball in that node. Furthermore, a nonnegative integer $d$ is given, and at level $d$ or greater one ball is removed from the leftmost node before the balls move down to the next level. These steps are repeated until all balls are removed (i.e., after $n+d$ steps). Observe that when $d=∞$ the above tree can be modeled as a $\textit{trie}$ that stores $n$ independent sequences generated by a memoryless source with parameter $p$. Therefore, we coin the name $(n,d)$-tries for the tree just described, and to which we often refer simply as $d$-tries. Parameters of such a tree (e.g., path length, depth, size) are described by an interesting two-dimensional recurrence (in terms of $n$ and $d$) that – to the best of our knowledge – was not analyzed before. We study it, and show how much parameters of such a $(n,d)$-trie differ from the corresponding parameters of regular tries. We use methods of analytic algorithmics, from Mellin transforms to analytic poissonization.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2979 https://doi.org/10.46298/dmtcs.2979 Choi, Yongwook Knessl, Charles Szpankowski, Wojciech Choi, Yongwook Knessl, Charles Szpankowski, Wojciech <![CDATA[In a recently proposed graphical compression algorithm by Choi and Szpankowski (2009), the following tree arose in the course of the analysis. The root contains n balls that are consequently distributed between two subtrees according to a simple rule: In each step, all balls independently move down to the left subtree (say with probability $p$) or the right subtree (with probability 1-$p$). A new node is created as long as there is at least one ball in that node. Furthermore, a nonnegative integer $d$ is given, and at level $d$ or greater one ball is removed from the leftmost node before the balls move down to the next level. These steps are repeated until all balls are removed (i.e., after $n+d$ steps). Observe that when $d=∞$ the above tree can be modeled as a $\textit{trie}$ that stores $n$ independent sequences generated by a memoryless source with parameter $p$. Therefore, we coin the name $(n,d)$-tries for the tree just described, and to which we often refer simply as $d$-tries. Parameters of such a tree (e.g., path length, depth, size) are described by an interesting two-dimensional recurrence (in terms of $n$ and $d$) that – to the best of our knowledge – was not analyzed before. We study it, and show how much parameters of such a $(n,d)$-trie differ from the corresponding parameters of regular tries. We use methods of analytic algorithmics, from Mellin transforms to analytic poissonization.]]> 0 <![CDATA[The alcove model of the first author and Postnikov describes highest weight crystals of semisimple Lie algebras. We present a generalization, called the quantum alcove model, and conjecture that it uniformly describes tensor products of column shape Kirillov-Reshetikhin crystals, for all untwisted affine types. We prove the conjecture in types $A$ and $C$. We also present evidence for the fact that a related statistic computes the energy function.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3090 https://doi.org/10.46298/dmtcs.3090 Lenart, Cristian Lubovsky, Arthur Lenart, Cristian Lubovsky, Arthur <![CDATA[The alcove model of the first author and Postnikov describes highest weight crystals of semisimple Lie algebras. We present a generalization, called the quantum alcove model, and conjecture that it uniformly describes tensor products of column shape Kirillov-Reshetikhin crystals, for all untwisted affine types. We prove the conjecture in types $A$ and $C$. We also present evidence for the fact that a related statistic computes the energy function.]]> 0 <![CDATA[We show that the Galois group of any Schubert problem involving lines in projective space contains the alternating group. Using a criterion of Vakil and a special position argument due to Schubert, this follows from a particular inequality among Kostka numbers of two-rowed tableaux. In most cases, an easy combinatorial injection proves the inequality. For the remaining cases, we use that these Kostka numbers appear in tensor product decompositions of $\mathfrak{sl}_2\mathbb{C}$ -modules. Interpreting the tensor product as the action of certain commuting Toeplitz matrices and using a spectral analysis and Fourier series rewrites the inequality as the positivity of an integral. We establish the inequality by estimating this integral.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3099 https://doi.org/10.46298/dmtcs.3099 Brooks, Christopher J. Campo, Abraham Martín, Sottile, Frank Brooks, Christopher J. Campo, Abraham Martín, Sottile, Frank <![CDATA[We show that the Galois group of any Schubert problem involving lines in projective space contains the alternating group. Using a criterion of Vakil and a special position argument due to Schubert, this follows from a particular inequality among Kostka numbers of two-rowed tableaux. In most cases, an easy combinatorial injection proves the inequality. For the remaining cases, we use that these Kostka numbers appear in tensor product decompositions of $\mathfrak{sl}_2\mathbb{C}$ -modules. Interpreting the tensor product as the action of certain commuting Toeplitz matrices and using a spectral analysis and Fourier series rewrites the inequality as the positivity of an integral. We establish the inequality by estimating this integral.]]> 0 <![CDATA[We consider two recent open problems stating that certain statistics on various sets of combinatorial objects are equidistributed. The first, posed by Anders Claesson and Svante Linusson, relates nestings in matchings on $\{1,2,\ldots,2n\}$ to occurrences of a certain pattern in permutations in $S_n$. The second, posed by Miles Jones and Jeffrey Remmel, relates occurrences of a large class of consecutive permutation patterns to occurrences of the same pattern in the cycles of permutations. We develop a general method that solves both of these problems and many more. We further employ the Garsia-Milne involution principle to obtain purely bijective proofs of these results.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3098 https://doi.org/10.46298/dmtcs.3098 Remmel, Jeffrey Tiefenbruck, Mark Remmel, Jeffrey Tiefenbruck, Mark <![CDATA[We consider two recent open problems stating that certain statistics on various sets of combinatorial objects are equidistributed. The first, posed by Anders Claesson and Svante Linusson, relates nestings in matchings on $\{1,2,\ldots,2n\}$ to occurrences of a certain pattern in permutations in $S_n$. The second, posed by Miles Jones and Jeffrey Remmel, relates occurrences of a large class of consecutive permutation patterns to occurrences of the same pattern in the cycles of permutations. We develop a general method that solves both of these problems and many more. We further employ the Garsia-Milne involution principle to obtain purely bijective proofs of these results.]]> 0 <![CDATA[We describe a perturbation method that can be used to compute the multivariate generating function (MGF) of a non-simple polyhedron, and then construct a perturbation that works for any transportation polytope. Applying this perturbation to the family of central transportation polytopes of order $kn \times n$, we obtain formulas for the MGF of the polytope. The formulas we obtain are enumerated by combinatorial objects. A special case of the formulas recovers the results on Birkhoff polytopes given by the author and De Loera and Yoshida. We also recover the formula for the number of maximum vertices of transportation polytopes of order $kn \times n$.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3097 https://doi.org/10.46298/dmtcs.3097 Liu, Fu Liu, Fu <![CDATA[We describe a perturbation method that can be used to compute the multivariate generating function (MGF) of a non-simple polyhedron, and then construct a perturbation that works for any transportation polytope. Applying this perturbation to the family of central transportation polytopes of order $kn \times n$, we obtain formulas for the MGF of the polytope. The formulas we obtain are enumerated by combinatorial objects. A special case of the formulas recovers the results on Birkhoff polytopes given by the author and De Loera and Yoshida. We also recover the formula for the number of maximum vertices of transportation polytopes of order $kn \times n$.]]> 0 <![CDATA[We establish the relationship between volumes of flow polytopes associated to signed graphs and the Kostant partition function. A special case of this relationship, namely, when the graphs are signless, has been studied in detail by Baldoni and Vergne using techniques of residues. In contrast with their approach, we provide combinatorial proofs inspired by the work of Postnikov and Stanley on flow polytopes. As an application of our results we study a distinguished family of flow polytopes: the Chan-Robbins-Yuen polytopes. Inspired by their beautiful volume formula $\prod_{k=0}^{n-2} Cat(k)$ for the type $A_n$ case, where $Cat(k)$ is the $k^{th}$ Catalan number, we introduce type $C_{n+1}$ and $D_{n+1}$ Chan-Robbins-Yuen polytopes along with intriguing conjectures about their volumes.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3096 https://doi.org/10.46298/dmtcs.3096 Mészáros, Karola Morales, Alejandro H. Mészáros, Karola Morales, Alejandro H. <![CDATA[We establish the relationship between volumes of flow polytopes associated to signed graphs and the Kostant partition function. A special case of this relationship, namely, when the graphs are signless, has been studied in detail by Baldoni and Vergne using techniques of residues. In contrast with their approach, we provide combinatorial proofs inspired by the work of Postnikov and Stanley on flow polytopes. As an application of our results we study a distinguished family of flow polytopes: the Chan-Robbins-Yuen polytopes. Inspired by their beautiful volume formula $\prod_{k=0}^{n-2} Cat(k)$ for the type $A_n$ case, where $Cat(k)$ is the $k^{th}$ Catalan number, we introduce type $C_{n+1}$ and $D_{n+1}$ Chan-Robbins-Yuen polytopes along with intriguing conjectures about their volumes.]]> 0 <![CDATA[We give a new description of the Pieri rule for $k$-Schur functions using the Bruhat order on the affine type-$A$ Weyl group. In doing so, we prove a new combinatorial formula for representatives of the Schubert classes for the cohomology of affine Grassmannians. We show how new combinatorics involved in our formulas gives the Kostka-Foulkes polynomials and discuss how this can be applied to study the transition matrices between Hall-Littlewood and $k$-Schur functions.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3095 https://doi.org/10.46298/dmtcs.3095 Dalal, Avinash J. Morse, Jennifer Dalal, Avinash J. Morse, Jennifer <![CDATA[We give a new description of the Pieri rule for $k$-Schur functions using the Bruhat order on the affine type-$A$ Weyl group. In doing so, we prove a new combinatorial formula for representatives of the Schubert classes for the cohomology of affine Grassmannians. We show how new combinatorics involved in our formulas gives the Kostka-Foulkes polynomials and discuss how this can be applied to study the transition matrices between Hall-Littlewood and $k$-Schur functions.]]> 0 <![CDATA[We introduce a "lifting'' construction for generalized permutohedra, which turns an $n$-dimensional generalized permutahedron into an $(n+1)$-dimensional one. We prove that this construction gives rise to Stasheff's multiplihedron from homotopy theory, and to the more general "nestomultiplihedra,'' answering two questions of Devadoss and Forcey. We construct a subdivision of any lifted generalized permutahedron whose pieces are indexed by compositions. The volume of each piece is given by a polynomial whose combinatorial properties we investigate. We show how this "composition polynomial'' arises naturally in the polynomial interpolation of an exponential function. We prove that its coefficients are positive integers, and conjecture that they are unimodal.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3094 https://doi.org/10.46298/dmtcs.3094 Ardila, Federico Doker, Jeffrey Ardila, Federico Doker, Jeffrey <![CDATA[We introduce a "lifting'' construction for generalized permutohedra, which turns an $n$-dimensional generalized permutahedron into an $(n+1)$-dimensional one. We prove that this construction gives rise to Stasheff's multiplihedron from homotopy theory, and to the more general "nestomultiplihedra,'' answering two questions of Devadoss and Forcey. We construct a subdivision of any lifted generalized permutahedron whose pieces are indexed by compositions. The volume of each piece is given by a polynomial whose combinatorial properties we investigate. We show how this "composition polynomial'' arises naturally in the polynomial interpolation of an exponential function. We prove that its coefficients are positive integers, and conjecture that they are unimodal.]]> 0 <![CDATA[The Severi degree is the degree of the Severi variety parametrizing plane curves of degree $d$ with $\delta$ nodes. Recently, Göttsche and Shende gave two refinements of Severi degrees, polynomials in a variable $q$, which are conjecturally equal, for large $d$. At $q=1$, one of the refinements, the relative Severi degree, specializes to the (non-relative) Severi degree. We give a combinatorial description of the refined Severi degrees, in terms of a $q$-analog count of Brugallé and Mikhalkin's floor diagrams. Our description implies that, for fixed $\delta$, the refined Severi degrees are polynomials in $d$ and $q$, for large $d$. As a consequence, we show that, for $\delta \leq 4$ and all $d$, both refinements of Göttsche and Shende agree and equal our $q$-count of floor diagrams.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3093 https://doi.org/10.46298/dmtcs.3093 Block, Florian Block, Florian <![CDATA[The Severi degree is the degree of the Severi variety parametrizing plane curves of degree $d$ with $\delta$ nodes. Recently, Göttsche and Shende gave two refinements of Severi degrees, polynomials in a variable $q$, which are conjecturally equal, for large $d$. At $q=1$, one of the refinements, the relative Severi degree, specializes to the (non-relative) Severi degree. We give a combinatorial description of the refined Severi degrees, in terms of a $q$-analog count of Brugallé and Mikhalkin's floor diagrams. Our description implies that, for fixed $\delta$, the refined Severi degrees are polynomials in $d$ and $q$, for large $d$. As a consequence, we show that, for $\delta \leq 4$ and all $d$, both refinements of Göttsche and Shende agree and equal our $q$-count of floor diagrams.]]> 0 <![CDATA[This paper discusses a surprising relationship between the quantum cohomology of the variety of complete flags and the partially ordered set of Newton polygons associated to an element in the affine Weyl group. One primary key to establishing this connection is the fact that paths in the quantum Bruhat graph, which is a weighted directed graph with vertices indexed by elements in the finite Weyl group, encode saturated chains in the strong Bruhat order on the affine Weyl group. This correspondence is also fundamental in the work of Lam and Shimozono establishing Peterson's isomorphism between the quantum cohomology of the finite flag variety and the homology of the affine Grassmannian. In addition, using some geometry associated to the poset of Newton polygons, one obtains independent proofs for several combinatorial statements about paths in the quantum Bruhat graph and its symmetries, which were originally proved by Postnikov using the tilted Bruhat order. An important geometric application of this work is an inequality which provides a necessary condition for non-emptiness of certain affine Deligne-Lusztig varieties in the affine flag variety.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3092 https://doi.org/10.46298/dmtcs.3092 Beazley, Elizabeth T. Beazley, Elizabeth T. <![CDATA[This paper discusses a surprising relationship between the quantum cohomology of the variety of complete flags and the partially ordered set of Newton polygons associated to an element in the affine Weyl group. One primary key to establishing this connection is the fact that paths in the quantum Bruhat graph, which is a weighted directed graph with vertices indexed by elements in the finite Weyl group, encode saturated chains in the strong Bruhat order on the affine Weyl group. This correspondence is also fundamental in the work of Lam and Shimozono establishing Peterson's isomorphism between the quantum cohomology of the finite flag variety and the homology of the affine Grassmannian. In addition, using some geometry associated to the poset of Newton polygons, one obtains independent proofs for several combinatorial statements about paths in the quantum Bruhat graph and its symmetries, which were originally proved by Postnikov using the tilted Bruhat order. An important geometric application of this work is an inequality which provides a necessary condition for non-emptiness of certain affine Deligne-Lusztig varieties in the affine flag variety.]]> 0 <![CDATA[Let $M$ be a finite monoid. In this paper we describe how the Cartan invariant matrix of the monoid algebra of $M$ over a field $\mathbb{K}$ of characteristic zero can be expressed using characters and some simple combinatorial statistic. In particular, it can be computed efficiently from the composition factors of the left and right class modules of $M$. When $M$ is aperiodic, this approach works in any characteristic, and generalizes to $\mathbb{K}$ a principal ideal domain like $\mathbb{Z}$. When $M$ is $\mathcal{R}$-trivial, we retrieve the formerly known purely combinatorial description of the Cartan matrix.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3091 https://doi.org/10.46298/dmtcs.3091 Thiéry, Nicolas M. Thiéry, Nicolas M. <![CDATA[Let $M$ be a finite monoid. In this paper we describe how the Cartan invariant matrix of the monoid algebra of $M$ over a field $\mathbb{K}$ of characteristic zero can be expressed using characters and some simple combinatorial statistic. In particular, it can be computed efficiently from the composition factors of the left and right class modules of $M$. When $M$ is aperiodic, this approach works in any characteristic, and generalizes to $\mathbb{K}$ a principal ideal domain like $\mathbb{Z}$. When $M$ is $\mathcal{R}$-trivial, we retrieve the formerly known purely combinatorial description of the Cartan matrix.]]> 0 <![CDATA[We characterize by pattern avoidance the Schubert varieties for $\mathrm{GL}_n$ which are local complete intersections (lci). For those Schubert varieties which are local complete intersections, we give an explicit minimal set of equations cutting out their neighbourhoods at the identity. Although the statement of our characterization only requires ordinary pattern avoidance, showing that the Schubert varieties not satisfying our conditions are not lci appears to require working with more general notions of pattern avoidance. The Schubert varieties defined by inclusions, originally introduced by Gasharov and Reiner, turn out to be an important subclass, and we further develop some of their combinatorics. One application is a new formula for certain specializations of Schubert polynomials.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3079 https://doi.org/10.46298/dmtcs.3079 Úlfarsson, Henning Woo, Alexander Úlfarsson, Henning Woo, Alexander <![CDATA[We characterize by pattern avoidance the Schubert varieties for $\mathrm{GL}_n$ which are local complete intersections (lci). For those Schubert varieties which are local complete intersections, we give an explicit minimal set of equations cutting out their neighbourhoods at the identity. Although the statement of our characterization only requires ordinary pattern avoidance, showing that the Schubert varieties not satisfying our conditions are not lci appears to require working with more general notions of pattern avoidance. The Schubert varieties defined by inclusions, originally introduced by Gasharov and Reiner, turn out to be an important subclass, and we further develop some of their combinatorics. One application is a new formula for certain specializations of Schubert polynomials.]]> 0 <![CDATA[The Severi variety parametrizes plane curves of degree $d$ with $\delta$ nodes. Its degree is called the Severi degree. For large enough $d$, the Severi degrees coincide with the Gromov-Witten invariants of $\mathbb{CP}^2$. Fomin and Mikhalkin (2009) proved the 1995 conjecture that for fixed $\delta$, Severi degrees are eventually polynomial in $d$. In this paper, we study the Severi varieties corresponding to a large family of toric surfaces. We prove the analogous result that the Severi degrees are eventually polynomial as a function of the multidegree. More surprisingly, we show that the Severi degrees are also eventually polynomial "as a function of the surface". Our strategy is to use tropical geometry to express Severi degrees in terms of Brugallé and Mikhalkin's floor diagrams, and study those combinatorial objects in detail. An important ingredient in the proof is the polynomiality of the discrete volume of a variable facet-unimodular polytope.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3089 https://doi.org/10.46298/dmtcs.3089 Ardila, Federico Block, Florian Ardila, Federico Block, Florian <![CDATA[The Severi variety parametrizes plane curves of degree $d$ with $\delta$ nodes. Its degree is called the Severi degree. For large enough $d$, the Severi degrees coincide with the Gromov-Witten invariants of $\mathbb{CP}^2$. Fomin and Mikhalkin (2009) proved the 1995 conjecture that for fixed $\delta$, Severi degrees are eventually polynomial in $d$. In this paper, we study the Severi varieties corresponding to a large family of toric surfaces. We prove the analogous result that the Severi degrees are eventually polynomial as a function of the multidegree. More surprisingly, we show that the Severi degrees are also eventually polynomial "as a function of the surface". Our strategy is to use tropical geometry to express Severi degrees in terms of Brugallé and Mikhalkin's floor diagrams, and study those combinatorial objects in detail. An important ingredient in the proof is the polynomiality of the discrete volume of a variable facet-unimodular polytope.]]> 0 <![CDATA[Let $W$ be an infinite Coxeter group, and $\Phi$ be the root system constructed from its geometric representation. We study the set $E$ of limit points of "normalized'' roots (representing the directions of the roots). We show that $E$ is contained in the isotropic cone $Q$ of the bilinear form associated to $W$, and illustrate this property with numerous examples and pictures in rank $3$ and $4$. We also define a natural geometric action of $W$ on $E$, for which $E$ is stable. Then we exhibit a countable subset $E_2$ of $E$, formed by limit points for the dihedral reflection subgroups of $W$; we explain how $E_2$ can be built from the intersection with $Q$ of the lines passing through two roots, and we establish that $E_2$ is dense in $E$.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3088 https://doi.org/10.46298/dmtcs.3088 Hohlweg, Christophe Labbé, Jean-Philippe Ripoll, Vivien Hohlweg, Christophe Labbé, Jean-Philippe Ripoll, Vivien <![CDATA[Let $W$ be an infinite Coxeter group, and $\Phi$ be the root system constructed from its geometric representation. We study the set $E$ of limit points of "normalized'' roots (representing the directions of the roots). We show that $E$ is contained in the isotropic cone $Q$ of the bilinear form associated to $W$, and illustrate this property with numerous examples and pictures in rank $3$ and $4$. We also define a natural geometric action of $W$ on $E$, for which $E$ is stable. Then we exhibit a countable subset $E_2$ of $E$, formed by limit points for the dihedral reflection subgroups of $W$; we explain how $E_2$ can be built from the intersection with $Q$ of the lines passing through two roots, and we establish that $E_2$ is dense in $E$.]]> 0 <![CDATA[We present a new edge selection heuristic and vertex ordering heuristic that together enable one to compute the Tutte polynomial of much larger sparse graphs than was previously doable. As a specific example, we are able to compute the Tutte polynomial of the truncated icosahedron graph using our Maple implementation in under 4 minutes on a single CPU. This compares with a recent result of Haggard, Pearce and Royle whose special purpose C++ software took one week on 150 computers.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3087 https://doi.org/10.46298/dmtcs.3087 Monagan, Michael Monagan, Michael <![CDATA[We present a new edge selection heuristic and vertex ordering heuristic that together enable one to compute the Tutte polynomial of much larger sparse graphs than was previously doable. As a specific example, we are able to compute the Tutte polynomial of the truncated icosahedron graph using our Maple implementation in under 4 minutes on a single CPU. This compares with a recent result of Haggard, Pearce and Royle whose special purpose C++ software took one week on 150 computers.]]> 0 <![CDATA[We prove that on the set of lattice paths with steps $N=(0,1)$ and $E=(1,0)$ that lie between two boundaries $B$ and $T$, the two statistics `number of $E$ steps shared with $B$' and `number of $E$ steps shared with $T$' have a symmetric joint distribution. We give an involution that switches these statistics, preserves additional parameters, and generalizes to paths that contain steps $S=(0,-1)$ at prescribed $x$-coordinates. We also show that a similar equidistribution result for other path statistics follows from the fact that the Tutte polynomial of a matroid is independent of the order of its ground set. Finally, we extend the two theorems to $k$-tuples of paths between two boundaries, and we give some applications to Dyck paths, generalizing a result of Deutsch, and to pattern-avoiding permutations.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3086 https://doi.org/10.46298/dmtcs.3086 Elizalde, Sergi Rubey, Martin Elizalde, Sergi Rubey, Martin <![CDATA[We prove that on the set of lattice paths with steps $N=(0,1)$ and $E=(1,0)$ that lie between two boundaries $B$ and $T$, the two statistics `number of $E$ steps shared with $B$' and `number of $E$ steps shared with $T$' have a symmetric joint distribution. We give an involution that switches these statistics, preserves additional parameters, and generalizes to paths that contain steps $S=(0,-1)$ at prescribed $x$-coordinates. We also show that a similar equidistribution result for other path statistics follows from the fact that the Tutte polynomial of a matroid is independent of the order of its ground set. Finally, we extend the two theorems to $k$-tuples of paths between two boundaries, and we give some applications to Dyck paths, generalizing a result of Deutsch, and to pattern-avoiding permutations.]]> 0 <![CDATA[Given a graph $G$, the number of nowhere-zero $\mathbb{Z}_q$-flows $\phi _G(q)$ is known to be a polynomial in $q$. We extend the definition of nowhere-zero $\mathbb{Z} _q$-flows to simplicial complexes $\Delta$ of dimension greater than one, and prove the polynomiality of the corresponding function $\phi_{\Delta}(q)$ for certain $q$ and certain subclasses of simplicial complexes.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3085 https://doi.org/10.46298/dmtcs.3085 Beck, Matthias Kemper, Yvonne Beck, Matthias Kemper, Yvonne <![CDATA[Given a graph $G$, the number of nowhere-zero $\mathbb{Z}_q$-flows $\phi _G(q)$ is known to be a polynomial in $q$. We extend the definition of nowhere-zero $\mathbb{Z} _q$-flows to simplicial complexes $\Delta$ of dimension greater than one, and prove the polynomiality of the corresponding function $\phi_{\Delta}(q)$ for certain $q$ and certain subclasses of simplicial complexes.]]> 0 <![CDATA[We consider the problem of enumerating planar constellations with two points at a prescribed distance. Our approach relies on a combinatorial correspondence between this family of constellations and the simpler family of rooted constellations, which we may formulate algebraically in terms of multicontinued fractions and generalized Hankel determinants. As an application, we provide a combinatorial derivation of the generating function of Eulerian triangulations with two points at a prescribed distance.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3084 https://doi.org/10.46298/dmtcs.3084 Albenque, Marie Bouttier, Jérémie Albenque, Marie Bouttier, Jérémie <![CDATA[We consider the problem of enumerating planar constellations with two points at a prescribed distance. Our approach relies on a combinatorial correspondence between this family of constellations and the simpler family of rooted constellations, which we may formulate algebraically in terms of multicontinued fractions and generalized Hankel determinants. As an application, we provide a combinatorial derivation of the generating function of Eulerian triangulations with two points at a prescribed distance.]]> 0 <![CDATA[Cylindric plane partitions may be thought of as a natural generalization of reverse plane partitions. A generating series for the enumeration of cylindric plane partitions was recently given by Borodin. As in the reverse plane partition case, the right hand side of this identity admits a simple factorization form in terms of the "hook lengths'' of the individual boxes in the underlying shape. The first result of this paper is a new bijective proof of Borodin's identity which makes use of Fomin's growth diagram framework for generalized RSK correspondences. The second result of this paper is a $(q,t)$-analog of Borodin's identity which extends previous work by Okada in the reverse plane partition case. The third result of this paper is an explicit combinatorial interpretation of the Macdonald weight occurring in the $(q,t)$-analog in terms of the non-intersecting lattice path model for cylindric plane partitions.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3083 https://doi.org/10.46298/dmtcs.3083 Langer, Robin Langer, Robin <![CDATA[Cylindric plane partitions may be thought of as a natural generalization of reverse plane partitions. A generating series for the enumeration of cylindric plane partitions was recently given by Borodin. As in the reverse plane partition case, the right hand side of this identity admits a simple factorization form in terms of the "hook lengths'' of the individual boxes in the underlying shape. The first result of this paper is a new bijective proof of Borodin's identity which makes use of Fomin's growth diagram framework for generalized RSK correspondences. The second result of this paper is a $(q,t)$-analog of Borodin's identity which extends previous work by Okada in the reverse plane partition case. The third result of this paper is an explicit combinatorial interpretation of the Macdonald weight occurring in the $(q,t)$-analog in terms of the non-intersecting lattice path model for cylindric plane partitions.]]> 0 <![CDATA[This article presents a methodology that automatically derives a combinatorial specification for the permutation class $\mathcal{C} = Av(B)$, given its basis $B$ of excluded patterns and the set of simple permutations in $\mathcal{C}$, when these sets are both finite. This is achieved considering both pattern avoidance and pattern containment constraints in permutations.The obtained specification yields a system of equations satisfied by the generating function of $\mathcal{C}$, this system being always positive and algebraic. It also yields a uniform random sampler of permutations in $\mathcal{C}$. The method presented is fully algorithmic.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3082 https://doi.org/10.46298/dmtcs.3082 Bassino, Frédérique Bouvel, Mathilde Pierrot, Adeline Pivoteau, Carine Rossin, Dominique Bassino, Frédérique Bouvel, Mathilde Pierrot, Adeline Pivoteau, Carine Rossin, Dominique <![CDATA[This article presents a methodology that automatically derives a combinatorial specification for the permutation class $\mathcal{C} = Av(B)$, given its basis $B$ of excluded patterns and the set of simple permutations in $\mathcal{C}$, when these sets are both finite. This is achieved considering both pattern avoidance and pattern containment constraints in permutations.The obtained specification yields a system of equations satisfied by the generating function of $\mathcal{C}$, this system being always positive and algebraic. It also yields a uniform random sampler of permutations in $\mathcal{C}$. The method presented is fully algorithmic.]]> 0 <![CDATA[Dyck tilings were introduced by Kenyon and Wilson in their study of double-dimer pairings. They are certain kinds of tilings of skew Young diagrams with ribbon tiles shaped like Dyck paths. We give two bijections between "cover-inclusive'' Dyck tilings and linear extensions of tree posets. The first bijection maps the statistic (area + tiles)/2 to inversions of the linear extension, and the second bijection maps the "discrepancy'' between the upper and lower boundary of the tiling to descents of the linear extension.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3081 https://doi.org/10.46298/dmtcs.3081 Kim, Jang Soo Mészáros, Karola Panova, Greta Wilson, David B. Kim, Jang Soo Mészáros, Karola Panova, Greta Wilson, David B. <![CDATA[Dyck tilings were introduced by Kenyon and Wilson in their study of double-dimer pairings. They are certain kinds of tilings of skew Young diagrams with ribbon tiles shaped like Dyck paths. We give two bijections between "cover-inclusive'' Dyck tilings and linear extensions of tree posets. The first bijection maps the statistic (area + tiles)/2 to inversions of the linear extension, and the second bijection maps the "discrepancy'' between the upper and lower boundary of the tiling to descents of the linear extension.]]> 0 <![CDATA[We examine the sets of permutations that are sorted by two passes through a stack with a $D_8$ operation performed in between. From a characterization of these in terms of generalized excluded patterns, we prove two conjectures on their enumeration, that can be refined with the distribution of some statistics. The results are obtained by generating trees.]]> Sat, 31 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3080 https://doi.org/10.46298/dmtcs.3080 Bouvel, Mathilde Guibert, Olivier Bouvel, Mathilde Guibert, Olivier <![CDATA[We examine the sets of permutations that are sorted by two passes through a stack with a $D_8$ operation performed in between. From a characterization of these in terms of generalized excluded patterns, we prove two conjectures on their enumeration, that can be refined with the distribution of some statistics. The results are obtained by generating trees.]]> 0 <![CDATA[A proper vertex coloring of a graphGis called a star-coloring if there is no path on four vertices assigned to two colors. The graph G is L-star-colorable if for a given list assignment L there is a star-coloring c such that c(v) epsilon L(v). If G is L-star-colorable for any list assignment L with vertical bar L(v)vertical bar \textgreater= k for all v epsilon V(G), then G is called k-star-choosable. The star list chromatic number of G, denoted by X-s(l)(G), is the smallest integer k such that G is k-star-choosable. In this article, we prove that every graph G with maximum average degree less than 3 is 8-star-choosable. This extends a result that planar graphs of girth at least 6 are 8-star-choosable [A. Kundgen, C. Timmons, Star coloring planar graphs from small lists, J. Graph Theory, 63(4): 324-337, 2010].]]> Fri, 30 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.561 https://doi.org/10.46298/dmtcs.561 Chen, Min Raspaud, André Wang, Weifan Chen, Min Raspaud, André Wang, Weifan <![CDATA[A proper vertex coloring of a graphGis called a star-coloring if there is no path on four vertices assigned to two colors. The graph G is L-star-colorable if for a given list assignment L there is a star-coloring c such that c(v) epsilon L(v). If G is L-star-colorable for any list assignment L with vertical bar L(v)vertical bar \textgreater= k for all v epsilon V(G), then G is called k-star-choosable. The star list chromatic number of G, denoted by X-s(l)(G), is the smallest integer k such that G is k-star-choosable. In this article, we prove that every graph G with maximum average degree less than 3 is 8-star-choosable. This extends a result that planar graphs of girth at least 6 are 8-star-choosable [A. Kundgen, C. Timmons, Star coloring planar graphs from small lists, J. Graph Theory, 63(4): 324-337, 2010].]]> 0 <![CDATA[A Nordhaus-Gaddum-type result is a (tight) lower or upper bound on the sum or product of a parameter of a graph and its complement. In this paper we study Nordhaus-Gaddum-type results for total domination. We examine the sum and product of γt(G1) and γt(G2) where G1 ⊕G2 = K(s,s), and γt is the total domination number. We show that the maximum value of the sum of the total domination numbers of G1 and G2 is 2s+4, with equality if and only if G1 = sK2 or G2 = sK2, while the maximum value of the product of the total domination numbers of G1 and G2 is max{8s,⌊(s+6)2/4 ⌋}.]]> Thu, 29 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.530 https://doi.org/10.46298/dmtcs.530 Henning, Michael, Joubert, Ernst Southey, Justin Henning, Michael, Joubert, Ernst Southey, Justin <![CDATA[A Nordhaus-Gaddum-type result is a (tight) lower or upper bound on the sum or product of a parameter of a graph and its complement. In this paper we study Nordhaus-Gaddum-type results for total domination. We examine the sum and product of γt(G1) and γt(G2) where G1 ⊕G2 = K(s,s), and γt is the total domination number. We show that the maximum value of the sum of the total domination numbers of G1 and G2 is 2s+4, with equality if and only if G1 = sK2 or G2 = sK2, while the maximum value of the product of the total domination numbers of G1 and G2 is max{8s,⌊(s+6)2/4 ⌋}.]]> 0 <![CDATA[Recently, Deutsch and Elizalde studied the largest fixed points of permutations. Motivated by their work, we consider the analogous problems in weighted set partitions. Let A (n,k) (t) denote the total weight of partitions on [n + 1] = \1,2,..., n + 1\ with the largest singleton \k + 1\. In this paper, explicit formulas for A (n,k) (t) and many combinatorial identities involving A (n,k) (t) are obtained by umbral operators and combinatorial methods. In particular, the permutation case leads to an identity related to tree enumerations, namely, [GRAPHICS] where D-k is the number of permutations of [k] with no fixed points.]]> Mon, 26 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.535 https://doi.org/10.46298/dmtcs.535 Sun, Yidong Xu, Yanjie Sun, Yidong Xu, Yanjie <![CDATA[Recently, Deutsch and Elizalde studied the largest fixed points of permutations. Motivated by their work, we consider the analogous problems in weighted set partitions. Let A (n,k) (t) denote the total weight of partitions on [n + 1] = \1,2,..., n + 1\ with the largest singleton \k + 1\. In this paper, explicit formulas for A (n,k) (t) and many combinatorial identities involving A (n,k) (t) are obtained by umbral operators and combinatorial methods. In particular, the permutation case leads to an identity related to tree enumerations, namely, [GRAPHICS] where D-k is the number of permutations of [k] with no fixed points.]]> 0 <![CDATA[A 2-packing of a hypergraph H is a permutation sigma on V (H) such that if an edge e belongs to epsilon(H), then sigma(e) does not belong to epsilon(H). Let H be a hypergraph of order n which contains edges of cardinality at least 2 and at most n - 2. We prove that if H has at most n - 2 edges then it is 2-packable.]]> Thu, 08 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.537 https://doi.org/10.46298/dmtcs.537 Pilsniak, Monika WoźniaK, Mariusz Pilsniak, Monika WoźniaK, Mariusz <![CDATA[A 2-packing of a hypergraph H is a permutation sigma on V (H) such that if an edge e belongs to epsilon(H), then sigma(e) does not belong to epsilon(H). Let H be a hypergraph of order n which contains edges of cardinality at least 2 and at most n - 2. We prove that if H has at most n - 2 edges then it is 2-packable.]]> 0 <![CDATA[For an integer constant d \textgreater 0, let Gamma(d) denote the class of finite groups all of whose nonabelian composition factors lie in S-d; in particular, Gamma(d) includes all solvable groups. Motivated by applications to graph-isomorphism testing, there has been extensive study of the complexity of computation for permutation groups in this class. In particular, the problems of finding set stabilizers, intersections and centralizers have all been shown to be polynomial-time computable. A notable open issue for the class Gamma(d) has been the question of whether normalizers can be found in polynomial time. We resolve this question in the affirmative. We prove that, given permutation groups G, H \textless= Sym(Omega) such that G is an element of Gamma(d), the normalizer of H in G can be found in polynomial time. Among other new procedures, our method includes a key subroutine to solve the problem of finding stabilizers of subspaces in linear representations of permutation groups in Gamma(d).]]> Thu, 08 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.531 https://doi.org/10.46298/dmtcs.531 Luks, Eugene M. Miyazaki, Takunari Luks, Eugene M. Miyazaki, Takunari <![CDATA[For an integer constant d \textgreater 0, let Gamma(d) denote the class of finite groups all of whose nonabelian composition factors lie in S-d; in particular, Gamma(d) includes all solvable groups. Motivated by applications to graph-isomorphism testing, there has been extensive study of the complexity of computation for permutation groups in this class. In particular, the problems of finding set stabilizers, intersections and centralizers have all been shown to be polynomial-time computable. A notable open issue for the class Gamma(d) has been the question of whether normalizers can be found in polynomial time. We resolve this question in the affirmative. We prove that, given permutation groups G, H \textless= Sym(Omega) such that G is an element of Gamma(d), the normalizer of H in G can be found in polynomial time. Among other new procedures, our method includes a key subroutine to solve the problem of finding stabilizers of subspaces in linear representations of permutation groups in Gamma(d).]]> 0 <![CDATA[In this paper we construct a cyclically invariant Boolean function whose sensitivity is Theta(n(1/3)). This result answers two previously published questions. Turan (1984) asked if any Boolean function, invariant under some transitive group of permutations, has sensitivity Omega(root n). Kenyon and Kutin (2004) asked whether for a "nice" function the product of 0-sensitivity and 1-sensitivity is Omega(n). Our function answers both questions in the negative. We also prove that for minterm-transitive functions (a natural class of Boolean functions including our example) the sensitivity is Omega(n(1/3)). Hence for this class of functions sensitivity and block sensitivity are polynomially related.]]> Sat, 03 Dec 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.552 https://doi.org/10.46298/dmtcs.552 Chakraborty, Sourav Chakraborty, Sourav <![CDATA[In this paper we construct a cyclically invariant Boolean function whose sensitivity is Theta(n(1/3)). This result answers two previously published questions. Turan (1984) asked if any Boolean function, invariant under some transitive group of permutations, has sensitivity Omega(root n). Kenyon and Kutin (2004) asked whether for a "nice" function the product of 0-sensitivity and 1-sensitivity is Omega(n). Our function answers both questions in the negative. We also prove that for minterm-transitive functions (a natural class of Boolean functions including our example) the sensitivity is Omega(n(1/3)). Hence for this class of functions sensitivity and block sensitivity are polynomially related.]]> 0 <![CDATA[In this paper, we study the residual solvability of the generalized free product of solvable groups.]]> Thu, 17 Nov 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.540 https://doi.org/10.46298/dmtcs.540 Kahrobaei, Delaram Majewicz, Stephen Kahrobaei, Delaram Majewicz, Stephen <![CDATA[In this paper, we study the residual solvability of the generalized free product of solvable groups.]]> 0 <![CDATA[We prove that the extended equivalence problem is solvable in polynomial time for finite nilpotent groups, and coNP-complete, otherwise. We prove that the extended equation solvability problem is solvable in polynomial time for finite nilpotent groups, and NP-complete, otherwise.]]> Tue, 15 Nov 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.536 https://doi.org/10.46298/dmtcs.536 Horvath, Gabor Szabo, Csaba Horvath, Gabor Szabo, Csaba <![CDATA[We prove that the extended equivalence problem is solvable in polynomial time for finite nilpotent groups, and coNP-complete, otherwise. We prove that the extended equation solvability problem is solvable in polynomial time for finite nilpotent groups, and NP-complete, otherwise.]]> 0 <![CDATA[For a polynomial f(x) is an element of Z(2)[x] it is natural to consider the near-ring code generated by the polynomials f circle x, f circle x(2) ,..., f circle x(k) as a vectorspace. It is a 19 year old conjecture of Gunter Pilz that for the polynomial f (x) - x(n) broken vertical bar x(n-1) broken vertical bar ... broken vertical bar x the minimal distance of this code is n. The conjecture is equivalent to the following purely number theoretical problem. Let (m) under bar = \1, 2 ,..., m\ and A subset of N be an arbitrary finite subset of N. Show that the number of products that occur odd many times in (n) under bar. A is at least n. Pilz also formulated the conjecture for the special case when A = (k) under bar. We show that for A = (k) under bar the conjecture holds and that the minimal distance of the code is at least n/(log n)(0.223). While proving the case A = (k) under bar we use different number theoretical methods depending on the size of k (respect to n). Furthermore, we apply several estimates on the distribution of primes.]]> Tue, 15 Nov 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.556 https://doi.org/10.46298/dmtcs.556 Pach, Peter Pal Szabo, Csaba Pach, Peter Pal Szabo, Csaba <![CDATA[For a polynomial f(x) is an element of Z(2)[x] it is natural to consider the near-ring code generated by the polynomials f circle x, f circle x(2) ,..., f circle x(k) as a vectorspace. It is a 19 year old conjecture of Gunter Pilz that for the polynomial f (x) - x(n) broken vertical bar x(n-1) broken vertical bar ... broken vertical bar x the minimal distance of this code is n. The conjecture is equivalent to the following purely number theoretical problem. Let (m) under bar = \1, 2 ,..., m\ and A subset of N be an arbitrary finite subset of N. Show that the number of products that occur odd many times in (n) under bar. A is at least n. Pilz also formulated the conjecture for the special case when A = (k) under bar. We show that for A = (k) under bar the conjecture holds and that the minimal distance of the code is at least n/(log n)(0.223). While proving the case A = (k) under bar we use different number theoretical methods depending on the size of k (respect to n). Furthermore, we apply several estimates on the distribution of primes.]]> 0 <![CDATA[For every positive integer k, we construct an explicit family of functions f : \0, 1\(n) -\textgreater \0, 1\ which has (k + 1) - party communication complexity O(k) under every partition of the input bits into k + 1 parts of equal size, and k-party communication complexity Omega (n/k(4)2(k)) under every partition of the input bits into k parts. This improves an earlier hierarchy theorem due to V. Grolmusz. Our construction relies on known explicit constructions for a famous open problem of K. Zarankiewicz, namely, to find the maximum number of edges in a graph on n vertices that does not contain K-s,K-t as a subgraph.]]> Sat, 12 Nov 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.546 https://doi.org/10.46298/dmtcs.546 Hayes, Thomas P. Hayes, Thomas P. <![CDATA[For every positive integer k, we construct an explicit family of functions f : \0, 1\(n) -\textgreater \0, 1\ which has (k + 1) - party communication complexity O(k) under every partition of the input bits into k + 1 parts of equal size, and k-party communication complexity Omega (n/k(4)2(k)) under every partition of the input bits into k parts. This improves an earlier hierarchy theorem due to V. Grolmusz. Our construction relies on known explicit constructions for a famous open problem of K. Zarankiewicz, namely, to find the maximum number of edges in a graph on n vertices that does not contain K-s,K-t as a subgraph.]]> 0 <![CDATA[We describe an algorithm to compute tensor decompositions of central products of groups. The novelty over previous algorithms is that in the case of matrix groups that are both tensor decomposable and imprimitive, the new algorithm more often outputs the more desirable tensor decomposition.]]> Wed, 09 Nov 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.562 https://doi.org/10.46298/dmtcs.562 Ankaralioglu, Nurullah Seress, Akos Ankaralioglu, Nurullah Seress, Akos <![CDATA[We describe an algorithm to compute tensor decompositions of central products of groups. The novelty over previous algorithms is that in the case of matrix groups that are both tensor decomposable and imprimitive, the new algorithm more often outputs the more desirable tensor decomposition.]]> 0 <![CDATA[We consider exchange of three intervals with permutation (3, 2, 1). The aim of this paper is to count the cardinality of the set 3iet (N) of all words of length N which appear as factors in infinite words coding such transformations. We use the strong relation of 3iet words and words coding exchange of two intervals, i.e., Sturmian words. The known asymptotic formula #2iet(N)/N-3 similar to 1/pi(2) for the number of Sturmian factors allows us to find bounds 1/3 pi(2) +o(1) \textless= #3iet(N)N-4 \textless= 2 pi(2) + o(1)]]> Tue, 08 Nov 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.553 https://doi.org/10.46298/dmtcs.553 Ambrož, Petr Frid, Anna Masáková, Zuzana Pelantová, Edita Ambrož, Petr Frid, Anna Masáková, Zuzana Pelantová, Edita <![CDATA[We consider exchange of three intervals with permutation (3, 2, 1). The aim of this paper is to count the cardinality of the set 3iet (N) of all words of length N which appear as factors in infinite words coding such transformations. We use the strong relation of 3iet words and words coding exchange of two intervals, i.e., Sturmian words. The known asymptotic formula #2iet(N)/N-3 similar to 1/pi(2) for the number of Sturmian factors allows us to find bounds 1/3 pi(2) +o(1) \textless= #3iet(N)N-4 \textless= 2 pi(2) + o(1)]]> 0 <![CDATA[Given a graph G = (V; E) and a weight function omega : E -\textgreater R, a coloring of vertices of G, induced by omega, is defined by chi(omega) (nu) = Sigma(e(sic)nu) omega (e) for all nu is an element of V. In this paper, we show that determining whether a particular graph has a weighting of the edges from \1, 2\ that induces a proper vertex coloring is NP-complete.]]> Sat, 05 Nov 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.548 https://doi.org/10.46298/dmtcs.548 Dudek, Andrzej Wajc, David Dudek, Andrzej Wajc, David <![CDATA[Given a graph G = (V; E) and a weight function omega : E -\textgreater R, a coloring of vertices of G, induced by omega, is defined by chi(omega) (nu) = Sigma(e(sic)nu) omega (e) for all nu is an element of V. In this paper, we show that determining whether a particular graph has a weighting of the edges from \1, 2\ that induces a proper vertex coloring is NP-complete.]]> 0 <![CDATA[Every k-tree has book thickness at most k + 1, and this bound is best possible for all k \textgreater= 3. Vandenbussche et al. [SIAM J. Discrete Math., 2009] proved that every k-tree that has a smooth degree-3 tree decomposition with width k has book thickness at most k. We prove this result is best possible for k \textgreater= 4, by constructing a k-tree with book thickness k + 1 that has a smooth degree-4 tree decomposition with width k. This solves an open problem of Vandenbussche et al.]]> Sun, 16 Oct 2011 22:00:00 +0000 https://doi.org/10.46298/dmtcs.550 https://doi.org/10.46298/dmtcs.550 Dujmović, Vida Wood, David R. Dujmović, Vida Wood, David R. <![CDATA[Every k-tree has book thickness at most k + 1, and this bound is best possible for all k \textgreater= 3. Vandenbussche et al. [SIAM J. Discrete Math., 2009] proved that every k-tree that has a smooth degree-3 tree decomposition with width k has book thickness at most k. We prove this result is best possible for k \textgreater= 4, by constructing a k-tree with book thickness k + 1 that has a smooth degree-4 tree decomposition with width k. This solves an open problem of Vandenbussche et al.]]> 0 <![CDATA[What today we call digital search tree (DST) is Coffman and Eve's sequence tree introduced in 1970. A digital search tree is a binary tree whose ordering of nodes is based on the values of bits in the binary representation of a node's key. In fact, a digital search tree is a digital tree in which strings (keys, words) are stored directly in internal nodes. The profile of a digital search tree is a parameter that counts the number of nodes at the same distance from the root. In this paper we concentrate on external profile, i.e., the number of external nodes at level k when n strings are sorted. By assuming that the n input strings are independent and follow a (binary) memoryless source the asymptotic behaviour of the average profile was determined by Drmota and Szpankowski (2011). The purpose of the present paper is to extend their analysis and to provide a precise analysis of variance of the profile. The main (technical) difference is that we have to deal with an inhomogeneous part in a proper functional-differential equations satisfied by the second moment and Poisson variance. However, we show that the variance is asymptotically of the same order as the expected value which implies concentration. These results are derived by methods of analytic combinatorics such as generating functions, Mellin transform, Poissonization, the saddle point method and singularity analysis.]]> Thu, 15 Sep 2011 22:00:00 +0000 https://doi.org/10.46298/dmtcs.555 https://doi.org/10.46298/dmtcs.555 Kazemi, Ramin Vahidi-Asl, Mohammad Q. Kazemi, Ramin Vahidi-Asl, Mohammad Q. <![CDATA[What today we call digital search tree (DST) is Coffman and Eve's sequence tree introduced in 1970. A digital search tree is a binary tree whose ordering of nodes is based on the values of bits in the binary representation of a node's key. In fact, a digital search tree is a digital tree in which strings (keys, words) are stored directly in internal nodes. The profile of a digital search tree is a parameter that counts the number of nodes at the same distance from the root. In this paper we concentrate on external profile, i.e., the number of external nodes at level k when n strings are sorted. By assuming that the n input strings are independent and follow a (binary) memoryless source the asymptotic behaviour of the average profile was determined by Drmota and Szpankowski (2011). The purpose of the present paper is to extend their analysis and to provide a precise analysis of variance of the profile. The main (technical) difference is that we have to deal with an inhomogeneous part in a proper functional-differential equations satisfied by the second moment and Poisson variance. However, we show that the variance is asymptotically of the same order as the expected value which implies concentration. These results are derived by methods of analytic combinatorics such as generating functions, Mellin transform, Poissonization, the saddle point method and singularity analysis.]]> 0 <![CDATA[Miller and Muller (1960) and independently Moon and Moser (1965) determined the maximum number of maximal independent sets in an n-vertex graph. We give a new and simple proof of this result.]]> Fri, 09 Sep 2011 22:00:00 +0000 https://doi.org/10.46298/dmtcs.543 https://doi.org/10.46298/dmtcs.543 Wood, David R. Wood, David R. <![CDATA[Miller and Muller (1960) and independently Moon and Moser (1965) determined the maximum number of maximal independent sets in an n-vertex graph. We give a new and simple proof of this result.]]> 0 <![CDATA[A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, it is shown that each 1-planar graph of minimum degree 6 contains a copy of 4-cycle with all vertices of degree at most 19. In addition, we also show that the complete graph K 4 is light in the family of 1-planar graphs of minimum degree 7, with its height at most 11.]]> Fri, 09 Sep 2011 22:00:00 +0000 https://doi.org/10.46298/dmtcs.560 https://doi.org/10.46298/dmtcs.560 Zhang, Xin Wu, Jian-Liang Liu, Guizhen Zhang, Xin Wu, Jian-Liang Liu, Guizhen <![CDATA[A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, it is shown that each 1-planar graph of minimum degree 6 contains a copy of 4-cycle with all vertices of degree at most 19. In addition, we also show that the complete graph K 4 is light in the family of 1-planar graphs of minimum degree 7, with its height at most 11.]]> 0 <![CDATA[We adapt a novel idea of Cichon's related to Approximate Counting to the present instance of Digital Search Trees, by using m (instead of one) such trees. We investigate the level polynomials, which have as coefficients the expected numbers of data on a given level, and the insertion costs. The level polynomials can be precisely described, thanks to formulae from q-analysis. The asymptotics of expectation and variance of the insertion cost are fairly standard these days and done with Rice's method.]]> Mon, 29 Aug 2011 22:00:00 +0000 https://doi.org/10.46298/dmtcs.557 https://doi.org/10.46298/dmtcs.557 Prodinger, Helmut Prodinger, Helmut <![CDATA[We adapt a novel idea of Cichon's related to Approximate Counting to the present instance of Digital Search Trees, by using m (instead of one) such trees. We investigate the level polynomials, which have as coefficients the expected numbers of data on a given level, and the insertion costs. The level polynomials can be precisely described, thanks to formulae from q-analysis. The asymptotics of expectation and variance of the insertion cost are fairly standard these days and done with Rice's method.]]> 0 <![CDATA[We prove upper and lower bounds on the chromatic number of the square of the cartesian product of trees. The bounds are equal if each tree has even maximum degree.]]> Wed, 17 Aug 2011 22:00:00 +0000 https://doi.org/10.46298/dmtcs.549 https://doi.org/10.46298/dmtcs.549 Wood, David R. Wood, David R. <![CDATA[We prove upper and lower bounds on the chromatic number of the square of the cartesian product of trees. The bounds are equal if each tree has even maximum degree.]]> 0 <![CDATA[The Tanny sequence T (i) is a sequence defined recursively as T(i) = T(i - 1 - T(i - 1)) + T(i - 2 - T(i - 2)), T(0) = T(1) = T(2) = 1. In the first part of this paper we give combinatorial proofs of all the results regarding T(i), that Tanny proved in his paper "A well-behaved cousin of the Hofstadter sequence", Discrete Mathematics, 105(1992), pp. 227-239, using algebraic means. In most cases our proofs turn out to be simpler and shorter. Moreover, they give a "visual" appeal to the theory developed by Tanny. We also generalize most of Tanny's results. In the second part of the paper we present many new results regarding T(i) and prove them combinatorially. Given two integers n and k, it is interesting to know if T(n) = k or not. In this paper we characterize such numbers.]]> Fri, 15 Jul 2011 22:00:00 +0000 https://doi.org/10.46298/dmtcs.547 https://doi.org/10.46298/dmtcs.547 Das, Anita Das, Anita <![CDATA[The Tanny sequence T (i) is a sequence defined recursively as T(i) = T(i - 1 - T(i - 1)) + T(i - 2 - T(i - 2)), T(0) = T(1) = T(2) = 1. In the first part of this paper we give combinatorial proofs of all the results regarding T(i), that Tanny proved in his paper "A well-behaved cousin of the Hofstadter sequence", Discrete Mathematics, 105(1992), pp. 227-239, using algebraic means. In most cases our proofs turn out to be simpler and shorter. Moreover, they give a "visual" appeal to the theory developed by Tanny. We also generalize most of Tanny's results. In the second part of the paper we present many new results regarding T(i) and prove them combinatorially. Given two integers n and k, it is interesting to know if T(n) = k or not. In this paper we characterize such numbers.]]> 0 <![CDATA[The irredundant Ramsey number s - s(m, n) [upper domination Ramsey number u - u(m, n), respectively] is the smallest natural number s [u, respectively] such that in any red-blue edge colouring (R, B) of the complete graph of order s [u, respectively], it holds that IR(B) \textgreater= m or IR(R) \textgreater= n [Gamma (B) \textgreater= m or Gamma(R) \textgreater= n, respectively], where Gamma and IR denote respectively the upper domination number and the irredundance number of a graph. Furthermore, the mixed irredundant Ramsey number t = t(m, n) [mixed domination Ramsey number v = v(m, n), respectively] is the smallest natural number t [v, respectively] such that in any red-blue edge colouring (R, B) of the complete graph of order t [v, respectively], it holds that IR(B) \textgreater= m or beta(R) \textgreater= n [Gamma(B) \textgreater= m or beta(R) \textgreater= n, respectively], where beta denotes the independent domination number of a graph. These four classes of non-classical Ramsey numbers have previously been studied in the literature. In this paper we introduce a new Ramsey number w = w(m, n), called the irredundant-domination Ramsey number, which is the smallest natural number w such that in any red-blue edge colouring (R, B) of the complete graph of order w, it holds that IR(B) \textgreater= m or Gamma(R) \textgreater= n. A computer search is employed to determine complete sets of avoidance colourings of small order for these five classes of nonclassical Ramsey numbers. In the process the fifteen previously unknown Ramsey numbers t(4, 4) = 14, t(6, 3) = 17, u(4, 4) = 13, v(4, 3) = 8, v(4, 4) = 14, v(5, 3) = 13, v(6, 3) = 17, w(3, 3) = 6, w(3, 4) = w(4, 3) = 8, w(4, 4) = 13, w(3, 5) = w(5, 3) = 12 and w(3, 6) = w(6, 3) = 15 are established.]]> Wed, 13 Jul 2011 22:00:00 +0000 https://doi.org/10.46298/dmtcs.559 https://doi.org/10.46298/dmtcs.559 Burger, Alewyn Petrus Vuuren, Jan H., Burger, Alewyn Petrus Vuuren, Jan H., <![CDATA[The irredundant Ramsey number s - s(m, n) [upper domination Ramsey number u - u(m, n), respectively] is the smallest natural number s [u, respectively] such that in any red-blue edge colouring (R, B) of the complete graph of order s [u, respectively], it holds that IR(B) \textgreater= m or IR(R) \textgreater= n [Gamma (B) \textgreater= m or Gamma(R) \textgreater= n, respectively], where Gamma and IR denote respectively the upper domination number and the irredundance number of a graph. Furthermore, the mixed irredundant Ramsey number t = t(m, n) [mixed domination Ramsey number v = v(m, n), respectively] is the smallest natural number t [v, respectively] such that in any red-blue edge colouring (R, B) of the complete graph of order t [v, respectively], it holds that IR(B) \textgreater= m or beta(R) \textgreater= n [Gamma(B) \textgreater= m or beta(R) \textgreater= n, respectively], where beta denotes the independent domination number of a graph. These four classes of non-classical Ramsey numbers have previously been studied in the literature. In this paper we introduce a new Ramsey number w = w(m, n), called the irredundant-domination Ramsey number, which is the smallest natural number w such that in any red-blue edge colouring (R, B) of the complete graph of order w, it holds that IR(B) \textgreater= m or Gamma(R) \textgreater= n. A computer search is employed to determine complete sets of avoidance colourings of small order for these five classes of nonclassical Ramsey numbers. In the process the fifteen previously unknown Ramsey numbers t(4, 4) = 14, t(6, 3) = 17, u(4, 4) = 13, v(4, 3) = 8, v(4, 4) = 14, v(5, 3) = 13, v(6, 3) = 17, w(3, 3) = 6, w(3, 4) = w(4, 3) = 8, w(4, 4) = 13, w(3, 5) = w(5, 3) = 12 and w(3, 6) = w(6, 3) = 15 are established.]]> 0 <![CDATA[We describe a new algorithm for the efficient generation of all non-isomorphic connected cubic graphs. Our implementation of this algorithm is more than 4 times faster than previous generators. The generation can also be efficiently restricted to cubic graphs with girth at least 4 or 5.]]> Fri, 01 Jul 2011 22:00:00 +0000 https://doi.org/10.46298/dmtcs.551 https://doi.org/10.46298/dmtcs.551 Brinkmann, Gunnar Goedgebeur, Jan Mckay, Brendan D. Brinkmann, Gunnar Goedgebeur, Jan Mckay, Brendan D. <![CDATA[We describe a new algorithm for the efficient generation of all non-isomorphic connected cubic graphs. Our implementation of this algorithm is more than 4 times faster than previous generators. The generation can also be efficiently restricted to cubic graphs with girth at least 4 or 5.]]> 0 <![CDATA[It is known that a Sturmian sequence S can be defined as a coding of the orbit of rho (called the intercept of S) under a rotation of irrational angle alpha (called the slope). On the other hand, a fixed point of an invertible substitution is Sturmian. Naturally, there are two interrelated questions: (1) Given an invertible substitution, we know that its fixed point is Sturmian. What is the slope and intercept? (2) Which kind of Sturmian sequences can be fixed by certain non-trivial invertible substitutions? In this paper we give a unified treatment to the two questions. We remark that though the results are known, our proof is very elementary and concise.]]> Thu, 30 Jun 2011 22:00:00 +0000 https://doi.org/10.46298/dmtcs.554 https://doi.org/10.46298/dmtcs.554 Peng, Li Tan, Bo Peng, Li Tan, Bo <![CDATA[It is known that a Sturmian sequence S can be defined as a coding of the orbit of rho (called the intercept of S) under a rotation of irrational angle alpha (called the slope). On the other hand, a fixed point of an invertible substitution is Sturmian. Naturally, there are two interrelated questions: (1) Given an invertible substitution, we know that its fixed point is Sturmian. What is the slope and intercept? (2) Which kind of Sturmian sequences can be fixed by certain non-trivial invertible substitutions? In this paper we give a unified treatment to the two questions. We remark that though the results are known, our proof is very elementary and concise.]]> 0 <![CDATA[We describe a limiting distribution for the number of connected components in the subgraph of the discrete cube induced by the satisfying assignments to a random 2-SAT formula. We show that, for the probability range where formulas are likely to be satisfied, the random number of components converges weakly (in the number of variables) to a distribution determined by a Poisson random variable. The number of satisfying assignments or solutions is known to grow exponentially in the number of variables. Thus, our result implies that exponentially many solutions are organized into a stochastically bounded number of components. We also describe an application to biological evolution; in particular, to a type of fitness landscape where satisfying assignments represent viable genotypes and connectivity of genotypes is limited by single site mutations. The biological result is that, with probability approaching 1, each viable genotype is connected by single site mutations to an exponential number of other viable genotypes while the number of viable clusters is finite.]]> Sun, 12 Jun 2011 22:00:00 +0000 https://doi.org/10.46298/dmtcs.534 https://doi.org/10.46298/dmtcs.534 Pitman, Damien Pitman, Damien <![CDATA[We describe a limiting distribution for the number of connected components in the subgraph of the discrete cube induced by the satisfying assignments to a random 2-SAT formula. We show that, for the probability range where formulas are likely to be satisfied, the random number of components converges weakly (in the number of variables) to a distribution determined by a Poisson random variable. The number of satisfying assignments or solutions is known to grow exponentially in the number of variables. Thus, our result implies that exponentially many solutions are organized into a stochastically bounded number of components. We also describe an application to biological evolution; in particular, to a type of fitness landscape where satisfying assignments represent viable genotypes and connectivity of genotypes is limited by single site mutations. The biological result is that, with probability approaching 1, each viable genotype is connected by single site mutations to an exponential number of other viable genotypes while the number of viable clusters is finite.]]> 0 <![CDATA[Seidel's switching is a graph operation which makes a given vertex adjacent to precisely those vertices to which it was non-adjacent before, while keeping the rest of the graph unchanged. Two graphs are called switching-equivalent if one can be made isomorphic to the other by a sequence of switches. In this paper, we continue the study of computational complexity aspects of Seidel's switching, concentrating on Fixed Parameter Complexity. Among other results we show that switching to a graph with at most k edges, to a graph of maximum degree at most k, to a k-regular graph, or to a graph with minimum degree at least k are fixed parameter tractable problems, where k is the parameter. On the other hand, switching to a graph that contains a given fixed graph as an induced subgraph is W [1]-complete. We also show the NP-completeness of switching to a graph with a clique of linear size, and of switching to a graph with small number of edges. A consequence of the latter result is the NP-completeness of Maximum Likelihood Decoding of graph theoretic codes based on complete graphs.]]> Mon, 09 May 2011 22:00:00 +0000 https://doi.org/10.46298/dmtcs.542 https://doi.org/10.46298/dmtcs.542 Jelinkova, Eva Suchy, Ondrej Hlineny, Petr Kratochvil, Jan Jelinkova, Eva Suchy, Ondrej Hlineny, Petr Kratochvil, Jan <![CDATA[Seidel's switching is a graph operation which makes a given vertex adjacent to precisely those vertices to which it was non-adjacent before, while keeping the rest of the graph unchanged. Two graphs are called switching-equivalent if one can be made isomorphic to the other by a sequence of switches. In this paper, we continue the study of computational complexity aspects of Seidel's switching, concentrating on Fixed Parameter Complexity. Among other results we show that switching to a graph with at most k edges, to a graph of maximum degree at most k, to a k-regular graph, or to a graph with minimum degree at least k are fixed parameter tractable problems, where k is the parameter. On the other hand, switching to a graph that contains a given fixed graph as an induced subgraph is W [1]-complete. We also show the NP-completeness of switching to a graph with a clique of linear size, and of switching to a graph with small number of edges. A consequence of the latter result is the NP-completeness of Maximum Likelihood Decoding of graph theoretic codes based on complete graphs.]]> 0 <![CDATA[We present an algorithm that for 2-colorable 3-uniform hypergraphs, finds a 2-coloring in average running time O (n(5) log(2) n).]]> Wed, 04 May 2011 22:00:00 +0000 https://doi.org/10.46298/dmtcs.558 https://doi.org/10.46298/dmtcs.558 Person, Yury Schacht, Mathias Person, Yury Schacht, Mathias <![CDATA[We present an algorithm that for 2-colorable 3-uniform hypergraphs, finds a 2-coloring in average running time O (n(5) log(2) n).]]> 0 <![CDATA[Monitoring physical phenomena in Sensor Networks requires guaranteeing permanent communication between nodes. Moreover, in an effective implementation of such infrastructure, the delay between any two consecutive communications should be minimized. The problem is challenging because, in a restricted Sensor Network, the communication is carried out through a single and shared radio channel without collision detection. Dealing with collisions is crucial to ensure effective communication between nodes. Additionally, minimizing them yields energy consumption minimization, given that sensing and computational costs in terms of energy are negligible with respect to radio communication. In this work, we present a deterministic recurrent-communication protocol for Sensor Networks. After an initial negotiation phase of the access pattern to the channel, each node running this protocol reaches a steady state, which is asymptotically optimal in terms of energy and time efficiency. As a by-product, a protocol for the synchronization of a Sensor Network is also proposed. Furthermore, the protocols are resilient to an arbitrary node power-up schedule and a general node failure model.]]> Thu, 21 Apr 2011 22:00:00 +0000 https://doi.org/10.46298/dmtcs.528 https://doi.org/10.46298/dmtcs.528 Fernández Anta, Antonio Mosteiro, Miguel Thraves Caro, Christopher Fernández Anta, Antonio Mosteiro, Miguel Thraves Caro, Christopher <![CDATA[Monitoring physical phenomena in Sensor Networks requires guaranteeing permanent communication between nodes. Moreover, in an effective implementation of such infrastructure, the delay between any two consecutive communications should be minimized. The problem is challenging because, in a restricted Sensor Network, the communication is carried out through a single and shared radio channel without collision detection. Dealing with collisions is crucial to ensure effective communication between nodes. Additionally, minimizing them yields energy consumption minimization, given that sensing and computational costs in terms of energy are negligible with respect to radio communication. In this work, we present a deterministic recurrent-communication protocol for Sensor Networks. After an initial negotiation phase of the access pattern to the channel, each node running this protocol reaches a steady state, which is asymptotically optimal in terms of energy and time efficiency. As a by-product, a protocol for the synchronization of a Sensor Network is also proposed. Furthermore, the protocols are resilient to an arbitrary node power-up schedule and a general node failure model.]]> 0 <![CDATA[We study expansions in non-integer negative base -beta introduced by Ito and Sadahiro. Using countable automata associated with (-beta)-expansions, we characterize the case where the (-beta)-shift is a system of finite type. We prove that, if beta is a Pisot number, then the (-beta)-shift is a sofic system. In that case, addition (and more generally normalization on any alphabet) is realizable by a finite transducer. We then give an on-line algorithm for the conversion from positive base beta to negative base -beta. When beta is a Pisot number, the conversion can be realized by a finite on-line transducer.]]> Sun, 10 Apr 2011 22:00:00 +0000 https://doi.org/10.46298/dmtcs.538 https://doi.org/10.46298/dmtcs.538 Frougny, Christiane Lai, Anna Chiara Frougny, Christiane Lai, Anna Chiara <![CDATA[We study expansions in non-integer negative base -beta introduced by Ito and Sadahiro. Using countable automata associated with (-beta)-expansions, we characterize the case where the (-beta)-shift is a system of finite type. We prove that, if beta is a Pisot number, then the (-beta)-shift is a sofic system. In that case, addition (and more generally normalization on any alphabet) is realizable by a finite transducer. We then give an on-line algorithm for the conversion from positive base beta to negative base -beta. When beta is a Pisot number, the conversion can be realized by a finite on-line transducer.]]> 0 <![CDATA[De Bruijn and Erdos proved that every noncollinear set of n points in the plane determines at least n distinct lines. Chen and Chvatal suggested a possible generalization of this theorem in the framework of metric spaces. We provide partial results in this direction.]]> Tue, 22 Mar 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.545 https://doi.org/10.46298/dmtcs.545 Chiniforooshan, Ehsan Chvatal, Vasek Chiniforooshan, Ehsan Chvatal, Vasek <![CDATA[De Bruijn and Erdos proved that every noncollinear set of n points in the plane determines at least n distinct lines. Chen and Chvatal suggested a possible generalization of this theorem in the framework of metric spaces. We provide partial results in this direction.]]> 0 <![CDATA[In this work we investigate the algorithmic complexity of computing a minimum C(k)-transversal, i.e., a subset of vertices that intersects all the chordless cycles with k vertices of the input graph, for a fixed k \textgreater= 3. For graphs of maximum degree at most three, we prove that this problem is polynomial-time solvable when k \textless= 4, and NP-hard otherwise. For graphs of maximum degree at most four, we prove that this problem is NP-hard for any fixed k \textgreater= 3. We also discuss polynomial-time approximation algorithms for computing C(3)-transversals in graphs of maximum degree at most four, based on a new decomposition theorem for such graphs that leads to useful reduction rules.]]> Thu, 03 Mar 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.533 https://doi.org/10.46298/dmtcs.533 Groshaus, Marina Hell, Pavol Klein, Sulamita Nogueira, Loana Tito Protti, Fábio Groshaus, Marina Hell, Pavol Klein, Sulamita Nogueira, Loana Tito Protti, Fábio <![CDATA[In this work we investigate the algorithmic complexity of computing a minimum C(k)-transversal, i.e., a subset of vertices that intersects all the chordless cycles with k vertices of the input graph, for a fixed k \textgreater= 3. For graphs of maximum degree at most three, we prove that this problem is polynomial-time solvable when k \textless= 4, and NP-hard otherwise. For graphs of maximum degree at most four, we prove that this problem is NP-hard for any fixed k \textgreater= 3. We also discuss polynomial-time approximation algorithms for computing C(3)-transversals in graphs of maximum degree at most four, based on a new decomposition theorem for such graphs that leads to useful reduction rules.]]> 0 <![CDATA[For each integer partition mu, let e (F) over tilde (mu)(q; t) be the coefficient of x(1) ... x(n) in the modified Macdonald polynomial (H) over tilde (mu). The polynomial (F) over tilde (mu)(q; t) can be regarded as the Hilbert series of a certain doubly-graded S(n)-module M(mu), or as a q, t-analogue of n! based on permutation statistics inv(mu) and maj(mu) that generalize the classical inversion and major index statistics. This paper uses the combinatorial definition of (F) over tilde (mu) to prove some recursions characterizing these polynomials, and other related ones, when mu is a two-column shape. Our result provides a complement to recent work of Garsia and Haglund, who proved a different recursion for two-column shapes by representation-theoretical methods. For all mu, we show that e (F) over tilde (mu)(q, t) is divisible by certain q-factorials and t-factorials depending on mu. We use our recursion and related tools to explain some of these factors bijectively. Finally, we present fermionic formulas that express e (F) over tilde ((2n)) (q, t) as a sum of q, t-analogues of n!2(n) indexed by perfect matchings.]]> Mon, 14 Feb 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.541 https://doi.org/10.46298/dmtcs.541 Loehr, Nicholas A. Niese, Elizabeth Loehr, Nicholas A. Niese, Elizabeth <![CDATA[For each integer partition mu, let e (F) over tilde (mu)(q; t) be the coefficient of x(1) ... x(n) in the modified Macdonald polynomial (H) over tilde (mu). The polynomial (F) over tilde (mu)(q; t) can be regarded as the Hilbert series of a certain doubly-graded S(n)-module M(mu), or as a q, t-analogue of n! based on permutation statistics inv(mu) and maj(mu) that generalize the classical inversion and major index statistics. This paper uses the combinatorial definition of (F) over tilde (mu) to prove some recursions characterizing these polynomials, and other related ones, when mu is a two-column shape. Our result provides a complement to recent work of Garsia and Haglund, who proved a different recursion for two-column shapes by representation-theoretical methods. For all mu, we show that e (F) over tilde (mu)(q, t) is divisible by certain q-factorials and t-factorials depending on mu. We use our recursion and related tools to explain some of these factors bijectively. Finally, we present fermionic formulas that express e (F) over tilde ((2n)) (q, t) as a sum of q, t-analogues of n!2(n) indexed by perfect matchings.]]> 0 <![CDATA[We show that for a planar graph with no g-grid minor there exists a tree-decomposition of width at most 5g - 6. The proof is constructive and simple. The underlying algorithm for the tree-decomposition runs in O(n(2) log n) time.]]> Tue, 08 Feb 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.539 https://doi.org/10.46298/dmtcs.539 Grigoriev, Alexander Grigoriev, Alexander <![CDATA[We show that for a planar graph with no g-grid minor there exists a tree-decomposition of width at most 5g - 6. The proof is constructive and simple. The underlying algorithm for the tree-decomposition runs in O(n(2) log n) time.]]> 0 <![CDATA[Let G be a simple graph and let us color its edges so that the multisets of colors around each vertex are distinct. The smallest number of colors for which such a coloring exists is called the irregular coloring number of G and is denoted by c(G). We determine the exact value of the irregular coloring number for almost all 2-regular graphs. The results obtained provide new examples demonstrating that a conjecture by Burris is false. As another consequence, we also determine the value of a graph invariant called the point distinguishing index (where sets, instead of multisets, are required to be distinct) for the same family of graphs.]]> Fri, 04 Feb 2011 23:00:00 +0000 https://doi.org/10.46298/dmtcs.544 https://doi.org/10.46298/dmtcs.544 Cichacz, Sylwia Przybylo, Jakub Cichacz, Sylwia Przybylo, Jakub <![CDATA[Let G be a simple graph and let us color its edges so that the multisets of colors around each vertex are distinct. The smallest number of colors for which such a coloring exists is called the irregular coloring number of G and is denoted by c(G). We determine the exact value of the irregular coloring number for almost all 2-regular graphs. The results obtained provide new examples demonstrating that a conjecture by Burris is false. As another consequence, we also determine the value of a graph invariant called the point distinguishing index (where sets, instead of multisets, are required to be distinct) for the same family of graphs.]]> 0 <![CDATA[In recent work, Mansour [Discrete Math. Theoret. Computer Science 11, 2009, 173--186] considers the problem of enumerating all words of length n over {1,2,...,k} (where k is a given integer), that have the total variation equal to a given integer m. In the present paper we study various types of random words over the infinite alphabet ℕ, where the letters have geometric probabilities. We are interested in the probabilities of words of given type to have a given total variation.]]> Sat, 01 Jan 2011 07:00:00 +0000 https://doi.org/10.46298/dmtcs.477 https://doi.org/10.46298/dmtcs.477 Cristea, Ligia Loreta Prodinger, Helmut Cristea, Ligia Loreta Prodinger, Helmut <![CDATA[In recent work, Mansour [Discrete Math. Theoret. Computer Science 11, 2009, 173--186] considers the problem of enumerating all words of length n over {1,2,...,k} (where k is a given integer), that have the total variation equal to a given integer m. In the present paper we study various types of random words over the infinite alphabet ℕ, where the letters have geometric probabilities. We are interested in the probabilities of words of given type to have a given total variation.]]> 0 <![CDATA[Consider spanning trees on the two-dimensional Sierpinski gasket SG(n) where stage n is a non-negative integer. For any given vertex x of SG(n), we derive rigorously the probability distribution of the degree j ∈{1,2,3,4} at the vertex and its value in the infinite n limit. Adding up such probabilities of all the vertices divided by the number of vertices, we obtain the average probability distribution of the degree j. The corresponding limiting distribution φj gives the average probability that a vertex is connected by 1, 2, 3 or 4 bond(s) among all the spanning tree configurations. They are rational numbers given as φ1=10957/40464, φ2=6626035/13636368, φ3=2943139/13636368, φ4=124895/4545456.]]> Sat, 01 Jan 2011 07:00:00 +0000 https://doi.org/10.46298/dmtcs.476 https://doi.org/10.46298/dmtcs.476 Chang, Shu-Chiuan Chen, Lung-Chi Chang, Shu-Chiuan Chen, Lung-Chi <![CDATA[Consider spanning trees on the two-dimensional Sierpinski gasket SG(n) where stage n is a non-negative integer. For any given vertex x of SG(n), we derive rigorously the probability distribution of the degree j ∈{1,2,3,4} at the vertex and its value in the infinite n limit. Adding up such probabilities of all the vertices divided by the number of vertices, we obtain the average probability distribution of the degree j. The corresponding limiting distribution φj gives the average probability that a vertex is connected by 1, 2, 3 or 4 bond(s) among all the spanning tree configurations. They are rational numbers given as φ1=10957/40464, φ2=6626035/13636368, φ3=2943139/13636368, φ4=124895/4545456.]]> 0 <![CDATA[Motivated by the bijection between Schnyder labelings of a plane triangulation and partitions of its inner edges into three trees, we look for binary labelings for quadrangulations (whose edges can be partitioned into two trees). Our labeling resembles many of the properties of Schnyder's one for triangulations: Apart from being in bijection with tree decompositions, paths in these trees allow to define the regions of a vertex such that counting faces in them yields an algorithm for embedding the quadrangulation, in this case on a 2-book. Furthermore, as Schnyder labelings have been extended to 3-connected plane graphs, we are able to extend our labeling from quadrangulations to a larger class of 2-connected bipartite graphs.]]> Sat, 01 Jan 2011 07:00:00 +0000 https://doi.org/10.46298/dmtcs.475 https://doi.org/10.46298/dmtcs.475 Felsner, Stefan Huemer, Clemens Kappes, Sarah Orden, David Felsner, Stefan Huemer, Clemens Kappes, Sarah Orden, David <![CDATA[Motivated by the bijection between Schnyder labelings of a plane triangulation and partitions of its inner edges into three trees, we look for binary labelings for quadrangulations (whose edges can be partitioned into two trees). Our labeling resembles many of the properties of Schnyder's one for triangulations: Apart from being in bijection with tree decompositions, paths in these trees allow to define the regions of a vertex such that counting faces in them yields an algorithm for embedding the quadrangulation, in this case on a 2-book. Furthermore, as Schnyder labelings have been extended to 3-connected plane graphs, we are able to extend our labeling from quadrangulations to a larger class of 2-connected bipartite graphs.]]> 0 <![CDATA[This work considers a cellular automaton (CA) with two particles: a stationary particle $1$ and left-going one $\overline{1}$. When a $\overline{1}$ encounters a $1$, both particles annihilate. We derive asymptotic distribution of appearence of particles at a given site when the CA is initialized with the Bernoulli measure with the probabilities of both particles equal to $1/2$.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2976 https://doi.org/10.46298/dmtcs.2976 Kůrka, Petr Formenti, Enrico Dennunzio, Alberto Kůrka, Petr Formenti, Enrico Dennunzio, Alberto <![CDATA[This work considers a cellular automaton (CA) with two particles: a stationary particle $1$ and left-going one $\overline{1}$. When a $\overline{1}$ encounters a $1$, both particles annihilate. We derive asymptotic distribution of appearence of particles at a given site when the CA is initialized with the Bernoulli measure with the probabilities of both particles equal to $1/2$.]]> 0 <![CDATA[We are interested in fixed points in Boolean networks, $\textit{i.e.}$ functions $f$ from $\{0,1\}^n$ to itself. We define the subnetworks of $f$ as the restrictions of $f$ to the hypercubes contained in $\{0,1\}^n$, and we exhibit a class $\mathcal{F}$ of Boolean networks, called even or odd self-dual networks, satisfying the following property: if a network $f$ has no subnetwork in $\mathcal{F}$, then it has a unique fixed point. We then discuss this "forbidden subnetworks theorem''. We show that it generalizes the following fixed point theorem of Shih and Dong: if, for every $x$ in $\{0,1\}^n$, there is no directed cycle in the directed graph whose the adjacency matrix is the discrete Jacobian matrix of $f$ evaluated at point $x$, then $f$ has a unique fixed point. We also show that $\mathcal{F}$ contains the class $\mathcal{F'}$ of networks whose the interaction graph is a directed cycle, but that the absence of subnetwork in $\mathcal{F'}$ does not imply the existence and the uniqueness of a fixed point.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2978 https://doi.org/10.46298/dmtcs.2978 Richard, Adrien Richard, Adrien <![CDATA[We are interested in fixed points in Boolean networks, $\textit{i.e.}$ functions $f$ from $\{0,1\}^n$ to itself. We define the subnetworks of $f$ as the restrictions of $f$ to the hypercubes contained in $\{0,1\}^n$, and we exhibit a class $\mathcal{F}$ of Boolean networks, called even or odd self-dual networks, satisfying the following property: if a network $f$ has no subnetwork in $\mathcal{F}$, then it has a unique fixed point. We then discuss this "forbidden subnetworks theorem''. We show that it generalizes the following fixed point theorem of Shih and Dong: if, for every $x$ in $\{0,1\}^n$, there is no directed cycle in the directed graph whose the adjacency matrix is the discrete Jacobian matrix of $f$ evaluated at point $x$, then $f$ has a unique fixed point. We also show that $\mathcal{F}$ contains the class $\mathcal{F'}$ of networks whose the interaction graph is a directed cycle, but that the absence of subnetwork in $\mathcal{F'}$ does not imply the existence and the uniqueness of a fixed point.]]> 0 <![CDATA[Studying cellular automata with methods from communication complexity appears to be a promising approach. In the past, interesting connections between communication complexity and intrinsic universality in cellular automata were shown. One of the last extensions of this theory was its generalization to various "communication problems'', or "questions'' one might ask about the dynamics of cellular automata. In this article, we aim at structuring these problems, and find what makes them interesting for the study of intrinsic universality and quasi-orders induced by simulation relations.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2977 https://doi.org/10.46298/dmtcs.2977 Briceño, Raimundo Meunier, Pierre-Etienne Briceño, Raimundo Meunier, Pierre-Etienne <![CDATA[Studying cellular automata with methods from communication complexity appears to be a promising approach. In the past, interesting connections between communication complexity and intrinsic universality in cellular automata were shown. One of the last extensions of this theory was its generalization to various "communication problems'', or "questions'' one might ask about the dynamics of cellular automata. In this article, we aim at structuring these problems, and find what makes them interesting for the study of intrinsic universality and quasi-orders induced by simulation relations.]]> 0 <![CDATA[Philippe Flajolet, mathematician and computer scientist extraordinaire, suddenly passed away on March 22, 2011, at the prime of his career. He is celebrated for opening new lines of research in analysis of algo- rithms, developing powerful new methods, and solving difficult open problems. His research contributions will have impact for generations, and his approach to research, based on curiosity, a discriminating taste, broad knowledge and interest, intellectual integrity, and a genuine sense of camaraderie, will serve as an inspiration to those who knew him for years to come]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2966 https://doi.org/10.46298/dmtcs.2966 Salvy, Bruno Sedgewick, Bob Soria, Michèle Szpankowski, Wojtek Vallée, Brigitte Salvy, Bruno Sedgewick, Bob Soria, Michèle Szpankowski, Wojtek Vallée, Brigitte <![CDATA[Philippe Flajolet, mathematician and computer scientist extraordinaire, suddenly passed away on March 22, 2011, at the prime of his career. He is celebrated for opening new lines of research in analysis of algo- rithms, developing powerful new methods, and solving difficult open problems. His research contributions will have impact for generations, and his approach to research, based on curiosity, a discriminating taste, broad knowledge and interest, intellectual integrity, and a genuine sense of camaraderie, will serve as an inspiration to those who knew him for years to come]]> 0 <![CDATA[This paper characterizes the attractor structure of synchronous and asynchronous Boolean networks induced by bi-threshold functions. Bi-threshold functions are generalizations of standard threshold functions and have separate threshold values for the transitions $0 \rightarrow $1 (up-threshold) and $1 \rightarrow 0$ (down-threshold). We show that synchronous bi-threshold systems may, just like standard threshold systems, only have fixed points and 2-cycles as attractors. Asynchronous bi-threshold systems (fixed permutation update sequence), on the other hand, undergo a bifurcation. When the difference $\Delta$ of the down- and up-threshold is less than 2 they only have fixed points as limit sets. However, for $\Delta \geq 2$ they may have long periodic orbits. The limiting case of $\Delta = 2$ is identified using a potential function argument. Finally, we present a series of results on the dynamics of bi-threshold systems for families of graphs.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2975 https://doi.org/10.46298/dmtcs.2975 Kuhlman, Chris, Mortveit, Henning, Murrugarra, David Kumar, Anil, Kuhlman, Chris, Mortveit, Henning, Murrugarra, David Kumar, Anil, <![CDATA[This paper characterizes the attractor structure of synchronous and asynchronous Boolean networks induced by bi-threshold functions. Bi-threshold functions are generalizations of standard threshold functions and have separate threshold values for the transitions $0 \rightarrow $1 (up-threshold) and $1 \rightarrow 0$ (down-threshold). We show that synchronous bi-threshold systems may, just like standard threshold systems, only have fixed points and 2-cycles as attractors. Asynchronous bi-threshold systems (fixed permutation update sequence), on the other hand, undergo a bifurcation. When the difference $\Delta$ of the down- and up-threshold is less than 2 they only have fixed points as limit sets. However, for $\Delta \geq 2$ they may have long periodic orbits. The limiting case of $\Delta = 2$ is identified using a potential function argument. Finally, we present a series of results on the dynamics of bi-threshold systems for families of graphs.]]> 0 <![CDATA[Sand Pile Models are discrete dynamical systems emphasizing the phenomenon of $\textit{Self-Organized Criticality}$. From a configuration composed of a finite number of stacked grains, we apply on every possible positions (in parallel) two grain moving transition rules. The transition rules permit one grain to fall to its right or left (symmetric) neighboring column if the difference of height between those columns is larger than 2. The model is nondeterministic and grains always fall downward. We propose a study of the set of fixed points reachable in the Parallel Symmetric Sand Pile Model (PSSPM). Using a comparison with the Symmetric Sand Pile Model (SSPM) on which rules are applied once at each iteration, we get a continuity property. This property states that within PSSPM we can't reach every fixed points of SSPM, but a continuous subset according to the lexicographic order. Moreover we define a successor relation to browse exhaustively the sets of fixed points of those models.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2974 https://doi.org/10.46298/dmtcs.2974 Perrot, Kévin Phan, Thi Ha Duong Pham, Trung Van Perrot, Kévin Phan, Thi Ha Duong Pham, Trung Van <![CDATA[Sand Pile Models are discrete dynamical systems emphasizing the phenomenon of $\textit{Self-Organized Criticality}$. From a configuration composed of a finite number of stacked grains, we apply on every possible positions (in parallel) two grain moving transition rules. The transition rules permit one grain to fall to its right or left (symmetric) neighboring column if the difference of height between those columns is larger than 2. The model is nondeterministic and grains always fall downward. We propose a study of the set of fixed points reachable in the Parallel Symmetric Sand Pile Model (PSSPM). Using a comparison with the Symmetric Sand Pile Model (SSPM) on which rules are applied once at each iteration, we get a continuity property. This property states that within PSSPM we can't reach every fixed points of SSPM, but a continuous subset according to the lexicographic order. Moreover we define a successor relation to browse exhaustively the sets of fixed points of those models.]]> 0 <![CDATA[We study intrinsic simulations between cellular automata and introduce a new necessary condition for a CA to simulate another one. Although expressed for general CA, this condition is targeted towards surjective CA and especially linear ones. Following the approach introduced by the first author in an earlier paper, we develop proof techniques to tell whether some linear CA can simulate another linear CA. Besides rigorous proofs, the necessary condition for the simulation to occur can be heuristically checked via simple observations of typical space-time diagrams generated from finite configurations. As an illustration, we give an example of linear reversible CA which cannot simulate the identity and which is 'time-asymmetric', i.e. which can neither simulate its own inverse, nor the mirror of its own inverse.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2973 https://doi.org/10.46298/dmtcs.2973 Nesme, Vincent Theyssier, Guillaume Nesme, Vincent Theyssier, Guillaume <![CDATA[We study intrinsic simulations between cellular automata and introduce a new necessary condition for a CA to simulate another one. Although expressed for general CA, this condition is targeted towards surjective CA and especially linear ones. Following the approach introduced by the first author in an earlier paper, we develop proof techniques to tell whether some linear CA can simulate another linear CA. Besides rigorous proofs, the necessary condition for the simulation to occur can be heuristically checked via simple observations of typical space-time diagrams generated from finite configurations. As an illustration, we give an example of linear reversible CA which cannot simulate the identity and which is 'time-asymmetric', i.e. which can neither simulate its own inverse, nor the mirror of its own inverse.]]> 0 <![CDATA[We study iterations of the Bernoulli measure under nearest-neighbour asynchronous binary cellular automata (CA) with a single transition. For these CA, we show that a coarse-level description of the orbit of the Bernoulli measure can be obtained, that is, one can explicitly compute measures of short cylinder sets after arbitrary number of iterations of the CA. In particular, we give expressions for probabilities of ones for all three minimal single-transition rules, as well as expressions for probabilities of blocks of length 3 for some of them. These expressions can be interpreted as "response curves'', that is, curves describing the dependence of the final density of ones on the initial density of ones.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2972 https://doi.org/10.46298/dmtcs.2972 Fukś, Henryk Skelton, Andrew Fukś, Henryk Skelton, Andrew <![CDATA[We study iterations of the Bernoulli measure under nearest-neighbour asynchronous binary cellular automata (CA) with a single transition. For these CA, we show that a coarse-level description of the orbit of the Bernoulli measure can be obtained, that is, one can explicitly compute measures of short cylinder sets after arbitrary number of iterations of the CA. In particular, we give expressions for probabilities of ones for all three minimal single-transition rules, as well as expressions for probabilities of blocks of length 3 for some of them. These expressions can be interpreted as "response curves'', that is, curves describing the dependence of the final density of ones on the initial density of ones.]]> 0 <![CDATA[In this paper we define products of one-dimensional Number Conserving Cellular Automata (NCCA) and show that surjective NCCA with 2 blocks (i.e radius 1/2) can always be represented as products of shifts and identites. In particular, this shows that surjective 2-block NCCA are injective.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2971 https://doi.org/10.46298/dmtcs.2971 García-Ramos, Felipe García-Ramos, Felipe <![CDATA[In this paper we define products of one-dimensional Number Conserving Cellular Automata (NCCA) and show that surjective NCCA with 2 blocks (i.e radius 1/2) can always be represented as products of shifts and identites. In particular, this shows that surjective 2-block NCCA are injective.]]> 0 <![CDATA[LFSR and NFSR are the basic building blocks in almost all the state of the art stream ciphers like Trivium and Grain-128. However, a number of attacks are mounted on these type of ciphers. Cellular Automata (CA) has recently been chosen as a suitable structure for crypto-primitives. In this work, a stream cipher is presented based on hybrid CA. The stream cipher takes 128 bit key and 128 bit initialization vector (IV) as input. It is designed to produce $\mathrm{2^{128}}$ random keystream bits and initialization phase is made faster 4 times than that of Grain-128. We also analyze the cryptographic strength of this cipher. Finally, the proposed cipher is shown to be resistant against known existing attacks.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2970 https://doi.org/10.46298/dmtcs.2970 Karmakar, Sandip Chowdhury, Dipanwita Roy Karmakar, Sandip Chowdhury, Dipanwita Roy <![CDATA[LFSR and NFSR are the basic building blocks in almost all the state of the art stream ciphers like Trivium and Grain-128. However, a number of attacks are mounted on these type of ciphers. Cellular Automata (CA) has recently been chosen as a suitable structure for crypto-primitives. In this work, a stream cipher is presented based on hybrid CA. The stream cipher takes 128 bit key and 128 bit initialization vector (IV) as input. It is designed to produce $\mathrm{2^{128}}$ random keystream bits and initialization phase is made faster 4 times than that of Grain-128. We also analyze the cryptographic strength of this cipher. Finally, the proposed cipher is shown to be resistant against known existing attacks.]]> 0 <![CDATA[We study the projective subdynamics of two-dimensional shifts of finite type, which is the set of one-dimensional configurations that appear as columns in them. We prove that a large class of one-dimensional shifts can be obtained as such, namely the effective subshifts which contain positive-entropy sofic subshifts. The proof involves some simple notions of simulation that may be of interest for other constructions. As an example, it allows us to prove the undecidability of all non-trivial properties of projective subdynamics.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2969 https://doi.org/10.46298/dmtcs.2969 Guillon, Pierre Guillon, Pierre <![CDATA[We study the projective subdynamics of two-dimensional shifts of finite type, which is the set of one-dimensional configurations that appear as columns in them. We prove that a large class of one-dimensional shifts can be obtained as such, namely the effective subshifts which contain positive-entropy sofic subshifts. The proof involves some simple notions of simulation that may be of interest for other constructions. As an example, it allows us to prove the undecidability of all non-trivial properties of projective subdynamics.]]> 0 <![CDATA[We discuss a close link between two seemingly different topics studied in the cellular automata literature: additive conservation laws and invariant probability measures. We provide an elementary proof of a simple correspondence between invariant full-support Bernoulli measures and interaction-free conserved quantities in the case of one-dimensional surjective cellular automata. We also discuss a generalization of this fact to Markov measures and higher-range conservation laws in arbitrary dimension. As a corollary, we show that the uniform Bernoulli measure is the only shift-invariant, full-support Markov measure that is invariant under a strongly transitive cellular automaton.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2968 https://doi.org/10.46298/dmtcs.2968 Kari, Jarkko Taati, Siamak Kari, Jarkko Taati, Siamak <![CDATA[We discuss a close link between two seemingly different topics studied in the cellular automata literature: additive conservation laws and invariant probability measures. We provide an elementary proof of a simple correspondence between invariant full-support Bernoulli measures and interaction-free conserved quantities in the case of one-dimensional surjective cellular automata. We also discuss a generalization of this fact to Markov measures and higher-range conservation laws in arbitrary dimension. As a corollary, we show that the uniform Bernoulli measure is the only shift-invariant, full-support Markov measure that is invariant under a strongly transitive cellular automaton.]]> 0 <![CDATA[We identify two seemingly disparate structures: supercharacters, a useful way of doing Fourier analysis on the group of unipotent uppertriangular matrices with coefficients in a finite field, and the ring of symmetric functions in noncommuting variables. Each is a Hopf algebra and the two are isomorphic as such. This allows developments in each to be transferred. The identification suggests a rich class of examples for the emerging field of combinatorial Hopf algebras.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2967 https://doi.org/10.46298/dmtcs.2967 Aguiar, Marcelo André, Carlos Benedetti, Carolina Bergeron, Nantel Chen, Zhi Diaconis, Persi Hendrickson, Anders Hsiao, Samuel Isaacs, I. Martin Jedwab, Andrea Johnson, Kenneth Karaali, Gizem Lauve, Aaron Le, Tung Lewis, Stephen Li, Huilan Magaard, Kay Marberg, Eric Novelli, Jean-Christophe Pang, Amy Saliola, Franco Tevlin, Lenny Thibon, Jean-Yves Thiem, Nathaniel Venkateswaran, Vidya Vinroot, C. Ryan Yan, Ning Zabrocki, Mike Aguiar, Marcelo André, Carlos Benedetti, Carolina Bergeron, Nantel Chen, Zhi Diaconis, Persi Hendrickson, Anders Hsiao, Samuel Isaacs, I. Martin Jedwab, Andrea Johnson, Kenneth Karaali, Gizem Lauve, Aaron Le, Tung Lewis, Stephen Li, Huilan Magaard, Kay Marberg, Eric Novelli, Jean-Christophe Pang, Amy Saliola, Franco Tevlin, Lenny Thibon, Jean-Yves Thiem, Nathaniel Venkateswaran, Vidya Vinroot, C. Ryan Yan, Ning Zabrocki, Mike <![CDATA[We identify two seemingly disparate structures: supercharacters, a useful way of doing Fourier analysis on the group of unipotent uppertriangular matrices with coefficients in a finite field, and the ring of symmetric functions in noncommuting variables. Each is a Hopf algebra and the two are isomorphic as such. This allows developments in each to be transferred. The identification suggests a rich class of examples for the emerging field of combinatorial Hopf algebras.]]> 0 <![CDATA[We show that maximal 0-1-fillings of moon polynomials, with restricted chain lengths, can be identified with certain rc-graphs, also known as pipe dreams. In particular, this exhibits a connection between maximal 0-1-fillings of Ferrers shapes and Schubert polynomials. Moreover, it entails a bijective proof showing that the number of maximal fillings of a stack polyomino $S$ with no north-east chains longer than $k$ depends only on $k$ and the multiset of column heights of $S$. Our main contribution is a slightly stronger theorem, which in turn leads us to conjecture that the poset of rc-graphs with covering relation given by generalised chute moves is in fact a lattice.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2957 https://doi.org/10.46298/dmtcs.2957 Rubey, Martin Rubey, Martin <![CDATA[We show that maximal 0-1-fillings of moon polynomials, with restricted chain lengths, can be identified with certain rc-graphs, also known as pipe dreams. In particular, this exhibits a connection between maximal 0-1-fillings of Ferrers shapes and Schubert polynomials. Moreover, it entails a bijective proof showing that the number of maximal fillings of a stack polyomino $S$ with no north-east chains longer than $k$ depends only on $k$ and the multiset of column heights of $S$. Our main contribution is a slightly stronger theorem, which in turn leads us to conjecture that the poset of rc-graphs with covering relation given by generalised chute moves is in fact a lattice.]]> 0 <![CDATA[We investigate a conjecture of Haglund that asserts that certain graph polynomials have only real roots. We prove a multivariate generalization of this conjecture for the special case of threshold graphs.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2965 https://doi.org/10.46298/dmtcs.2965 Visontai, Mirkó Visontai, Mirkó <![CDATA[We investigate a conjecture of Haglund that asserts that certain graph polynomials have only real roots. We prove a multivariate generalization of this conjecture for the special case of threshold graphs.]]> 0 <![CDATA[Noncommutative symmetric functions have many properties analogous to those of classical (commutative) symmetric functions. For instance, ribbon Schur functions (analogs of the classical Schur basis) expand positively in noncommutative monomial basis. More of the classical properties extend to noncommutative setting as I will demonstrate introducing a new family of noncommutative symmetric functions, depending on one parameter. It seems to be an appropriate noncommutative analog of the Hall-Littlewood polynomials.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2964 https://doi.org/10.46298/dmtcs.2964 Tevlin, Lenny Tevlin, Lenny <![CDATA[Noncommutative symmetric functions have many properties analogous to those of classical (commutative) symmetric functions. For instance, ribbon Schur functions (analogs of the classical Schur basis) expand positively in noncommutative monomial basis. More of the classical properties extend to noncommutative setting as I will demonstrate introducing a new family of noncommutative symmetric functions, depending on one parameter. It seems to be an appropriate noncommutative analog of the Hall-Littlewood polynomials.]]> 0 <![CDATA[We combinatorially construct the complex cohomology (equivariant and ordinary) of a family of algebraic varieties called regular semisimple Hessenberg varieties. This construction is purely in terms of the Bruhat order on the symmetric group. From this a representation of the symmetric group on the cohomology is defined. This representation generalizes work of Procesi, Stembridge and Tymoczko. Here a partial answer to an open question of Tymoczko is provided in our two main result. The first states, when the variety has multiple connected components, this representation is made up by inducing through a parabolic subgroup of the symmetric group. Using this, our second result obtains, for a special family of varieties, an explicit formula for this representation via Young's rule, giving the multiplicity of the irreducible representations in terms of the classical Kostka numbers.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2963 https://doi.org/10.46298/dmtcs.2963 Teff, Nicholas Teff, Nicholas <![CDATA[We combinatorially construct the complex cohomology (equivariant and ordinary) of a family of algebraic varieties called regular semisimple Hessenberg varieties. This construction is purely in terms of the Bruhat order on the symmetric group. From this a representation of the symmetric group on the cohomology is defined. This representation generalizes work of Procesi, Stembridge and Tymoczko. Here a partial answer to an open question of Tymoczko is provided in our two main result. The first states, when the variety has multiple connected components, this representation is made up by inducing through a parabolic subgroup of the symmetric group. Using this, our second result obtains, for a special family of varieties, an explicit formula for this representation via Young's rule, giving the multiplicity of the irreducible representations in terms of the classical Kostka numbers.]]> 0 <![CDATA[We prove a $q$-analog of a classical binomial congruence due to Ljunggren which states that $\binom{ap}{bp} \equiv \binom{a}{b}$ modulo $p^3$ for primes $p \geq 5$. This congruence subsumes and builds on earlier congruences by Babbage, Wolstenholme and Glaisher for which we recall existing $q$-analogs. Our congruence generalizes an earlier result of Clark.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2962 https://doi.org/10.46298/dmtcs.2962 Straub, Armin Straub, Armin <![CDATA[We prove a $q$-analog of a classical binomial congruence due to Ljunggren which states that $\binom{ap}{bp} \equiv \binom{a}{b}$ modulo $p^3$ for primes $p \geq 5$. This congruence subsumes and builds on earlier congruences by Babbage, Wolstenholme and Glaisher for which we recall existing $q$-analogs. Our congruence generalizes an earlier result of Clark.]]> 0 <![CDATA[We exhibit a canonical connection between maximal $(0,1)$-fillings of a moon polyomino avoiding north-east chains of a given length and reduced pipe dreams of a certain permutation. Following this approach we show that the simplicial complex of such maximal fillings is a vertex-decomposable and thus a shellable sphere. In particular, this implies a positivity result for Schubert polynomials. For Ferrers shapes, we moreover construct a bijection to maximal fillings avoiding south-east chains of the same length which specializes to a bijection between $k$-triangulations of the $n$-gon and $k$-fans of Dyck paths. Using this, we translate a conjectured cyclic sieving phenomenon for $k$-triangulations with rotation to $k$-flagged tableaux with promotion.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2961 https://doi.org/10.46298/dmtcs.2961 Serrano, Luis Stump, Christian Serrano, Luis Stump, Christian <![CDATA[We exhibit a canonical connection between maximal $(0,1)$-fillings of a moon polyomino avoiding north-east chains of a given length and reduced pipe dreams of a certain permutation. Following this approach we show that the simplicial complex of such maximal fillings is a vertex-decomposable and thus a shellable sphere. In particular, this implies a positivity result for Schubert polynomials. For Ferrers shapes, we moreover construct a bijection to maximal fillings avoiding south-east chains of the same length which specializes to a bijection between $k$-triangulations of the $n$-gon and $k$-fans of Dyck paths. Using this, we translate a conjectured cyclic sieving phenomenon for $k$-triangulations with rotation to $k$-flagged tableaux with promotion.]]> 0 <![CDATA[Schrijver introduced the stable Kneser graph $SG_{n,k}, n \geq 1, k \geq 0$. This graph is a vertex critical graph with chromatic number $k+2$, its vertices are certain subsets of a set of cardinality $m=2n+k$. Björner and de Longueville have shown that its box complex is homotopy equivalent to a sphere, $\mathrm{Hom}(K_2,SG_{n,k}) \simeq \mathbb{S}^k$. The dihedral group $D_{2m}$ acts canonically on $SG_{n,k}$. We study the $D_{2m}$ action on $\mathrm{Hom}(K_2,SG_{n,k})$ and define a corresponding orthogonal action on $\mathbb{R}^{k+1} \supset \mathbb{S}^k$. We establish a close equivariant relationship between the graphs $SG_{n,k}$ and Borsuk graphs of the $k$-sphere and use this together with calculations in the $\mathbb{Z}_2$-cohomology ring of $D_{2m}$ to tell which stable Kneser graphs are test graphs in the sense of Babson and Kozlov. The graphs $SG_{2s,4}$ are test graphs, i.e. for every graph $H$ and $r \geq 0$ such that $\mathrm{Hom}(SG_{2s,4},H)$ is $(r-1)$-connected, the chromatic number $\chi (H)$ is at least $r+6$. On the other hand, if $k \notin \{0,1,2,4,8\}$ and $n \geq N(k)$ then $SG_{n,k}$ is not a homotopy test graph, i.e. there are a graph $G$ and an $r \geq 1$ such that $\mathrm{Hom}(SG_{n,k}, G)$ is $(r-1)$-connected and $\chi (G) < r+k+2$. The latter result also depends on a new necessary criterion for being a test graph, which involves the automorphism group of the graph.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2960 https://doi.org/10.46298/dmtcs.2960 Schultz, Carsten Schultz, Carsten <![CDATA[Schrijver introduced the stable Kneser graph $SG_{n,k}, n \geq 1, k \geq 0$. This graph is a vertex critical graph with chromatic number $k+2$, its vertices are certain subsets of a set of cardinality $m=2n+k$. Björner and de Longueville have shown that its box complex is homotopy equivalent to a sphere, $\mathrm{Hom}(K_2,SG_{n,k}) \simeq \mathbb{S}^k$. The dihedral group $D_{2m}$ acts canonically on $SG_{n,k}$. We study the $D_{2m}$ action on $\mathrm{Hom}(K_2,SG_{n,k})$ and define a corresponding orthogonal action on $\mathbb{R}^{k+1} \supset \mathbb{S}^k$. We establish a close equivariant relationship between the graphs $SG_{n,k}$ and Borsuk graphs of the $k$-sphere and use this together with calculations in the $\mathbb{Z}_2$-cohomology ring of $D_{2m}$ to tell which stable Kneser graphs are test graphs in the sense of Babson and Kozlov. The graphs $SG_{2s,4}$ are test graphs, i.e. for every graph $H$ and $r \geq 0$ such that $\mathrm{Hom}(SG_{2s,4},H)$ is $(r-1)$-connected, the chromatic number $\chi (H)$ is at least $r+6$. On the other hand, if $k \notin \{0,1,2,4,8\}$ and $n \geq N(k)$ then $SG_{n,k}$ is not a homotopy test graph, i.e. there are a graph $G$ and an $r \geq 1$ such that $\mathrm{Hom}(SG_{n,k}, G)$ is $(r-1)$-connected and $\chi (G) < r+k+2$. The latter result also depends on a new necessary criterion for being a test graph, which involves the automorphism group of the graph.]]> 0 <![CDATA[There is a close connection between Demazure crystals and tensor products of Kirillov–Reshetikhin crystals. For example, certain Demazure crystals are isomorphic as classical crystals to tensor products of Kirillov–Reshetikhin crystals via a canonically chosen isomorphism. Here we show that this isomorphism intertwines the natural affine grading on Demazure crystals with a combinatorially defined energy function. As a consequence, we obtain a formula of the Demazure character in terms of the energy function, which has applications to nonsymmetric Macdonald polynomials and $q$-deformed Whittaker functions.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2959 https://doi.org/10.46298/dmtcs.2959 Schilling, Anne Tingley, Peter Schilling, Anne Tingley, Peter <![CDATA[There is a close connection between Demazure crystals and tensor products of Kirillov–Reshetikhin crystals. For example, certain Demazure crystals are isomorphic as classical crystals to tensor products of Kirillov–Reshetikhin crystals via a canonically chosen isomorphism. Here we show that this isomorphism intertwines the natural affine grading on Demazure crystals with a combinatorially defined energy function. As a consequence, we obtain a formula of the Demazure character in terms of the energy function, which has applications to nonsymmetric Macdonald polynomials and $q$-deformed Whittaker functions.]]> 0 <![CDATA[We derive in this paper the asymptotics of several-partition Hurwitz numbers, relying on a theorem of Kazarian for the one-partition case and on an induction carried on by Zvonkine. Essentially, the asymptotics for several partitions is the same as the one-partition asymptotics obtained by concatenating the partitions.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2958 https://doi.org/10.46298/dmtcs.2958 Sage, Marc Sage, Marc <![CDATA[We derive in this paper the asymptotics of several-partition Hurwitz numbers, relying on a theorem of Kazarian for the one-partition case and on an induction carried on by Zvonkine. Essentially, the asymptotics for several partitions is the same as the one-partition asymptotics obtained by concatenating the partitions.]]> 0 <![CDATA[We consider the family of 3D minimal polyominoes inscribed in a rectanglar prism. These objects are polyominos and so they are connected sets of unitary cubic cells inscribed in a given rectangular prism of size $b\times k \times h$ and of minimal volume equal to $b+k+h-2$. They extend the concept of minimal 2D polyominoes inscribed in a rectangle studied in a previous work. Using their geometric structure and elementary combinatorial principles, we construct rational generating functions of minimal 3D polyominoes. We also obtain a number of exact formulas and recurrences for sub-families of these polyominoes.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2922 https://doi.org/10.46298/dmtcs.2922 Goupil, Alain Cloutier, Hugo Goupil, Alain Cloutier, Hugo <![CDATA[We consider the family of 3D minimal polyominoes inscribed in a rectanglar prism. These objects are polyominos and so they are connected sets of unitary cubic cells inscribed in a given rectangular prism of size $b\times k \times h$ and of minimal volume equal to $b+k+h-2$. They extend the concept of minimal 2D polyominoes inscribed in a rectangle studied in a previous work. Using their geometric structure and elementary combinatorial principles, we construct rational generating functions of minimal 3D polyominoes. We also obtain a number of exact formulas and recurrences for sub-families of these polyominoes.]]> 0 <![CDATA[In this paper, we develop a new method to compute generating functions of the form $NM_τ (t,x,y) = \sum\limits_{n ≥0} {\frac{t^n} {n!}}∑_{σ ∈\mathcal{lNM_{n}(τ )}} x^{LRMin(σ)} y^{1+des(σ )}$ where $τ$ is a permutation that starts with $1, \mathcal{NM_n}(τ )$ is the set of permutations in the symmetric group $S_n$ with no $τ$ -matches, and for any permutation $σ ∈S_n$, $LRMin(σ )$ is the number of left-to-right minima of $σ$ and $des(σ )$ is the number of descents of $σ$ . Our method does not compute $NM_τ (t,x,y)$ directly, but assumes that $NM_τ (t,x,y) = \frac{1}{/ (U_τ (t,y))^x}$ where $U_τ (t,y) = \sum_{n ≥0} U_τ ,n(y) \frac{t^n}{ n!}$ so that $U_τ (t,y) = \frac{1}{ NM_τ (t,1,y)}$. We then use the so-called homomorphism method and the combinatorial interpretation of $NM_τ (t,1,y)$ to develop recursions for the coefficient of $U_τ (t,y)$.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2933 https://doi.org/10.46298/dmtcs.2933 Jones, Miles Eli Remmel, Jeffrey Jones, Miles Eli Remmel, Jeffrey <![CDATA[In this paper, we develop a new method to compute generating functions of the form $NM_τ (t,x,y) = \sum\limits_{n ≥0} {\frac{t^n} {n!}}∑_{σ ∈\mathcal{lNM_{n}(τ )}} x^{LRMin(σ)} y^{1+des(σ )}$ where $τ$ is a permutation that starts with $1, \mathcal{NM_n}(τ )$ is the set of permutations in the symmetric group $S_n$ with no $τ$ -matches, and for any permutation $σ ∈S_n$, $LRMin(σ )$ is the number of left-to-right minima of $σ$ and $des(σ )$ is the number of descents of $σ$ . Our method does not compute $NM_τ (t,x,y)$ directly, but assumes that $NM_τ (t,x,y) = \frac{1}{/ (U_τ (t,y))^x}$ where $U_τ (t,y) = \sum_{n ≥0} U_τ ,n(y) \frac{t^n}{ n!}$ so that $U_τ (t,y) = \frac{1}{ NM_τ (t,1,y)}$. We then use the so-called homomorphism method and the combinatorial interpretation of $NM_τ (t,1,y)$ to develop recursions for the coefficient of $U_τ (t,y)$.]]> 0 <![CDATA[In this paper, we first derive an explicit formula for the generating function that counts unlabeled interval orders (a.k.a. (2+2)-free posets) with respect to several natural statistics, including their size, magnitude, and the number of minimal and maximal elements. In the second part of the paper, we derive a generating function for the number of self-dual unlabeled interval orders, with respect to the same statistics. Our method is based on a bijective correspondence between interval orders and upper-triangular matrices in which each row and column has a positive entry.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2932 https://doi.org/10.46298/dmtcs.2932 Jelínek, Vít Jelínek, Vít <![CDATA[In this paper, we first derive an explicit formula for the generating function that counts unlabeled interval orders (a.k.a. (2+2)-free posets) with respect to several natural statistics, including their size, magnitude, and the number of minimal and maximal elements. In the second part of the paper, we derive a generating function for the number of self-dual unlabeled interval orders, with respect to the same statistics. Our method is based on a bijective correspondence between interval orders and upper-triangular matrices in which each row and column has a positive entry.]]> 0 <![CDATA[We present an insertion algorithm of Robinson–Schensted type that applies to set-valued shifted Young tableaux. Our algorithm is a generalization of both set-valued non-shifted tableaux by Buch and non set-valued shifted tableaux by Worley and Sagan. As an application, we obtain a Pieri rule for a K-theoretic analogue of the Schur Q-functions.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2931 https://doi.org/10.46298/dmtcs.2931 Ikeda, Takeshi Naruse, Hiroshi Numata, Yasuhide Ikeda, Takeshi Naruse, Hiroshi Numata, Yasuhide <![CDATA[We present an insertion algorithm of Robinson–Schensted type that applies to set-valued shifted Young tableaux. Our algorithm is a generalization of both set-valued non-shifted tableaux by Buch and non set-valued shifted tableaux by Worley and Sagan. As an application, we obtain a Pieri rule for a K-theoretic analogue of the Schur Q-functions.]]> 0 <![CDATA[The graph algebra is a commutative, cocommutative, graded, connected incidence Hopf algebra, whose basis elements correspond to finite simple graphs and whose Hopf product and coproduct admit simple combinatorial descriptions. We give a new formula for the antipode in the graph algebra in terms of acyclic orientations; our formula contains many fewer terms than Schmitt's more general formula for the antipode in an incidence Hopf algebra. Applications include several formulas (some old and some new) for evaluations of the Tutte polynomial.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2930 https://doi.org/10.46298/dmtcs.2930 Humpert, Brandon Martin, Jeremy L. Humpert, Brandon Martin, Jeremy L. <![CDATA[The graph algebra is a commutative, cocommutative, graded, connected incidence Hopf algebra, whose basis elements correspond to finite simple graphs and whose Hopf product and coproduct admit simple combinatorial descriptions. We give a new formula for the antipode in the graph algebra in terms of acyclic orientations; our formula contains many fewer terms than Schmitt's more general formula for the antipode in an incidence Hopf algebra. Applications include several formulas (some old and some new) for evaluations of the Tutte polynomial.]]> 0 <![CDATA[By investigating the action of the 0-Hecke algebra on the coinvariant algebra and the complete flag variety, we interpret generating functions counting the permutations with fixed inverse descent set by their inversion number and major index.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2929 https://doi.org/10.46298/dmtcs.2929 Huang, Jia Huang, Jia <![CDATA[By investigating the action of the 0-Hecke algebra on the coinvariant algebra and the complete flag variety, we interpret generating functions counting the permutations with fixed inverse descent set by their inversion number and major index.]]> 0 <![CDATA[In this paper we present a work in progress on a conjectural new combinatorial model for the Genocchi numbers. This new model called irreducible k-shapes has a strong algebraic background in the theory of symmetric functions and leads to seemingly new features on the theory of Genocchi numbers. In particular, the natural q-analogue coming from the degree of symmetric functions seems to be unknown so far.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2928 https://doi.org/10.46298/dmtcs.2928 Hivert, Florent Mallet, Olivier Hivert, Florent Mallet, Olivier <![CDATA[In this paper we present a work in progress on a conjectural new combinatorial model for the Genocchi numbers. This new model called irreducible k-shapes has a strong algebraic background in the theory of symmetric functions and leads to seemingly new features on the theory of Genocchi numbers. In particular, the natural q-analogue coming from the degree of symmetric functions seems to be unknown so far.]]> 0 <![CDATA[We introduce the short toric polynomial associated to a graded Eulerian poset. This polynomial contains the same information as Stanley's pair of toric polynomials, but allows different algebraic manipulations. Stanley's intertwined recurrence may be replaced by a single recurrence, in which the degree of the discarded terms is independent of the rank. A short toric variant of the formula by Bayer and Ehrenborg, expressing the toric h-vector in terms of the cd-index, may be stated in a rank-independent form, and it may be shown using weighted lattice path enumeration and the reflection principle. We use our techniques to derive a formula expressing the toric h-vector of a dual simplicial Eulerian poset in terms of its f-vector. This formula implies Gessel's formula for the toric h-vector of a cube, and may be used to prove that the nonnegativity of the toric h-vector of a simple polytope is a consequence of the Generalized Lower Bound Theorem holding for simplicial polytopes.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2927 https://doi.org/10.46298/dmtcs.2927 Hetyei, Gábor Hetyei, Gábor <![CDATA[We introduce the short toric polynomial associated to a graded Eulerian poset. This polynomial contains the same information as Stanley's pair of toric polynomials, but allows different algebraic manipulations. Stanley's intertwined recurrence may be replaced by a single recurrence, in which the degree of the discarded terms is independent of the rank. A short toric variant of the formula by Bayer and Ehrenborg, expressing the toric h-vector in terms of the cd-index, may be stated in a rank-independent form, and it may be shown using weighted lattice path enumeration and the reflection principle. We use our techniques to derive a formula expressing the toric h-vector of a dual simplicial Eulerian poset in terms of its f-vector. This formula implies Gessel's formula for the toric h-vector of a cube, and may be used to prove that the nonnegativity of the toric h-vector of a simple polytope is a consequence of the Generalized Lower Bound Theorem holding for simplicial polytopes.]]> 0 <![CDATA[We consider a Markov chain Monte Carlo approach to the uniform sampling of meanders. Combinatorially, a meander $M = [A:B]$ is formed by two noncrossing perfect matchings, above $A$ and below $B$ the same endpoints, which form a single closed loop. We prove that meanders are connected under appropriate pairs of balanced local moves, one operating on $A$ and the other on $B$. We also prove that the subset of meanders with a fixed $B$ is connected under a suitable local move operating on an appropriately defined meandric triple in $A$. We provide diameter bounds under such moves, tight up to a (worst case) factor of two. The mixing times of the Markov chains remain open.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2926 https://doi.org/10.46298/dmtcs.2926 Heitsch, Christine E. Tetali, Prasad Heitsch, Christine E. Tetali, Prasad <![CDATA[We consider a Markov chain Monte Carlo approach to the uniform sampling of meanders. Combinatorially, a meander $M = [A:B]$ is formed by two noncrossing perfect matchings, above $A$ and below $B$ the same endpoints, which form a single closed loop. We prove that meanders are connected under appropriate pairs of balanced local moves, one operating on $A$ and the other on $B$. We also prove that the subset of meanders with a fixed $B$ is connected under a suitable local move operating on an appropriately defined meandric triple in $A$. We provide diameter bounds under such moves, tight up to a (worst case) factor of two. The mixing times of the Markov chains remain open.]]> 0 <![CDATA[We give a generating function for the fully commutative affine permutations enumerated by rank and Coxeter length, extending formulas due to Stembridge and Barcucci–Del Lungo–Pergola–Pinzani. For fixed rank, the length generating functions have coefficients that are periodic with period dividing the rank. In the course of proving these formulas, we obtain results that elucidate the structure of the fully commutative affine permutations. This is a summary of the results; the full version appears elsewhere.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2925 https://doi.org/10.46298/dmtcs.2925 Hanusa, Christopher R. H. Jones, Brant C. Hanusa, Christopher R. H. Jones, Brant C. <![CDATA[We give a generating function for the fully commutative affine permutations enumerated by rank and Coxeter length, extending formulas due to Stembridge and Barcucci–Del Lungo–Pergola–Pinzani. For fixed rank, the length generating functions have coefficients that are periodic with period dividing the rank. In the course of proving these formulas, we obtain results that elucidate the structure of the fully commutative affine permutations. This is a summary of the results; the full version appears elsewhere.]]> 0 <![CDATA[A special case of Haiman's identity [Invent. Math. 149 (2002), pp. 371–407] for the character of the quotient ring of diagonal coinvariants under the diagonal action of the symmetric group yields a formula for the bigraded Hilbert series as a sum of rational functions in $q,t$. In this paper we show how a summation identity of Garsia and Zabrocki for Macdonald polynomial Pieri coefficients can be used to transform Haiman's formula for the Hilbert series into an explicit polynomial in $q,t$ with integer coefficients. We also provide an equivalent formula for the Hilbert series as the constant term in a multivariate Laurent series.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2924 https://doi.org/10.46298/dmtcs.2924 Haglund, J. Haglund, J. <![CDATA[A special case of Haiman's identity [Invent. Math. 149 (2002), pp. 371–407] for the character of the quotient ring of diagonal coinvariants under the diagonal action of the symmetric group yields a formula for the bigraded Hilbert series as a sum of rational functions in $q,t$. In this paper we show how a summation identity of Garsia and Zabrocki for Macdonald polynomial Pieri coefficients can be used to transform Haiman's formula for the Hilbert series into an explicit polynomial in $q,t$ with integer coefficients. We also provide an equivalent formula for the Hilbert series as the constant term in a multivariate Laurent series.]]> 0 <![CDATA[A non-crossing connected graph is a connected graph on vertices arranged in a circle such that its edges do not cross. The count for such graphs can be made naturally into a q-binomial generating function. We prove that this generating function exhibits the cyclic sieving phenomenon, as conjectured by S.-P. Eu.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2923 https://doi.org/10.46298/dmtcs.2923 Guo, Alan Guo, Alan <![CDATA[A non-crossing connected graph is a connected graph on vertices arranged in a circle such that its edges do not cross. The count for such graphs can be made naturally into a q-binomial generating function. We prove that this generating function exhibits the cyclic sieving phenomenon, as conjectured by S.-P. Eu.]]> 0 <![CDATA[For a real number $β >1$, we say that a permutation $π$ of length $n$ is allowed (or realized) by the $β$-shift if there is some $x∈[0,1]$ such that the relative order of the sequence $x,f(x),\ldots,f^n-1(x)$, where $f(x)$ is the fractional part of $βx$, is the same as that of the entries of $π$ . Widely studied from such diverse fields as number theory and automata theory, $β$-shifts are prototypical examples of one-dimensional chaotic dynamical systems. When $β$ is an integer, permutations realized by shifts have been recently characterized. In this paper we generalize some of the results to arbitrary $β$-shifts. We describe a method to compute, for any given permutation $π$ , the smallest $β$ such that $π$ is realized by the $β$-shift.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2911 https://doi.org/10.46298/dmtcs.2911 Elizalde, Sergi Elizalde, Sergi <![CDATA[For a real number $β >1$, we say that a permutation $π$ of length $n$ is allowed (or realized) by the $β$-shift if there is some $x∈[0,1]$ such that the relative order of the sequence $x,f(x),\ldots,f^n-1(x)$, where $f(x)$ is the fractional part of $βx$, is the same as that of the entries of $π$ . Widely studied from such diverse fields as number theory and automata theory, $β$-shifts are prototypical examples of one-dimensional chaotic dynamical systems. When $β$ is an integer, permutations realized by shifts have been recently characterized. In this paper we generalize some of the results to arbitrary $β$-shifts. We describe a method to compute, for any given permutation $π$ , the smallest $β$ such that $π$ is realized by the $β$-shift.]]> 0 <![CDATA[C. Merino [Electron. J. Combin. 15 (2008)] showed that the Tutte polynomial of a complete graph satisfies $t(K_{n+2};2,-1)=t(K_n;1,-1)$. We first give a bijective proof of this identity based on the relationship between the Tutte polynomial and the inversion polynomial for trees. Next we move to our main result, a sufficient condition for a graph G to have two vertices u and v such that $t(G;2,-1)=t(G-\{u,v\};1,-1)$; the condition is satisfied in particular by the class of threshold graphs. Finally, we give a formula for the evaluation of $t(K_{n,m};2,-1)$ involving up-down permutations.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2921 https://doi.org/10.46298/dmtcs.2921 Goodall, Andrew Merino, Criel de Mier, Anna Noy, Marc Goodall, Andrew Merino, Criel de Mier, Anna Noy, Marc <![CDATA[C. Merino [Electron. J. Combin. 15 (2008)] showed that the Tutte polynomial of a complete graph satisfies $t(K_{n+2};2,-1)=t(K_n;1,-1)$. We first give a bijective proof of this identity based on the relationship between the Tutte polynomial and the inversion polynomial for trees. Next we move to our main result, a sufficient condition for a graph G to have two vertices u and v such that $t(G;2,-1)=t(G-\{u,v\};1,-1)$; the condition is satisfied in particular by the class of threshold graphs. Finally, we give a formula for the evaluation of $t(K_{n,m};2,-1)$ involving up-down permutations.]]> 0 <![CDATA[The pentagram map, introduced by R. Schwartz, is defined by the following construction: given a polygon as input, draw all of its ``shortest'' diagonals, and output the smaller polygon which they cut out. We employ the machinery of cluster algebras to obtain explicit formulas for the iterates of the pentagram map.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2920 https://doi.org/10.46298/dmtcs.2920 Glick, Max Glick, Max <![CDATA[The pentagram map, introduced by R. Schwartz, is defined by the following construction: given a polygon as input, draw all of its ``shortest'' diagonals, and output the smaller polygon which they cut out. We employ the machinery of cluster algebras to obtain explicit formulas for the iterates of the pentagram map.]]> 0 <![CDATA[We give a new construction of a Hopf subalgebra of the Hopf algebra of Free quasi-symmetric functions whose bases are indexed by objects belonging to the Baxter combinatorial family (\emphi.e. Baxter permutations, pairs of twin binary trees, \emphetc.). This construction relies on the definition of the Baxter monoid, analog of the plactic monoid and the sylvester monoid, and on a Robinson-Schensted-like insertion algorithm. The algebraic properties of this Hopf algebra are studied. This Hopf algebra appeared for the first time in the work of Reading [Lattice congruences, fans and Hopf algebras, \textitJournal of Combinatorial Theory Series A, 110:237–273, 2005].]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2919 https://doi.org/10.46298/dmtcs.2919 Giraudo, Samuele Giraudo, Samuele <![CDATA[We give a new construction of a Hopf subalgebra of the Hopf algebra of Free quasi-symmetric functions whose bases are indexed by objects belonging to the Baxter combinatorial family (\emphi.e. Baxter permutations, pairs of twin binary trees, \emphetc.). This construction relies on the definition of the Baxter monoid, analog of the plactic monoid and the sylvester monoid, and on a Robinson-Schensted-like insertion algorithm. The algebraic properties of this Hopf algebra are studied. This Hopf algebra appeared for the first time in the work of Reading [Lattice congruences, fans and Hopf algebras, \textitJournal of Combinatorial Theory Series A, 110:237–273, 2005].]]> 0 <![CDATA[We study the combinatorics of weighted trees from the point of view of tropical algebraic geometry and tropical linear spaces. The set of dissimilarity vectors of weighted trees is contained in the tropical Grassmannian, so we describe here the tropical linear space of a dissimilarity vector and its associated family of matroids. This gives a family of complete flags of tropical linear spaces, where each flag is described by a weighted tree.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2918 https://doi.org/10.46298/dmtcs.2918 Iriarte Giraldo, Benjamin Iriarte Giraldo, Benjamin <![CDATA[We study the combinatorics of weighted trees from the point of view of tropical algebraic geometry and tropical linear spaces. The set of dissimilarity vectors of weighted trees is contained in the tropical Grassmannian, so we describe here the tropical linear space of a dissimilarity vector and its associated family of matroids. This gives a family of complete flags of tropical linear spaces, where each flag is described by a weighted tree.]]> 0 <![CDATA[We develop the notion of the composition of two coalgebras, which arises naturally in higher category theory and the theory of species. We prove that the composition of two cofree coalgebras is cofree and give conditions which imply that the composition is a one-sided Hopf algebra. These conditions hold when one coalgebra is a graded Hopf operad $\mathcal{D}$ and the other is a connected graded coalgebra with coalgebra map to $\mathcal{D}$. We conclude with examples of these structures, where the factor coalgebras have bases indexed by the vertices of multiplihedra, composihedra, and hypercubes.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2917 https://doi.org/10.46298/dmtcs.2917 Forcey, Stefan Lauve, Aaron Sottile, Frank Forcey, Stefan Lauve, Aaron Sottile, Frank <![CDATA[We develop the notion of the composition of two coalgebras, which arises naturally in higher category theory and the theory of species. We prove that the composition of two cofree coalgebras is cofree and give conditions which imply that the composition is a one-sided Hopf algebra. These conditions hold when one coalgebra is a graded Hopf operad $\mathcal{D}$ and the other is a connected graded coalgebra with coalgebra map to $\mathcal{D}$. We conclude with examples of these structures, where the factor coalgebras have bases indexed by the vertices of multiplihedra, composihedra, and hypercubes.]]> 0 <![CDATA[Athanasiadis introduced separating walls for a region in the extended Shi arrangement and used them to generalize the Narayana numbers. In this paper, we fix a hyperplane in the extended Shi arrangement for type A and calculate the number of dominant regions which have the fixed hyperplane as a separating wall; that is, regions where the hyperplane supports a facet of the region and separates the region from the origin.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2916 https://doi.org/10.46298/dmtcs.2916 Fishel, Susanna Tzanaki, Eleni Vazirani, Monica Fishel, Susanna Tzanaki, Eleni Vazirani, Monica <![CDATA[Athanasiadis introduced separating walls for a region in the extended Shi arrangement and used them to generalize the Narayana numbers. In this paper, we fix a hyperplane in the extended Shi arrangement for type A and calculate the number of dominant regions which have the fixed hyperplane as a separating wall; that is, regions where the hyperplane supports a facet of the region and separates the region from the origin.]]> 0 <![CDATA[To every matroid, we associate a class in the K-theory of the Grassmannian. We study this class using the method of equivariant localization. In particular, we provide a geometric interpretation of the Tutte polynomial. We also extend results of the second author concerning the behavior of such classes under direct sum, series and parallel connection and two-sum; these results were previously only established for realizable matroids, and their earlier proofs were more difficult.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2915 https://doi.org/10.46298/dmtcs.2915 Fink, Alex Speyer, David Fink, Alex Speyer, David <![CDATA[To every matroid, we associate a class in the K-theory of the Grassmannian. We study this class using the method of equivariant localization. In particular, we provide a geometric interpretation of the Tutte polynomial. We also extend results of the second author concerning the behavior of such classes under direct sum, series and parallel connection and two-sum; these results were previously only established for realizable matroids, and their earlier proofs were more difficult.]]> 0 <![CDATA[We establish several properties of an algorithm defined by Mason and Remmel (2010) which inserts a positive integer into a row-strict composition tableau. These properties lead to a Littlewood-Richardson type rule for expanding the product of a row-strict quasisymmetric Schur function and a symmetric Schur function in terms of row-strict quasisymmetric Schur functions.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2914 https://doi.org/10.46298/dmtcs.2914 Ferreira, Jeffrey Ferreira, Jeffrey <![CDATA[We establish several properties of an algorithm defined by Mason and Remmel (2010) which inserts a positive integer into a row-strict composition tableau. These properties lead to a Littlewood-Richardson type rule for expanding the product of a row-strict quasisymmetric Schur function and a symmetric Schur function in terms of row-strict quasisymmetric Schur functions.]]> 0 <![CDATA[In this paper we establish a new combinatorial formula for zonal polynomials in terms of power-sums. The proof relies on the sign-reversing involution principle. We deduce from it formulas for zonal characters, which are defined as suitably normalized coefficients in the expansion of zonal polynomials in terms of power-sum symmetric functions. These formulas are analogs of recent developments on irreducible character values of symmetric groups. The existence of such formulas could have been predicted from the work of M. Lassalle who formulated two positivity conjectures for Jack characters, which we prove in the special case of zonal polynomials.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2913 https://doi.org/10.46298/dmtcs.2913 Féray, Valentin Sniady, Piotr Féray, Valentin Sniady, Piotr <![CDATA[In this paper we establish a new combinatorial formula for zonal polynomials in terms of power-sums. The proof relies on the sign-reversing involution principle. We deduce from it formulas for zonal characters, which are defined as suitably normalized coefficients in the expansion of zonal polynomials in terms of power-sum symmetric functions. These formulas are analogs of recent developments on irreducible character values of symmetric groups. The existence of such formulas could have been predicted from the work of M. Lassalle who formulated two positivity conjectures for Jack characters, which we prove in the special case of zonal polynomials.]]> 0 <![CDATA[We study polytopes that are convex hulls of vectors of subgraph densities. Many graph theoretical questions can be expressed in terms of these polytopes, and statisticians use them to understand exponential random graph models. Relations among their Ehrhart polynomials are described, their duals are applied to certify that polynomials are non-negative, and we find some of their faces. For the general picture we inscribe cyclic polytopes in them and calculate volumes. From the volume calculations we conjecture that a variation of the Selberg integral indexed by Schur polynomials has a combinatorial formula. We inscribe polynomially parametrized sets, called curvy zonotopes, in the polytopes and show that they approximate the polytopes arbitrarily close.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2912 https://doi.org/10.46298/dmtcs.2912 Engström, Alexander Norén, Patrik Engström, Alexander Norén, Patrik <![CDATA[We study polytopes that are convex hulls of vectors of subgraph densities. Many graph theoretical questions can be expressed in terms of these polytopes, and statisticians use them to understand exponential random graph models. Relations among their Ehrhart polynomials are described, their duals are applied to certify that polynomials are non-negative, and we find some of their faces. For the general picture we inscribe cyclic polytopes in them and calculate volumes. From the volume calculations we conjecture that a variation of the Selberg integral indexed by Schur polynomials has a combinatorial formula. We inscribe polynomially parametrized sets, called curvy zonotopes, in the polytopes and show that they approximate the polytopes arbitrarily close.]]> 0 <![CDATA[Any continuous map of an $N$-dimensional simplex $Δ _N$ with colored vertices to a $d$-dimensional manifold $M$ must map $r$ points from disjoint rainbow faces of $Δ _N$ to the same point in $M$, assuming that $N≥(r-1)(d+1)$, no $r$ vertices of $Δ _N$ get the same color, and our proof needs that $r$ is a prime. A face of $Δ _N$ is called a rainbow face if all vertices have different colors. This result is an extension of our recent "new colored Tverberg theorem'', the special case of $M=ℝ^d$. It is also a generalization of Volovikov's 1996 topological Tverberg theorem for maps to manifolds, which arises when all color classes have size 1 (i.e., without color constraints); for this special case Volovikov's proofs, as well as ours, work when r is a prime power.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2901 https://doi.org/10.46298/dmtcs.2901 Blagojević, Pavle V. M. Matschke, Benjamin Ziegler, Günter M. Blagojević, Pavle V. M. Matschke, Benjamin Ziegler, Günter M. <![CDATA[Any continuous map of an $N$-dimensional simplex $Δ _N$ with colored vertices to a $d$-dimensional manifold $M$ must map $r$ points from disjoint rainbow faces of $Δ _N$ to the same point in $M$, assuming that $N≥(r-1)(d+1)$, no $r$ vertices of $Δ _N$ get the same color, and our proof needs that $r$ is a prime. A face of $Δ _N$ is called a rainbow face if all vertices have different colors. This result is an extension of our recent "new colored Tverberg theorem'', the special case of $M=ℝ^d$. It is also a generalization of Volovikov's 1996 topological Tverberg theorem for maps to manifolds, which arises when all color classes have size 1 (i.e., without color constraints); for this special case Volovikov's proofs, as well as ours, work when r is a prime power.]]> 0 <![CDATA[For each composition $\vec{c}$ we show that the order complex of the poset of pointed set partitions $Π ^• _{\vec{c}}$ is a wedge of $β\vec{c}$ spheres of the same dimensions, where $β\vec{c}$ is the number of permutations with descent composition ^$\vec{c}$. Furthermore, the action of the symmetric group on the top homology is isomorphic to the Specht module $S^B$ where $B$ is a border strip associated to the composition $\vec{c}$. We also study the filter of pointed set partitions generated by a knapsack integer partitions and show the analogous results on homotopy type and action on the top homology.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2910 https://doi.org/10.46298/dmtcs.2910 Ehrenborg, Richard Jung, JiYoon Ehrenborg, Richard Jung, JiYoon <![CDATA[For each composition $\vec{c}$ we show that the order complex of the poset of pointed set partitions $Π ^• _{\vec{c}}$ is a wedge of $β\vec{c}$ spheres of the same dimensions, where $β\vec{c}$ is the number of permutations with descent composition ^$\vec{c}$. Furthermore, the action of the symmetric group on the top homology is isomorphic to the Specht module $S^B$ where $B$ is a border strip associated to the composition $\vec{c}$. We also study the filter of pointed set partitions generated by a knapsack integer partitions and show the analogous results on homotopy type and action on the top homology.]]> 0 <![CDATA[We generalize the theory of critical groups from graphs to simplicial complexes. Specifically, given a simplicial complex, we define a family of abelian groups in terms of combinatorial Laplacian operators, generalizing the construction of the critical group of a graph. We show how to realize these critical groups explicitly as cokernels of reduced Laplacians, and prove that they are finite, with orders given by weighted enumerators of simplicial spanning trees. We describe how the critical groups of a complex represent flow along its faces, and sketch another potential interpretation as analogues of Chow groups.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2909 https://doi.org/10.46298/dmtcs.2909 Duval, Art M. Klivans, Caroline J. Martin, Jeremy L. Duval, Art M. Klivans, Caroline J. Martin, Jeremy L. <![CDATA[We generalize the theory of critical groups from graphs to simplicial complexes. Specifically, given a simplicial complex, we define a family of abelian groups in terms of combinatorial Laplacian operators, generalizing the construction of the critical group of a graph. We show how to realize these critical groups explicitly as cokernels of reduced Laplacians, and prove that they are finite, with orders given by weighted enumerators of simplicial spanning trees. We describe how the critical groups of a complex represent flow along its faces, and sketch another potential interpretation as analogues of Chow groups.]]> 0 <![CDATA[We study the class of functions on the set of (generalized) Young diagrams arising as the number of embeddings of bipartite graphs. We give a criterion for checking when such a function is a polynomial function on Young diagrams (in the sense of Kerov and Olshanski) in terms of combinatorial properties of the corresponding bipartite graphs. Our method involves development of a differential calculus of functions on the set of generalized Young diagrams.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2908 https://doi.org/10.46298/dmtcs.2908 Dolęga, Maciej Sniady, Piotr Dolęga, Maciej Sniady, Piotr <![CDATA[We study the class of functions on the set of (generalized) Young diagrams arising as the number of embeddings of bipartite graphs. We give a criterion for checking when such a function is a polynomial function on Young diagrams (in the sense of Kerov and Olshanski) in terms of combinatorial properties of the corresponding bipartite graphs. Our method involves development of a differential calculus of functions on the set of generalized Young diagrams.]]> 0 <![CDATA[We give a simple bijection between some staircase tableaux and tables of inversion. Some nice properties of the bijection allows us to define some q-Eulerian polynomials related to the staircase tableaux. We also give a combinatorial interpretation of these q-Eulerian polynomials in terms of permutations.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2907 https://doi.org/10.46298/dmtcs.2907 Corteel, Sylvie Dasse-Hartaut, Sandrine Corteel, Sylvie Dasse-Hartaut, Sandrine <![CDATA[We give a simple bijection between some staircase tableaux and tables of inversion. Some nice properties of the bijection allows us to define some q-Eulerian polynomials related to the staircase tableaux. We also give a combinatorial interpretation of these q-Eulerian polynomials in terms of permutations.]]> 0 <![CDATA[Let $χ ^λ$ be the irreducible $S_n$-character corresponding to the partition $λ$ of $n$, equivalently, the preimage of the Schur function $s_λ$ under the Frobenius characteristic map. Let $\phi ^λ$ be the function $S_n →ℂ$ which is the preimage of the monomial symmetric function $m_λ$ under the Frobenius characteristic map. The irreducible character immanant $Imm_λ {(x)} = ∑_w ∈S_n χ ^λ (w) x_1,w_1 ⋯x_n,w_n$ evaluates nonnegatively on each totally nonnegative matrix $A$. We provide a combinatorial interpretation for the value $Imm_λ (A)$ in the case that $λ$ is a hook partition. The monomial immanant $Imm_{{\phi} ^λ} (x) = ∑_w ∈S_n φ ^λ (w) x_1,w_1 ⋯x_n,w_n$ is conjectured to evaluate nonnegatively on each totally nonnegative matrix $A$. We confirm this conjecture in the case that $λ$ is a two-column partition by providing a combinatorial interpretation for the value $Imm_{{\phi} ^λ} (A)$.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2906 https://doi.org/10.46298/dmtcs.2906 Clearman, Sam Shelton, Brittany Skandera, Mark Clearman, Sam Shelton, Brittany Skandera, Mark <![CDATA[Let $χ ^λ$ be the irreducible $S_n$-character corresponding to the partition $λ$ of $n$, equivalently, the preimage of the Schur function $s_λ$ under the Frobenius characteristic map. Let $\phi ^λ$ be the function $S_n →ℂ$ which is the preimage of the monomial symmetric function $m_λ$ under the Frobenius characteristic map. The irreducible character immanant $Imm_λ {(x)} = ∑_w ∈S_n χ ^λ (w) x_1,w_1 ⋯x_n,w_n$ evaluates nonnegatively on each totally nonnegative matrix $A$. We provide a combinatorial interpretation for the value $Imm_λ (A)$ in the case that $λ$ is a hook partition. The monomial immanant $Imm_{{\phi} ^λ} (x) = ∑_w ∈S_n φ ^λ (w) x_1,w_1 ⋯x_n,w_n$ is conjectured to evaluate nonnegatively on each totally nonnegative matrix $A$. We confirm this conjecture in the case that $λ$ is a two-column partition by providing a combinatorial interpretation for the value $Imm_{{\phi} ^λ} (A)$.]]> 0 <![CDATA[This paper introduces two matrix analogues for set partitions; partition and composition matrices. These two analogues are the natural result of lifting the mapping between ascent sequences and integer matrices given in Dukes & Parviainen (2010). We prove that partition matrices are in one-to-one correspondence with inversion tables. Non-decreasing inversion tables are shown to correspond to partition matrices with a row ordering relation. Partition matrices which are s-diagonal are classified in terms of inversion tables. Bidiagonal partition matrices are enumerated using the transfer-matrix method and are equinumerous with permutations which are sortable by two pop-stacks in parallel. We show that composition matrices on the set $X$ are in one-to-one correspondence with (2+2)-free posets on $X$.We show that pairs of ascent sequences and permutations are in one-to-one correspondence with (2+2)-free posets whose elements are the cycles of a permutation, and use this relation to give an expression for the number of (2+2)-free posets on $\{1,\ldots,n\}$.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2905 https://doi.org/10.46298/dmtcs.2905 Claesson, Anders Dukes, Mark Kubitzke, Martina Claesson, Anders Dukes, Mark Kubitzke, Martina <![CDATA[This paper introduces two matrix analogues for set partitions; partition and composition matrices. These two analogues are the natural result of lifting the mapping between ascent sequences and integer matrices given in Dukes & Parviainen (2010). We prove that partition matrices are in one-to-one correspondence with inversion tables. Non-decreasing inversion tables are shown to correspond to partition matrices with a row ordering relation. Partition matrices which are s-diagonal are classified in terms of inversion tables. Bidiagonal partition matrices are enumerated using the transfer-matrix method and are equinumerous with permutations which are sortable by two pop-stacks in parallel. We show that composition matrices on the set $X$ are in one-to-one correspondence with (2+2)-free posets on $X$.We show that pairs of ascent sequences and permutations are in one-to-one correspondence with (2+2)-free posets whose elements are the cycles of a permutation, and use this relation to give an expression for the number of (2+2)-free posets on $\{1,\ldots,n\}$.]]> 0 <![CDATA[Arc spaces have been introduced in algebraic geometry as a tool to study singularities but they show strong connections with combinatorics as well. Exploiting these relations we obtain a new approach to the classical Rogers-Ramanujan Identities. The linking object is the Hilbert-Poincaré series of the arc space over a point of the base variety. In the case of the double point this is precisely the generating series for the integer partitions without equal or consecutive parts.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2904 https://doi.org/10.46298/dmtcs.2904 Bruschek, Clemens Mourtada, Hussein Schepers, Jan Bruschek, Clemens Mourtada, Hussein Schepers, Jan <![CDATA[Arc spaces have been introduced in algebraic geometry as a tool to study singularities but they show strong connections with combinatorics as well. Exploiting these relations we obtain a new approach to the classical Rogers-Ramanujan Identities. The linking object is the Hilbert-Poincaré series of the arc space over a point of the base variety. In the case of the double point this is precisely the generating series for the integer partitions without equal or consecutive parts.]]> 0 <![CDATA[We generalize the recent work of Fomin and Mikhalkin on polynomial formulas for Severi degrees. The degree of the Severi variety of plane curves of degree d and δ nodes is given by a polynomial in d, provided δ is fixed and d is large enough. We extend this result to generalized Severi varieties parametrizing plane curves which, in addition, satisfy tangency conditions of given orders with respect to a given line. We show that the degrees of these varieties, appropriately rescaled, are given by a combinatorially defined ``relative node polynomial'' in the tangency orders, provided the latter are large enough. We describe a method to compute these polynomials for arbitrary δ , and use it to present explicit formulas for δ ≤ 6. We also give a threshold for polynomiality, and compute the first few leading terms for any δ .]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2903 https://doi.org/10.46298/dmtcs.2903 Block, Florian Block, Florian <![CDATA[We generalize the recent work of Fomin and Mikhalkin on polynomial formulas for Severi degrees. The degree of the Severi variety of plane curves of degree d and δ nodes is given by a polynomial in d, provided δ is fixed and d is large enough. We extend this result to generalized Severi varieties parametrizing plane curves which, in addition, satisfy tangency conditions of given orders with respect to a given line. We show that the degrees of these varieties, appropriately rescaled, are given by a combinatorially defined ``relative node polynomial'' in the tangency orders, provided the latter are large enough. We describe a method to compute these polynomials for arbitrary δ , and use it to present explicit formulas for δ ≤ 6. We also give a threshold for polynomiality, and compute the first few leading terms for any δ .]]> 0 <![CDATA[Let $[u,v]$ be a Bruhat interval and $B(u,v)$ be its corresponding Bruhat graph. The combinatorial and topological structure of the longest $u-v$ paths of $B(u,v)$ has been extensively studied and is well-known. Nevertheless, not much is known of the remaining paths. Here we describe combinatorial properties of the shortest $u-v$ paths of $B(u,v)$. We also derive the non-negativity of some coefficients of the complete mcd-index of $[u,v]$.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2902 https://doi.org/10.46298/dmtcs.2902 Blanco, Saúl A. Blanco, Saúl A. <![CDATA[Let $[u,v]$ be a Bruhat interval and $B(u,v)$ be its corresponding Bruhat graph. The combinatorial and topological structure of the longest $u-v$ paths of $B(u,v)$ has been extensively studied and is well-known. Nevertheless, not much is known of the remaining paths. Here we describe combinatorial properties of the shortest $u-v$ paths of $B(u,v)$. We also derive the non-negativity of some coefficients of the complete mcd-index of $[u,v]$.]]> 0 <![CDATA[In this work we introduce and study tree-like tableaux, which are certain fillings of Ferrers diagrams in simple bijection with permutation tableaux and alternative tableaux. We exhibit an elementary insertion procedure on our tableaux which gives a clear proof that tableaux of size n are counted by n!, and which moreover respects most of the well-known statistics studied originally on alternative and permutation tableaux. Our insertion procedure allows to define in particular two simple new bijections between tree-like tableaux and permutations: the first one is conceived specifically to respect the generalized pattern 2-31, while the second one respects the underlying tree of a tree-like tableau.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2891 https://doi.org/10.46298/dmtcs.2891 Aval, Jean-Christophe Boussicault, Adrien Nadeau, Philippe Aval, Jean-Christophe Boussicault, Adrien Nadeau, Philippe <![CDATA[In this work we introduce and study tree-like tableaux, which are certain fillings of Ferrers diagrams in simple bijection with permutation tableaux and alternative tableaux. We exhibit an elementary insertion procedure on our tableaux which gives a clear proof that tableaux of size n are counted by n!, and which moreover respects most of the well-known statistics studied originally on alternative and permutation tableaux. Our insertion procedure allows to define in particular two simple new bijections between tree-like tableaux and permutations: the first one is conceived specifically to respect the generalized pattern 2-31, while the second one respects the underlying tree of a tree-like tableau.]]> 0 <![CDATA[The study of Schubert varieties in G/B has led to numerous advances in algebraic combinatorics and algebraic geometry. These varieties are indexed by elements of the corresponding Weyl group, an affine Weyl group, or one of their parabolic quotients. Often times, the goal is to determine which of the algebraic and topological properties of the Schubert variety can be described in terms of the combinatorics of its corresponding Weyl group element. A celebrated example of this occurs when G/B is of type A, due to Lakshmibai and Sandhya. They showed that the smooth Schubert varieties are precisely those indexed by permutations that avoid the patterns 3412 and 4231. Our main result is a characterization of the rationally smooth Schubert varieties corresponding to affine permutations in terms of the patterns 4231 and 3412 and the twisted spiral permutations.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2900 https://doi.org/10.46298/dmtcs.2900 Billey, Sara Crites, Andrew Billey, Sara Crites, Andrew <![CDATA[The study of Schubert varieties in G/B has led to numerous advances in algebraic combinatorics and algebraic geometry. These varieties are indexed by elements of the corresponding Weyl group, an affine Weyl group, or one of their parabolic quotients. Often times, the goal is to determine which of the algebraic and topological properties of the Schubert variety can be described in terms of the combinatorics of its corresponding Weyl group element. A celebrated example of this occurs when G/B is of type A, due to Lakshmibai and Sandhya. They showed that the smooth Schubert varieties are precisely those indexed by permutations that avoid the patterns 3412 and 4231. Our main result is a characterization of the rationally smooth Schubert varieties corresponding to affine permutations in terms of the patterns 4231 and 3412 and the twisted spiral permutations.]]> 0 <![CDATA[In this paper we study finite Eulerian posets which are binomial or Sheffer. These important classes of posets are related to the theory of generating functions and to geometry. The results of this paper are organized as follows: (1) We completely determine the structure of Eulerian binomial posets and, as a conclusion, we are able to classify factorial functions of Eulerian binomial posets; (2) We give an almost complete classification of factorial functions of Eulerian Sheffer posets by dividing the original question into several cases; (3) In most cases above, we completely determine the structure of Eulerian Sheffer posets, a result stronger than just classifying factorial functions of these Eulerian Sheffer posets. We also study Eulerian triangular posets. This paper answers questions posed by R. Ehrenborg and M. Readdy. This research is also motivated by the work of R. Stanley about recognizing the \emphboolean lattice by looking at smaller intervals.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2899 https://doi.org/10.46298/dmtcs.2899 Bidkhori, Hoda Bidkhori, Hoda <![CDATA[In this paper we study finite Eulerian posets which are binomial or Sheffer. These important classes of posets are related to the theory of generating functions and to geometry. The results of this paper are organized as follows: (1) We completely determine the structure of Eulerian binomial posets and, as a conclusion, we are able to classify factorial functions of Eulerian binomial posets; (2) We give an almost complete classification of factorial functions of Eulerian Sheffer posets by dividing the original question into several cases; (3) In most cases above, we completely determine the structure of Eulerian Sheffer posets, a result stronger than just classifying factorial functions of these Eulerian Sheffer posets. We also study Eulerian triangular posets. This paper answers questions posed by R. Ehrenborg and M. Readdy. This research is also motivated by the work of R. Stanley about recognizing the \emphboolean lattice by looking at smaller intervals.]]> 0 <![CDATA[Projective reflection groups have been recently defined by the second author. They include a special class of groups denoted G(r,p,s,n) which contains all classical Weyl groups and more generally all the complex reflection groups of type G(r,p,n). In this paper we define some statistics analogous to descent number and major index over the projective reflection groups G(r,p,s,n), and we compute several generating functions concerning these parameters. Some aspects of the representation theory of G(r,p,s,n), as distribution of one-dimensional characters and computation of Hilbert series of some invariant algebras, are also treated.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2898 https://doi.org/10.46298/dmtcs.2898 Biagioli, Riccardo Caselli, Fabrizio Biagioli, Riccardo Caselli, Fabrizio <![CDATA[Projective reflection groups have been recently defined by the second author. They include a special class of groups denoted G(r,p,s,n) which contains all classical Weyl groups and more generally all the complex reflection groups of type G(r,p,n). In this paper we define some statistics analogous to descent number and major index over the projective reflection groups G(r,p,s,n), and we compute several generating functions concerning these parameters. Some aspects of the representation theory of G(r,p,s,n), as distribution of one-dimensional characters and computation of Hilbert series of some invariant algebras, are also treated.]]> 0 <![CDATA[We introduce deformations of the space of (multi-diagonal) harmonic polynomials for any finite complex reflection group of the form W=G(m,p,n), and give supporting evidence that this space seems to always be isomorphic, as a graded W-module, to the undeformed version.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2897 https://doi.org/10.46298/dmtcs.2897 Bergeron, François Borie, Nicolas Thiéry, Nicolas M. Bergeron, François Borie, Nicolas Thiéry, Nicolas M. <![CDATA[We introduce deformations of the space of (multi-diagonal) harmonic polynomials for any finite complex reflection group of the form W=G(m,p,n), and give supporting evidence that this space seems to always be isomorphic, as a graded W-module, to the undeformed version.]]> 0 <![CDATA[We construct a recursive formula for a complete system of primitive orthogonal idempotents for any R-trivial monoid. This uses the newly proved equivalence between the notions of R-trivial monoid and weakly ordered monoid.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2896 https://doi.org/10.46298/dmtcs.2896 Berg, Chris Bergeron, Nantel Bhargava, Sandeep Saliola, Franco Berg, Chris Bergeron, Nantel Bhargava, Sandeep Saliola, Franco <![CDATA[We construct a recursive formula for a complete system of primitive orthogonal idempotents for any R-trivial monoid. This uses the newly proved equivalence between the notions of R-trivial monoid and weakly ordered monoid.]]> 0 <![CDATA[We study the enumeration of \emphcolumn-convex permutominoes, i.e. column-convex polyominoes defined by a pair of permutations. We provide a direct recursive construction for the column-convex permutominoes of a given size, based on the application of the ECO method and generating trees, which leads to a functional equation. Then we obtain some upper and lower bounds for the number of column-convex permutominoes, and conjecture its asymptotic behavior using numerical analysis.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2895 https://doi.org/10.46298/dmtcs.2895 Beaton, Nicholas R. Disanto, Filippo Guttmann, Anthony J. Rinaldi, Simone Beaton, Nicholas R. Disanto, Filippo Guttmann, Anthony J. Rinaldi, Simone <![CDATA[We study the enumeration of \emphcolumn-convex permutominoes, i.e. column-convex polyominoes defined by a pair of permutations. We provide a direct recursive construction for the column-convex permutominoes of a given size, based on the application of the ECO method and generating trees, which leads to a functional equation. Then we obtain some upper and lower bounds for the number of column-convex permutominoes, and conjecture its asymptotic behavior using numerical analysis.]]> 0 <![CDATA[We prove a Murnaghan–Nakayama rule for k-Schur functions of Lapointe and Morse. That is, we give an explicit formula for the expansion of the product of a power sum symmetric function and a k-Schur function in terms of k-Schur functions. This is proved using the noncommutative k-Schur functions in terms of the nilCoxeter algebra introduced by Lam and the affine analogue of noncommutative symmetric functions of Fomin and Greene.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2894 https://doi.org/10.46298/dmtcs.2894 Bandlow, Jason Schilling, Anne Zabrocki, Mike Bandlow, Jason Schilling, Anne Zabrocki, Mike <![CDATA[We prove a Murnaghan–Nakayama rule for k-Schur functions of Lapointe and Morse. That is, we give an explicit formula for the expansion of the product of a power sum symmetric function and a k-Schur function in terms of k-Schur functions. This is proved using the noncommutative k-Schur functions in terms of the nilCoxeter algebra introduced by Lam and the affine analogue of noncommutative symmetric functions of Fomin and Greene.]]> 0 <![CDATA[Since every even power of the Vandermonde determinant is a symmetric polynomial, we want to understand its decomposition in terms of the basis of Schur functions. We investigate several combinatorial properties of the coefficients in the decomposition. In particular, I will give a recursive approach for computing the coefficient of the Schur function $s_μ$ in the decomposition of an even power of the Vandermonde determinant in $n+1$ variables in terms of the coefficient of the Schur function $s_λ$ in the decomposition of the same even power of the Vandermonde determinant in $n$ variables if the Young diagram of $μ$ is obtained from the Young diagram of $λ$ by adding a tetris type shape to the top or to the left.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2893 https://doi.org/10.46298/dmtcs.2893 Ballantine, Cristina Ballantine, Cristina <![CDATA[Since every even power of the Vandermonde determinant is a symmetric polynomial, we want to understand its decomposition in terms of the basis of Schur functions. We investigate several combinatorial properties of the coefficients in the decomposition. In particular, I will give a recursive approach for computing the coefficient of the Schur function $s_μ$ in the decomposition of an even power of the Vandermonde determinant in $n+1$ variables in terms of the coefficient of the Schur function $s_λ$ in the decomposition of the same even power of the Vandermonde determinant in $n$ variables if the Young diagram of $μ$ is obtained from the Young diagram of $λ$ by adding a tetris type shape to the top or to the left.]]> 0 <![CDATA[We show that the # product of binary trees introduced by Aval and Viennot (2008) is in fact defined at the level of the free associative algebra, and can be extended to most of the classical combinatorial Hopf algebras.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2892 https://doi.org/10.46298/dmtcs.2892 Aval, Jean-Christophe Novelli, Jean-Christophe Thibon, Jean-Yves Aval, Jean-Christophe Novelli, Jean-Christophe Thibon, Jean-Yves <![CDATA[We show that the # product of binary trees introduced by Aval and Viennot (2008) is in fact defined at the level of the free associative algebra, and can be extended to most of the classical combinatorial Hopf algebras.]]> 0 <![CDATA[This paper is about two arrangements of hyperplanes. The first — the Shi arrangement — was introduced by Jian-Yi Shi to describe the Kazhdan-Lusztig cells in the affine Weyl group of type A. The second — the Ish arrangement — was recently defined by the first author who used the two arrangements together to give a new interpretation of the q,t-Catalan numbers of Garsia and Haiman. In the present paper we will define a mysterious "combinatorial symmetry'' between the two arrangements and show that this symmetry preserves a great deal of information. For example, the Shi and Ish arrangements share the same characteristic polynomial, the same numbers of regions, bounded regions, dominant regions, regions with c "ceilings'' and d "degrees of freedom'', etc. Moreover, all of these results hold in the greater generality of "deleted'' Shi and Ish arrangements corresponding to an arbitrary subgraph of the complete graph. Our proofs are based on nice combinatorial labellings of Shi and Ish regions and a new set partition-valued statistic on these regions.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2890 https://doi.org/10.46298/dmtcs.2890 Armstrong, Drew Rhoades, Brendon Armstrong, Drew Rhoades, Brendon <![CDATA[This paper is about two arrangements of hyperplanes. The first — the Shi arrangement — was introduced by Jian-Yi Shi to describe the Kazhdan-Lusztig cells in the affine Weyl group of type A. The second — the Ish arrangement — was recently defined by the first author who used the two arrangements together to give a new interpretation of the q,t-Catalan numbers of Garsia and Haiman. In the present paper we will define a mysterious "combinatorial symmetry'' between the two arrangements and show that this symmetry preserves a great deal of information. For example, the Shi and Ish arrangements share the same characteristic polynomial, the same numbers of regions, bounded regions, dominant regions, regions with c "ceilings'' and d "degrees of freedom'', etc. Moreover, all of these results hold in the greater generality of "deleted'' Shi and Ish arrangements corresponding to an arbitrary subgraph of the complete graph. Our proofs are based on nice combinatorial labellings of Shi and Ish regions and a new set partition-valued statistic on these regions.]]> 0 <![CDATA[In 2003, Haglund's bounce statistic gave the first combinatorial interpretation of the q,t-Catalan numbers and the Hilbert series of diagonal harmonics. In this paper we propose a new combinatorial interpretation in terms of the affine Weyl group of type A. In particular, we define two statistics on affine permutations; one in terms of the Shi hyperplane arrangement, and one in terms of a new arrangement — which we call the Ish arrangement. We prove that our statistics are equivalent to the area' and bounce statistics of Haglund and Loehr. In this setting, we observe that bounce is naturally expressed as a statistic on the root lattice. We extend our statistics in two directions: to "extended'' Shi arrangements and to the bounded chambers of these arrangements. This leads to a (conjectural) combinatorial interpretation for all integral powers of the Bergeron-Garsia nabla operator applied to elementary symmetric functions.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2889 https://doi.org/10.46298/dmtcs.2889 Armstrong, Drew Armstrong, Drew <![CDATA[In 2003, Haglund's bounce statistic gave the first combinatorial interpretation of the q,t-Catalan numbers and the Hilbert series of diagonal harmonics. In this paper we propose a new combinatorial interpretation in terms of the affine Weyl group of type A. In particular, we define two statistics on affine permutations; one in terms of the Shi hyperplane arrangement, and one in terms of a new arrangement — which we call the Ish arrangement. We prove that our statistics are equivalent to the area' and bounce statistics of Haglund and Loehr. In this setting, we observe that bounce is naturally expressed as a statistic on the root lattice. We extend our statistics in two directions: to "extended'' Shi arrangements and to the bounded chambers of these arrangements. This leads to a (conjectural) combinatorial interpretation for all integral powers of the Bergeron-Garsia nabla operator applied to elementary symmetric functions.]]> 0 <![CDATA[Stanley (1986) showed how a finite partially ordered set gives rise to two polytopes, called the order polytope and chain polytope, which have the same Ehrhart polynomial despite being quite different combinatorially. We generalize his result to a wider family of polytopes constructed from a poset P with integers assigned to some of its elements. Through this construction, we explain combinatorially the relationship between the Gelfand–Tsetlin polytopes (1950) and the Feigin–Fourier–Littelmann–Vinberg polytopes (2010, 2005), which arise in the representation theory of the special linear Lie algebra. We then use the generalized Gelfand–Tsetlin polytopes of Berenstein and Zelevinsky (1989) to propose conjectural analogues of the Feigin–Fourier–Littelmann–Vinberg polytopes corresponding to the symplectic and odd orthogonal Lie algebras.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2888 https://doi.org/10.46298/dmtcs.2888 Ardila, Federico Bliem, Thomas Salazar, Dido Ardila, Federico Bliem, Thomas Salazar, Dido <![CDATA[Stanley (1986) showed how a finite partially ordered set gives rise to two polytopes, called the order polytope and chain polytope, which have the same Ehrhart polynomial despite being quite different combinatorially. We generalize his result to a wider family of polytopes constructed from a poset P with integers assigned to some of its elements. Through this construction, we explain combinatorially the relationship between the Gelfand–Tsetlin polytopes (1950) and the Feigin–Fourier–Littelmann–Vinberg polytopes (2010, 2005), which arise in the representation theory of the special linear Lie algebra. We then use the generalized Gelfand–Tsetlin polytopes of Berenstein and Zelevinsky (1989) to propose conjectural analogues of the Feigin–Fourier–Littelmann–Vinberg polytopes corresponding to the symplectic and odd orthogonal Lie algebras.]]> 0 <![CDATA[We prove Lagrange's theorem for Hopf monoids in the category of connected species. We deduce necessary conditions for a given subspecies $\textrm{k}$ of a Hopf monoid $\textrm{h}$ to be a Hopf submonoid: each of the generating series of $\textrm{k}$ must divide the corresponding generating series of $\textrm{k}$ in ℕ〚x〛. Among other corollaries we obtain necessary inequalities for a sequence of nonnegative integers to be the sequence of dimensions of a Hopf monoid. In the set-theoretic case the inequalities are linear and demand the non negativity of the binomial transform of the sequence.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2887 https://doi.org/10.46298/dmtcs.2887 Aguiar, Marcelo Lauve, Aaron Aguiar, Marcelo Lauve, Aaron <![CDATA[We prove Lagrange's theorem for Hopf monoids in the category of connected species. We deduce necessary conditions for a given subspecies $\textrm{k}$ of a Hopf monoid $\textrm{h}$ to be a Hopf submonoid: each of the generating series of $\textrm{k}$ must divide the corresponding generating series of $\textrm{k}$ in ℕ〚x〛. Among other corollaries we obtain necessary inequalities for a sequence of nonnegative integers to be the sequence of dimensions of a Hopf monoid. In the set-theoretic case the inequalities are linear and demand the non negativity of the binomial transform of the sequence.]]> 0 <![CDATA[We interpret the coefficients of the cyclotomic polynomial in terms of simplicial homology.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2945 https://doi.org/10.46298/dmtcs.2945 Musiker, Gregg Reiner, Victor Musiker, Gregg Reiner, Victor <![CDATA[We interpret the coefficients of the cyclotomic polynomial in terms of simplicial homology.]]> 0 <![CDATA[Let $\textbf{as}_n$ denote the length of a longest alternating subsequence in a uniformly random permutation of order $n$. Stanley studied the distribution of $\textbf{as}_n$ using algebraic methods, and showed in particular that $\mathbb{E}(\textbf{as}_n) = (4n+1)/6$ and $\textrm{Var}(\textbf{as}_n) = (32n-13)/180$. From Stanley's result it can be shown that after rescaling, $\textbf{as}_n$ converges in the limit to the Gaussian distribution. In this extended abstract we present a new approach to the study of $\textbf{as}_n$ by relating it to the sequence of local extrema of a random permutation, which is shown to form a "canonical'' longest alternating subsequence. Using this connection we reprove the abovementioned results in a more probabilistic and transparent way. We also study the distribution of the values of the local minima and maxima, and prove that in the limit the joint distribution of successive minimum-maximum pairs converges to the two-dimensional distribution whose density function is given by $f(s,t) = 3(1-s)t e^{t-s}$.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2956 https://doi.org/10.46298/dmtcs.2956 Romik, Dan Romik, Dan <![CDATA[Let $\textbf{as}_n$ denote the length of a longest alternating subsequence in a uniformly random permutation of order $n$. Stanley studied the distribution of $\textbf{as}_n$ using algebraic methods, and showed in particular that $\mathbb{E}(\textbf{as}_n) = (4n+1)/6$ and $\textrm{Var}(\textbf{as}_n) = (32n-13)/180$. From Stanley's result it can be shown that after rescaling, $\textbf{as}_n$ converges in the limit to the Gaussian distribution. In this extended abstract we present a new approach to the study of $\textbf{as}_n$ by relating it to the sequence of local extrema of a random permutation, which is shown to form a "canonical'' longest alternating subsequence. Using this connection we reprove the abovementioned results in a more probabilistic and transparent way. We also study the distribution of the values of the local minima and maxima, and prove that in the limit the joint distribution of successive minimum-maximum pairs converges to the two-dimensional distribution whose density function is given by $f(s,t) = 3(1-s)t e^{t-s}$.]]> 0 <![CDATA[When $W$ is a finite reflection group, the noncrossing partition lattice $NC(W)$ of type $W$ is a very rich combinatorial object, extending the notion of noncrossing partitions of an $n$-gon. A formula (for which the only known proofs are case-by-case) expresses the number of multichains of a given length in $NC(W)$ as a generalized Fuß-Catalan number, depending on the invariant degrees of $W$. We describe how to understand some specifications of this formula in a case-free way, using an interpretation of the chains of $NC(W)$ as fibers of a "Lyashko-Looijenga covering''. This covering is constructed from the geometry of the discriminant hypersurface of $W$. We deduce new enumeration formulas for certain factorizations of a Coxeter element of $W$.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2955 https://doi.org/10.46298/dmtcs.2955 Ripoll, Vivien Ripoll, Vivien <![CDATA[When $W$ is a finite reflection group, the noncrossing partition lattice $NC(W)$ of type $W$ is a very rich combinatorial object, extending the notion of noncrossing partitions of an $n$-gon. A formula (for which the only known proofs are case-by-case) expresses the number of multichains of a given length in $NC(W)$ as a generalized Fuß-Catalan number, depending on the invariant degrees of $W$. We describe how to understand some specifications of this formula in a case-free way, using an interpretation of the chains of $NC(W)$ as fibers of a "Lyashko-Looijenga covering''. This covering is constructed from the geometry of the discriminant hypersurface of $W$. We deduce new enumeration formulas for certain factorizations of a Coxeter element of $W$.]]> 0 <![CDATA[The spinor variety is cut out by the quadratic Wick relations among the principal Pfaffians of an $n \times n$ skew-symmetric matrix. Its points correspond to $n$-dimensional isotropic subspaces of a $2n$-dimensional vector space. In this paper we tropicalize this picture, and we develop a combinatorial theory of tropical Wick vectors and tropical linear spaces that are tropically isotropic. We characterize tropical Wick vectors in terms of subdivisions of Delta-matroid polytopes, and we examine to what extent the Wick relations form a tropical basis. Our theory generalizes several results for tropical linear spaces and valuated matroids to the class of Coxeter matroids of type $D$.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2954 https://doi.org/10.46298/dmtcs.2954 Rincón, Felipe Rincón, Felipe <![CDATA[The spinor variety is cut out by the quadratic Wick relations among the principal Pfaffians of an $n \times n$ skew-symmetric matrix. Its points correspond to $n$-dimensional isotropic subspaces of a $2n$-dimensional vector space. In this paper we tropicalize this picture, and we develop a combinatorial theory of tropical Wick vectors and tropical linear spaces that are tropically isotropic. We characterize tropical Wick vectors in terms of subdivisions of Delta-matroid polytopes, and we examine to what extent the Wick relations form a tropical basis. Our theory generalizes several results for tropical linear spaces and valuated matroids to the class of Coxeter matroids of type $D$.]]> 0 <![CDATA[We prove the cyclic sieving phenomenon for non-crossing forests and non-crossing graphs. More precisely, the cyclic group acts on these graphs naturally by rotation and we show that the orbit structure of this action is encoded by certain polynomials. Our results confirm two conjectures of Alan Guo.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2953 https://doi.org/10.46298/dmtcs.2953 Poznanović, Svetlana Poznanović, Svetlana <![CDATA[We prove the cyclic sieving phenomenon for non-crossing forests and non-crossing graphs. More precisely, the cyclic group acts on these graphs naturally by rotation and we show that the orbit structure of this action is encoded by certain polynomials. Our results confirm two conjectures of Alan Guo.]]> 0 <![CDATA[The associahedron is a polytope whose graph is the graph of flips on triangulations of a convex polygon. Pseudotriangulations and multitriangulations generalize triangulations in two different ways, which have been unified by Pilaud and Pocchiola in their study of pseudoline arrangements with contacts supported by a given network. In this paper, we construct the "brick polytope'' of a network, obtained as the convex hull of the "brick vectors'' associated to each pseudoline arrangement supported by the network. We characterize its vertices, describe its faces, and decompose it as a Minkowski sum of simpler polytopes. Our brick polytopes include Hohlweg and Lange's many realizations of the associahedron, which arise as brick polytopes of certain well-chosen networks.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2952 https://doi.org/10.46298/dmtcs.2952 Pilaud, Vincent Santos, Francisco Pilaud, Vincent Santos, Francisco <![CDATA[The associahedron is a polytope whose graph is the graph of flips on triangulations of a convex polygon. Pseudotriangulations and multitriangulations generalize triangulations in two different ways, which have been unified by Pilaud and Pocchiola in their study of pseudoline arrangements with contacts supported by a given network. In this paper, we construct the "brick polytope'' of a network, obtained as the convex hull of the "brick vectors'' associated to each pseudoline arrangement supported by the network. We characterize its vertices, describe its faces, and decompose it as a Minkowski sum of simpler polytopes. Our brick polytopes include Hohlweg and Lange's many realizations of the associahedron, which arise as brick polytopes of certain well-chosen networks.]]> 0 <![CDATA[We continue a study of the equivalence class induced on $S_n$ when one is permitted to replace a consecutive set of elements in a permutation with the same elements in a different order. For each possible set of allowed replacements, we characterise and/or enumerate the set of permutations reachable from the identity. In some cases we also count the number of equivalence classes.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2951 https://doi.org/10.46298/dmtcs.2951 Pierrot, Adeline Rossin, Dominique West, Julian Pierrot, Adeline Rossin, Dominique West, Julian <![CDATA[We continue a study of the equivalence class induced on $S_n$ when one is permitted to replace a consecutive set of elements in a permutation with the same elements in a different order. For each possible set of allowed replacements, we characterise and/or enumerate the set of permutations reachable from the identity. In some cases we also count the number of equivalence classes.]]> 0 <![CDATA[We consider a new kind of straight and shifted plane partitions/Young tableaux — ones whose diagrams are no longer of partition shape, but rather Young diagrams with boxes erased from their upper right ends. We find formulas for the number of standard tableaux in certain cases, namely a shifted staircase without the box in its upper right corner, i.e. truncated by a box, a rectangle truncated by a staircase and a rectangle truncated by a square minus a box. The proofs involve finding the generating function of the corresponding plane partitions using interpretations and formulas for sums of restricted Schur functions and their specializations. The number of standard tableaux is then found as a certain limit of this function.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2950 https://doi.org/10.46298/dmtcs.2950 Panova, Greta Panova, Greta <![CDATA[We consider a new kind of straight and shifted plane partitions/Young tableaux — ones whose diagrams are no longer of partition shape, but rather Young diagrams with boxes erased from their upper right ends. We find formulas for the number of standard tableaux in certain cases, namely a shifted staircase without the box in its upper right corner, i.e. truncated by a box, a rectangle truncated by a staircase and a rectangle truncated by a square minus a box. The proofs involve finding the generating function of the corresponding plane partitions using interpretations and formulas for sums of restricted Schur functions and their specializations. The number of standard tableaux is then found as a certain limit of this function.]]> 0 <![CDATA[Words $a_1 a_2 \ldots a_n$ with independent letters $a_k$ taken from the set of natural numbers, and a weight (probability) attached via the geometric distribution $pq^{i-1}(p+q=1)$ are considered. A consecutive record (motivated by the analysis of a skip list structure) can only advance from $k$ to $k+1$, thus ignoring perhaps some larger (=superior) values. We investigate the number of these rejected superior values. Further, we study the probability that there is a single consecutive maximum and show that (apart from fluctuations) it tends to a constant.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2949 https://doi.org/10.46298/dmtcs.2949 Oliver, Kamilla Prodinger, Helmut Oliver, Kamilla Prodinger, Helmut <![CDATA[Words $a_1 a_2 \ldots a_n$ with independent letters $a_k$ taken from the set of natural numbers, and a weight (probability) attached via the geometric distribution $pq^{i-1}(p+q=1)$ are considered. A consecutive record (motivated by the analysis of a skip list structure) can only advance from $k$ to $k+1$, thus ignoring perhaps some larger (=superior) values. We investigate the number of these rejected superior values. Further, we study the probability that there is a single consecutive maximum and show that (apart from fluctuations) it tends to a constant.]]> 0 <![CDATA[For an affine algebra of nonexceptional type in the large rank we show the fermionic formula depends only on the attachment of the node 0 of the Dynkin diagram to the rest, and the fermionic formula of not type $A$ can be expressed as a sum of that of type $A$ with Littlewood–Richardson coefficients. Combining this result with theorems of Kirillov–Schilling–Shimozono and Lecouvey–Okado–Shimozono, we settle the $X=M$ conjecture under the large rank hypothesis.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2948 https://doi.org/10.46298/dmtcs.2948 Okado, Masato Sakamoto, Reiho Okado, Masato Sakamoto, Reiho <![CDATA[For an affine algebra of nonexceptional type in the large rank we show the fermionic formula depends only on the attachment of the node 0 of the Dynkin diagram to the rest, and the fermionic formula of not type $A$ can be expressed as a sum of that of type $A$ with Littlewood–Richardson coefficients. Combining this result with theorems of Kirillov–Schilling–Shimozono and Lecouvey–Okado–Shimozono, we settle the $X=M$ conjecture under the large rank hypothesis.]]> 0 <![CDATA[Develin and Sturmfels showed that regular triangulations of $\Delta_{n-1} \times \Delta_{d-1}$ can be thought of as tropical polytopes. Tropical oriented matroids were defined by Ardila and Develin, and were conjectured to be in bijection with all subdivisions of $\Delta_{n-1} \times \Delta_{d-1}$. In this paper, we show that any triangulation of $\Delta_{n-1} \times \Delta_{d-1}$ encodes a tropical oriented matroid. We also suggest a new class of combinatorial objects that may describe all subdivisions of a bigger class of polytopes.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2947 https://doi.org/10.46298/dmtcs.2947 Oh, Suho Yoo, Hwanchul Oh, Suho Yoo, Hwanchul <![CDATA[Develin and Sturmfels showed that regular triangulations of $\Delta_{n-1} \times \Delta_{d-1}$ can be thought of as tropical polytopes. Tropical oriented matroids were defined by Ardila and Develin, and were conjectured to be in bijection with all subdivisions of $\Delta_{n-1} \times \Delta_{d-1}$. In this paper, we show that any triangulation of $\Delta_{n-1} \times \Delta_{d-1}$ encodes a tropical oriented matroid. We also suggest a new class of combinatorial objects that may describe all subdivisions of a bigger class of polytopes.]]> 0 <![CDATA[Stanley has conjectured that the h-vector of a matroid complex is a pure O-sequence. We will prove this for cotransversal matroids by using generalized permutohedra. We construct a bijection between lattice points inside a $r$-dimensional convex polytope and bases of a rank $r$ transversal matroid.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2946 https://doi.org/10.46298/dmtcs.2946 Oh, Suho Oh, Suho <![CDATA[Stanley has conjectured that the h-vector of a matroid complex is a pure O-sequence. We will prove this for cotransversal matroids by using generalized permutohedra. We construct a bijection between lattice points inside a $r$-dimensional convex polytope and bases of a rank $r$ transversal matroid.]]> 0 <![CDATA[We give a combinatorial proof of a Touchard-Riordan-like formula discovered by the first author. As a consequence we find a connection between his formula and Jacobi's triple product identity. We then give a combinatorial analog of Jacobi's triple product identity by showing that a finite sum can be interpreted as a generating function of weighted Schröder paths, so that the triple product identity is recovered by taking the limit. This can be stated in terms of some continued fractions called T-fractions, whose important property is the fact that they satisfy some functional equation. We show that this result permits to explain and generalize some Touchard-Riordan-like formulas appearing in enumerative problems.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2934 https://doi.org/10.46298/dmtcs.2934 Josuat-Vergès, Matthieu Kim, Jang-Soo Josuat-Vergès, Matthieu Kim, Jang-Soo <![CDATA[We give a combinatorial proof of a Touchard-Riordan-like formula discovered by the first author. As a consequence we find a connection between his formula and Jacobi's triple product identity. We then give a combinatorial analog of Jacobi's triple product identity by showing that a finite sum can be interpreted as a generating function of weighted Schröder paths, so that the triple product identity is recovered by taking the limit. This can be stated in terms of some continued fractions called T-fractions, whose important property is the fact that they satisfy some functional equation. We show that this result permits to explain and generalize some Touchard-Riordan-like formulas appearing in enumerative problems.]]> 0 <![CDATA[This paper is devoted to the evaluation of the generating series of the connection coefficients of the double cosets of the hyperoctahedral group. Hanlon, Stanley, Stembridge (1992) showed that this series, indexed by a partition $ν$, gives the spectral distribution of some random matrices that are of interest in random matrix theory. We provide an explicit evaluation of this series when $ν =(n)$ in terms of monomial symmetric functions. Our development relies on an interpretation of the connection coefficients in terms of locally orientable hypermaps and a new bijective construction between partitioned locally orientable hypermaps and some permuted forests.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2944 https://doi.org/10.46298/dmtcs.2944 Morales, Alejandro H. Vassilieva, Ekaterina A. Morales, Alejandro H. Vassilieva, Ekaterina A. <![CDATA[This paper is devoted to the evaluation of the generating series of the connection coefficients of the double cosets of the hyperoctahedral group. Hanlon, Stanley, Stembridge (1992) showed that this series, indexed by a partition $ν$, gives the spectral distribution of some random matrices that are of interest in random matrix theory. We provide an explicit evaluation of this series when $ν =(n)$ in terms of monomial symmetric functions. Our development relies on an interpretation of the connection coefficients in terms of locally orientable hypermaps and a new bijective construction between partitioned locally orientable hypermaps and some permuted forests.]]> 0 <![CDATA[We show that the shapes of integer partitions chosen randomly according to Schur-Weyl measures of parameter $\alpha =1/2$ and Gelfand measures satisfy Kerov's central limit theorem. Thus, there is a gaussian process $\Delta$ such that under Plancherel, Schur-Weyl or Gelfand measures, the deviations $\Delta_n(s)=\lambda _n(\sqrt{n} s)-\sqrt{n} \lambda _{\infty}^{\ast}(s)$ converge in law towards $\Delta (s)$, up to a translation along the $x$-axis in the case of Schur-Weyl measures, and up to a factor $\sqrt{2}$ and a deterministic remainder in the case of Gelfand measures. The proofs of these results follow the one given by Ivanov and Olshanski for Plancherel measures; hence, one uses a "method of noncommutative moments''.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2943 https://doi.org/10.46298/dmtcs.2943 Méliot, Pierre-Loïc Méliot, Pierre-Loïc <![CDATA[We show that the shapes of integer partitions chosen randomly according to Schur-Weyl measures of parameter $\alpha =1/2$ and Gelfand measures satisfy Kerov's central limit theorem. Thus, there is a gaussian process $\Delta$ such that under Plancherel, Schur-Weyl or Gelfand measures, the deviations $\Delta_n(s)=\lambda _n(\sqrt{n} s)-\sqrt{n} \lambda _{\infty}^{\ast}(s)$ converge in law towards $\Delta (s)$, up to a translation along the $x$-axis in the case of Schur-Weyl measures, and up to a factor $\sqrt{2}$ and a deterministic remainder in the case of Gelfand measures. The proofs of these results follow the one given by Ivanov and Olshanski for Plancherel measures; hence, one uses a "method of noncommutative moments''.]]> 0 <![CDATA[Haglund, Luoto, Mason, and van Willigenburg introduced a basis for quasisymmetric functions called the $\textit{quasisymmetric Schur function basis}$ which are generated combinatorially through fillings of composition diagrams in much the same way as Schur functions are generated through reverse column-strict tableaux. We introduce a new basis for quasisymmetric functions called the $\textit{row-strict quasisymmetric Schur function basis}$ which are generated combinatorially through fillings of composition diagrams in much the same way as Schur functions are generated through row-strict tableaux. We describe the relationship between this new basis and other known bases for quasisymmetric functions, as well as its relationship to Schur polynomials. We obtain a refinement of the omega transform operator as a result of these relationships.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2942 https://doi.org/10.46298/dmtcs.2942 Mason, Sarah K Remmel, Jeffrey Mason, Sarah K Remmel, Jeffrey <![CDATA[Haglund, Luoto, Mason, and van Willigenburg introduced a basis for quasisymmetric functions called the $\textit{quasisymmetric Schur function basis}$ which are generated combinatorially through fillings of composition diagrams in much the same way as Schur functions are generated through reverse column-strict tableaux. We introduce a new basis for quasisymmetric functions called the $\textit{row-strict quasisymmetric Schur function basis}$ which are generated combinatorially through fillings of composition diagrams in much the same way as Schur functions are generated through row-strict tableaux. We describe the relationship between this new basis and other known bases for quasisymmetric functions, as well as its relationship to Schur polynomials. We obtain a refinement of the omega transform operator as a result of these relationships.]]> 0 <![CDATA[We study the functions that count matrices of given rank over a finite field with specified positions equal to zero. We show that these matrices are $q$-analogues of permutations with certain restricted values. We obtain a simple closed formula for the number of invertible matrices with zero diagonal, a $q$-analogue of derangements, and a curious relationship between invertible skew-symmetric matrices and invertible symmetric matrices with zero diagonal. In addition, we provide recursions to enumerate matrices and symmetric matrices with zero diagonal by rank. Finally, we provide a brief exposition of polynomiality results for enumeration questions related to those mentioned, and give several open questions.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2941 https://doi.org/10.46298/dmtcs.2941 Lewis, Joel Brewster Liu, Ricky Ini Morales, Alejandro H. Panova, Greta Sam, Steven V Zhang, Yan Lewis, Joel Brewster Liu, Ricky Ini Morales, Alejandro H. Panova, Greta Sam, Steven V Zhang, Yan <![CDATA[We study the functions that count matrices of given rank over a finite field with specified positions equal to zero. We show that these matrices are $q$-analogues of permutations with certain restricted values. We obtain a simple closed formula for the number of invertible matrices with zero diagonal, a $q$-analogue of derangements, and a curious relationship between invertible skew-symmetric matrices and invertible symmetric matrices with zero diagonal. In addition, we provide recursions to enumerate matrices and symmetric matrices with zero diagonal by rank. Finally, we provide a brief exposition of polynomiality results for enumeration questions related to those mentioned, and give several open questions.]]> 0 <![CDATA[We examine the $q=1$ and $t=0$ special cases of the parking functions conjecture. The parking functions conjecture states that the Hilbert series for the space of diagonal harmonics is equal to the bivariate generating function of $area$ and $dinv$ over the set of parking functions. Haglund recently proved that the Hilbert series for the space of diagonal harmonics is equal to a bivariate generating function over the set of Tesler matrices–upper-triangular matrices with every hook sum equal to one. We give a combinatorial interpretation of the Haglund generating function at $q=1$ and prove the corresponding case of the parking functions conjecture (first proven by Garsia and Haiman). We also discuss a possible proof of the $t = 0$ case consistent with this combinatorial interpretation. We conclude by briefly discussing possible refinements of the parking functions conjecture arising from this research and point of view. $\textbf{Note added in proof}$: We have since found such a proof of the $t = 0$ case and conjectured more detailed refinements. This research will most likely be presented in full in a forthcoming article.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2940 https://doi.org/10.46298/dmtcs.2940 Levande, Paul Levande, Paul <![CDATA[We examine the $q=1$ and $t=0$ special cases of the parking functions conjecture. The parking functions conjecture states that the Hilbert series for the space of diagonal harmonics is equal to the bivariate generating function of $area$ and $dinv$ over the set of parking functions. Haglund recently proved that the Hilbert series for the space of diagonal harmonics is equal to a bivariate generating function over the set of Tesler matrices–upper-triangular matrices with every hook sum equal to one. We give a combinatorial interpretation of the Haglund generating function at $q=1$ and prove the corresponding case of the parking functions conjecture (first proven by Garsia and Haiman). We also discuss a possible proof of the $t = 0$ case consistent with this combinatorial interpretation. We conclude by briefly discussing possible refinements of the parking functions conjecture arising from this research and point of view. $\textbf{Note added in proof}$: We have since found such a proof of the $t = 0$ case and conjectured more detailed refinements. This research will most likely be presented in full in a forthcoming article.]]> 0 <![CDATA[Zonotopal algebra deals with ideals and vector spaces of polynomials that are related to several combinatorial and geometric structures defined by a finite sequence of vectors. Given such a sequence $X$, an integer $k \geq -1$ and an upper set in the lattice of flats of the matroid defined by $X$, we define and study the associated $\textit{hierarchical zonotopal power ideal}$. This ideal is generated by powers of linear forms. Its Hilbert series depends only on the matroid structure of $X$. It is related to various other matroid invariants, $\textit{e. g.}$ the shelling polynomial and the characteristic polynomial. This work unifies and generalizes results by Ardila-Postnikov on power ideals and by Holtz-Ron and Holtz-Ron-Xu on (hierarchical) zonotopal algebra. We also generalize a result on zonotopal Cox modules due to Sturmfels-Xu.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2939 https://doi.org/10.46298/dmtcs.2939 Lenz, Matthias Lenz, Matthias <![CDATA[Zonotopal algebra deals with ideals and vector spaces of polynomials that are related to several combinatorial and geometric structures defined by a finite sequence of vectors. Given such a sequence $X$, an integer $k \geq -1$ and an upper set in the lattice of flats of the matroid defined by $X$, we define and study the associated $\textit{hierarchical zonotopal power ideal}$. This ideal is generated by powers of linear forms. Its Hilbert series depends only on the matroid structure of $X$. It is related to various other matroid invariants, $\textit{e. g.}$ the shelling polynomial and the characteristic polynomial. This work unifies and generalizes results by Ardila-Postnikov on power ideals and by Holtz-Ron and Holtz-Ron-Xu on (hierarchical) zonotopal algebra. We also generalize a result on zonotopal Cox modules due to Sturmfels-Xu.]]> 0 <![CDATA[Realisations of associahedra can be obtained from the classical permutahedron by removing some of its facets and the set of facets is determined by the diagonals of certain labeled convex planar $n$-gons as shown by Hohlweg and Lange (2007). Ardila, Benedetti, and Doker (2010) expressed polytopes of this type as Minkowski sums and differences of scaled faces of a standard simplex and computed the corresponding coefficients $y_I$ by Möbius inversion from the $z_I$ if tight right-hand sides $z_I$ for all inequalities of the permutahedron are assumed. Given an associahedron of Hohlweg and Lange, we first characterise all tight values $z_I$ in terms of non-crossing diagonals of the associated labeled $n$-gon, simplify the formula of Ardila et al., and characterise the remaining terms combinatorially.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2938 https://doi.org/10.46298/dmtcs.2938 Lange, Carsten Lange, Carsten <![CDATA[Realisations of associahedra can be obtained from the classical permutahedron by removing some of its facets and the set of facets is determined by the diagonals of certain labeled convex planar $n$-gons as shown by Hohlweg and Lange (2007). Ardila, Benedetti, and Doker (2010) expressed polytopes of this type as Minkowski sums and differences of scaled faces of a standard simplex and computed the corresponding coefficients $y_I$ by Möbius inversion from the $z_I$ if tight right-hand sides $z_I$ for all inequalities of the permutahedron are assumed. Given an associahedron of Hohlweg and Lange, we first characterise all tight values $z_I$ in terms of non-crossing diagonals of the associated labeled $n$-gon, simplify the formula of Ardila et al., and characterise the remaining terms combinatorially.]]> 0 <![CDATA[We give explicit formulas for the number $U_n(N)$ of closed polygonal paths of length $N$ (starting from the origin) whose steps are $n^{\textrm{th}}$ roots of unity, as well as asymptotic expressions for these numbers when $N \rightarrow \infty$. We also prove that the sequences $(U_n(N))_{N \geq 0}$ are $P$-recursive for each fixed $n \geq 1$ and leave open the problem of determining the values of $N$ for which the $\textit{dual}$ sequences $(U_n(N))_{n \geq 1}$ are $P$-recursive.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2937 https://doi.org/10.46298/dmtcs.2937 Labelle, Gilbert Lacasse, Annie Labelle, Gilbert Lacasse, Annie <![CDATA[We give explicit formulas for the number $U_n(N)$ of closed polygonal paths of length $N$ (starting from the origin) whose steps are $n^{\textrm{th}}$ roots of unity, as well as asymptotic expressions for these numbers when $N \rightarrow \infty$. We also prove that the sequences $(U_n(N))_{N \geq 0}$ are $P$-recursive for each fixed $n \geq 1$ and leave open the problem of determining the values of $N$ for which the $\textit{dual}$ sequences $(U_n(N))_{n \geq 1}$ are $P$-recursive.]]> 0 <![CDATA[In this extended abstract, we extend recent results of Assaf and McNamara, the skew Pieri rule and the skew Murnaghan-Nakayama rule, to a more general identity, which gives an elegant expansion of the product of a skew Schur function with a quantum power sum function in terms of skew Schur functions. We give two proofs, one completely bijective in the spirit of Assaf-McNamara's original proof, and one via Lam-Lauve-Sotille's skew Littlewood-Richardson rule.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2936 https://doi.org/10.46298/dmtcs.2936 Konvalinka, Matjaž Konvalinka, Matjaž <![CDATA[In this extended abstract, we extend recent results of Assaf and McNamara, the skew Pieri rule and the skew Murnaghan-Nakayama rule, to a more general identity, which gives an elegant expansion of the product of a skew Schur function with a quantum power sum function in terms of skew Schur functions. We give two proofs, one completely bijective in the spirit of Assaf-McNamara's original proof, and one via Lam-Lauve-Sotille's skew Littlewood-Richardson rule.]]> 0 <![CDATA[In this paper we study topological properties of the poset of injective words and the lattice of classical non-crossing partitions. Specifically, it is shown that after the removal of the bottom and top elements (if existent) these posets are doubly Cohen-Macaulay. This extends the well-known result that those posets are shellable. Both results rely on a new poset fiber theorem, for doubly homotopy Cohen-Macaulay posets, which can be considered as an extension of the classical poset fiber theorem for homotopy Cohen-Macaulay posets.]]> Fri, 31 Dec 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2935 https://doi.org/10.46298/dmtcs.2935 Kallipoliti, Myrto Kubitzke, Martina Kallipoliti, Myrto Kubitzke, Martina <![CDATA[In this paper we study topological properties of the poset of injective words and the lattice of classical non-crossing partitions. Specifically, it is shown that after the removal of the bottom and top elements (if existent) these posets are doubly Cohen-Macaulay. This extends the well-known result that those posets are shellable. Both results rely on a new poset fiber theorem, for doubly homotopy Cohen-Macaulay posets, which can be considered as an extension of the classical poset fiber theorem for homotopy Cohen-Macaulay posets.]]> 0 <![CDATA[We construct explicit solutions of a number of Stieltjes moment problems based on moments of the form ${\rho}_{1}^{(r)}(n)=(2rn)!$ and ${\rho}_{2}^{(r)}(n)=[(rn)!]^{2}$, $r=1,2,\dots$, $n=0,1,2,\dots$, \textit{i.e.} we find functions $W^{(r)}_{1,2}(x)>0$ satisfying $\int_{0}^{\infty}x^{n}W^{(r)}_{1,2}(x)dx = {\rho}_{1,2}^{(r)}(n)$. It is shown using criteria for uniqueness and non-uniqueness (Carleman, Krein, Berg, Pakes, Stoyanov) that for $r>1$ both ${\rho}_{1,2}^{(r)}(n)$ give rise to non-unique solutions. Examples of such solutions are constructed using the technique of the inverse Mellin transform supplemented by a Mellin convolution. We outline a general method of generating non-unique solutions for moment problems generalizing ${\rho}_{1,2}^{(r)}(n)$, such as the product ${\rho}_{1}^{(r)}(n)\cdot{\rho}_{2}^{(r)}(n)$ and $[(rn)!]^{p}$, $p=3,4,\dots$.]]> Wed, 15 Sep 2010 06:00:00 +0000 https://doi.org/10.46298/dmtcs.507 https://doi.org/10.46298/dmtcs.507 Penson, K. A. Blasiak, Pawel Duchamp, Gérard, Horzela, A. Solomon, A. I. Penson, K. A. Blasiak, Pawel Duchamp, Gérard, Horzela, A. Solomon, A. I. <![CDATA[We construct explicit solutions of a number of Stieltjes moment problems based on moments of the form ${\rho}_{1}^{(r)}(n)=(2rn)!$ and ${\rho}_{2}^{(r)}(n)=[(rn)!]^{2}$, $r=1,2,\dots$, $n=0,1,2,\dots$, \textit{i.e.} we find functions $W^{(r)}_{1,2}(x)>0$ satisfying $\int_{0}^{\infty}x^{n}W^{(r)}_{1,2}(x)dx = {\rho}_{1,2}^{(r)}(n)$. It is shown using criteria for uniqueness and non-uniqueness (Carleman, Krein, Berg, Pakes, Stoyanov) that for $r>1$ both ${\rho}_{1,2}^{(r)}(n)$ give rise to non-unique solutions. Examples of such solutions are constructed using the technique of the inverse Mellin transform supplemented by a Mellin convolution. We outline a general method of generating non-unique solutions for moment problems generalizing ${\rho}_{1,2}^{(r)}(n)$, such as the product ${\rho}_{1}^{(r)}(n)\cdot{\rho}_{2}^{(r)}(n)$ and $[(rn)!]^{p}$, $p=3,4,\dots$.]]> 0 <![CDATA[Honoring László (Laci) Babai's 60th birthday, the conference "Combinatorics, Groups, Algorithms, and Complexity" (Ohio State University, March 15-25, 2010) explored the links between the areas mentioned in the title. These areas represent Laci's wide interests in mathematics and theoretical computer science; his work has revealed and enriched many of the interconnections between them. The conference had 109 participants from North America, Europe, Asia, and Australia (31 of them from overseas), including 3 Nevanlinna prize winners, 32 students, 13 postdocs, 20 females, and 18 former and current students of Laci Babai. The program consisted of 73 talks and a problem session. The full list of talks can be found in the introductory article by the guest editors of this special issue who also served as the organizers of the conference. We thank all participants and speakers for the success of the conference. We wish to express our gratitude to the National Science Foundation, National Security Agency, and The Ohio State Mathematical Research Institute for their generous support. This special issue contains papers in the conference topics, but not necessarily coinciding with the authors' talks at the conference. Each paper has been peer-reviewed. Toniann Pitassi, László Pyber, Uwe Schöning, Jiří Sgall, and Aner Shalev served with us as editors of this special issue. We thank for their work as well as for the assistance of the anonymous referees.]]> Sat, 20 Mar 2010 23:00:00 +0000 https://doi.org/10.46298/dmtcs.529 https://doi.org/10.46298/dmtcs.529 Seress, Akos Szegedy, Mario Seress, Akos Szegedy, Mario <![CDATA[Honoring László (Laci) Babai's 60th birthday, the conference "Combinatorics, Groups, Algorithms, and Complexity" (Ohio State University, March 15-25, 2010) explored the links between the areas mentioned in the title. These areas represent Laci's wide interests in mathematics and theoretical computer science; his work has revealed and enriched many of the interconnections between them. The conference had 109 participants from North America, Europe, Asia, and Australia (31 of them from overseas), including 3 Nevanlinna prize winners, 32 students, 13 postdocs, 20 females, and 18 former and current students of Laci Babai. The program consisted of 73 talks and a problem session. The full list of talks can be found in the introductory article by the guest editors of this special issue who also served as the organizers of the conference. We thank all participants and speakers for the success of the conference. We wish to express our gratitude to the National Science Foundation, National Security Agency, and The Ohio State Mathematical Research Institute for their generous support. This special issue contains papers in the conference topics, but not necessarily coinciding with the authors' talks at the conference. Each paper has been peer-reviewed. Toniann Pitassi, László Pyber, Uwe Schöning, Jiří Sgall, and Aner Shalev served with us as editors of this special issue. We thank for their work as well as for the assistance of the anonymous referees.]]> 0 <![CDATA[The polynomial ring Z[x(11), ..., x(33)] has a basis called the dual canonical basis whose quantization facilitates the study of representations of the quantum group U-q(sl(3) (C)). On the other hand, Z[x(1 1), ... , x(33)] inherits a basis from the cluster monomial basis of a geometric model of the type D-4 cluster algebra. We prove that these two bases are equal. This extends work of Skandera and proves a conjecture of Fomin and Zelevinsky.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.515 https://doi.org/10.46298/dmtcs.515 Rhoades, Brendon Rhoades, Brendon <![CDATA[The polynomial ring Z[x(11), ..., x(33)] has a basis called the dual canonical basis whose quantization facilitates the study of representations of the quantum group U-q(sl(3) (C)). On the other hand, Z[x(1 1), ... , x(33)] inherits a basis from the cluster monomial basis of a geometric model of the type D-4 cluster algebra. We prove that these two bases are equal. This extends work of Skandera and proves a conjecture of Fomin and Zelevinsky.]]> 0 <![CDATA[Following the model of Bondesson, Nilsson, and Wikstrand, we consider randomly filled urns, where the probability of falling into urn i is the geometric probability (1-q)qi-1. Assuming n independent random entries, and a fixed parameter k, the interest is in the following parameters: Let T be the smallest index, such that urn T is non-empty, but the following k are empty, then: XT= number of balls in urn T, ST= number of balls in urns with index larger than T, and finally T itself..]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.527 https://doi.org/10.46298/dmtcs.527 Louchard, Guy Prodinger, Helmut Louchard, Guy Prodinger, Helmut <![CDATA[Following the model of Bondesson, Nilsson, and Wikstrand, we consider randomly filled urns, where the probability of falling into urn i is the geometric probability (1-q)qi-1. Assuming n independent random entries, and a fixed parameter k, the interest is in the following parameters: Let T be the smallest index, such that urn T is non-empty, but the following k are empty, then: XT= number of balls in urn T, ST= number of balls in urns with index larger than T, and finally T itself..]]> 0 <![CDATA[In this work, we focus our attention to algorithmic solutions for problems where the instances are presented as straight-line programs on a given algebra. In our exposition, we try to survey general results by presenting some meaningful examples; moreover, where possible, we outline the proofs in order to give an insight of the methods and the techniques. We recall some recent results for the problem PosSLP, consisting of deciding if the integer defined by a straight-line program on the ring Z is greater than zero; we discuss some implications in the areas of numerical analysis and strategic games. Furthermore, we propose some methods for reducing Compressed Word Problem from an algebra to another; reductions from trace monoids to the semiring of nonnegative integers are exhibited and polynomial time algorithms for compressed equivalence in monoids related to Dyck reductions are shown. Finally, we consider inclusion problems for context-free languages, proving how in some cases efficient algorithms for these problems benefit from the ability to work with compressed data.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.526 https://doi.org/10.46298/dmtcs.526 Bertoni, Alberto Radicioni, Roberto Bertoni, Alberto Radicioni, Roberto <![CDATA[In this work, we focus our attention to algorithmic solutions for problems where the instances are presented as straight-line programs on a given algebra. In our exposition, we try to survey general results by presenting some meaningful examples; moreover, where possible, we outline the proofs in order to give an insight of the methods and the techniques. We recall some recent results for the problem PosSLP, consisting of deciding if the integer defined by a straight-line program on the ring Z is greater than zero; we discuss some implications in the areas of numerical analysis and strategic games. Furthermore, we propose some methods for reducing Compressed Word Problem from an algebra to another; reductions from trace monoids to the semiring of nonnegative integers are exhibited and polynomial time algorithms for compressed equivalence in monoids related to Dyck reductions are shown. Finally, we consider inclusion problems for context-free languages, proving how in some cases efficient algorithms for these problems benefit from the ability to work with compressed data.]]> 0 <![CDATA[A geometric graph is a graph G = (V, E) drawn in the plane, such that V is a point set in general position and E is a set of straight-line segments whose endpoints belong to V. We study the following extremal problem for geometric graphs: How many arbitrary edges can be removed from a complete geometric graph with n vertices such that the remaining graph still contains a certain non-crossing subgraph. The non-crossing subgraphs that we consider are perfect matchings, subtrees of a given size, and triangulations. In each case, we obtain tight bounds on the maximum number of removable edges.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.525 https://doi.org/10.46298/dmtcs.525 Aichholzer, Oswin Cabello, Sergio Fabila-Monroy, Ruy Flores-Peñaloza, David Hackl, Thomas Huemer, Clemens Hurtado, Ferran Wood, David R. Aichholzer, Oswin Cabello, Sergio Fabila-Monroy, Ruy Flores-Peñaloza, David Hackl, Thomas Huemer, Clemens Hurtado, Ferran Wood, David R. <![CDATA[A geometric graph is a graph G = (V, E) drawn in the plane, such that V is a point set in general position and E is a set of straight-line segments whose endpoints belong to V. We study the following extremal problem for geometric graphs: How many arbitrary edges can be removed from a complete geometric graph with n vertices such that the remaining graph still contains a certain non-crossing subgraph. The non-crossing subgraphs that we consider are perfect matchings, subtrees of a given size, and triangulations. In each case, we obtain tight bounds on the maximum number of removable edges.]]> 0 <![CDATA[We prove that for each partition of the Lobachevsky plane into finitely many Borel pieces one of the cells of the partition contains an unbounded centrally symmetric subset.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.524 https://doi.org/10.46298/dmtcs.524 Banakh, Taras Dudko, Artem Repovs, Dusan Banakh, Taras Dudko, Artem Repovs, Dusan <![CDATA[We prove that for each partition of the Lobachevsky plane into finitely many Borel pieces one of the cells of the partition contains an unbounded centrally symmetric subset.]]> 0 <![CDATA[We give an instance of a class of morphisms for which it is easy to prove that their equality set is regular, but its emptiness is still undecidable. The class is that of bounded delay 2 morphisms.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.523 https://doi.org/10.46298/dmtcs.523 Karhumaki, Juhani Saarela, Aleksi Karhumaki, Juhani Saarela, Aleksi <![CDATA[We give an instance of a class of morphisms for which it is easy to prove that their equality set is regular, but its emptiness is still undecidable. The class is that of bounded delay 2 morphisms.]]> 0 <![CDATA[The direct product of two words is a naturally defined word on the alphabet of pairs of symbols. An infinite word is uniformly recurrent if each its subword occurs in it with bounded gaps. An infinite word is strongly recurrent if the direct product of it with each uniformly recurrent word is also uniformly recurrent. We prove that fixed points of the expanding binary symmetric morphisms are strongly recurrent. In particular, such is the Thue-Morse word.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.522 https://doi.org/10.46298/dmtcs.522 Salimov, Pavel Vadimovich Salimov, Pavel Vadimovich <![CDATA[The direct product of two words is a naturally defined word on the alphabet of pairs of symbols. An infinite word is uniformly recurrent if each its subword occurs in it with bounded gaps. An infinite word is strongly recurrent if the direct product of it with each uniformly recurrent word is also uniformly recurrent. We prove that fixed points of the expanding binary symmetric morphisms are strongly recurrent. In particular, such is the Thue-Morse word.]]> 0 <![CDATA[M.-P. Schutzenberger asked to determine the support of the free Lie algebra L(Zm) (A) on a finite alphabet A over the ring Z(m) of integers mod m and all pairs of twin and anti-twin words, i.e., words that appear with equal (resp. opposite) coefficients in each Lie polynomial. We characterize the complement of the support of L(Zm) (A) in A* as the set of all words w such that m divides all the coefficients appearing in the monomials of l* (w), where l* is the adjoint endomorphism of the left normed Lie bracketing l of the free Lie ring. Calculating l* (w) via the shuffle product, we recover the well known result of Duchamp and Thibon (Discrete Math. 76 (1989) 123-132) for the support of the free Lie ring in a much more natural way. We conjecture that two words u and v of common length n, which lie in the support of the free Lie ring, are twin (resp. anti-twin) if and only if either u = v or n is odd and u = (v) over tilde (resp. if n is even and u = (v) over tilde), where (v) over tilde denotes the reversal of v and we prove that it suffices to show this for a two-lettered alphabet. These problems can be rephrased, for words of length n, in terms of the action of the Dynkin operator l(n) on lambda-tabloids, where lambda is a partition of n. Representing a word w in two letters by the subset I of [n] = \1, 2, ... , n\ that consists of all positions that one of the letters occurs in w, the computation of l* (w) leads us to the notion of the Pascal descent polynomial p(n)(I), a particular commutative multi-linear polynomial which is equal to the signed binomial coefficient when vertical bar I vertical bar = 1. We provide a recursion formula for p(n) (I) and show that if m inverted iota Sigma(i is an element of I)(1)(i-1) (n - 1 i - 1), then w lies in the support of L(Zm) (A).]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.521 https://doi.org/10.46298/dmtcs.521 Michos, Ioannis C. Michos, Ioannis C. <![CDATA[M.-P. Schutzenberger asked to determine the support of the free Lie algebra L(Zm) (A) on a finite alphabet A over the ring Z(m) of integers mod m and all pairs of twin and anti-twin words, i.e., words that appear with equal (resp. opposite) coefficients in each Lie polynomial. We characterize the complement of the support of L(Zm) (A) in A* as the set of all words w such that m divides all the coefficients appearing in the monomials of l* (w), where l* is the adjoint endomorphism of the left normed Lie bracketing l of the free Lie ring. Calculating l* (w) via the shuffle product, we recover the well known result of Duchamp and Thibon (Discrete Math. 76 (1989) 123-132) for the support of the free Lie ring in a much more natural way. We conjecture that two words u and v of common length n, which lie in the support of the free Lie ring, are twin (resp. anti-twin) if and only if either u = v or n is odd and u = (v) over tilde (resp. if n is even and u = (v) over tilde), where (v) over tilde denotes the reversal of v and we prove that it suffices to show this for a two-lettered alphabet. These problems can be rephrased, for words of length n, in terms of the action of the Dynkin operator l(n) on lambda-tabloids, where lambda is a partition of n. Representing a word w in two letters by the subset I of [n] = \1, 2, ... , n\ that consists of all positions that one of the letters occurs in w, the computation of l* (w) leads us to the notion of the Pascal descent polynomial p(n)(I), a particular commutative multi-linear polynomial which is equal to the signed binomial coefficient when vertical bar I vertical bar = 1. We provide a recursion formula for p(n) (I) and show that if m inverted iota Sigma(i is an element of I)(1)(i-1) (n - 1 i - 1), then w lies in the support of L(Zm) (A).]]> 0 <![CDATA[We study a recurrence relation, originating in combinatorial problems, where the generating function, as a formal power series, satisfies a differential equation that can be solved in a suitable domain; this yields an analytic function in a domain, but the solution is singular at the origin and the generating function has radius of convergence 0. Nevertheless, the solution to the recurrence can be obtained from the analytic solution by finding an asymptotic series expansion. Conversely, the analytic solution can be obtained by summing the generating function by the Borel summation method. This is an explicit example, which we study detail, of a behaviour known to be typical for a large class of holonomic functions. We also express the solution using Bessel functions and Lommel polynomials.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.520 https://doi.org/10.46298/dmtcs.520 Janson, Svante Janson, Svante <![CDATA[We study a recurrence relation, originating in combinatorial problems, where the generating function, as a formal power series, satisfies a differential equation that can be solved in a suitable domain; this yields an analytic function in a domain, but the solution is singular at the origin and the generating function has radius of convergence 0. Nevertheless, the solution to the recurrence can be obtained from the analytic solution by finding an asymptotic series expansion. Conversely, the analytic solution can be obtained by summing the generating function by the Borel summation method. This is an explicit example, which we study detail, of a behaviour known to be typical for a large class of holonomic functions. We also express the solution using Bessel functions and Lommel polynomials.]]> 0 <![CDATA[This paper presents the first distributional analysis of both, a parking problem and a linear probing hashing scheme with buckets of size b. The exact distribution of the cost of successful searches for a b alpha-full table is obtained, and moments and asymptotic results are derived. With the use of the Poisson transform distributional results are also obtained for tables of size m and n elements. A key element in the analysis is the use of a new family of numbers, called Tuba Numbers, that satisfies a recurrence resembling that of the Bernoulli numbers. These numbers may prove helpful in studying recurrences involving truncated generating functions, as well as in other problems related with buckets.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.519 https://doi.org/10.46298/dmtcs.519 Viola, Alfredo Viola, Alfredo <![CDATA[This paper presents the first distributional analysis of both, a parking problem and a linear probing hashing scheme with buckets of size b. The exact distribution of the cost of successful searches for a b alpha-full table is obtained, and moments and asymptotic results are derived. With the use of the Poisson transform distributional results are also obtained for tables of size m and n elements. A key element in the analysis is the use of a new family of numbers, called Tuba Numbers, that satisfies a recurrence resembling that of the Bernoulli numbers. These numbers may prove helpful in studying recurrences involving truncated generating functions, as well as in other problems related with buckets.]]> 0 <![CDATA[Cellular automata are usually associated with synchronous deterministic dynamics, and their asynchronous or stochastic versions have been far less studied although significant for modeling purposes. This paper analyzes the dynamics of a two-dimensional cellular automaton, 2D Minority, for the Moore neighborhood (eight closest neighbors of each cell) under fully asynchronous dynamics (where one single random cell updates at each time step). 2D Minority may appear as a simple rule, but It is known from the experience of Ising models and Hopfield nets that 2D models with negative feedback are hard to study. This automaton actually presents a rich variety of behaviors, even more complex that what has been observed and analyzed in a previous work on 2D Minority for the von Neumann neighborhood (four neighbors to each cell) (2007) This paper confirms the relevance of the later approach (definition of energy functions and identification of competing regions) Switching to the Moot e neighborhood however strongly complicates the description of intermediate configurations. New phenomena appear (particles, wider range of stable configurations) Nevertheless our methods allow to analyze different stages of the dynamics It suggests that predicting the behavior of this automaton although difficult is possible, opening the way to the analysis of the whole class of totalistic automata]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.518 https://doi.org/10.46298/dmtcs.518 Regnault, Damien Schabanel, Nicolas Thierry, Eric Regnault, Damien Schabanel, Nicolas Thierry, Eric <![CDATA[Cellular automata are usually associated with synchronous deterministic dynamics, and their asynchronous or stochastic versions have been far less studied although significant for modeling purposes. This paper analyzes the dynamics of a two-dimensional cellular automaton, 2D Minority, for the Moore neighborhood (eight closest neighbors of each cell) under fully asynchronous dynamics (where one single random cell updates at each time step). 2D Minority may appear as a simple rule, but It is known from the experience of Ising models and Hopfield nets that 2D models with negative feedback are hard to study. This automaton actually presents a rich variety of behaviors, even more complex that what has been observed and analyzed in a previous work on 2D Minority for the von Neumann neighborhood (four neighbors to each cell) (2007) This paper confirms the relevance of the later approach (definition of energy functions and identification of competing regions) Switching to the Moot e neighborhood however strongly complicates the description of intermediate configurations. New phenomena appear (particles, wider range of stable configurations) Nevertheless our methods allow to analyze different stages of the dynamics It suggests that predicting the behavior of this automaton although difficult is possible, opening the way to the analysis of the whole class of totalistic automata]]> 0 <![CDATA[We contribute to combinatorics and algorithmics of words by introducing new types of periodicities in words. A tiling period of a word w is partial word u such that w can be decomposed into several disjoint parallel copies of u, e.g. a lozenge b is a tiling period of a a b b. We investigate properties of tiling periodicities and design an algorithm working in O(n log (n) log log (n)) time which finds a tiling period of minimal size, the number of such minimal periods and their compact representation. The combinatorics of tiling periods differs significantly from that for classical full periods, for example unlike the classical case the same word can have many different primitive tiling periods. We consider also a related new type of periods called in the paper multi-periods. As a side product of the paper we solve an open problem posted by T. Harju (2003).]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.517 https://doi.org/10.46298/dmtcs.517 Karhumaki, Juhani Lifshits, Yury Rytter, Wojciech Karhumaki, Juhani Lifshits, Yury Rytter, Wojciech <![CDATA[We contribute to combinatorics and algorithmics of words by introducing new types of periodicities in words. A tiling period of a word w is partial word u such that w can be decomposed into several disjoint parallel copies of u, e.g. a lozenge b is a tiling period of a a b b. We investigate properties of tiling periodicities and design an algorithm working in O(n log (n) log log (n)) time which finds a tiling period of minimal size, the number of such minimal periods and their compact representation. The combinatorics of tiling periods differs significantly from that for classical full periods, for example unlike the classical case the same word can have many different primitive tiling periods. We consider also a related new type of periods called in the paper multi-periods. As a side product of the paper we solve an open problem posted by T. Harju (2003).]]> 0 <![CDATA[Block-transitive Steiner t-designs form a central part of the study of highly symmetric combinatorial configurations at the interface of several disciplines, including group theory, geometry, combinatorics, coding and information theory, and cryptography. The main result of the paper settles an important open question: There exist no non-trivial examples with t = 7 (or larger). The proof is based on the classification of the finite 3-homogeneous permutation groups, itself relying on the finite simple group classification.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.516 https://doi.org/10.46298/dmtcs.516 Huber, Michael Huber, Michael <![CDATA[Block-transitive Steiner t-designs form a central part of the study of highly symmetric combinatorial configurations at the interface of several disciplines, including group theory, geometry, combinatorics, coding and information theory, and cryptography. The main result of the paper settles an important open question: There exist no non-trivial examples with t = 7 (or larger). The proof is based on the classification of the finite 3-homogeneous permutation groups, itself relying on the finite simple group classification.]]> 0 <![CDATA[Using the saddle point method, we obtain from the generating function of the Stirling numbers of the first kind [n j] and Cauchy's integral formula, asymptotic results in central and non-central regions. In the central region, we revisit the celebrated Goncharov theorem with more precision. In the region j = n - n(alpha); alpha > 1/2, we analyze the dependence of [n j] on alpha.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.501 https://doi.org/10.46298/dmtcs.501 Louchard, Guy Louchard, Guy <![CDATA[Using the saddle point method, we obtain from the generating function of the Stirling numbers of the first kind [n j] and Cauchy's integral formula, asymptotic results in central and non-central regions. In the central region, we revisit the celebrated Goncharov theorem with more precision. In the region j = n - n(alpha); alpha > 1/2, we analyze the dependence of [n j] on alpha.]]> 0 <![CDATA[Conjecture that any synchronizing automaton with n states has a reset word of length (n - 1)(2) was made by. Cerny in 1964. Notwithstanding the numerous attempts made by various researchers this conjecture hasn't been definitively proven yet. In this paper we study a random automaton that is sampled uniformly at random from the set of all automata with n states and m(n) letters. We show that for m(n) > 18 ln n any random automaton is synchronizing with high probability. For m(n) > n(beta), beta > 1/2 we also show that any random automaton with high probability satisfies the. Cerny conjecture.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.514 https://doi.org/10.46298/dmtcs.514 Skvortsov, Evgeny Zaks, Yulia Skvortsov, Evgeny Zaks, Yulia <![CDATA[Conjecture that any synchronizing automaton with n states has a reset word of length (n - 1)(2) was made by. Cerny in 1964. Notwithstanding the numerous attempts made by various researchers this conjecture hasn't been definitively proven yet. In this paper we study a random automaton that is sampled uniformly at random from the set of all automata with n states and m(n) letters. We show that for m(n) > 18 ln n any random automaton is synchronizing with high probability. For m(n) > n(beta), beta > 1/2 we also show that any random automaton with high probability satisfies the. Cerny conjecture.]]> 0 <![CDATA[In 1961, Erdos asked whether or not there exist words of arbitrary length over a fixed finite alphabet that avoid patterns of the form XX' where X' is a permutation of X (called abelian squares). This problem has since been solved in the affirmative in a series of papers from 1968 to 1992. Much less is known in the case of abelian k-th powers, i.e., words of the form X1X2 ... X-k where X-i is a permutation of X-1 for 2 <= i <= k. In this paper, we consider crucial words for abelian k-th powers, i. e., finite words that avoid abelian k-th powers, but which cannot be extended to the right by any letter of their own alphabets without creating an abelian k-th power. More specifically, we consider the problem of determining the minimal length of a crucial word avoiding abelian k-th powers. This problem has already been solved for abelian squares by Evdokimov and Kitaev (2004), who showed that a minimal crucial word over an n-letter alphabet A(n) = \1, 2, ..., n\ avoiding abelian squares has length 4n - 7 for n >= 3. Extending this result, we prove that a minimal crucial word over A(n) avoiding abelian cubes has length 9n - 13 for n >= 5, and it has length 2, 5, 11, and 20 for n = 1, 2, 3, and 4, respectively. Moreover, for n >= 4 and k >= 2, we give a construction of length k(2) (n - 1) - k - 1 of a crucial word over A(n) avoiding abelian k-th powers. This construction gives the minimal length for k = 2 and k = 3. For k >= 4 and n >= 5, we provide a lower bound for the length of crucial words over A(n) avoiding abelian k-th powers.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.513 https://doi.org/10.46298/dmtcs.513 Glen, Amy Halldorsson, Bjarni V. Kitaev, Sergey Glen, Amy Halldorsson, Bjarni V. Kitaev, Sergey <![CDATA[In 1961, Erdos asked whether or not there exist words of arbitrary length over a fixed finite alphabet that avoid patterns of the form XX' where X' is a permutation of X (called abelian squares). This problem has since been solved in the affirmative in a series of papers from 1968 to 1992. Much less is known in the case of abelian k-th powers, i.e., words of the form X1X2 ... X-k where X-i is a permutation of X-1 for 2 <= i <= k. In this paper, we consider crucial words for abelian k-th powers, i. e., finite words that avoid abelian k-th powers, but which cannot be extended to the right by any letter of their own alphabets without creating an abelian k-th power. More specifically, we consider the problem of determining the minimal length of a crucial word avoiding abelian k-th powers. This problem has already been solved for abelian squares by Evdokimov and Kitaev (2004), who showed that a minimal crucial word over an n-letter alphabet A(n) = \1, 2, ..., n\ avoiding abelian squares has length 4n - 7 for n >= 3. Extending this result, we prove that a minimal crucial word over A(n) avoiding abelian cubes has length 9n - 13 for n >= 5, and it has length 2, 5, 11, and 20 for n = 1, 2, 3, and 4, respectively. Moreover, for n >= 4 and k >= 2, we give a construction of length k(2) (n - 1) - k - 1 of a crucial word over A(n) avoiding abelian k-th powers. This construction gives the minimal length for k = 2 and k = 3. For k >= 4 and n >= 5, we provide a lower bound for the length of crucial words over A(n) avoiding abelian k-th powers.]]> 0 <![CDATA[We simplify the known formula for the asymptotic estimate of the number of deterministic and accessible automata with n states over a k-letter alphabet. The proof relies on the theory of Lagrange inversion applied in the context of generalized binomial series.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.512 https://doi.org/10.46298/dmtcs.512 Lebensztayn, Elcio Lebensztayn, Elcio <![CDATA[We simplify the known formula for the asymptotic estimate of the number of deterministic and accessible automata with n states over a k-letter alphabet. The proof relies on the theory of Lagrange inversion applied in the context of generalized binomial series.]]> 0 <![CDATA[We present a model of automaton for picture language recognition, called Wang automaton, which is based on labeled Wang tiles. Wang automata combine features of both online tessellation acceptors and 4-way automata: as in online tessellation acceptors, computation assigns states to each picture position; as in 4-way automata, the input head visits the picture moving from one pixel to an adjacent one, according to some scanning strategy. Wang automata recognize the class REC, i.e. they are equivalent to tiling systems or online tessellation acceptors, and hence strictly more powerful than 4-way automata. We also introduce a natural notion of determinism for Wang automata, and study the resulting class, extending the more traditional approach of diagonal-based determinism, used e. g. by deterministic tiling systems. In particular, we prove that the concept of row (or column) ambiguity defines the class of languages recognized by Wang automata directed by boustrophedonic scanning strategies.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.511 https://doi.org/10.46298/dmtcs.511 Lonati, Violetta Pradella, Matteo Lonati, Violetta Pradella, Matteo <![CDATA[We present a model of automaton for picture language recognition, called Wang automaton, which is based on labeled Wang tiles. Wang automata combine features of both online tessellation acceptors and 4-way automata: as in online tessellation acceptors, computation assigns states to each picture position; as in 4-way automata, the input head visits the picture moving from one pixel to an adjacent one, according to some scanning strategy. Wang automata recognize the class REC, i.e. they are equivalent to tiling systems or online tessellation acceptors, and hence strictly more powerful than 4-way automata. We also introduce a natural notion of determinism for Wang automata, and study the resulting class, extending the more traditional approach of diagonal-based determinism, used e. g. by deterministic tiling systems. In particular, we prove that the concept of row (or column) ambiguity defines the class of languages recognized by Wang automata directed by boustrophedonic scanning strategies.]]> 0 <![CDATA[We characterize the relations which are first-order definable in the model of the group of integers with the constant 1. This allows us to show that given a relation defined by a first-order formula in this model enriched with the usual ordering, it is recursively decidable whether or not it is first-order definable without the ordering.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.510 https://doi.org/10.46298/dmtcs.510 Choffrut, Christian Frigeri, Achille Choffrut, Christian Frigeri, Achille <![CDATA[We characterize the relations which are first-order definable in the model of the group of integers with the constant 1. This allows us to show that given a relation defined by a first-order formula in this model enriched with the usual ordering, it is recursively decidable whether or not it is first-order definable without the ordering.]]> 0 <![CDATA[We introduce a new model of two-way finite automaton, which is endowed with the capability of resetting the position of the tape head to the left end of the tape in a single move during the computation. Several variants of this model are examined, with the following results: The weakest known model of computation where quantum computers recognize more languages with bounded error than their classical counterparts is identified. We prove that two-way probabilistic and quantum finite automata (2PFAs and 2QFAs) can be considerably more concise than both their one-way versions (1PFAs and 1QFAs), and two-way nondeterministic finite automata (2NFAs). For this purpose, we demonstrate several infinite families of regular languages which can be recognized with some fixed probability greater than 1 2 by just tuning the transition amplitudes of a 2QFA (and, in one case, a 2PFA) with a constant number of states, whereas the sizes of the corresponding 1PFAs, 1QFAs and 2NFAs grow without bound. We also show that 2QFAs with mixed states can support highly efficient probability amplification.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.509 https://doi.org/10.46298/dmtcs.509 Yakaryilmaz, Abuzer Say, A. C. Cem Yakaryilmaz, Abuzer Say, A. C. Cem <![CDATA[We introduce a new model of two-way finite automaton, which is endowed with the capability of resetting the position of the tape head to the left end of the tape in a single move during the computation. Several variants of this model are examined, with the following results: The weakest known model of computation where quantum computers recognize more languages with bounded error than their classical counterparts is identified. We prove that two-way probabilistic and quantum finite automata (2PFAs and 2QFAs) can be considerably more concise than both their one-way versions (1PFAs and 1QFAs), and two-way nondeterministic finite automata (2NFAs). For this purpose, we demonstrate several infinite families of regular languages which can be recognized with some fixed probability greater than 1 2 by just tuning the transition amplitudes of a 2QFA (and, in one case, a 2PFA) with a constant number of states, whereas the sizes of the corresponding 1PFAs, 1QFAs and 2NFAs grow without bound. We also show that 2QFAs with mixed states can support highly efficient probability amplification.]]> 0 <![CDATA[For a graph G and integers a and b, an (a, b)-code of G is a set C of vertices such that any vertex from C has exactly a neighbors in C and any vertex not in C has exactly b neighbors in C. In this paper we classify integers a and b for which there exist (a, b)-codes in Sierpinski graphs.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.508 https://doi.org/10.46298/dmtcs.508 Beaudou, Laurent Gravier, Sylvain Klavžar, Sandi Kovse, Matjaz Mollard, Michel Beaudou, Laurent Gravier, Sylvain Klavžar, Sandi Kovse, Matjaz Mollard, Michel <![CDATA[For a graph G and integers a and b, an (a, b)-code of G is a set C of vertices such that any vertex from C has exactly a neighbors in C and any vertex not in C has exactly b neighbors in C. In this paper we classify integers a and b for which there exist (a, b)-codes in Sierpinski graphs.]]> 0 <![CDATA[Let X(1),X(2),...,X(n) be independent, identically distributed uniform random variables on [0, 1]. We can observe the outcomes sequentially and must select online at least r of them, and, moreover, in expectation at least mu >= r. Here mu need not be integer. We see X(k) as the cost of selecting item k and want to minimize the expected total cost under the described combined (r, mu)-constraint. We will see that an optimal selection strategy exists on the set S(n) of all selection strategies for which the decision at instant k may depend on the value X(k), on the number N(k) of selections up to time k and of the number n - k of forthcoming observations. Let sigma(r,mu)(n) be the corresponding S(n)-optimal selection strategy and v(r,mu)(n) its value. The main goal of this paper is to determine these and to understand the limiting behavior of v(r,mu)(n). After discussion of the specific character of this combination of two types of constraints we conclude that the S(n)-problem has a recursive structure and solve it in terms of a double recursion. Our interest will then focus on the limiting behavior of nv(r,mu)(n) as n -> infinity. This sequence converges and its limit allows for the interpretation of a normalized limiting cost L (r, mu) of the (r, mu)-constraint. Our main result is that L(r, mu) = g(r) ((mu - r)(2)/(2)) where g(r) is the r(th) iterate of the function g(x) = 1 + x + root 1 + 2x. Our motivation to study mixed-constraints problems is indicated by several examples of possible applications. We also shortly discuss the intricacy of the expectational part of the constraint if we try to extend the class of strategies S n to the set of full-history-dependent and/or randomized strategies.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.506 https://doi.org/10.46298/dmtcs.506 Bruss, F. Thomas Bruss, F. Thomas <![CDATA[Let X(1),X(2),...,X(n) be independent, identically distributed uniform random variables on [0, 1]. We can observe the outcomes sequentially and must select online at least r of them, and, moreover, in expectation at least mu >= r. Here mu need not be integer. We see X(k) as the cost of selecting item k and want to minimize the expected total cost under the described combined (r, mu)-constraint. We will see that an optimal selection strategy exists on the set S(n) of all selection strategies for which the decision at instant k may depend on the value X(k), on the number N(k) of selections up to time k and of the number n - k of forthcoming observations. Let sigma(r,mu)(n) be the corresponding S(n)-optimal selection strategy and v(r,mu)(n) its value. The main goal of this paper is to determine these and to understand the limiting behavior of v(r,mu)(n). After discussion of the specific character of this combination of two types of constraints we conclude that the S(n)-problem has a recursive structure and solve it in terms of a double recursion. Our interest will then focus on the limiting behavior of nv(r,mu)(n) as n -> infinity. This sequence converges and its limit allows for the interpretation of a normalized limiting cost L (r, mu) of the (r, mu)-constraint. Our main result is that L(r, mu) = g(r) ((mu - r)(2)/(2)) where g(r) is the r(th) iterate of the function g(x) = 1 + x + root 1 + 2x. Our motivation to study mixed-constraints problems is indicated by several examples of possible applications. We also shortly discuss the intricacy of the expectational part of the constraint if we try to extend the class of strategies S n to the set of full-history-dependent and/or randomized strategies.]]> 0 <![CDATA[We find the asymptotic number of 2-orientations of quadrangulations with n inner faces, and of 3-orientations of triangulations with n inner vertices. We also find the asymptotic number of prime 2-orientations (no separating quadrangle) and prime 3-orientations (no separating triangle). The estimates we find are of the form c . n(-alpha)gamma(n), for suitable constants c, alpha, gamma with alpha = 4 for 2-orientations and alpha = 5 for 3-orientations. The proofs are based on singularity analysis of D-finite generating functions, using the Fuchsian theory of complex linear differential equations.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.505 https://doi.org/10.46298/dmtcs.505 Felsner, Stefan Fusy, Eric Noy, Marc Felsner, Stefan Fusy, Eric Noy, Marc <![CDATA[We find the asymptotic number of 2-orientations of quadrangulations with n inner faces, and of 3-orientations of triangulations with n inner vertices. We also find the asymptotic number of prime 2-orientations (no separating quadrangle) and prime 3-orientations (no separating triangle). The estimates we find are of the form c . n(-alpha)gamma(n), for suitable constants c, alpha, gamma with alpha = 4 for 2-orientations and alpha = 5 for 3-orientations. The proofs are based on singularity analysis of D-finite generating functions, using the Fuchsian theory of complex linear differential equations.]]> 0 <![CDATA[We prove generating function identities producing fast convergent series for the sequences beta(2n + 1); beta(2n + 2) and beta(2n + 3), where beta is Dirichlet's beta function. In particular, we obtain a new accelerated series for Catalan's constant convergent at a geometric rate with ratio 2(-10); which can be considered as an analog of Amdeberhan-Zeilberger's series for zeta(3)]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.504 https://doi.org/10.46298/dmtcs.504 Pilehrood, Khodabakhsh Hessami Pilehrood, Tatiana Hessami Pilehrood, Khodabakhsh Hessami Pilehrood, Tatiana Hessami <![CDATA[We prove generating function identities producing fast convergent series for the sequences beta(2n + 1); beta(2n + 2) and beta(2n + 3), where beta is Dirichlet's beta function. In particular, we obtain a new accelerated series for Catalan's constant convergent at a geometric rate with ratio 2(-10); which can be considered as an analog of Amdeberhan-Zeilberger's series for zeta(3)]]> 0 <![CDATA[A decomposable combinatorial structure consists of simpler objects called components which by thems elves cannot be further decomposed. We focus on the multi-set construction where the component generating function C(z) is of alg-log type, that is, C(z) behaves like c + d(1 -z/rho)(alpha) (ln1/1-z/rho)(beta) (1 + o(1)) when z is near the dominant singularity rho. We provide asymptotic results about the size of thes mallest components in random combinatorial structures for the cases 0 < alpha < 1 and any beta, and alpha < 0 and beta=0. The particular case alpha=0 and beta=1, the so-called exp-log class, has been treated in previous papers. We also provide similar asymptotic estimates for combinatorial objects with a restricted pattern, that is, when part of its factorization patterns is known. We extend our results to include certain type of integers partitions. partitions]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.503 https://doi.org/10.46298/dmtcs.503 Dong, Li Gao, Zhicheng Panario, Daniel Richmond, Bruce Dong, Li Gao, Zhicheng Panario, Daniel Richmond, Bruce <![CDATA[A decomposable combinatorial structure consists of simpler objects called components which by thems elves cannot be further decomposed. We focus on the multi-set construction where the component generating function C(z) is of alg-log type, that is, C(z) behaves like c + d(1 -z/rho)(alpha) (ln1/1-z/rho)(beta) (1 + o(1)) when z is near the dominant singularity rho. We provide asymptotic results about the size of thes mallest components in random combinatorial structures for the cases 0 < alpha < 1 and any beta, and alpha < 0 and beta=0. The particular case alpha=0 and beta=1, the so-called exp-log class, has been treated in previous papers. We also provide similar asymptotic estimates for combinatorial objects with a restricted pattern, that is, when part of its factorization patterns is known. We extend our results to include certain type of integers partitions. partitions]]> 0 <![CDATA[A set C of vertices of a graph G is P(3)-convex if v is an element of C for every path uvw in G with u, w is an element of C. We prove that it is NP-complete to decide for a given graph G and a given integer p whether the vertex set of G can be partitioned into p non-empty disjoint P(3)-convex sets. Furthermore, we study such partitions for a variety of graph classes.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.502 https://doi.org/10.46298/dmtcs.502 Centeno, C. C. Dantas, S. Dourado, M. C. Rautenbach, Dieter Szwarcfiter, Jayme Luiz Centeno, C. C. Dantas, S. Dourado, M. C. Rautenbach, Dieter Szwarcfiter, Jayme Luiz <![CDATA[A set C of vertices of a graph G is P(3)-convex if v is an element of C for every path uvw in G with u, w is an element of C. We prove that it is NP-complete to decide for a given graph G and a given integer p whether the vertex set of G can be partitioned into p non-empty disjoint P(3)-convex sets. Furthermore, we study such partitions for a variety of graph classes.]]> 0 <![CDATA[Connected graphs in which any two of its cycles have at most one common vertex are called cactuses. In this paper, we continue the work on Laplacian spread of graphs, and determine the graph with maximal Laplacian spread in all cactuses with n vertices.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.488 https://doi.org/10.46298/dmtcs.488 Liu, Ying Liu, Ying <![CDATA[Connected graphs in which any two of its cycles have at most one common vertex are called cactuses. In this paper, we continue the work on Laplacian spread of graphs, and determine the graph with maximal Laplacian spread in all cactuses with n vertices.]]> 0 <![CDATA[We show that there exist series-parallel graphs requiring Omega(n2(root log n)) area in any straight-line or poly-line grid drawing. Such a result is achieved in two steps. First, we show that, in any straight-line or poly-line drawing of K(2,n), one side of the bounding box has length Omega(n), thus answering two questions posed by Biedl et al. Second, we show a family of series-parallel graphs requiring Omega(2(root log n)) width and Omega(2(root log n)) height in any straight-line or poly-line grid drawing. Combining the two results, the Omega(n2(root log n)) area lower bound is achieved.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.500 https://doi.org/10.46298/dmtcs.500 Frati, Fabrizio Frati, Fabrizio <![CDATA[We show that there exist series-parallel graphs requiring Omega(n2(root log n)) area in any straight-line or poly-line grid drawing. Such a result is achieved in two steps. First, we show that, in any straight-line or poly-line drawing of K(2,n), one side of the bounding box has length Omega(n), thus answering two questions posed by Biedl et al. Second, we show a family of series-parallel graphs requiring Omega(2(root log n)) width and Omega(2(root log n)) height in any straight-line or poly-line grid drawing. Combining the two results, the Omega(n2(root log n)) area lower bound is achieved.]]> 0 <![CDATA[A graph with degree set \r, r + 1\ is said to be semiregular. A semiregular cage is a semiregular graph with given girth g and the least possible order. First, an upper bound on the diameter of semiregular graphs with girth g and order close enough to the minimum possible value is given in this work. As a consequence, these graphs are proved to be maximally connected when the girth g >= 7 is odd. Moreover an upper bound for the order of semiregular cages is given and, as an application, every semiregular cage with degree set \r, r + 1\ is proved to be maximally connected for g is an element of \6, 8\, and when g = 12 for r >= 7 and r not equal 20. Finally it is also shown that every (\r, r + 1\; g)-cage is 3-connected.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.499 https://doi.org/10.46298/dmtcs.499 Balbuena, Camino Marcote, Xavier Gonzalez-Moreno, Diego Balbuena, Camino Marcote, Xavier Gonzalez-Moreno, Diego <![CDATA[A graph with degree set \r, r + 1\ is said to be semiregular. A semiregular cage is a semiregular graph with given girth g and the least possible order. First, an upper bound on the diameter of semiregular graphs with girth g and order close enough to the minimum possible value is given in this work. As a consequence, these graphs are proved to be maximally connected when the girth g >= 7 is odd. Moreover an upper bound for the order of semiregular cages is given and, as an application, every semiregular cage with degree set \r, r + 1\ is proved to be maximally connected for g is an element of \6, 8\, and when g = 12 for r >= 7 and r not equal 20. Finally it is also shown that every (\r, r + 1\; g)-cage is 3-connected.]]> 0 <![CDATA[Asymptotics of the variances of many cost measures in random digital search trees are often notoriously messy and involved to obtain. A new approach is proposed to facilitate such an analysis for several shape parameters on random symmetric digital search trees. Our approach starts from a more careful normalization at the level of Poisson generating functions, which then provides an asymptotically equivalent approximation to the variance in question. Several new ingredients are also introduced such as a combined use of the Laplace and Mellin transforms and a simple, mechanical technique for justifying the analytic de-Poissonization procedures involved. The methodology we develop can be easily adapted to many other problems with an underlying binomial distribution. In particular, the less expected and somewhat surprising n (logn)(2)-variance for certain notions of total path-length is also clarified.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.498 https://doi.org/10.46298/dmtcs.498 Hwang, Hsien-Kuei Fuchs, Michael Zacharovas, Vytas Hwang, Hsien-Kuei Fuchs, Michael Zacharovas, Vytas <![CDATA[Asymptotics of the variances of many cost measures in random digital search trees are often notoriously messy and involved to obtain. A new approach is proposed to facilitate such an analysis for several shape parameters on random symmetric digital search trees. Our approach starts from a more careful normalization at the level of Poisson generating functions, which then provides an asymptotically equivalent approximation to the variance in question. Several new ingredients are also introduced such as a combined use of the Laplace and Mellin transforms and a simple, mechanical technique for justifying the analytic de-Poissonization procedures involved. The methodology we develop can be easily adapted to many other problems with an underlying binomial distribution. In particular, the less expected and somewhat surprising n (logn)(2)-variance for certain notions of total path-length is also clarified.]]> 0 <![CDATA[We propose a coloring algorithm for sparse random graphs generated by the geographical threshold graph (GTG) model, a generalization of random geometric graphs (RGG). In a GTG, nodes are distributed in a Euclidean space, and edges are assigned according to a threshold function involving the distance between nodes as well as randomly chosen node weights. The motivation for analyzing this model is that many real networks (e. g., wireless networks, the Internet, etc.) need to be studied by using a ''richer'' stochastic model (which in this case includes both a distance between nodes and weights on the nodes). Here, we analyze the GTG coloring algorithm together with the graph's clique number, showing formally that in spite of the differences in structure between GTG and RGG, the asymptotic behavior of the chromatic number is identical: chi = ln n/ln ln n(1 +o(1)). Finally, we consider the leading corrections to this expression, again using the coloring algorithm and clique number to provide bounds on the chromatic number. We show that the gap between the lower and upper bound is within C ln n/(ln ln n)(2), and specify the constant C.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.497 https://doi.org/10.46298/dmtcs.497 Bradonjic, Milan Mueller, Tobias Percus, Allon G. Bradonjic, Milan Mueller, Tobias Percus, Allon G. <![CDATA[We propose a coloring algorithm for sparse random graphs generated by the geographical threshold graph (GTG) model, a generalization of random geometric graphs (RGG). In a GTG, nodes are distributed in a Euclidean space, and edges are assigned according to a threshold function involving the distance between nodes as well as randomly chosen node weights. The motivation for analyzing this model is that many real networks (e. g., wireless networks, the Internet, etc.) need to be studied by using a ''richer'' stochastic model (which in this case includes both a distance between nodes and weights on the nodes). Here, we analyze the GTG coloring algorithm together with the graph's clique number, showing formally that in spite of the differences in structure between GTG and RGG, the asymptotic behavior of the chromatic number is identical: chi = ln n/ln ln n(1 +o(1)). Finally, we consider the leading corrections to this expression, again using the coloring algorithm and clique number to provide bounds on the chromatic number. We show that the gap between the lower and upper bound is within C ln n/(ln ln n)(2), and specify the constant C.]]> 0 <![CDATA[Following a question of J. Cooper, we study the expected number of occurrences of a given permutation pattern q in permutations that avoid another given pattern r. In some cases, we find the pattern that occurs least often, (resp. most often) in all r-avoiding permutations. We also prove a few exact enumeration formulae, some of which are surprising.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.496 https://doi.org/10.46298/dmtcs.496 Bóna, Miklós Bóna, Miklós <![CDATA[Following a question of J. Cooper, we study the expected number of occurrences of a given permutation pattern q in permutations that avoid another given pattern r. In some cases, we find the pattern that occurs least often, (resp. most often) in all r-avoiding permutations. We also prove a few exact enumeration formulae, some of which are surprising.]]> 0 <![CDATA[Cuckoo hashing is a hash table data structure offering constant access time, even in the worst case. As a drawback, the construction fails with small, but practically significant probability. However, Kirsch et al. (2008) showed that a constant-sized additional memory, the so called stash, is sufficient to reduce the failure rate drastically. But so far, using a modified insertion procedure that demands additional running time to look for an admissible key is required. As a major contribution of this paper, we show that the same bounds on the failure probability hold even without this search process and thus, the performance increases. Second, we extend the analysis to simplified cuckoo hashing, a variant of the original algorithm offering increased performance. Further, we derive some explicit asymptotic approximations concerning the number of usual resp. bipartite graphs related to the data structures. Using these results, we obtain much more precise asymptotic expansions of the success rate. These calculations are based on a generating function approach and applying the saddle point method. Finally, we provide numerical results to support the theoretical analysis.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.495 https://doi.org/10.46298/dmtcs.495 Kutzelnigg, Reinhard Kutzelnigg, Reinhard <![CDATA[Cuckoo hashing is a hash table data structure offering constant access time, even in the worst case. As a drawback, the construction fails with small, but practically significant probability. However, Kirsch et al. (2008) showed that a constant-sized additional memory, the so called stash, is sufficient to reduce the failure rate drastically. But so far, using a modified insertion procedure that demands additional running time to look for an admissible key is required. As a major contribution of this paper, we show that the same bounds on the failure probability hold even without this search process and thus, the performance increases. Second, we extend the analysis to simplified cuckoo hashing, a variant of the original algorithm offering increased performance. Further, we derive some explicit asymptotic approximations concerning the number of usual resp. bipartite graphs related to the data structures. Using these results, we obtain much more precise asymptotic expansions of the success rate. These calculations are based on a generating function approach and applying the saddle point method. Finally, we provide numerical results to support the theoretical analysis.]]> 0 <![CDATA[A graph is a probe interval graph if its vertices can be partitioned into probes and nonprobes with an interval associated to each vertex so that vertices are adjacent if and only if their corresponding intervals intersect and at least one of them is a probe. A graph G = (V, E) is a tolerance graph if each vertex v is an element of V can be associated to an interval I(v) of the real line and a positive real number t(v) such that uv is an element of E if and only if vertical bar I(u) boolean AND I(v)vertical bar >= min \t(u), t(v)\. In this paper we present O(vertical bar V vertical bar + vertical bar E vertical bar) recognition algorithms for both bipartite probe interval graphs and bipartite tolerance graphs. We also give a new structural characterization for each class which follows from the algorithms.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.494 https://doi.org/10.46298/dmtcs.494 Brown, David E. Busch, Arthur H. Isaak, Garth Brown, David E. Busch, Arthur H. Isaak, Garth <![CDATA[A graph is a probe interval graph if its vertices can be partitioned into probes and nonprobes with an interval associated to each vertex so that vertices are adjacent if and only if their corresponding intervals intersect and at least one of them is a probe. A graph G = (V, E) is a tolerance graph if each vertex v is an element of V can be associated to an interval I(v) of the real line and a positive real number t(v) such that uv is an element of E if and only if vertical bar I(u) boolean AND I(v)vertical bar >= min \t(u), t(v)\. In this paper we present O(vertical bar V vertical bar + vertical bar E vertical bar) recognition algorithms for both bipartite probe interval graphs and bipartite tolerance graphs. We also give a new structural characterization for each class which follows from the algorithms.]]> 0 <![CDATA[We consider the monoid T with the presentation < a, b; aab = aba > which is ''close'' to trace monoids. We prove two different types of results. First, we give a combinatorial description of the lexicographically minimum and maximum representatives of their congruence classes in the free monoid \a, b\* and solve the classical equations, such as commutation and conjugacy in T. Then we study the closure properties of the two subfamilies of the rational subsets of T whose lexicographically minimum and maximum cross-sections respectively, are rational in \a, b\*.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.493 https://doi.org/10.46298/dmtcs.493 Choffrut, Christian Mercas, Robert Choffrut, Christian Mercas, Robert <![CDATA[We consider the monoid T with the presentation < a, b; aab = aba > which is ''close'' to trace monoids. We prove two different types of results. First, we give a combinatorial description of the lexicographically minimum and maximum representatives of their congruence classes in the free monoid \a, b\* and solve the classical equations, such as commutation and conjugacy in T. Then we study the closure properties of the two subfamilies of the rational subsets of T whose lexicographically minimum and maximum cross-sections respectively, are rational in \a, b\*.]]> 0 <![CDATA[For 3 different versions of q-tangent resp. q-cotangent functions, we compute the continued fraction expansion explicitly, by guessing the relative quantities and proving the recursive relation afterwards. It is likely that these are the only instances with a ''nice'' expansion. Additional formulae of a similar type are also provided.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.492 https://doi.org/10.46298/dmtcs.492 Prodinger, Helmut Prodinger, Helmut <![CDATA[For 3 different versions of q-tangent resp. q-cotangent functions, we compute the continued fraction expansion explicitly, by guessing the relative quantities and proving the recursive relation afterwards. It is likely that these are the only instances with a ''nice'' expansion. Additional formulae of a similar type are also provided.]]> 0 <![CDATA[Let us assign positive integers to the edges and vertices of a simple graph G. As a result we obtain a vertex-colouring of G with integers, where a vertex colour is simply a sum of the weight assigned to the vertex itself and the weights of its incident edges. Can we obtain a proper colouring using only weights 1 and 2 for an arbitrary G? We give a positive answer when G is a 3-colourable, complete or 4-regular graph. We also show that it is enough to C use weights from 1 to 11, as well as from 1 to 11 [chi(G)/2] + 1, for an arbitrary graph G.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.491 https://doi.org/10.46298/dmtcs.491 Przybylo, Jakub WoźniaK, Mariusz Przybylo, Jakub WoźniaK, Mariusz <![CDATA[Let us assign positive integers to the edges and vertices of a simple graph G. As a result we obtain a vertex-colouring of G with integers, where a vertex colour is simply a sum of the weight assigned to the vertex itself and the weights of its incident edges. Can we obtain a proper colouring using only weights 1 and 2 for an arbitrary G? We give a positive answer when G is a 3-colourable, complete or 4-regular graph. We also show that it is enough to C use weights from 1 to 11, as well as from 1 to 11 [chi(G)/2] + 1, for an arbitrary graph G.]]> 0 <![CDATA[In a recent paper we gave a counterexample to a longstanding conjecture concerning the characterization of regular languages of level 2 in the Straubing-Therien concatenation hierarchy of star-free languages. In that paper a new upper bound for the corresponding pseudovariety of monoids was implicitly given. In this paper we show that it is decidable whether a given monoid belongs to the new upper bound. We also prove that this new upper bound is incomparable with the previous upper bound.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.490 https://doi.org/10.46298/dmtcs.490 Almeida, Jorge Klima, Ondrej Almeida, Jorge Klima, Ondrej <![CDATA[In a recent paper we gave a counterexample to a longstanding conjecture concerning the characterization of regular languages of level 2 in the Straubing-Therien concatenation hierarchy of star-free languages. In that paper a new upper bound for the corresponding pseudovariety of monoids was implicitly given. In this paper we show that it is decidable whether a given monoid belongs to the new upper bound. We also prove that this new upper bound is incomparable with the previous upper bound.]]> 0 <![CDATA[This article studies the fundamental trade-off between delay and communication cost in networks. We consider an online optimization problem where nodes are organized in a tree topology. The nodes seek to minimize the time until the root is informed about the changes of their states and to use as few transmissions as possible. We derive an upper bound on the competitive ratio of O(min (h, c)) where h is the tree's height, and c is the transmission cost per edge. Moreover, we prove that this upper bound is tight in the sense that any oblivious algorithm has a ratio of at least Omega(min (h, c)). For chain networks, we prove a tight competitive ratio of Theta(min (root h, c)). Furthermore, we introduce a model for value-sensitive aggregation, where the cost depends on the number of transmissions and the error at the root.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.489 https://doi.org/10.46298/dmtcs.489 Pignolet, Yvonne Anne Schmid, Stefan Wattenhofer, Roger Pignolet, Yvonne Anne Schmid, Stefan Wattenhofer, Roger <![CDATA[This article studies the fundamental trade-off between delay and communication cost in networks. We consider an online optimization problem where nodes are organized in a tree topology. The nodes seek to minimize the time until the root is informed about the changes of their states and to use as few transmissions as possible. We derive an upper bound on the competitive ratio of O(min (h, c)) where h is the tree's height, and c is the transmission cost per edge. Moreover, we prove that this upper bound is tight in the sense that any oblivious algorithm has a ratio of at least Omega(min (h, c)). For chain networks, we prove a tight competitive ratio of Theta(min (root h, c)). Furthermore, we introduce a model for value-sensitive aggregation, where the cost depends on the number of transmissions and the error at the root.]]> 0 <![CDATA[We proved that every planar triangle-free graph of order n has a subset of vertices that induces a forest of size at least (71n + 72)/128. This improves the earlier work of Salavatipour (2006). We also pose some questions regarding planar graphs of higher girth.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.487 https://doi.org/10.46298/dmtcs.487 Kowalik, Lukasz Luzar, Borut Skrekovski, Riste Kowalik, Lukasz Luzar, Borut Skrekovski, Riste <![CDATA[We proved that every planar triangle-free graph of order n has a subset of vertices that induces a forest of size at least (71n + 72)/128. This improves the earlier work of Salavatipour (2006). We also pose some questions regarding planar graphs of higher girth.]]> 0 <![CDATA[We construct infinite cubefree binary words containing exponentially many distinct squares of length n. We also show that for every positive integer n, there is a cubefree binary square of length 2n.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.486 https://doi.org/10.46298/dmtcs.486 Currie, James Rampersad, Narad Currie, James Rampersad, Narad <![CDATA[We construct infinite cubefree binary words containing exponentially many distinct squares of length n. We also show that for every positive integer n, there is a cubefree binary square of length 2n.]]> 0 <![CDATA[In a recent paper, Brlek, Jamet and Paquin showed that some extremal infinite smooth words are also infinite Lyndon words. This result raises a natural question: are they the only ones? If no, what do the infinite smooth words that are also Lyndon words look like? In this paper, we give the answer, proving that the only infinite smooth Lyndon words are m(\a ...]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.485 https://doi.org/10.46298/dmtcs.485 Paquin, Geneviève Paquin, Geneviève <![CDATA[In a recent paper, Brlek, Jamet and Paquin showed that some extremal infinite smooth words are also infinite Lyndon words. This result raises a natural question: are they the only ones? If no, what do the infinite smooth words that are also Lyndon words look like? In this paper, we give the answer, proving that the only infinite smooth Lyndon words are m(\a ...]]> 0 <![CDATA[The split-coloring problem is a generalized vertex coloring problem where we partition the vertices into a minimum number of split graphs. In this paper, we study some notions which are extensively studied for the usual vertex coloring and the cocoloring problem from the point of view of split-coloring, such as criticality and the uniqueness of the minimum split-coloring. We discuss some properties of split-critical and uniquely split-colorable graphs. We describe constructions of such graphs with some additional properties. We also study the effect of the addition and the removal of some edge sets on the value of the split-chromatic number. All these results are compared with their cochromatic counterparts. We conclude with several research directions on the topic.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.484 https://doi.org/10.46298/dmtcs.484 Ekim, Tınaz Ries, Bernard De Werra, Dominique Ekim, Tınaz Ries, Bernard De Werra, Dominique <![CDATA[The split-coloring problem is a generalized vertex coloring problem where we partition the vertices into a minimum number of split graphs. In this paper, we study some notions which are extensively studied for the usual vertex coloring and the cocoloring problem from the point of view of split-coloring, such as criticality and the uniqueness of the minimum split-coloring. We discuss some properties of split-critical and uniquely split-colorable graphs. We describe constructions of such graphs with some additional properties. We also study the effect of the addition and the removal of some edge sets on the value of the split-chromatic number. All these results are compared with their cochromatic counterparts. We conclude with several research directions on the topic.]]> 0 <![CDATA[A k-colouring of a graph G is called acyclic if for every two distinct colours i and j, the subgraph induced in G by all the edges linking a vertex coloured with i and a vertex coloured with j is acyclic. In other words, there are no bichromatic alternating cycles. In 1999 Boiron et al. conjectured that a graph G with maximum degree at most 3 has an acyclic 2-colouring such that the set of vertices in each colour induces a subgraph with maximum degree at most 2. In this paper we prove this conjecture and show that such a colouring of a cubic graph can be determined in polynomial time. We also prove that it is an NP-complete problem to decide if a graph with maximum degree 4 has the above mentioned colouring.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.483 https://doi.org/10.46298/dmtcs.483 Borowiecki, Mieczyslaw Fiedorowicz, Anna Jesse-Jozefczyk, Katarzyna Sidorowicz, Elzbieta Borowiecki, Mieczyslaw Fiedorowicz, Anna Jesse-Jozefczyk, Katarzyna Sidorowicz, Elzbieta <![CDATA[A k-colouring of a graph G is called acyclic if for every two distinct colours i and j, the subgraph induced in G by all the edges linking a vertex coloured with i and a vertex coloured with j is acyclic. In other words, there are no bichromatic alternating cycles. In 1999 Boiron et al. conjectured that a graph G with maximum degree at most 3 has an acyclic 2-colouring such that the set of vertices in each colour induces a subgraph with maximum degree at most 2. In this paper we prove this conjecture and show that such a colouring of a cubic graph can be determined in polynomial time. We also prove that it is an NP-complete problem to decide if a graph with maximum degree 4 has the above mentioned colouring.]]> 0 <![CDATA[Edge-intersection graphs of paths in grids are graphs that can be represented such that vertices are paths in a grid and edges between vertices of the graph exist whenever two grid paths share a grid edge. This type of graphs is motivated by applications in conflict resolution of paths in grid networks. In this paper, we continue the study of edge-intersection graphs of paths in a grid, which was initiated by Golumbic, Lipshteyn and Stern. We show that for any k, if the number of bends in each path is restricted to be at most k, then not all graphs can be represented. Then we study some graph classes that can be represented with k-bend paths, for small k. We show that every planar graph has a representation with 5-bend paths, every outerplanar graph has a representation with 3-bend paths, and every planar bipartite graph has a representation with 2-bend paths. We also study line graphs, graphs of bounded pathwidth, and graphs with -regular edge orientations.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.482 https://doi.org/10.46298/dmtcs.482 Biedl, Therese Stern, Michal Biedl, Therese Stern, Michal <![CDATA[Edge-intersection graphs of paths in grids are graphs that can be represented such that vertices are paths in a grid and edges between vertices of the graph exist whenever two grid paths share a grid edge. This type of graphs is motivated by applications in conflict resolution of paths in grid networks. In this paper, we continue the study of edge-intersection graphs of paths in a grid, which was initiated by Golumbic, Lipshteyn and Stern. We show that for any k, if the number of bends in each path is restricted to be at most k, then not all graphs can be represented. Then we study some graph classes that can be represented with k-bend paths, for small k. We show that every planar graph has a representation with 5-bend paths, every outerplanar graph has a representation with 3-bend paths, and every planar bipartite graph has a representation with 2-bend paths. We also study line graphs, graphs of bounded pathwidth, and graphs with -regular edge orientations.]]> 0 <![CDATA[We show how the ECO method can be applied to exhaustively generate some classes of permutations. A previous work initiating this technique and motivating our research was published in Ac ta Informatica, 2004, by S. Bacchelli, E. Barcucci, E. Grazzini and E. Pergola.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.479 https://doi.org/10.46298/dmtcs.479 Vajnovszki, Vincent Vajnovszki, Vincent <![CDATA[We show how the ECO method can be applied to exhaustively generate some classes of permutations. A previous work initiating this technique and motivating our research was published in Ac ta Informatica, 2004, by S. Bacchelli, E. Barcucci, E. Grazzini and E. Pergola.]]> 0 <![CDATA[We exhibit a recurrence on the number of discrete line segments joining two integer points in the plane using an encoding of such segments as balanced words of given length and height over the two-letter alphabet $\{0,1\}$. We give generating functions and study the asymptotic behaviour. As a particular case, we focus on the symmetrical discrete segments which are encoded by balanced palindromes.]]> Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/dmtcs.474 https://doi.org/10.46298/dmtcs.474 Bédaride, Nicolas Domenjoud, Eric Jamet, Damien Rémy, Jean-Luc Bédaride, Nicolas Domenjoud, Eric Jamet, Damien Rémy, Jean-Luc <![CDATA[We exhibit a recurrence on the number of discrete line segments joining two integer points in the plane using an encoding of such segments as balanced words of given length and height over the two-letter alphabet $\{0,1\}$. We give generating functions and study the asymptotic behaviour. As a particular case, we focus on the symmetrical discrete segments which are encoded by balanced palindromes.]]> 0 <![CDATA[We present examples of flag homology spheres whose $\gamma$-vectors satisfy the Kruskal-Katona inequalities. This includes several families of well-studied simplicial complexes, including Coxeter complexes and the simplicial complexes dual to the associahedron and to the cyclohedron. In these cases, we construct explicit flag simplicial complexes whose $f$-vectors are the $\gamma$-vectors in question, and so a result of Frohmader shows that the $\gamma$-vectors satisfy not only the Kruskal-Katona inequalities but also the stronger Frankl-Füredi-Kalai inequalities. In another direction, we show that if a flag $(d-1)$-sphere has at most $2d+3$ vertices its $\gamma$-vector satisfies the Frankl-Füredi-Kalai inequalities. We conjecture that if $\Delta$ is a flag homology sphere then $\gamma (\Delta)$ satisfies the Kruskal-Katona, and further, the Frankl-Füredi-Kalai inequalities. This conjecture is a significant refinement of Gal's conjecture, which asserts that such $\gamma$-vectors are nonnegative.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2842 https://doi.org/10.46298/dmtcs.2842 Nevo, E. Petersen, T. K. Nevo, E. Petersen, T. K. <![CDATA[We present examples of flag homology spheres whose $\gamma$-vectors satisfy the Kruskal-Katona inequalities. This includes several families of well-studied simplicial complexes, including Coxeter complexes and the simplicial complexes dual to the associahedron and to the cyclohedron. In these cases, we construct explicit flag simplicial complexes whose $f$-vectors are the $\gamma$-vectors in question, and so a result of Frohmader shows that the $\gamma$-vectors satisfy not only the Kruskal-Katona inequalities but also the stronger Frankl-Füredi-Kalai inequalities. In another direction, we show that if a flag $(d-1)$-sphere has at most $2d+3$ vertices its $\gamma$-vector satisfies the Frankl-Füredi-Kalai inequalities. We conjecture that if $\Delta$ is a flag homology sphere then $\gamma (\Delta)$ satisfies the Kruskal-Katona, and further, the Frankl-Füredi-Kalai inequalities. This conjecture is a significant refinement of Gal's conjecture, which asserts that such $\gamma$-vectors are nonnegative.]]> 0 <![CDATA[The purpose of this paper is to present an algorithm which generates linear extensions for a generalized Young diagram, in the sense of D. Peterson and R. A. Proctor, with uniform probability. This gives a proof of a D. Peterson's hook formula for the number of reduced decompositions of a given minuscule elements. \par]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2843 https://doi.org/10.46298/dmtcs.2843 Nakada, Kento Okamura, Shuji Nakada, Kento Okamura, Shuji <![CDATA[The purpose of this paper is to present an algorithm which generates linear extensions for a generalized Young diagram, in the sense of D. Peterson and R. A. Proctor, with uniform probability. This gives a proof of a D. Peterson's hook formula for the number of reduced decompositions of a given minuscule elements. \par]]> 0 <![CDATA[We show the $q$-analog of a well-known result of Farahat and Higman: in the center of the Iwahori-Hecke algebra $\mathscr{H}_{n,q}$, if $(a_{\lambda \mu}^ν (n,q))_ν$ is the set of structure constants involved in the product of two Geck-Rouquier conjugacy classes $\Gamma_{\lambda, n}$ and $\Gamma_{\mu,n}$, then each coefficient $a_{\lambda \mu}^ν (n,q)$ depend on $n$ and $q$ in a polynomial way. Our proof relies on the construction of a projective limit of the Hecke algebras; this projective limit is inspired by the Ivanov-Kerov algebra of partial permutations.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2844 https://doi.org/10.46298/dmtcs.2844 Méliot, Pierre-Loïc Méliot, Pierre-Loïc <![CDATA[We show the $q$-analog of a well-known result of Farahat and Higman: in the center of the Iwahori-Hecke algebra $\mathscr{H}_{n,q}$, if $(a_{\lambda \mu}^ν (n,q))_ν$ is the set of structure constants involved in the product of two Geck-Rouquier conjugacy classes $\Gamma_{\lambda, n}$ and $\Gamma_{\mu,n}$, then each coefficient $a_{\lambda \mu}^ν (n,q)$ depend on $n$ and $q$ in a polynomial way. Our proof relies on the construction of a projective limit of the Hecke algebras; this projective limit is inspired by the Ivanov-Kerov algebra of partial permutations.]]> 0 <![CDATA[We establish a stronger symmetry between the numbers of northeast and southeast chains in the context of $01$-fillings of moon polyominoes. Let $\mathcal{M}$ be a moon polyomino. Consider all the $01$-fillings of $\mathcal{M}$ in which every row has at most one $1$. We introduce four mixed statistics with respect to a bipartition of rows or columns of $\mathcal{M}$. More precisely, let $S$ be a subset of rows of $\mathcal{M}$. For any filling $M$, the top-mixed (resp. bottom-mixed) statistic $\alpha (S; M)$ (resp. $\beta (S; M)$) is the sum of the number of northeast chains whose top (resp. bottom) cell is in $S$, together with the number of southeast chains whose top (resp. bottom) cell is in the complement of $S$. Similarly, we define the left-mixed and right-mixed statistics $\gamma (T; M)$ and $\delta (T; M)$, where $T$ is a subset of the columns. Let $\lambda (A; M)$ be any of these four statistics $\alpha (S; M)$, $\beta (S; M)$, $\gamma (T; M)$ and $\delta (T; M)$. We show that the joint distribution of the pair $(\lambda (A; M), \lambda (M/A; M))$ is symmetric and independent of the subsets $S, T$. In particular, the pair of statistics $(\lambda (A;M), \lambda (M/A; M))$ is equidistributed with $(\mathrm{se}(M), \mathrm{ne}(M))$, where $\mathrm{se}(M)$ and $\mathrm{ne}(M)$ are the numbers of southeast chains and northeast chains of $M$, respectively.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2845 https://doi.org/10.46298/dmtcs.2845 Chen, William Y. C. Wang, Andrew Y. Z. Yan, Catherine H. Zhao, Alina F. Y. Chen, William Y. C. Wang, Andrew Y. Z. Yan, Catherine H. Zhao, Alina F. Y. <![CDATA[We establish a stronger symmetry between the numbers of northeast and southeast chains in the context of $01$-fillings of moon polyominoes. Let $\mathcal{M}$ be a moon polyomino. Consider all the $01$-fillings of $\mathcal{M}$ in which every row has at most one $1$. We introduce four mixed statistics with respect to a bipartition of rows or columns of $\mathcal{M}$. More precisely, let $S$ be a subset of rows of $\mathcal{M}$. For any filling $M$, the top-mixed (resp. bottom-mixed) statistic $\alpha (S; M)$ (resp. $\beta (S; M)$) is the sum of the number of northeast chains whose top (resp. bottom) cell is in $S$, together with the number of southeast chains whose top (resp. bottom) cell is in the complement of $S$. Similarly, we define the left-mixed and right-mixed statistics $\gamma (T; M)$ and $\delta (T; M)$, where $T$ is a subset of the columns. Let $\lambda (A; M)$ be any of these four statistics $\alpha (S; M)$, $\beta (S; M)$, $\gamma (T; M)$ and $\delta (T; M)$. We show that the joint distribution of the pair $(\lambda (A; M), \lambda (M/A; M))$ is symmetric and independent of the subsets $S, T$. In particular, the pair of statistics $(\lambda (A;M), \lambda (M/A; M))$ is equidistributed with $(\mathrm{se}(M), \mathrm{ne}(M))$, where $\mathrm{se}(M)$ and $\mathrm{ne}(M)$ are the numbers of southeast chains and northeast chains of $M$, respectively.]]> 0 <![CDATA[We prove several conjectures of Eriksen regarding the joint distribution on permutations of the number of adjacencies (descents with consecutive values in consecutive positions), descents and some Mahonian statistics. We also prove Eriksen's conjecture that a certain bistatistic on Viennot's alternative tableaux is Euler-Mahonian.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2846 https://doi.org/10.46298/dmtcs.2846 Burstein, Alexander Burstein, Alexander <![CDATA[We prove several conjectures of Eriksen regarding the joint distribution on permutations of the number of adjacencies (descents with consecutive values in consecutive positions), descents and some Mahonian statistics. We also prove Eriksen's conjecture that a certain bistatistic on Viennot's alternative tableaux is Euler-Mahonian.]]> 0 <![CDATA[A tropical curve $\Gamma$ is a metric graph with possibly unbounded edges, and tropical rational functions are continuous piecewise linear functions with integer slopes. We define the complete linear system $|D|$ of a divisor $D$ on a tropical curve $\Gamma$ analogously to the classical counterpart. We investigate the structure of $|D|$ as a cell complex and show that linear systems are quotients of tropical modules, finitely generated by vertices of the cell complex. Using a finite set of generators, $|D|$ defines a map from $\Gamma$ to a tropical projective space, and the image can be modified to a tropical curve of degree equal to $\mathrm{deg}(D)$. The tropical convex hull of the image realizes the linear system $|D|$ as a polyhedral complex.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2847 https://doi.org/10.46298/dmtcs.2847 Haase, Christian Musiker, Gregg Yu, Josephine Haase, Christian Musiker, Gregg Yu, Josephine <![CDATA[A tropical curve $\Gamma$ is a metric graph with possibly unbounded edges, and tropical rational functions are continuous piecewise linear functions with integer slopes. We define the complete linear system $|D|$ of a divisor $D$ on a tropical curve $\Gamma$ analogously to the classical counterpart. We investigate the structure of $|D|$ as a cell complex and show that linear systems are quotients of tropical modules, finitely generated by vertices of the cell complex. Using a finite set of generators, $|D|$ defines a map from $\Gamma$ to a tropical projective space, and the image can be modified to a tropical curve of degree equal to $\mathrm{deg}(D)$. The tropical convex hull of the image realizes the linear system $|D|$ as a polyhedral complex.]]> 0 <![CDATA[It is well-known that Catalan numbers $C_n = \frac{1}{ n+1} \binom{2n}{n}$ count the number of dominant regions in the Shi arrangement of type $A$, and that they also count partitions which are both n-cores as well as $(n+1)$-cores. These concepts have natural extensions, which we call here the $m$-Catalan numbers and $m$-Shi arrangement. In this paper, we construct a bijection between dominant regions of the $m$-Shi arrangement and partitions which are both $n$-cores as well as $(mn+1)$-cores. We also modify our construction to produce a bijection between bounded dominant regions of the $m$-Shi arrangement and partitions which are both $n$-cores as well as $(mn-1)$-cores. The bijections are natural in the sense that they commute with the action of the affine symmetric group.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2848 https://doi.org/10.46298/dmtcs.2848 Fishel, Susanna Vazirani, Monica Fishel, Susanna Vazirani, Monica <![CDATA[It is well-known that Catalan numbers $C_n = \frac{1}{ n+1} \binom{2n}{n}$ count the number of dominant regions in the Shi arrangement of type $A$, and that they also count partitions which are both n-cores as well as $(n+1)$-cores. These concepts have natural extensions, which we call here the $m$-Catalan numbers and $m$-Shi arrangement. In this paper, we construct a bijection between dominant regions of the $m$-Shi arrangement and partitions which are both $n$-cores as well as $(mn+1)$-cores. We also modify our construction to produce a bijection between bounded dominant regions of the $m$-Shi arrangement and partitions which are both $n$-cores as well as $(mn-1)$-cores. The bijections are natural in the sense that they commute with the action of the affine symmetric group.]]> 0 <![CDATA[Many important invariants for matroids and polymatroids, such as the Tutte polynomial, the Billera-Jia-Reiner quasi-symmetric function, and the invariant $\mathcal{G}$ introduced by the first author, are valuative. In this paper we construct the $\mathbb{Z}$-modules of all $\mathbb{Z}$-valued valuative functions for labelled matroids and polymatroids on a fixed ground set, and their unlabelled counterparts, the $\mathbb{Z}$-modules of valuative invariants. We give explicit bases for these modules and for their dual modules generated by indicator functions of polytopes, and explicit formulas for their ranks. Our results confirm a conjecture of the first author that $\mathcal{G}$ is universal for valuative invariants.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2849 https://doi.org/10.46298/dmtcs.2849 Derksen, Harm Fink, Alex Derksen, Harm Fink, Alex <![CDATA[Many important invariants for matroids and polymatroids, such as the Tutte polynomial, the Billera-Jia-Reiner quasi-symmetric function, and the invariant $\mathcal{G}$ introduced by the first author, are valuative. In this paper we construct the $\mathbb{Z}$-modules of all $\mathbb{Z}$-valued valuative functions for labelled matroids and polymatroids on a fixed ground set, and their unlabelled counterparts, the $\mathbb{Z}$-modules of valuative invariants. We give explicit bases for these modules and for their dual modules generated by indicator functions of polytopes, and explicit formulas for their ranks. Our results confirm a conjecture of the first author that $\mathcal{G}$ is universal for valuative invariants.]]> 0 <![CDATA[The famous hook-length formula is a simple consequence of the branching rule for the hook lengths. While the Greene-Nijenhuis-Wilf probabilistic proof is the most famous proof of the rule, it is not completely combinatorial, and a simple bijection was an open problem for a long time. In this extended abstract, we show an elegant bijective argument that proves a stronger, weighted analogue of the branching rule. Variants of the bijection prove seven other interesting formulas. Another important approach to the formulas is via weighted hook walks; we discuss some results in this area. We present another motivation for our work: $J$-functions of the Hilbert scheme of points.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2850 https://doi.org/10.46298/dmtcs.2850 Ciocan-Fontanine, Ionuţ Konvalinka, Matjaž Pak, Igor Ciocan-Fontanine, Ionuţ Konvalinka, Matjaž Pak, Igor <![CDATA[The famous hook-length formula is a simple consequence of the branching rule for the hook lengths. While the Greene-Nijenhuis-Wilf probabilistic proof is the most famous proof of the rule, it is not completely combinatorial, and a simple bijection was an open problem for a long time. In this extended abstract, we show an elegant bijective argument that proves a stronger, weighted analogue of the branching rule. Variants of the bijection prove seven other interesting formulas. Another important approach to the formulas is via weighted hook walks; we discuss some results in this area. We present another motivation for our work: $J$-functions of the Hilbert scheme of points.]]> 0 <![CDATA[For any finite Coxeter group $W$, we introduce two new objects: its cutting poset and its biHecke monoid. The cutting poset, constructed using a generalization of the notion of blocks in permutation matrices, almost forms a lattice on $W$. The construction of the biHecke monoid relies on the usual combinatorial model for the $0-Hecke$ algebra $H_0(W)$, that is, for the symmetric group, the algebra (or monoid) generated by the elementary bubble sort operators. The authors previously introduced the Hecke group algebra, constructed as the algebra generated simultaneously by the bubble sort and antisort operators, and described its representation theory. In this paper, we consider instead the monoid generated by these operators. We prove that it admits |W| simple and projective modules. In order to construct the simple modules, we introduce for each $w∈W$ a combinatorial module $T_w$ whose support is the interval $[1,w]_R$ in right weak order. This module yields an algebra, whose representation theory generalizes that of the Hecke group algebra, with the combinatorics of descents replaced by that of blocks and of the cutting poset.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2851 https://doi.org/10.46298/dmtcs.2851 Hivert, Florent Schilling, Anne Thiéry, Nicolas M. Hivert, Florent Schilling, Anne Thiéry, Nicolas M. <![CDATA[For any finite Coxeter group $W$, we introduce two new objects: its cutting poset and its biHecke monoid. The cutting poset, constructed using a generalization of the notion of blocks in permutation matrices, almost forms a lattice on $W$. The construction of the biHecke monoid relies on the usual combinatorial model for the $0-Hecke$ algebra $H_0(W)$, that is, for the symmetric group, the algebra (or monoid) generated by the elementary bubble sort operators. The authors previously introduced the Hecke group algebra, constructed as the algebra generated simultaneously by the bubble sort and antisort operators, and described its representation theory. In this paper, we consider instead the monoid generated by these operators. We prove that it admits |W| simple and projective modules. In order to construct the simple modules, we introduce for each $w∈W$ a combinatorial module $T_w$ whose support is the interval $[1,w]_R$ in right weak order. This module yields an algebra, whose representation theory generalizes that of the Hecke group algebra, with the combinatorics of descents replaced by that of blocks and of the cutting poset.]]> 0 <![CDATA[Let $G$ be a connected reductive linear algebraic group over $ℂ$ with an involution $θ$ . Denote by $K$ the subgroup of fixed points. In certain cases, the $K-orbits$ in the flag variety $G/B$ are indexed by the twisted identities $ι (θ ) = {θ (w^{-1})w | w∈W}$ in the Weyl group $W$. Under this assumption, we establish a criterion for rational smoothness of orbit closures which generalises classical results of Carrell and Peterson for Schubert varieties. That is, whether an orbit closure is rationally smooth at a given point can be determined by examining the degrees in a "Bruhat graph'' whose vertices form a subset of $ι (θ )$. Moreover, an orbit closure is rationally smooth everywhere if and only if its corresponding interval in the Bruhat order on $ι (θ )$ is rank symmetric. In the special case $K=\mathrm{Sp}_{2n}(ℂ), G=\mathrm{SL}_{2n}(ℂ)$, we strengthen our criterion by showing that only the degree of a single vertex, the "bottom one'', needs to be examined. This generalises a result of Deodhar for type A Schubert varieties.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2852 https://doi.org/10.46298/dmtcs.2852 Hultman, Axel Hultman, Axel <![CDATA[Let $G$ be a connected reductive linear algebraic group over $ℂ$ with an involution $θ$ . Denote by $K$ the subgroup of fixed points. In certain cases, the $K-orbits$ in the flag variety $G/B$ are indexed by the twisted identities $ι (θ ) = {θ (w^{-1})w | w∈W}$ in the Weyl group $W$. Under this assumption, we establish a criterion for rational smoothness of orbit closures which generalises classical results of Carrell and Peterson for Schubert varieties. That is, whether an orbit closure is rationally smooth at a given point can be determined by examining the degrees in a "Bruhat graph'' whose vertices form a subset of $ι (θ )$. Moreover, an orbit closure is rationally smooth everywhere if and only if its corresponding interval in the Bruhat order on $ι (θ )$ is rank symmetric. In the special case $K=\mathrm{Sp}_{2n}(ℂ), G=\mathrm{SL}_{2n}(ℂ)$, we strengthen our criterion by showing that only the degree of a single vertex, the "bottom one'', needs to be examined. This generalises a result of Deodhar for type A Schubert varieties.]]> 0 <![CDATA[We use Hopf algebras to prove a version of the Littlewood―Richardson rule for skew Schur functions, which implies a conjecture of Assaf and McNamara. We also establish skew Littlewood―Richardson rules for Schur $P-$ and $Q-$functions and noncommutative ribbon Schur functions, as well as skew Pieri rules for k-Schur functions, dual k-Schur functions, and for the homology of the affine Grassmannian of the symplectic group.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2853 https://doi.org/10.46298/dmtcs.2853 Lam, Thomas Lauve, Aaron Sottile, Frank Lam, Thomas Lauve, Aaron Sottile, Frank <![CDATA[We use Hopf algebras to prove a version of the Littlewood―Richardson rule for skew Schur functions, which implies a conjecture of Assaf and McNamara. We also establish skew Littlewood―Richardson rules for Schur $P-$ and $Q-$functions and noncommutative ribbon Schur functions, as well as skew Pieri rules for k-Schur functions, dual k-Schur functions, and for the homology of the affine Grassmannian of the symplectic group.]]> 0 <![CDATA[We give an interpretation of the $t=1$ specialization of the modified Macdonald polynomial as a generating function of the energy statistics defined on the set of paths arising in the context of Box-Ball Systems (BBS-paths for short). We also introduce one parameter generalizations of the energy statistics on the set of BBS-paths which all, conjecturally, have the same distribution.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2854 https://doi.org/10.46298/dmtcs.2854 Kirillov, Anatol N. Sakamoto, Reiho Kirillov, Anatol N. Sakamoto, Reiho <![CDATA[We give an interpretation of the $t=1$ specialization of the modified Macdonald polynomial as a generating function of the energy statistics defined on the set of paths arising in the context of Box-Ball Systems (BBS-paths for short). We also introduce one parameter generalizations of the energy statistics on the set of BBS-paths which all, conjecturally, have the same distribution.]]> 0 <![CDATA[We consider a three-parameter PASEP model on $N$ sites. A closed formula for the partition function was obtained analytically by Blythe et al. We give a new formula which generalizes the one of Blythe et al, and is proved in two combinatorial ways. Moreover the first proof can be adapted to give the moments of Al-Salam-Chihara polynomials.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2855 https://doi.org/10.46298/dmtcs.2855 Josuat-Vergès, Matthieu Josuat-Vergès, Matthieu <![CDATA[We consider a three-parameter PASEP model on $N$ sites. A closed formula for the partition function was obtained analytically by Blythe et al. We give a new formula which generalizes the one of Blythe et al, and is proved in two combinatorial ways. Moreover the first proof can be adapted to give the moments of Al-Salam-Chihara polynomials.]]> 0 <![CDATA[Motivated by juggling sequences and bubble sort, we examine permutations on the set${1, 2, \ldots, n}$ with $d$ descents and maximum drop size $k$. We give explicit formulas for enumerating such permutations for given integers $k$ and $d$. We also derive the related generating functions and prove unimodality and symmetry of the coefficients.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2856 https://doi.org/10.46298/dmtcs.2856 Chung, Fan Claesson, Anders Dukes, Mark Graham, Ronald Chung, Fan Claesson, Anders Dukes, Mark Graham, Ronald <![CDATA[Motivated by juggling sequences and bubble sort, we examine permutations on the set${1, 2, \ldots, n}$ with $d$ descents and maximum drop size $k$. We give explicit formulas for enumerating such permutations for given integers $k$ and $d$. We also derive the related generating functions and prove unimodality and symmetry of the coefficients.]]> 0 <![CDATA[Let $P$ be a polytope with rational vertices. A classical theorem of Ehrhart states that the number of lattice points in the dilations $P(n) = nP$ is a quasi-polynomial in $n$. We generalize this theorem by allowing the vertices of $P(n)$ to be arbitrary rational functions in $n$. In this case we prove that the number of lattice points in $P(n)$ is a quasi-polynomial for $n$ sufficiently large. Our work was motivated by a conjecture of Ehrhart on the number of solutions to parametrized linear Diophantine equations whose coefficients are polynomials in $n$, and we explain how these two problems are related.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2857 https://doi.org/10.46298/dmtcs.2857 Chen, Sheng Li, Nan Sam, Steven V Chen, Sheng Li, Nan Sam, Steven V <![CDATA[Let $P$ be a polytope with rational vertices. A classical theorem of Ehrhart states that the number of lattice points in the dilations $P(n) = nP$ is a quasi-polynomial in $n$. We generalize this theorem by allowing the vertices of $P(n)$ to be arbitrary rational functions in $n$. In this case we prove that the number of lattice points in $P(n)$ is a quasi-polynomial for $n$ sufficiently large. Our work was motivated by a conjecture of Ehrhart on the number of solutions to parametrized linear Diophantine equations whose coefficients are polynomials in $n$, and we explain how these two problems are related.]]> 0 <![CDATA[The $k$-parabolic subspace arrangement, introduced by Barcelo, Severs and White, is a generalization of the well known $k$-equal arrangements of type-$A$ and type-$B$. In this paper we use the discrete Morse theory of Forman to study the homology of the complements of $k$-parabolic subspace arrangements. In doing so, we recover some known results of Björner et al. and provide a combinatorial interpretation of the Betti numbers for any $k$-parabolic subspace arrangement. The paper provides results for any $k$-parabolic subspace arrangement, however we also include an extended example of our methods applied to the $k$-equal arrangements of type-$A$ and type-$B$. In these cases, we obtain new formulas for the Betti numbers.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2882 https://doi.org/10.46298/dmtcs.2882 Severs, Christopher White, Jacob A. Severs, Christopher White, Jacob A. <![CDATA[The $k$-parabolic subspace arrangement, introduced by Barcelo, Severs and White, is a generalization of the well known $k$-equal arrangements of type-$A$ and type-$B$. In this paper we use the discrete Morse theory of Forman to study the homology of the complements of $k$-parabolic subspace arrangements. In doing so, we recover some known results of Björner et al. and provide a combinatorial interpretation of the Betti numbers for any $k$-parabolic subspace arrangement. The paper provides results for any $k$-parabolic subspace arrangement, however we also include an extended example of our methods applied to the $k$-equal arrangements of type-$A$ and type-$B$. In these cases, we obtain new formulas for the Betti numbers.]]> 0 <![CDATA[The Pieri rule expresses the product of a Schur function and a single row Schur function in terms of Schur functions. We extend the classical Pieri rule by expressing the product of a skew Schur function and a single row Schur function in terms of skew Schur functions. Like the classical rule, our rule involves simple additions of boxes to the original skew shape. Our proof is purely combinatorial and extends the combinatorial proof of the classical case.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2886 https://doi.org/10.46298/dmtcs.2886 Assaf, Sami H. McNamara, Peter R. W. Assaf, Sami H. McNamara, Peter R. W. <![CDATA[The Pieri rule expresses the product of a Schur function and a single row Schur function in terms of Schur functions. We extend the classical Pieri rule by expressing the product of a skew Schur function and a single row Schur function in terms of skew Schur functions. Like the classical rule, our rule involves simple additions of boxes to the original skew shape. Our proof is purely combinatorial and extends the combinatorial proof of the classical case.]]> 0 <![CDATA[We investigate the probability that a random composition (ordered partition) of the positive integer $n$ has no parts occurring exactly $j$ times, where $j$ belongs to a specified finite $\textit{`forbidden set'}$ $A$ of multiplicities. This probability is also studied in the related case of samples $\Gamma =(\Gamma_1,\Gamma_2,\ldots, \Gamma_n)$ of independent, identically distributed random variables with a geometric distribution.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2885 https://doi.org/10.46298/dmtcs.2885 Archibald, Margaret Knopfmacher, Arnold Mansour, Toufik Archibald, Margaret Knopfmacher, Arnold Mansour, Toufik <![CDATA[We investigate the probability that a random composition (ordered partition) of the positive integer $n$ has no parts occurring exactly $j$ times, where $j$ belongs to a specified finite $\textit{`forbidden set'}$ $A$ of multiplicities. This probability is also studied in the related case of samples $\Gamma =(\Gamma_1,\Gamma_2,\ldots, \Gamma_n)$ of independent, identically distributed random variables with a geometric distribution.]]> 0 <![CDATA[In recent work on nonequilibrium statistical physics, a certain Markovian exclusion model called an asymmetric annihilation process was studied by Ayyer and Mallick. In it they gave a precise conjecture for the eigenvalues (along with the multiplicities) of the transition matrix. They further conjectured that to each eigenvalue, there corresponds only one eigenvector. We prove the first of these conjectures by generalizing the original Markov matrix by introducing extra parameters, explicitly calculating its eigenvalues, and showing that the new matrix reduces to the original one by a suitable specialization. In addition, we outline a derivation of the partition function in the generalized model, which also reduces to the one obtained by Ayyer and Mallick in the original model.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2884 https://doi.org/10.46298/dmtcs.2884 Ayyer, Arvind Strehl, Volker Ayyer, Arvind Strehl, Volker <![CDATA[In recent work on nonequilibrium statistical physics, a certain Markovian exclusion model called an asymmetric annihilation process was studied by Ayyer and Mallick. In it they gave a precise conjecture for the eigenvalues (along with the multiplicities) of the transition matrix. They further conjectured that to each eigenvalue, there corresponds only one eigenvector. We prove the first of these conjectures by generalizing the original Markov matrix by introducing extra parameters, explicitly calculating its eigenvalues, and showing that the new matrix reduces to the original one by a suitable specialization. In addition, we outline a derivation of the partition function in the generalized model, which also reduces to the one obtained by Ayyer and Mallick in the original model.]]> 0 <![CDATA[We define a new family of self-avoiding walks (SAW) on the square lattice, called $\textit{weakly directed walks}$. These walks have a simple characterization in terms of the irreducible bridges that compose them. We determine their generating function. This series has a complex singularity structure and in particular, is not D-finite. The growth constant is approximately 2.54 and is thus larger than that of all natural families of SAW enumerated so far (but smaller than that of general SAW, which is about 2.64). We also prove that the end-to-end distance of weakly directed walks grows linearly. Finally, we study a diagonal variant of this model.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2883 https://doi.org/10.46298/dmtcs.2883 Bacher, Axel Bousquet-Mélou, Mireille Bacher, Axel Bousquet-Mélou, Mireille <![CDATA[We define a new family of self-avoiding walks (SAW) on the square lattice, called $\textit{weakly directed walks}$. These walks have a simple characterization in terms of the irreducible bridges that compose them. We determine their generating function. This series has a complex singularity structure and in particular, is not D-finite. The growth constant is approximately 2.54 and is thus larger than that of all natural families of SAW enumerated so far (but smaller than that of general SAW, which is about 2.64). We also prove that the end-to-end distance of weakly directed walks grows linearly. Finally, we study a diagonal variant of this model.]]> 0 <![CDATA[We use a quantum analog of the polynomial ring $\mathbb{Z}[x_{1,1},\ldots, x_{n,n}]$ to modify the Kazhdan-Lusztig construction of irreducible $H_n(q)$-modules. This modified construction produces exactly the same matrices as the original construction in [$\textit{Invent. Math.}$ $\textbf{53}$ (1979)], but does not employ the Kazhdan-Lusztig preorders. Our main result is dependent on new vanishing results for immanants in the quantum polynomial ring.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2874 https://doi.org/10.46298/dmtcs.2874 Buehrle, Charles Skandera, Mark Buehrle, Charles Skandera, Mark <![CDATA[We use a quantum analog of the polynomial ring $\mathbb{Z}[x_{1,1},\ldots, x_{n,n}]$ to modify the Kazhdan-Lusztig construction of irreducible $H_n(q)$-modules. This modified construction produces exactly the same matrices as the original construction in [$\textit{Invent. Math.}$ $\textbf{53}$ (1979)], but does not employ the Kazhdan-Lusztig preorders. Our main result is dependent on new vanishing results for immanants in the quantum polynomial ring.]]> 0 <![CDATA[We are interested in Fully Packed Loops in a triangle (TFPLs), as introduced by Caselli at al. and studied by Thapper. We show that for Fully Packed Loops with a fixed link pattern (refined FPL), there exist linear recurrence relations with coefficients computed from TFPL configurations. We then give constraints and enumeration results for certain classes of TFPL configurations. For special boundary conditions, we show that TFPLs are counted by the famous Littlewood Richardson coefficients.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2881 https://doi.org/10.46298/dmtcs.2881 Nadeau, Philippe Nadeau, Philippe <![CDATA[We are interested in Fully Packed Loops in a triangle (TFPLs), as introduced by Caselli at al. and studied by Thapper. We show that for Fully Packed Loops with a fixed link pattern (refined FPL), there exist linear recurrence relations with coefficients computed from TFPL configurations. We then give constraints and enumeration results for certain classes of TFPL configurations. For special boundary conditions, we show that TFPLs are counted by the famous Littlewood Richardson coefficients.]]> 0 <![CDATA[We give bijective proofs of pattern-avoidance results for a class of permutations generalizing alternating permutations. The bijections employed include a modified form of the RSK insertion algorithm and recursive bijections based on generating trees. As special cases, we show that the sets $A_{2n}(1234)$ and $A_{2n}(2143)$ are in bijection with standard Young tableaux of shape $\langle 3^n \rangle$. Alternating permutations may be viewed as the reading words of standard Young tableaux of a certain skew shape. In the last section of the paper, we study pattern avoidance in the reading words of standard Young tableaux of any skew shape. We show bijectively that the number of standard Young tableaux of shape $\lambda / \mu$ whose reading words avoid $213$ is a natural $\mu$-analogue of the Catalan numbers. Similar results for the patterns $132$, $231$ and $312$.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2880 https://doi.org/10.46298/dmtcs.2880 Lewis, Joel Brewster Lewis, Joel Brewster <![CDATA[We give bijective proofs of pattern-avoidance results for a class of permutations generalizing alternating permutations. The bijections employed include a modified form of the RSK insertion algorithm and recursive bijections based on generating trees. As special cases, we show that the sets $A_{2n}(1234)$ and $A_{2n}(2143)$ are in bijection with standard Young tableaux of shape $\langle 3^n \rangle$. Alternating permutations may be viewed as the reading words of standard Young tableaux of a certain skew shape. In the last section of the paper, we study pattern avoidance in the reading words of standard Young tableaux of any skew shape. We show bijectively that the number of standard Young tableaux of shape $\lambda / \mu$ whose reading words avoid $213$ is a natural $\mu$-analogue of the Catalan numbers. Similar results for the patterns $132$, $231$ and $312$.]]> 0 <![CDATA[A factorization of a permutation into transpositions is called "primitive'' if its factors are weakly ordered.We discuss the problem of enumerating primitive factorizations of permutations, and its place in the hierarchy of previously studied factorization problems. Several formulas enumerating minimal primitive and possibly non-minimal primitive factorizations are presented, and interesting connections with Jucys-Murphy elements, symmetric group characters, and matrix models are described.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2879 https://doi.org/10.46298/dmtcs.2879 Matsumoto, Sho Novak, Jonathan Matsumoto, Sho Novak, Jonathan <![CDATA[A factorization of a permutation into transpositions is called "primitive'' if its factors are weakly ordered.We discuss the problem of enumerating primitive factorizations of permutations, and its place in the hierarchy of previously studied factorization problems. Several formulas enumerating minimal primitive and possibly non-minimal primitive factorizations are presented, and interesting connections with Jucys-Murphy elements, symmetric group characters, and matrix models are described.]]> 0 <![CDATA[We introduce a multiplicity Tutte polynomial $M(x,y)$, which generalizes the ordinary one and has applications to zonotopes and toric arrangements. We prove that $M(x,y)$ satisfies a deletion-restriction recurrence and has positive coefficients. The characteristic polynomial and the Poincaré polynomial of a toric arrangement are shown to be specializations of the associated polynomial $M(x,y)$, likewise the corresponding polynomials for a hyperplane arrangement are specializations of the ordinary Tutte polynomial. Furthermore, $M(1,y)$ is the Hilbert series of the related discrete Dahmen-Micchelli space, while $M(x,1)$ computes the volume and the number of integral points of the associated zonotope.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2878 https://doi.org/10.46298/dmtcs.2878 Moci, Luca Moci, Luca <![CDATA[We introduce a multiplicity Tutte polynomial $M(x,y)$, which generalizes the ordinary one and has applications to zonotopes and toric arrangements. We prove that $M(x,y)$ satisfies a deletion-restriction recurrence and has positive coefficients. The characteristic polynomial and the Poincaré polynomial of a toric arrangement are shown to be specializations of the associated polynomial $M(x,y)$, likewise the corresponding polynomials for a hyperplane arrangement are specializations of the ordinary Tutte polynomial. Furthermore, $M(1,y)$ is the Hilbert series of the related discrete Dahmen-Micchelli space, while $M(x,1)$ computes the volume and the number of integral points of the associated zonotope.]]> 0 <![CDATA[We study the poset of Borel congruence classes of symmetric matrices ordered by containment of closures. We give a combinatorial description of this poset and calculate its rank function. We discuss the relation between this poset and the Bruhat poset of involutions of the symmetric group. Also we present the poset of Borel congruence classes of anti-symmetric matrices ordered by containment of closures. We show that there exists a bijection between the set of these classes and the set of involutions of the symmetric group. We give two formulas for the rank function of this poset.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2877 https://doi.org/10.46298/dmtcs.2877 Bagno, Eli Cherniavsky, Yonah Bagno, Eli Cherniavsky, Yonah <![CDATA[We study the poset of Borel congruence classes of symmetric matrices ordered by containment of closures. We give a combinatorial description of this poset and calculate its rank function. We discuss the relation between this poset and the Bruhat poset of involutions of the symmetric group. Also we present the poset of Borel congruence classes of anti-symmetric matrices ordered by containment of closures. We show that there exists a bijection between the set of these classes and the set of involutions of the symmetric group. We give two formulas for the rank function of this poset.]]> 0 <![CDATA[We explore in this paper the spaces of common zeros of several deformations of Steenrod operators. Proofs are omitted in view of pages limitation for the extended abstract. \par]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2876 https://doi.org/10.46298/dmtcs.2876 Bergeron, François Garsia, Adriano Wallach, Nolan Bergeron, François Garsia, Adriano Wallach, Nolan <![CDATA[We explore in this paper the spaces of common zeros of several deformations of Steenrod operators. Proofs are omitted in view of pages limitation for the extended abstract. \par]]> 0 <![CDATA[We take a geometric view of lecture hall partitions and anti-lecture hall compositions in order to settle some open questions about their enumeration. In the process, we discover an intrinsic connection between these families of partitions and certain quadratic permutation statistics. We define some unusual quadratic permutation statistics and derive results about their joint distributions with linear statistics. We show that certain specializations are equivalent to the lecture hall and anti-lecture hall theorems and another leads back to a special case of a Weyl group generating function that "ought to be better known.'']]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2875 https://doi.org/10.46298/dmtcs.2875 Bright, Katie L. Savage, Carla D. Bright, Katie L. Savage, Carla D. <![CDATA[We take a geometric view of lecture hall partitions and anti-lecture hall compositions in order to settle some open questions about their enumeration. In the process, we discover an intrinsic connection between these families of partitions and certain quadratic permutation statistics. We define some unusual quadratic permutation statistics and derive results about their joint distributions with linear statistics. We show that certain specializations are equivalent to the lecture hall and anti-lecture hall theorems and another leads back to a special case of a Weyl group generating function that "ought to be better known.'']]> 0 <![CDATA[We identify a subalgebra $\widehat{\mathscr{H}}^+_n$ of the extended affine Hecke algebra $\widehat{\mathscr{H}}_n$ of type $A$. The subalgebra $\widehat{\mathscr{H}}^+_n$ is a u-analogue of the monoid algebra of $\mathcal{S}_n ⋉ℤ_≥0^n$ and inherits a canonical basis from that of $\widehat{\mathscr{H}}_n$. We show that its left cells are naturally labeled by tableaux filled with positive integer entries having distinct residues mod $n$, which we term positive affine tableaux (PAT). We then exhibit a cellular subquotient $\mathscr{R}_1^n$ of $\widehat{\mathscr{H}}^+_n$ that is a $u$-analogue of the ring of coinvariants $ℂ[y_1,\ldots,y_n]/(e_1, \ldots,e_n)$ with left cells labeled by PAT that are essentially standard Young tableaux with cocharge labels. Multiplying canonical basis elements by a certain element $*π ∈ \widehat{\mathscr{H}}^+_n$ corresponds to rotations of words, and on cells corresponds to cocyclage. We further show that $\mathscr{R}_1^n$ has cellular quotients $\mathscr{R}_λ$ that are $u$-analogues of the Garsia-Procesi modules $R_λ$ with left cells labeled by (a PAT version of) the $λ$ -catabolizable tableaux.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2858 https://doi.org/10.46298/dmtcs.2858 Blasiak, Jonah Blasiak, Jonah <![CDATA[We identify a subalgebra $\widehat{\mathscr{H}}^+_n$ of the extended affine Hecke algebra $\widehat{\mathscr{H}}_n$ of type $A$. The subalgebra $\widehat{\mathscr{H}}^+_n$ is a u-analogue of the monoid algebra of $\mathcal{S}_n ⋉ℤ_≥0^n$ and inherits a canonical basis from that of $\widehat{\mathscr{H}}_n$. We show that its left cells are naturally labeled by tableaux filled with positive integer entries having distinct residues mod $n$, which we term positive affine tableaux (PAT). We then exhibit a cellular subquotient $\mathscr{R}_1^n$ of $\widehat{\mathscr{H}}^+_n$ that is a $u$-analogue of the ring of coinvariants $ℂ[y_1,\ldots,y_n]/(e_1, \ldots,e_n)$ with left cells labeled by PAT that are essentially standard Young tableaux with cocharge labels. Multiplying canonical basis elements by a certain element $*π ∈ \widehat{\mathscr{H}}^+_n$ corresponds to rotations of words, and on cells corresponds to cocyclage. We further show that $\mathscr{R}_1^n$ has cellular quotients $\mathscr{R}_λ$ that are $u$-analogues of the Garsia-Procesi modules $R_λ$ with left cells labeled by (a PAT version of) the $λ$ -catabolizable tableaux.]]> 0 <![CDATA[A unicellular map is the embedding of a connected graph in a surface in such a way that the complement of the graph is a topological disk. In this paper we give a bijective operation that relates unicellular maps on a non-orientable surface to unicellular maps of a lower topological type, with distinguished vertices. From that we obtain a recurrence equation that leads to (new) explicit counting formulas for non-orientable precubic (all vertices of degree 1 or 3) unicellular maps of fixed topology. We also determine asymptotic formulas for the number of all unicellular maps of fixed topology, when the number of edges goes to infinity. Our strategy is inspired by recent results obtained for the orientable case [Chapuy, PTRF 2010], but significant novelties are introduced: in particular we construct an involution which, in some sense, ``averages'' the effects of non-orientability. \par]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2859 https://doi.org/10.46298/dmtcs.2859 Bernardi, Olivier Chapuy, Guillaume Bernardi, Olivier Chapuy, Guillaume <![CDATA[A unicellular map is the embedding of a connected graph in a surface in such a way that the complement of the graph is a topological disk. In this paper we give a bijective operation that relates unicellular maps on a non-orientable surface to unicellular maps of a lower topological type, with distinguished vertices. From that we obtain a recurrence equation that leads to (new) explicit counting formulas for non-orientable precubic (all vertices of degree 1 or 3) unicellular maps of fixed topology. We also determine asymptotic formulas for the number of all unicellular maps of fixed topology, when the number of edges goes to infinity. Our strategy is inspired by recent results obtained for the orientable case [Chapuy, PTRF 2010], but significant novelties are introduced: in particular we construct an involution which, in some sense, ``averages'' the effects of non-orientability. \par]]> 0 <![CDATA[A combinatorial expansion of the Hall-Littlewood functions into the Schur basis of symmetric functions was first given by Lascoux and Schützenberger, with their discovery of the charge statistic. A combinatorial expansion of stable Grassmannian Grothendieck polynomials into monomials was first given by Buch, using set-valued tableaux. The dual basis of the stable Grothendieck polynomials was given a combinatorial expansion into monomials by Lam and Pylyavskyy using reverse plane partitions. We generalize charge to set-valued tableaux and use all of these combinatorial ideas to give a nice expansion of Hall-Littlewood polynomials into the dual Grothendieck basis. \par]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2860 https://doi.org/10.46298/dmtcs.2860 Bandlow, Jason Morse, Jennifer Bandlow, Jason Morse, Jennifer <![CDATA[A combinatorial expansion of the Hall-Littlewood functions into the Schur basis of symmetric functions was first given by Lascoux and Schützenberger, with their discovery of the charge statistic. A combinatorial expansion of stable Grassmannian Grothendieck polynomials into monomials was first given by Buch, using set-valued tableaux. The dual basis of the stable Grothendieck polynomials was given a combinatorial expansion into monomials by Lam and Pylyavskyy using reverse plane partitions. We generalize charge to set-valued tableaux and use all of these combinatorial ideas to give a nice expansion of Hall-Littlewood polynomials into the dual Grothendieck basis. \par]]> 0 <![CDATA[According to the Göttsche conjecture (now a theorem), the degree $N^{d, \delta}$ of the Severi variety of plane curves of degree $d$ with $\delta$ nodes is given by a polynomial in $d$, provided $d$ is large enough. These "node polynomials'' $N_{\delta} (d)$ were determined by Vainsencher and Kleiman―Piene for $\delta \leq 6$ and $\delta \leq 8$, respectively. Building on ideas of Fomin and Mikhalkin, we develop an explicit algorithm for computing all node polynomials, and use it to compute $N_{\delta} (d)$ for $\delta \leq 14$. Furthermore, we improve the threshold of polynomiality and verify Göttsche's conjecture on the optimal threshold up to $\delta \leq 14$. We also determine the first 9 coefficients of $N_{\delta} (d)$, for general $\delta$, settling and extending a 1994 conjecture of Di Francesco and Itzykson.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2861 https://doi.org/10.46298/dmtcs.2861 Block, Florian Block, Florian <![CDATA[According to the Göttsche conjecture (now a theorem), the degree $N^{d, \delta}$ of the Severi variety of plane curves of degree $d$ with $\delta$ nodes is given by a polynomial in $d$, provided $d$ is large enough. These "node polynomials'' $N_{\delta} (d)$ were determined by Vainsencher and Kleiman―Piene for $\delta \leq 6$ and $\delta \leq 8$, respectively. Building on ideas of Fomin and Mikhalkin, we develop an explicit algorithm for computing all node polynomials, and use it to compute $N_{\delta} (d)$ for $\delta \leq 14$. Furthermore, we improve the threshold of polynomiality and verify Göttsche's conjecture on the optimal threshold up to $\delta \leq 14$. We also determine the first 9 coefficients of $N_{\delta} (d)$, for general $\delta$, settling and extending a 1994 conjecture of Di Francesco and Itzykson.]]> 0 <![CDATA[We study the expected distance of a two-dimensional walk in the plane with unit steps in random directions. A series evaluation and recursions are obtained making it possible to explicitly formulate this distance for small number of steps. Formulae for all the moments of a 2-step and a 3-step walk are given, and an expression is conjectured for the 4-step walk. The paper makes use of the combinatorical features exhibited by the even moments which, for instance, lead to analytic continuations of the underlying integral.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2862 https://doi.org/10.46298/dmtcs.2862 Borwein, Jonathan M. Nuyens, Dirk Straub, Armin Wan, James Borwein, Jonathan M. Nuyens, Dirk Straub, Armin Wan, James <![CDATA[We study the expected distance of a two-dimensional walk in the plane with unit steps in random directions. A series evaluation and recursions are obtained making it possible to explicitly formulate this distance for small number of steps. Formulae for all the moments of a 2-step and a 3-step walk are given, and an expression is conjectured for the 4-step walk. The paper makes use of the combinatorical features exhibited by the even moments which, for instance, lead to analytic continuations of the underlying integral.]]> 0 <![CDATA[Double Hurwitz numbers count covers of the sphere by genus $g$ curves with assigned ramification profiles over $0$ and $\infty$, and simple ramification over a fixed branch divisor. Goulden, Jackson and Vakil (2005) have shown double Hurwitz numbers are piecewise polynomial in the orders of ramification, and Shadrin, Shapiro and Vainshtein (2008) have determined the chamber structure and wall crossing formulas for $g=0$. We provide new proofs of these results, and extend them in several directions. Most importantly we prove wall crossing formulas for all genera. The main tool is the authors' previous work expressing double Hurwitz number as a sum over labelled graphs. We identify the labels of the graphs with lattice points in the chambers of certain hyperplane arrangements, which give rise to piecewise polynomial functions. Our understanding of the wall crossing for these functions builds on the work of Varchenko (1987). This approach to wall crossing appears novel, and may be of broader interest. This extended abstract is based on a new preprint by the authors.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2863 https://doi.org/10.46298/dmtcs.2863 Cavalieri, Renzo JOHNSON, Paul Markwig, Hannah Cavalieri, Renzo JOHNSON, Paul Markwig, Hannah <![CDATA[Double Hurwitz numbers count covers of the sphere by genus $g$ curves with assigned ramification profiles over $0$ and $\infty$, and simple ramification over a fixed branch divisor. Goulden, Jackson and Vakil (2005) have shown double Hurwitz numbers are piecewise polynomial in the orders of ramification, and Shadrin, Shapiro and Vainshtein (2008) have determined the chamber structure and wall crossing formulas for $g=0$. We provide new proofs of these results, and extend them in several directions. Most importantly we prove wall crossing formulas for all genera. The main tool is the authors' previous work expressing double Hurwitz number as a sum over labelled graphs. We identify the labels of the graphs with lattice points in the chambers of certain hyperplane arrangements, which give rise to piecewise polynomial functions. Our understanding of the wall crossing for these functions builds on the work of Varchenko (1987). This approach to wall crossing appears novel, and may be of broader interest. This extended abstract is based on a new preprint by the authors.]]> 0 <![CDATA[A finite subgroup $G$ of $GL(n,\mathbb{C})$ is involutory if the sum of the dimensions of its irreducible complex representations is given by the number of absolute involutions in the group, i.e. elements $g \in G$ such that $g \bar{g}=1$, where the bar denotes complex conjugation. A uniform combinatorial model is constructed for all non-exceptional irreducible complex reflection groups which are involutory including, in particular, all infinite families of finite irreducible Coxeter groups. If $G$ is a classical Weyl group this result is much refined in a way which is compatible with the Robinson-Schensted correspondence on involutions.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2864 https://doi.org/10.46298/dmtcs.2864 Caselli, Fabrizio Fulci, Roberta Caselli, Fabrizio Fulci, Roberta <![CDATA[A finite subgroup $G$ of $GL(n,\mathbb{C})$ is involutory if the sum of the dimensions of its irreducible complex representations is given by the number of absolute involutions in the group, i.e. elements $g \in G$ such that $g \bar{g}=1$, where the bar denotes complex conjugation. A uniform combinatorial model is constructed for all non-exceptional irreducible complex reflection groups which are involutory including, in particular, all infinite families of finite irreducible Coxeter groups. If $G$ is a classical Weyl group this result is much refined in a way which is compatible with the Robinson-Schensted correspondence on involutions.]]> 0 <![CDATA[We introduce and study three different notions of tropical rank for symmetric matrices and dissimilarity matrices in terms of minimal decompositions into rank 1 symmetric matrices, star tree matrices, and tree matrices. Our results provide a close study of the tropical secant sets of certain nice tropical varieties, including the tropical Grassmannian. In particular, we determine the dimension of each secant set, the convex hull of the variety, and in most cases, the smallest secant set which is equal to the convex hull.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2865 https://doi.org/10.46298/dmtcs.2865 Cartwright, Dustin Chan, Melody Cartwright, Dustin Chan, Melody <![CDATA[We introduce and study three different notions of tropical rank for symmetric matrices and dissimilarity matrices in terms of minimal decompositions into rank 1 symmetric matrices, star tree matrices, and tree matrices. Our results provide a close study of the tropical secant sets of certain nice tropical varieties, including the tropical Grassmannian. In particular, we determine the dimension of each secant set, the convex hull of the variety, and in most cases, the smallest secant set which is equal to the convex hull.]]> 0 <![CDATA[We prove that the subset of quasisymmetric polynomials conjectured by Bergeron and Reutenauer to be a basis for the coinvariant space of quasisymmetric polynomials is indeed a basis. This provides the first constructive proof of the Garsia―Wallach result stating that quasisymmetric polynomials form a free module over symmetric polynomials and that the dimension of this module is $n!$.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2866 https://doi.org/10.46298/dmtcs.2866 Lauve, Aaron Mason, Sarah K Lauve, Aaron Mason, Sarah K <![CDATA[We prove that the subset of quasisymmetric polynomials conjectured by Bergeron and Reutenauer to be a basis for the coinvariant space of quasisymmetric polynomials is indeed a basis. This provides the first constructive proof of the Garsia―Wallach result stating that quasisymmetric polynomials form a free module over symmetric polynomials and that the dimension of this module is $n!$.]]> 0 <![CDATA[We generalize a theorem of Knuth relating the oriented spanning trees of a directed graph $G$ and its directed line graph $\mathcal{L} G$. The sandpile group is an abelian group associated to a directed graph, whose order is the number of oriented spanning trees rooted at a fixed vertex. In the case when $G$ is regular of degree $k$, we show that the sandpile group of $G$ is isomorphic to the quotient of the sandpile group of $\mathcal{L} G$ by its $k$-torsion subgroup. As a corollary we compute the sandpile groups of two families of graphs widely studied in computer science, the de Bruijn graphs and Kautz graphs.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2867 https://doi.org/10.46298/dmtcs.2867 Levine, Lionel Levine, Lionel <![CDATA[We generalize a theorem of Knuth relating the oriented spanning trees of a directed graph $G$ and its directed line graph $\mathcal{L} G$. The sandpile group is an abelian group associated to a directed graph, whose order is the number of oriented spanning trees rooted at a fixed vertex. In the case when $G$ is regular of degree $k$, we show that the sandpile group of $G$ is isomorphic to the quotient of the sandpile group of $\mathcal{L} G$ by its $k$-torsion subgroup. As a corollary we compute the sandpile groups of two families of graphs widely studied in computer science, the de Bruijn graphs and Kautz graphs.]]> 0 <![CDATA[In the late 30's, Maurits Cornelis Escher astonished the artistic world by producing some puzzling drawings. In particular, the tesselations of the plane obtained by using a single tile appear to be a major concern in his work, drawing attention from the mathematical community. Since a tile in the continuous world can be approximated by a path on a sufficiently small square grid - a widely used method in applications using computer displays - the natural combinatorial object that models the tiles is the polyomino. As polyominoes are encoded by paths on a four letter alphabet coding their contours, the use of combinatorics on words for the study of tiling properties becomes relevant. In this paper we present several results, ranging from recognition of these tiles to their generation, leading also to some surprising links with the well-known sequences of Fibonacci and Pell.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2868 https://doi.org/10.46298/dmtcs.2868 Massé, Alexandre Blondin Brlek, Srecko Labbé, Sébastien Massé, Alexandre Blondin Brlek, Srecko Labbé, Sébastien <![CDATA[In the late 30's, Maurits Cornelis Escher astonished the artistic world by producing some puzzling drawings. In particular, the tesselations of the plane obtained by using a single tile appear to be a major concern in his work, drawing attention from the mathematical community. Since a tile in the continuous world can be approximated by a path on a sufficiently small square grid - a widely used method in applications using computer displays - the natural combinatorial object that models the tiles is the polyomino. As polyominoes are encoded by paths on a four letter alphabet coding their contours, the use of combinatorics on words for the study of tiling properties becomes relevant. In this paper we present several results, ranging from recognition of these tiles to their generation, leading also to some surprising links with the well-known sequences of Fibonacci and Pell.]]> 0 <![CDATA[Based on a construction of the first author, we present a general bijection between certain decorated plane trees and certain orientations of planar maps with no counterclockwise circuit. Many natural classes of maps (e.g. Eulerian maps, simple triangulations,...) are in bijection with a subset of these orientations, and our construction restricts in a simple way on the subset. This gives a general bijective strategy for classes of maps. As a non-trivial application of our method we give the first bijective proofs for counting (rooted) simple triangulations and quadrangulations with a boundary of arbitrary size, recovering enumeration results found by Brown using Tutte's recursive method.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2869 https://doi.org/10.46298/dmtcs.2869 Bernardi, Olivier Fusy, Eric Bernardi, Olivier Fusy, Eric <![CDATA[Based on a construction of the first author, we present a general bijection between certain decorated plane trees and certain orientations of planar maps with no counterclockwise circuit. Many natural classes of maps (e.g. Eulerian maps, simple triangulations,...) are in bijection with a subset of these orientations, and our construction restricts in a simple way on the subset. This gives a general bijective strategy for classes of maps. As a non-trivial application of our method we give the first bijective proofs for counting (rooted) simple triangulations and quadrangulations with a boundary of arbitrary size, recovering enumeration results found by Brown using Tutte's recursive method.]]> 0 <![CDATA[Let $\mathcal{B}_n$ be the hyperoctahedral group acting on a complex vector space $\mathcal{V}$. We present a combinatorial method to decompose the tensor algebra $T(\mathcal{V})$ on $\mathcal{V}$ into simple modules via certain words in a particular Cayley graph of $\mathcal{B}_n$. We then give combinatorial interpretations for the graded dimension and the number of free generators of the subalgebra $T(\mathcal{V})^{\mathcal{B}_n}$ of invariants of $\mathcal{B}_n$, in terms of these words, and make explicit the case of the signed permutation module. To this end, we require a morphism from the Mantaci-Reutenauer algebra onto the algebra of characters due to Bonnafé and Hohlweg.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2870 https://doi.org/10.46298/dmtcs.2870 Bergeron-Brlek, Anouk Bergeron-Brlek, Anouk <![CDATA[Let $\mathcal{B}_n$ be the hyperoctahedral group acting on a complex vector space $\mathcal{V}$. We present a combinatorial method to decompose the tensor algebra $T(\mathcal{V})$ on $\mathcal{V}$ into simple modules via certain words in a particular Cayley graph of $\mathcal{B}_n$. We then give combinatorial interpretations for the graded dimension and the number of free generators of the subalgebra $T(\mathcal{V})^{\mathcal{B}_n}$ of invariants of $\mathcal{B}_n$, in terms of these words, and make explicit the case of the signed permutation module. To this end, we require a morphism from the Mantaci-Reutenauer algebra onto the algebra of characters due to Bonnafé and Hohlweg.]]> 0 <![CDATA[Steingrímsson (2001) showed that the chromatic polynomial of a graph is the Hilbert function of a relative Stanley-Reisner ideal. We approach this result from the point of view of Ehrhart theory and give a sufficient criterion for when the Ehrhart polynomial of a given relative polytopal complex is a Hilbert function in Steingrímsson's sense. We use this result to establish that the modular and integral flow and tension polynomials of a graph are Hilbert functions.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2871 https://doi.org/10.46298/dmtcs.2871 Breuer, Felix Dall, Aaron Breuer, Felix Dall, Aaron <![CDATA[Steingrímsson (2001) showed that the chromatic polynomial of a graph is the Hilbert function of a relative Stanley-Reisner ideal. We approach this result from the point of view of Ehrhart theory and give a sufficient criterion for when the Ehrhart polynomial of a given relative polytopal complex is a Hilbert function in Steingrímsson's sense. We use this result to establish that the modular and integral flow and tension polynomials of a graph are Hilbert functions.]]> 0 <![CDATA[In the late 1930's Murnaghan discovered the existence of a stabilization phenomenon for the Kronecker product of Schur functions. For $n$ large enough, the values of the Kronecker coefficients appearing in the product of two Schur functions of degree $n$ do not depend on the first part of the indexing partitions, but only on the values of their remaining parts. We compute the exact value of n when this stable expansion is reached. We also compute two new bounds for the stabilization of a particular coefficient of such a product. Given partitions $\alpha$ and $\beta$, we give bounds for all the parts of any partition $\gamma$ such that the corresponding Kronecker coefficient is nonzero. Finally, we also show that the reduced Kronecker coefficients are structure coefficients for the Heisenberg product introduced by Aguiar, Ferrer and Moreira.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2872 https://doi.org/10.46298/dmtcs.2872 Briand, Emmanuel Orellana, Rosa Rosas, Mercedes Briand, Emmanuel Orellana, Rosa Rosas, Mercedes <![CDATA[In the late 1930's Murnaghan discovered the existence of a stabilization phenomenon for the Kronecker product of Schur functions. For $n$ large enough, the values of the Kronecker coefficients appearing in the product of two Schur functions of degree $n$ do not depend on the first part of the indexing partitions, but only on the values of their remaining parts. We compute the exact value of n when this stable expansion is reached. We also compute two new bounds for the stabilization of a particular coefficient of such a product. Given partitions $\alpha$ and $\beta$, we give bounds for all the parts of any partition $\gamma$ such that the corresponding Kronecker coefficient is nonzero. Finally, we also show that the reduced Kronecker coefficients are structure coefficients for the Heisenberg product introduced by Aguiar, Ferrer and Moreira.]]> 0 <![CDATA[We introduce $k$-crossings and $k$-nestings of permutations. We show that the crossing number and the nesting number of permutations have a symmetric joint distribution. As a corollary, the number of $k$-noncrossing permutations is equal to the number of $k$-nonnesting permutations. We also provide some enumerative results for $k$-noncrossing permutations for some values of $k$.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2873 https://doi.org/10.46298/dmtcs.2873 Burrill, Sophie Mishna, Marni Post, Jacob Burrill, Sophie Mishna, Marni Post, Jacob <![CDATA[We introduce $k$-crossings and $k$-nestings of permutations. We show that the crossing number and the nesting number of permutations have a symmetric joint distribution. As a corollary, the number of $k$-noncrossing permutations is equal to the number of $k$-nonnesting permutations. We also provide some enumerative results for $k$-noncrossing permutations for some values of $k$.]]> 0 <![CDATA[The Bruhat order gives a poset structure to any Coxeter group. The ideal of elements in this poset having boolean principal order ideals forms a simplicial poset. This simplicial poset defines the boolean complex for the group. In a Coxeter system of rank n, we show that the boolean complex is homotopy equivalent to a wedge of (n-1)-dimensional spheres. The number of these spheres is the boolean number, which can be computed inductively from the unlabeled Coxeter system, thus defining a graph invariant. For certain families of graphs, the boolean numbers have intriguing combinatorial properties. This work involves joint efforts with Claesson, Kitaev, and Ragnarsson. \par]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2833 https://doi.org/10.46298/dmtcs.2833 Tenner, Bridget Eileen Tenner, Bridget Eileen <![CDATA[The Bruhat order gives a poset structure to any Coxeter group. The ideal of elements in this poset having boolean principal order ideals forms a simplicial poset. This simplicial poset defines the boolean complex for the group. In a Coxeter system of rank n, we show that the boolean complex is homotopy equivalent to a wedge of (n-1)-dimensional spheres. The number of these spheres is the boolean number, which can be computed inductively from the unlabeled Coxeter system, thus defining a graph invariant. For certain families of graphs, the boolean numbers have intriguing combinatorial properties. This work involves joint efforts with Claesson, Kitaev, and Ragnarsson. \par]]> 0 <![CDATA[We consider a large family of equivalence relations on permutations in $S_n$ that generalise those discovered by Knuth in his study of the Robinson-Schensted correspondence. In our most general setting, two permutations are equivalent if one can be obtained from the other by a sequence of pattern-replacing moves of prescribed form; however, we limit our focus to patterns where two elements are transposed, conditional upon the presence of a third element of suitable value and location. For some relations of this type, we compute the number of equivalence classes, determine how many $n$-permutations are equivalent to the identity permutation, or characterise this equivalence class. Although our results include familiar integer sequences (e.g., Catalan, Fibonacci, and Tribonacci numbers) and special classes of permutations (layered, connected, and $123$-avoiding), some of the sequences that arise appear to be new.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2841 https://doi.org/10.46298/dmtcs.2841 Linton, Stephen Propp, James Roby, Tom West, Julian Linton, Stephen Propp, James Roby, Tom West, Julian <![CDATA[We consider a large family of equivalence relations on permutations in $S_n$ that generalise those discovered by Knuth in his study of the Robinson-Schensted correspondence. In our most general setting, two permutations are equivalent if one can be obtained from the other by a sequence of pattern-replacing moves of prescribed form; however, we limit our focus to patterns where two elements are transposed, conditional upon the presence of a third element of suitable value and location. For some relations of this type, we compute the number of equivalence classes, determine how many $n$-permutations are equivalent to the identity permutation, or characterise this equivalence class. Although our results include familiar integer sequences (e.g., Catalan, Fibonacci, and Tribonacci numbers) and special classes of permutations (layered, connected, and $123$-avoiding), some of the sequences that arise appear to be new.]]> 0 <![CDATA[The standard supercharacter theory of the finite unipotent upper-triangular matrices $U_n(q)$ gives rise to a beautiful combinatorics based on set partitions. As with the representation theory of the symmetric group, embeddings of $U_m(q) \subseteq U_n(q)$ for $m \leq n$ lead to branching rules. Diaconis and Isaacs established that the restriction of a supercharacter of $U_n(q)$ is a nonnegative integer linear combination of supercharacters of $U_m(q)$ (in fact, it is polynomial in $q$). In a first step towards understanding the combinatorics of coefficients in the branching rules of the supercharacters of $U_n(q)$, this paper characterizes when a given coefficient is nonzero in the restriction of a supercharacter and the tensor product of two supercharacters. These conditions are given uniformly in terms of complete matchings in bipartite graphs.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2840 https://doi.org/10.46298/dmtcs.2840 Lewis, Stephen Thiem, Nathaniel Lewis, Stephen Thiem, Nathaniel <![CDATA[The standard supercharacter theory of the finite unipotent upper-triangular matrices $U_n(q)$ gives rise to a beautiful combinatorics based on set partitions. As with the representation theory of the symmetric group, embeddings of $U_m(q) \subseteq U_n(q)$ for $m \leq n$ lead to branching rules. Diaconis and Isaacs established that the restriction of a supercharacter of $U_n(q)$ is a nonnegative integer linear combination of supercharacters of $U_m(q)$ (in fact, it is polynomial in $q$). In a first step towards understanding the combinatorics of coefficients in the branching rules of the supercharacters of $U_n(q)$, this paper characterizes when a given coefficient is nonzero in the restriction of a supercharacter and the tensor product of two supercharacters. These conditions are given uniformly in terms of complete matchings in bipartite graphs.]]> 0 <![CDATA[We study the crystal structure on categories of graded modules over algebras which categorify the negative half of the quantum Kac-Moody algebra associated to a symmetrizable Cartan data. We identify this crystal with Kashiwara's crystal for the corresponding negative half of the quantum Kac-Moody algebra. As a consequence, we show the simple graded modules for certain cyclotomic quotients carry the structure of highest weight crystals, and hence compute the rank of the corresponding Grothendieck group.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2839 https://doi.org/10.46298/dmtcs.2839 Lauda, Aaron D. Vazirani, Monica Lauda, Aaron D. Vazirani, Monica <![CDATA[We study the crystal structure on categories of graded modules over algebras which categorify the negative half of the quantum Kac-Moody algebra associated to a symmetrizable Cartan data. We identify this crystal with Kashiwara's crystal for the corresponding negative half of the quantum Kac-Moody algebra. As a consequence, we show the simple graded modules for certain cyclotomic quotients carry the structure of highest weight crystals, and hence compute the rank of the corresponding Grothendieck group.]]> 0 <![CDATA[Let $I_n$ be the (big) diagonal ideal of $(\mathbb{C}^2)^n$. Haiman proved that the $q,t$-Catalan number is the Hilbert series of the graded vector space $M_n=\bigoplus_{d_1,d_2}(M_n)_{d_1,d_2}$ spanned by a minimal set of generators for $I_n$. We give simple upper bounds on $\textrm{dim} (M_n)_{d_1, d_2}$ in terms of partition numbers, and find all bi-degrees $(d_1,d_2)$ such that $\textrm{dim} (M_n)_{d_1, d_2}$ achieve the upper bounds. For such bi-degrees, we also find explicit bases for $(M_n)_{d_1, d_2}$.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2838 https://doi.org/10.46298/dmtcs.2838 Lee, Kyungyong Li, Li Lee, Kyungyong Li, Li <![CDATA[Let $I_n$ be the (big) diagonal ideal of $(\mathbb{C}^2)^n$. Haiman proved that the $q,t$-Catalan number is the Hilbert series of the graded vector space $M_n=\bigoplus_{d_1,d_2}(M_n)_{d_1,d_2}$ spanned by a minimal set of generators for $I_n$. We give simple upper bounds on $\textrm{dim} (M_n)_{d_1, d_2}$ in terms of partition numbers, and find all bi-degrees $(d_1,d_2)$ such that $\textrm{dim} (M_n)_{d_1, d_2}$ achieve the upper bounds. For such bi-degrees, we also find explicit bases for $(M_n)_{d_1, d_2}$.]]> 0 <![CDATA[We show that toric ideals of flow polytopes are generated in degree $3$. This was conjectured by Diaconis and Eriksson for the special case of the Birkhoff polytope. Our proof uses a hyperplane subdivision method developed by Haase and Paffenholz. It is known that reduced revlex Gröbner bases of the toric ideal of the Birkhoff polytope $B_n$ have at most degree $n$. We show that this bound is sharp for some revlex term orders. For $(m \times n)$-transportation polytopes, a similar result holds: they have Gröbner bases of at most degree $\lfloor mn/2 \rfloor$. We construct a family of examples, where this bound is sharp.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2837 https://doi.org/10.46298/dmtcs.2837 Lenz, Matthias Lenz, Matthias <![CDATA[We show that toric ideals of flow polytopes are generated in degree $3$. This was conjectured by Diaconis and Eriksson for the special case of the Birkhoff polytope. Our proof uses a hyperplane subdivision method developed by Haase and Paffenholz. It is known that reduced revlex Gröbner bases of the toric ideal of the Birkhoff polytope $B_n$ have at most degree $n$. We show that this bound is sharp for some revlex term orders. For $(m \times n)$-transportation polytopes, a similar result holds: they have Gröbner bases of at most degree $\lfloor mn/2 \rfloor$. We construct a family of examples, where this bound is sharp.]]> 0 <![CDATA[We consider the real and complex noncentral Wishart distributions. The moments of these distributions are shown to be expressed as weighted generating functions of graphs associated with the Wishart distributions. We give some bijections between sets of graphs related to moments of the real Wishart distribution and the complex noncentral Wishart distribution. By means of the bijections, we see that calculating these moments of a certain class the real Wishart distribution boils down to calculations for the case of complex Wishart distributions.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2836 https://doi.org/10.46298/dmtcs.2836 Numata, Yasuhide Kuriki, Satoshi Numata, Yasuhide Kuriki, Satoshi <![CDATA[We consider the real and complex noncentral Wishart distributions. The moments of these distributions are shown to be expressed as weighted generating functions of graphs associated with the Wishart distributions. We give some bijections between sets of graphs related to moments of the real Wishart distribution and the complex noncentral Wishart distribution. By means of the bijections, we see that calculating these moments of a certain class the real Wishart distribution boils down to calculations for the case of complex Wishart distributions.]]> 0 <![CDATA[We link Schubert varieties in the generalized flag manifolds with hyperplane arrangements. For an element of a Weyl group, we construct a certain graphical hyperplane arrangement. We show that the generating function for regions of this arrangement coincides with the Poincaré polynomial of the corresponding Schubert variety if and only if the Schubert variety is rationally smooth.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2835 https://doi.org/10.46298/dmtcs.2835 Oh, Suho Yoo, Hwanchul Oh, Suho Yoo, Hwanchul <![CDATA[We link Schubert varieties in the generalized flag manifolds with hyperplane arrangements. For an element of a Weyl group, we construct a certain graphical hyperplane arrangement. We show that the generating function for regions of this arrangement coincides with the Poincaré polynomial of the corresponding Schubert variety if and only if the Schubert variety is rationally smooth.]]> 0 <![CDATA[In 2004, Condon and coauthors gave a hierarchical classification of exact RNA structure prediction algorithms according to the generality of structure classes that they handle. We complete this classification by adding two recent prediction algorithms. More importantly, we precisely quantify the hierarchy by giving closed or asymptotic formulas for the theoretical number of structures of given size n in all the classes but one. This allows to assess the tradeoff between the expressiveness and the computational complexity of RNA structure prediction algorithms. \par]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2834 https://doi.org/10.46298/dmtcs.2834 Saule, Cédric Regnier, Mireille Steyaert, Jean-Marc Denise, Alain Saule, Cédric Regnier, Mireille Steyaert, Jean-Marc Denise, Alain <![CDATA[In 2004, Condon and coauthors gave a hierarchical classification of exact RNA structure prediction algorithms according to the generality of structure classes that they handle. We complete this classification by adding two recent prediction algorithms. More importantly, we precisely quantify the hierarchy by giving closed or asymptotic formulas for the theoretical number of structures of given size n in all the classes but one. This allows to assess the tradeoff between the expressiveness and the computational complexity of RNA structure prediction algorithms. \par]]> 0 <![CDATA[The well-known R-polynomials in ℤ[q], which appear in Hecke algebra computations, are closely related to certain modified R-polynomials in ℕ[q] whose coefficients have simple combinatorial interpretations. We generalize this second family of polynomials, providing combinatorial interpretations for expressions arising in a much broader class of computations. In particular, we extend results of Brenti, Deodhar, and Dyer to new settings which include parabolic Hecke algebra modules and the quantum polynomial ring.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2825 https://doi.org/10.46298/dmtcs.2825 Lambright, Justin Skandera, Mark Lambright, Justin Skandera, Mark <![CDATA[The well-known R-polynomials in ℤ[q], which appear in Hecke algebra computations, are closely related to certain modified R-polynomials in ℕ[q] whose coefficients have simple combinatorial interpretations. We generalize this second family of polynomials, providing combinatorial interpretations for expressions arising in a much broader class of computations. In particular, we extend results of Brenti, Deodhar, and Dyer to new settings which include parabolic Hecke algebra modules and the quantum polynomial ring.]]> 0 <![CDATA[We prove new connections between permutation patterns and singularities of Schubert varieties, by giving a new characterization of factorial and Gorenstein varieties in terms of so called bivincular patterns. These are generalizations of classical patterns where conditions are placed on the location of an occurrence in a permutation, as well as on the values in the occurrence. This clarifies what happens when the requirement of smoothness is weakened to factoriality and further to Gorensteinness, extending work of Bousquet-Mélou and Butler (2007), and Woo and Yong (2006). We also prove results that translate some known patterns in the literature into bivincular patterns.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2832 https://doi.org/10.46298/dmtcs.2832 Úlfarsson, Henning A. Úlfarsson, Henning A. <![CDATA[We prove new connections between permutation patterns and singularities of Schubert varieties, by giving a new characterization of factorial and Gorenstein varieties in terms of so called bivincular patterns. These are generalizations of classical patterns where conditions are placed on the location of an occurrence in a permutation, as well as on the values in the occurrence. This clarifies what happens when the requirement of smoothness is weakened to factoriality and further to Gorensteinness, extending work of Bousquet-Mélou and Butler (2007), and Woo and Yong (2006). We also prove results that translate some known patterns in the literature into bivincular patterns.]]> 0 <![CDATA[The classical Thom―Porteous formula expresses the homology class of the degeneracy locus of a generic map between two vector bundles as an alternating sum of Schur polynomials. A proof of this formula was given by Pragacz by expressing this alternating sum as the Euler characteristic of a Schur complex, which gives an explanation for the signs. Fulton later generalized this formula to the situation of flags of vector bundles by using alternating sums of Schubert polynomials. Building on the Schubert functors of Kraśkiewicz and Pragacz, we introduce Schubert complexes and show that Fulton's alternating sum can be realized as the Euler characteristic of this complex, thereby providing a conceptual proof for why an alternating sum appears. \par]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2831 https://doi.org/10.46298/dmtcs.2831 Sam, Steven V Sam, Steven V <![CDATA[The classical Thom―Porteous formula expresses the homology class of the degeneracy locus of a generic map between two vector bundles as an alternating sum of Schur polynomials. A proof of this formula was given by Pragacz by expressing this alternating sum as the Euler characteristic of a Schur complex, which gives an explanation for the signs. Fulton later generalized this formula to the situation of flags of vector bundles by using alternating sums of Schubert polynomials. Building on the Schubert functors of Kraśkiewicz and Pragacz, we introduce Schubert complexes and show that Fulton's alternating sum can be realized as the Euler characteristic of this complex, thereby providing a conceptual proof for why an alternating sum appears. \par]]> 0 <![CDATA[Let $G$ be a simple graph with $n$ vertices. The coloring complex$ Δ (G)$ was defined by Steingrímsson, and the homology of $Δ (G)$ was shown to be nonzero only in dimension $n-3$ by Jonsson. Hanlon recently showed that the Eulerian idempotents provide a decomposition of the homology group $H_{n-3}(Δ (G))$ where the dimension of the $j^th$ component in the decomposition, $H_{n-3}^{(j)}(Δ (G))$, equals the absolute value of the coefficient of $λ ^j$ in the chromatic polynomial of $G, _{\mathcal{χg}}(λ )$. Let $H$ be a hypergraph with $n$ vertices. In this paper, we define the coloring complex of a hypergraph, $Δ (H)$, and show that the coefficient of $λ ^j$ in $χ _H(λ )$ gives the Euler Characteristic of the $j^{th}$ Hodge subcomplex of the Hodge decomposition of $Δ (H)$. We also examine conditions on a hypergraph, $H$, for which its Hodge subcomplexes are Cohen-Macaulay, and thus where the absolute value of the coefficient of $λ ^j$ in $χ _H(λ )$ equals the dimension of the $j^{th}$ Hodge piece of the Hodge decomposition of $Δ (H)$.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2830 https://doi.org/10.46298/dmtcs.2830 Rundell, Sarah C Long, Jane H Rundell, Sarah C Long, Jane H <![CDATA[Let $G$ be a simple graph with $n$ vertices. The coloring complex$ Δ (G)$ was defined by Steingrímsson, and the homology of $Δ (G)$ was shown to be nonzero only in dimension $n-3$ by Jonsson. Hanlon recently showed that the Eulerian idempotents provide a decomposition of the homology group $H_{n-3}(Δ (G))$ where the dimension of the $j^th$ component in the decomposition, $H_{n-3}^{(j)}(Δ (G))$, equals the absolute value of the coefficient of $λ ^j$ in the chromatic polynomial of $G, _{\mathcal{χg}}(λ )$. Let $H$ be a hypergraph with $n$ vertices. In this paper, we define the coloring complex of a hypergraph, $Δ (H)$, and show that the coefficient of $λ ^j$ in $χ _H(λ )$ gives the Euler Characteristic of the $j^{th}$ Hodge subcomplex of the Hodge decomposition of $Δ (H)$. We also examine conditions on a hypergraph, $H$, for which its Hodge subcomplexes are Cohen-Macaulay, and thus where the absolute value of the coefficient of $λ ^j$ in $χ _H(λ )$ equals the dimension of the $j^{th}$ Hodge piece of the Hodge decomposition of $Δ (H)$.]]> 0 <![CDATA[We prove a formula for the number of permutations in $S_n$ such that their first $n-k$ entries are increasing and their longest increasing subsequence has length $n-k$. This formula first appeared as a consequence of character polynomial calculations in recent work of Adriano Garsia and Alain Goupil. We give two "elementary' bijective proofs of this result and of its q-analogue, one proof using the RSK correspondence and one only permutations.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2829 https://doi.org/10.46298/dmtcs.2829 Panova, Greta Panova, Greta <![CDATA[We prove a formula for the number of permutations in $S_n$ such that their first $n-k$ entries are increasing and their longest increasing subsequence has length $n-k$. This formula first appeared as a consequence of character polynomial calculations in recent work of Adriano Garsia and Alain Goupil. We give two "elementary' bijective proofs of this result and of its q-analogue, one proof using the RSK correspondence and one only permutations.]]> 0 <![CDATA[We show that the set $R(w_0)$ of reduced expressions for the longest element in the hyperoctahedral group exhibits the cyclic sieving phenomenon. More specifically, $R(w_0)$ possesses a natural cyclic action given by moving the first letter of a word to the end, and we show that the orbit structure of this action is encoded by the generating function for the major index on $R(w_0)$.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2828 https://doi.org/10.46298/dmtcs.2828 Petersen, T. K. Serrano, L. Petersen, T. K. Serrano, L. <![CDATA[We show that the set $R(w_0)$ of reduced expressions for the longest element in the hyperoctahedral group exhibits the cyclic sieving phenomenon. More specifically, $R(w_0)$ possesses a natural cyclic action given by moving the first letter of a word to the end, and we show that the orbit structure of this action is encoded by the generating function for the major index on $R(w_0)$.]]> 0 <![CDATA[The polynomial ring $\mathbb{Z}[x_{11}, . . . , x_{33}]$ has a basis called the dual canonical basis whose quantization facilitates the study of representations of the quantum group $U_q(\mathfrak{sl}3(\mathbb{C}))$. On the other hand, $\mathbb{Z}[x_{11}, . . . , x_{33}]$ inherits a basis from the cluster monomial basis of a geometric model of the type $D_4$ cluster algebra. We prove that these two bases are equal. This extends work of Skandera and proves a conjecture of Fomin and Zelevinsky. This also provides an explicit factorization of the dual canonical basis elements of $\mathbb{Z}[x_{11}, . . . , x_{33}]$ into irreducible polynomials.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2827 https://doi.org/10.46298/dmtcs.2827 Rhoades, Brendon Rhoades, Brendon <![CDATA[The polynomial ring $\mathbb{Z}[x_{11}, . . . , x_{33}]$ has a basis called the dual canonical basis whose quantization facilitates the study of representations of the quantum group $U_q(\mathfrak{sl}3(\mathbb{C}))$. On the other hand, $\mathbb{Z}[x_{11}, . . . , x_{33}]$ inherits a basis from the cluster monomial basis of a geometric model of the type $D_4$ cluster algebra. We prove that these two bases are equal. This extends work of Skandera and proves a conjecture of Fomin and Zelevinsky. This also provides an explicit factorization of the dual canonical basis elements of $\mathbb{Z}[x_{11}, . . . , x_{33}]$ into irreducible polynomials.]]> 0 <![CDATA[In this extended abstract, we investigate bijections on various classes of set partitions of classical types that preserve openers and closers. On the one hand we present bijections for types $B$ and $C$ that interchange crossings and nestings, which generalize a construction by Kasraoui and Zeng for type $A$. On the other hand we generalize a bijection to type $B$ and $C$ that interchanges the cardinality of a maximal crossing with the cardinality of a maximal nesting, as given by Chen, Deng, Du, Stanley and Yan for type $A$. For type $D$, we were only able to construct a bijection between non-crossing and non-nesting set partitions. For all classical types we show that the set of openers and the set of closers determine a non-crossing or non-nesting set partition essentially uniquely.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2826 https://doi.org/10.46298/dmtcs.2826 Rubey, Martin Stump, Christian Rubey, Martin Stump, Christian <![CDATA[In this extended abstract, we investigate bijections on various classes of set partitions of classical types that preserve openers and closers. On the one hand we present bijections for types $B$ and $C$ that interchange crossings and nestings, which generalize a construction by Kasraoui and Zeng for type $A$. On the other hand we generalize a bijection to type $B$ and $C$ that interchanges the cardinality of a maximal crossing with the cardinality of a maximal nesting, as given by Chen, Deng, Du, Stanley and Yan for type $A$. For type $D$, we were only able to construct a bijection between non-crossing and non-nesting set partitions. For all classical types we show that the set of openers and the set of closers determine a non-crossing or non-nesting set partition essentially uniquely.]]> 0 <![CDATA[Digital trees, also known as $\textit{"tries''}$, are fundamental to a number of algorithmic schemes, including radix-based searching and sorting, lossless text compression, dynamic hashing algorithms, communication protocols of the tree or stack type, distributed leader election, and so on. This extended abstract develops the asymptotic form of expectations of the main parameters of interest, such as tree size and path length. The analysis is conducted under the simplest of all probabilistic models; namely, the $\textit{memoryless source}$, under which letters that data items are comprised of are drawn independently from a fixed (finite) probability distribution. The precise asymptotic structure of the parameters' expectations is shown to depend on fine singular properties in the complex plane of a ubiquitous $\textit{Dirichlet series}$. Consequences include the characterization of a broad range of asymptotic regimes for error terms associated with trie parameters, as well as a classification that depends on specific $\textit{arithmetic properties}$, especially irrationality measures, of the sources under consideration.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2799 https://doi.org/10.46298/dmtcs.2799 Flajolet, Philippe Roux, Mathieu Vallée, Brigitte Flajolet, Philippe Roux, Mathieu Vallée, Brigitte <![CDATA[Digital trees, also known as $\textit{"tries''}$, are fundamental to a number of algorithmic schemes, including radix-based searching and sorting, lossless text compression, dynamic hashing algorithms, communication protocols of the tree or stack type, distributed leader election, and so on. This extended abstract develops the asymptotic form of expectations of the main parameters of interest, such as tree size and path length. The analysis is conducted under the simplest of all probabilistic models; namely, the $\textit{memoryless source}$, under which letters that data items are comprised of are drawn independently from a fixed (finite) probability distribution. The precise asymptotic structure of the parameters' expectations is shown to depend on fine singular properties in the complex plane of a ubiquitous $\textit{Dirichlet series}$. Consequences include the characterization of a broad range of asymptotic regimes for error terms associated with trie parameters, as well as a classification that depends on specific $\textit{arithmetic properties}$, especially irrationality measures, of the sources under consideration.]]> 0 <![CDATA[We study equations in groups $G$ with unique $m$-th roots for each positive integer $m$. A word equation in two letters is an expression of the form$ w(X,A) = B$, where $w$ is a finite word in the alphabet ${X,A}$. We think of $A,B ∈G$ as fixed coefficients, and $X ∈G$ as the unknown. Certain word equations, such as $XAXAX=B$, have solutions in terms of radicals: $X = A^-1/2(A^1/2BA^1/2)^1/3A^-1/2$, while others such as $X^2 A X = B$ do not. We obtain the first known infinite families of word equations not solvable by radicals, and conjecture a complete classification. To a word w we associate a polynomial $P_w ∈ℤ[x,y]$ in two commuting variables, which factors whenever $w$ is a composition of smaller words. We prove that if $P_w(x^2,y^2)$ has an absolutely irreducible factor in $ℤ[x,y]$, then the equation $w(X,A)=B$ is not solvable in terms of radicals.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2807 https://doi.org/10.46298/dmtcs.2807 Hillar, Christopher J. Levine, Lionel Rhea, Darren Hillar, Christopher J. Levine, Lionel Rhea, Darren <![CDATA[We study equations in groups $G$ with unique $m$-th roots for each positive integer $m$. A word equation in two letters is an expression of the form$ w(X,A) = B$, where $w$ is a finite word in the alphabet ${X,A}$. We think of $A,B ∈G$ as fixed coefficients, and $X ∈G$ as the unknown. Certain word equations, such as $XAXAX=B$, have solutions in terms of radicals: $X = A^-1/2(A^1/2BA^1/2)^1/3A^-1/2$, while others such as $X^2 A X = B$ do not. We obtain the first known infinite families of word equations not solvable by radicals, and conjecture a complete classification. To a word w we associate a polynomial $P_w ∈ℤ[x,y]$ in two commuting variables, which factors whenever $w$ is a composition of smaller words. We prove that if $P_w(x^2,y^2)$ has an absolutely irreducible factor in $ℤ[x,y]$, then the equation $w(X,A)=B$ is not solvable in terms of radicals.]]> 0 <![CDATA[Based on constant term evaluation, we present a new method to compute a closed form of the summation $∑_k=0^n-1 ∏_j=1^r F_j(a_jn+b_jk+c_j)$, where ${F_j(k)} are $C$-finite sequences and $a_j$ and $a_j+b_j$ are nonnegative integers. Our algorithm is much faster than that of Greene and Wilf.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2806 https://doi.org/10.46298/dmtcs.2806 Hou, Qing-Hu Xin, Guoce Hou, Qing-Hu Xin, Guoce <![CDATA[Based on constant term evaluation, we present a new method to compute a closed form of the summation $∑_k=0^n-1 ∏_j=1^r F_j(a_jn+b_jk+c_j)$, where ${F_j(k)} are $C$-finite sequences and $a_j$ and $a_j+b_j$ are nonnegative integers. Our algorithm is much faster than that of Greene and Wilf.]]> 0 <![CDATA[A permutation is separable if it can be generated from the permutation 1 by successive sums and skew sums or, equivalently, if it avoids the patterns 2413 and 3142. Using the notion of separating tree, we give a computationally efficient formula for the Möbius function of an interval $(q,p)$ in the poset of separable permutations ordered by pattern containment. A consequence of the formula is that the Möbius function of such an interval $(q,p)$ is bounded by the number of occurrences of $q$ as a pattern in $p$. The formula also implies that for any separable permutation $p$ the Möbius function of $(1,p)$ is either 0, 1 or -1.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2805 https://doi.org/10.46298/dmtcs.2805 Jelínek, Vít Jelínková, Eva Steingrímsson, Einar Jelínek, Vít Jelínková, Eva Steingrímsson, Einar <![CDATA[A permutation is separable if it can be generated from the permutation 1 by successive sums and skew sums or, equivalently, if it avoids the patterns 2413 and 3142. Using the notion of separating tree, we give a computationally efficient formula for the Möbius function of an interval $(q,p)$ in the poset of separable permutations ordered by pattern containment. A consequence of the formula is that the Möbius function of such an interval $(q,p)$ is bounded by the number of occurrences of $q$ as a pattern in $p$. The formula also implies that for any separable permutation $p$ the Möbius function of $(1,p)$ is either 0, 1 or -1.]]> 0 <![CDATA[In this paper we establish the cover time of a random graph $G(\textbf{d})$ chosen uniformly at random from the set of graphs with vertex set $[n]$ and degree sequence $\textbf{d}$. We show that under certain restrictions on $\textbf{d}$, the cover time of $G(\textbf{d})$ is with high probability asymptotic to $\frac{d-1}{ d-2} \frac{\theta}{ d}n \log n$. Here $\theta$ is the average degree and $d$ is the $\textit{effective minimum degree}$. The effective minimum degree is the first entry in the sorted degree sequence which occurs order $n$ times.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2804 https://doi.org/10.46298/dmtcs.2804 Abdullah, Mohammed Cooper, Colin Frieze, Alan Abdullah, Mohammed Cooper, Colin Frieze, Alan <![CDATA[In this paper we establish the cover time of a random graph $G(\textbf{d})$ chosen uniformly at random from the set of graphs with vertex set $[n]$ and degree sequence $\textbf{d}$. We show that under certain restrictions on $\textbf{d}$, the cover time of $G(\textbf{d})$ is with high probability asymptotic to $\frac{d-1}{ d-2} \frac{\theta}{ d}n \log n$. Here $\theta$ is the average degree and $d$ is the $\textit{effective minimum degree}$. The effective minimum degree is the first entry in the sorted degree sequence which occurs order $n$ times.]]> 0 <![CDATA[This paper introduces a Markov process inspired by the problem of quasicrystal growth. It acts over dimer tilings of the triangular grid by randomly performing local transformations, called $\textit{flips}$, which do not increase the number of identical adjacent tiles (this number can be thought as the tiling energy). Fixed-points of such a process play the role of quasicrystals. We are here interested in the worst-case expected number of flips to converge towards a fixed-point. Numerical experiments suggest a $\Theta (n^2)$ bound, where $n$ is the number of tiles of the tiling. We prove a $O(n^{2.5})$ upper bound and discuss the gap between this bound and the previous one. We also briefly discuss the average-case.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2803 https://doi.org/10.46298/dmtcs.2803 Fernique, Thomas Regnault, Damien Fernique, Thomas Regnault, Damien <![CDATA[This paper introduces a Markov process inspired by the problem of quasicrystal growth. It acts over dimer tilings of the triangular grid by randomly performing local transformations, called $\textit{flips}$, which do not increase the number of identical adjacent tiles (this number can be thought as the tiling energy). Fixed-points of such a process play the role of quasicrystals. We are here interested in the worst-case expected number of flips to converge towards a fixed-point. Numerical experiments suggest a $\Theta (n^2)$ bound, where $n$ is the number of tiles of the tiling. We prove a $O(n^{2.5})$ upper bound and discuss the gap between this bound and the previous one. We also briefly discuss the average-case.]]> 0 <![CDATA[In the hidden clique problem, one needs to find the maximum clique in an $n$-vertex graph that has a clique of size $k$ but is otherwise random. An algorithm of Alon, Krivelevich and Sudakov that is based on spectral techniques is known to solve this problem (with high probability over the random choice of input graph) when $k \geq c \sqrt{n}$ for a sufficiently large constant $c$. In this manuscript we present a new algorithm for finding hidden cliques. It too provably works when $k > c \sqrt{n}$ for a sufficiently large constant $c$. However, our algorithm has the advantage of being much simpler (no use of spectral techniques), running faster (linear time), and experiments show that the leading constant $c$ is smaller than in the spectral approach. We also present linear time algorithms that experimentally find even smaller hidden cliques, though it remains open whether any of these algorithms finds hidden cliques of size $o(\sqrt{n})$.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2802 https://doi.org/10.46298/dmtcs.2802 Feige, Uriel Ron, Dorit Feige, Uriel Ron, Dorit <![CDATA[In the hidden clique problem, one needs to find the maximum clique in an $n$-vertex graph that has a clique of size $k$ but is otherwise random. An algorithm of Alon, Krivelevich and Sudakov that is based on spectral techniques is known to solve this problem (with high probability over the random choice of input graph) when $k \geq c \sqrt{n}$ for a sufficiently large constant $c$. In this manuscript we present a new algorithm for finding hidden cliques. It too provably works when $k > c \sqrt{n}$ for a sufficiently large constant $c$. However, our algorithm has the advantage of being much simpler (no use of spectral techniques), running faster (linear time), and experiments show that the leading constant $c$ is smaller than in the spectral approach. We also present linear time algorithms that experimentally find even smaller hidden cliques, though it remains open whether any of these algorithms finds hidden cliques of size $o(\sqrt{n})$.]]> 0 <![CDATA[We consider lattice walks in $\mathbb{R}^k$ confined to the region $0 < x_1 < x_2 \ldots < x_k$ with fixed (but arbitrary) starting and end points. The walks are required to be "reflectable", that is, we assume that the number of paths can be counted using the reflection principle. The main result is an asymptotic formula for the total number of walks of length $n$ with fixed but arbitrary starting and end point for a general class of walks as the number $n$ of steps tends to infinity. As applications, we find the asymptotics for the number of $k$-non-crossing tangled diagrams on the set $\{1,2, \ldots,n\}$ as $n$ tends to infinity, and asymptotics for the number of $k$-vicious walkers subject to a wall restriction in the random turns model as well as in the lock step model. Asymptotics for all of these objects were either known only for certain special cases, or have only been partially determined.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2801 https://doi.org/10.46298/dmtcs.2801 Feierl, Thomas Feierl, Thomas <![CDATA[We consider lattice walks in $\mathbb{R}^k$ confined to the region $0 < x_1 < x_2 \ldots < x_k$ with fixed (but arbitrary) starting and end points. The walks are required to be "reflectable", that is, we assume that the number of paths can be counted using the reflection principle. The main result is an asymptotic formula for the total number of walks of length $n$ with fixed but arbitrary starting and end point for a general class of walks as the number $n$ of steps tends to infinity. As applications, we find the asymptotics for the number of $k$-non-crossing tangled diagrams on the set $\{1,2, \ldots,n\}$ as $n$ tends to infinity, and asymptotics for the number of $k$-vicious walkers subject to a wall restriction in the random turns model as well as in the lock step model. Asymptotics for all of these objects were either known only for certain special cases, or have only been partially determined.]]> 0 <![CDATA[Most previous studies of the sorting algorithm $\mathtt{QuickSort}$ have used the number of key comparisons as a measure of the cost of executing the algorithm. Here we suppose that the $n$ independent and identically distributed (iid) keys are each represented as a sequence of symbols from a probabilistic source and that $\mathtt{QuickSort}$ operates on individual symbols, and we measure the execution cost as the number of symbol comparisons. Assuming only a mild "tameness'' condition on the source, we show that there is a limiting distribution for the number of symbol comparisons after normalization: first centering by the mean and then dividing by $n$. Additionally, under a condition that grows more restrictive as $p$ increases, we have convergence of moments of orders $p$ and smaller. In particular, we have convergence in distribution and convergence of moments of every order whenever the source is memoryless, i.e., whenever each key is generated as an infinite string of iid symbols. This is somewhat surprising: Even for the classical model that each key is an iid string of unbiased ("fair'') bits, the mean exhibits periodic fluctuations of order $n$.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2800 https://doi.org/10.46298/dmtcs.2800 fill, james Allen fill, james Allen <![CDATA[Most previous studies of the sorting algorithm $\mathtt{QuickSort}$ have used the number of key comparisons as a measure of the cost of executing the algorithm. Here we suppose that the $n$ independent and identically distributed (iid) keys are each represented as a sequence of symbols from a probabilistic source and that $\mathtt{QuickSort}$ operates on individual symbols, and we measure the execution cost as the number of symbol comparisons. Assuming only a mild "tameness'' condition on the source, we show that there is a limiting distribution for the number of symbol comparisons after normalization: first centering by the mean and then dividing by $n$. Additionally, under a condition that grows more restrictive as $p$ increases, we have convergence of moments of orders $p$ and smaller. In particular, we have convergence in distribution and convergence of moments of every order whenever the source is memoryless, i.e., whenever each key is generated as an infinite string of iid symbols. This is somewhat surprising: Even for the classical model that each key is an iid string of unbiased ("fair'') bits, the mean exhibits periodic fluctuations of order $n$.]]> 0 <![CDATA[We consider a stochastic version of the $k$-server problem in which $k$ servers move on a circle to satisfy stochastically generated requests. The requests are independent and identically distributed according to an arbitrary distribution on a circle, which is either discrete or continuous. The cost of serving a request is the distance that a server needs to move to reach the request. The goal is to minimize the steady-state expected cost induced by the requests. We study the performance of a greedy strategy, focusing, in particular, on its convergence properties and the interplay between the discrete and continuous versions of the process.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2791 https://doi.org/10.46298/dmtcs.2791 Anagnostopoulos, Aris Dombry, Clément Guillotin-Plantard, Nadine Kontoyiannis, Ioannis Upfal, Eli Anagnostopoulos, Aris Dombry, Clément Guillotin-Plantard, Nadine Kontoyiannis, Ioannis Upfal, Eli <![CDATA[We consider a stochastic version of the $k$-server problem in which $k$ servers move on a circle to satisfy stochastically generated requests. The requests are independent and identically distributed according to an arbitrary distribution on a circle, which is either discrete or continuous. The cost of serving a request is the distance that a server needs to move to reach the request. The goal is to minimize the steady-state expected cost induced by the requests. We study the performance of a greedy strategy, focusing, in particular, on its convergence properties and the interplay between the discrete and continuous versions of the process.]]> 0 <![CDATA[We consider the multiset construction of decomposable structures with component generating function $C(z)$ of alg-log type, $\textit{i.e.}$, $C(z) = (1-z)^{-\alpha} (\log \frac{1}{ 1-z})^{\beta}$. We provide asymptotic results for the number of labeled objects of size $n$ in the case when $\alpha$ is positive and $\beta$ is positive and in the case $\alpha = 0$ and $\beta \geq 2$. The case $0<-\alpha <1$ and any $\beta$ and the case $\alpha > 0$ and $\beta = 0$ have been treated in previous papers. Our results extend previous work of Wright.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2798 https://doi.org/10.46298/dmtcs.2798 Gao, Zhicheng Laferrière, David Panario, Daniel Gao, Zhicheng Laferrière, David Panario, Daniel <![CDATA[We consider the multiset construction of decomposable structures with component generating function $C(z)$ of alg-log type, $\textit{i.e.}$, $C(z) = (1-z)^{-\alpha} (\log \frac{1}{ 1-z})^{\beta}$. We provide asymptotic results for the number of labeled objects of size $n$ in the case when $\alpha$ is positive and $\beta$ is positive and in the case $\alpha = 0$ and $\beta \geq 2$. The case $0<-\alpha <1$ and any $\beta$ and the case $\alpha > 0$ and $\beta = 0$ have been treated in previous papers. Our results extend previous work of Wright.]]> 0 <![CDATA[We enumerate rooted 3-connected (2-connected) planar triangulations with respect to the vertices and 3-cuts (2-cuts). Consequently, we show that the distribution of the number of 3-cuts in a random rooted 3-connected planar triangulation with $n+3$ vertices is asymptotically normal with mean $(10/27)n$ and variance $(320/729)n$, and the distribution of the number of 2-cuts in a random 2-connected planar triangulation with $n+2$ vertices is asymptotically normal with mean $(8/27)n$ and variance $(152/729)n$. We also show that the distribution of the number of 3-connected components in a random 2-connected triangulation with $n+2$ vertices is asymptotically normal with mean $n/3$ and variance $\frac{8}{ 27}n$ .]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2797 https://doi.org/10.46298/dmtcs.2797 Gao, Zhicheng Schaeffer, Gilles Gao, Zhicheng Schaeffer, Gilles <![CDATA[We enumerate rooted 3-connected (2-connected) planar triangulations with respect to the vertices and 3-cuts (2-cuts). Consequently, we show that the distribution of the number of 3-cuts in a random rooted 3-connected planar triangulation with $n+3$ vertices is asymptotically normal with mean $(10/27)n$ and variance $(320/729)n$, and the distribution of the number of 2-cuts in a random 2-connected planar triangulation with $n+2$ vertices is asymptotically normal with mean $(8/27)n$ and variance $(152/729)n$. We also show that the distribution of the number of 3-connected components in a random 2-connected triangulation with $n+2$ vertices is asymptotically normal with mean $n/3$ and variance $\frac{8}{ 27}n$ .]]> 0 <![CDATA[The variance of partial match queries in $k$-dimensional tries was investigated in a couple of papers in the mid-nineties, the resulting analysis being long and complicated. In this paper, we are going to re-derive these results with a much easier approach. Moreover, our approach works for $k$-dimensional PATRICIA tries, $k$-dimensional digital search trees and bucket versions as well.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2796 https://doi.org/10.46298/dmtcs.2796 Fuchs, Michael Fuchs, Michael <![CDATA[The variance of partial match queries in $k$-dimensional tries was investigated in a couple of papers in the mid-nineties, the resulting analysis being long and complicated. In this paper, we are going to re-derive these results with a much easier approach. Moreover, our approach works for $k$-dimensional PATRICIA tries, $k$-dimensional digital search trees and bucket versions as well.]]> 0 <![CDATA[Given a simple directed graph $D = (V,A)$, let the size of the largest induced directed acyclic graph $\textit{(dag)}$ be denoted by $mas(D)$. Let $D \in \mathcal{D}(n,p)$ be a $\textit{random}$ instance, obtained by choosing each of the $\binom{n}{2}$ possible undirected edges independently with probability $2p$ and then orienting each chosen edge independently in one of two possible directions with probabibility $1/2$. We obtain improved bounds on the range of concentration, upper and lower bounds of $mas(D)$. Our main result is that $mas(D) \geq \lfloor 2\log_q np - X \rfloor$ where $q = (1-p)^{-1}, X=W$ if $p \geq n^{-1/3+\epsilon}$ ($\epsilon > 0$ is any constant), $X=W/(\ln q)$ if $p \geq n^{-1/2}(\ln n)^2$, and $W$ is a suitably large constant. where we have an $O(\ln \ln np/\ln q)$ term instead of $W$. This improves the previously known lower bound with an $O(\ln \ln np/\ln q)$ term instead of $W$. We also obtain a slight improvement on the upper bound, using an upper bound on the number of acyclic orientations of an undirected graph. We also analyze a polynomial-time heuristic to find a large induced dag and show that it produces a solution whose size is at least $\log _q np + \Theta (\sqrt{\log_q np})$.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2795 https://doi.org/10.46298/dmtcs.2795 Dutta, Kunal Subramanian, C. R. Dutta, Kunal Subramanian, C. R. <![CDATA[Given a simple directed graph $D = (V,A)$, let the size of the largest induced directed acyclic graph $\textit{(dag)}$ be denoted by $mas(D)$. Let $D \in \mathcal{D}(n,p)$ be a $\textit{random}$ instance, obtained by choosing each of the $\binom{n}{2}$ possible undirected edges independently with probability $2p$ and then orienting each chosen edge independently in one of two possible directions with probabibility $1/2$. We obtain improved bounds on the range of concentration, upper and lower bounds of $mas(D)$. Our main result is that $mas(D) \geq \lfloor 2\log_q np - X \rfloor$ where $q = (1-p)^{-1}, X=W$ if $p \geq n^{-1/3+\epsilon}$ ($\epsilon > 0$ is any constant), $X=W/(\ln q)$ if $p \geq n^{-1/2}(\ln n)^2$, and $W$ is a suitably large constant. where we have an $O(\ln \ln np/\ln q)$ term instead of $W$. This improves the previously known lower bound with an $O(\ln \ln np/\ln q)$ term instead of $W$. We also obtain a slight improvement on the upper bound, using an upper bound on the number of acyclic orientations of an undirected graph. We also analyze a polynomial-time heuristic to find a large induced dag and show that it produces a solution whose size is at least $\log _q np + \Theta (\sqrt{\log_q np})$.]]> 0 <![CDATA[We consider pyramids made of one-dimensional pieces of fixed integer length $a$ and which may have pairwise overlaps of integer length from $1$ to $a$. We give a combinatorial proof that the number of pyramids of size $m$, i.e., consisting of $m$ pieces, equals $\binom{am-1}{m-1}$ for each $a \geq 2$. This generalises a well known result for $a=2$. A bijective correspondence between so-called right (or left) pyramids and $a$-ary trees is pointed out, and it is shown that asymptotically the average width of pyramids equals $\sqrt{\frac{\pi}{2} a(a-1)m}$.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2794 https://doi.org/10.46298/dmtcs.2794 Durhuus, Bergfinnur Eilers, Søren Durhuus, Bergfinnur Eilers, Søren <![CDATA[We consider pyramids made of one-dimensional pieces of fixed integer length $a$ and which may have pairwise overlaps of integer length from $1$ to $a$. We give a combinatorial proof that the number of pyramids of size $m$, i.e., consisting of $m$ pieces, equals $\binom{am-1}{m-1}$ for each $a \geq 2$. This generalises a well known result for $a=2$. A bijective correspondence between so-called right (or left) pyramids and $a$-ary trees is pointed out, and it is shown that asymptotically the average width of pyramids equals $\sqrt{\frac{\pi}{2} a(a-1)m}$.]]> 0 <![CDATA[We address the uniform random generation of words from a context-free language (over an alphabet of size $k$), while constraining every letter to a targeted frequency of occurrence. Our approach consists in a multidimensional extension of Boltzmann samplers. We show that, under mostly $\textit{strong-connectivity}$ hypotheses, our samplers return a word of size in $[(1- \epsilon)n, (1+ \epsilon)n]$ and exact frequency in $\mathcal{O}(n^{1+k/2})$ expected time. Moreover, if we accept tolerance intervals of width in $\Omega (\sqrt{n})$ for the number of occurrences of each letters, our samplers perform an approximate-size generation of words in expected $\mathcal{O}(n)$ time. We illustrate our approach on the generation of Tetris tessellations with uniform statistics in the different types of tetraminoes.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2793 https://doi.org/10.46298/dmtcs.2793 Bodini, Olivier Ponty, Yann Bodini, Olivier Ponty, Yann <![CDATA[We address the uniform random generation of words from a context-free language (over an alphabet of size $k$), while constraining every letter to a targeted frequency of occurrence. Our approach consists in a multidimensional extension of Boltzmann samplers. We show that, under mostly $\textit{strong-connectivity}$ hypotheses, our samplers return a word of size in $[(1- \epsilon)n, (1+ \epsilon)n]$ and exact frequency in $\mathcal{O}(n^{1+k/2})$ expected time. Moreover, if we accept tolerance intervals of width in $\Omega (\sqrt{n})$ for the number of occurrences of each letters, our samplers perform an approximate-size generation of words in expected $\mathcal{O}(n)$ time. We illustrate our approach on the generation of Tetris tessellations with uniform statistics in the different types of tetraminoes.]]> 0 <![CDATA[This article tackles the enumeration and asymptotics of directed lattice paths (that are isomorphic to unidimensional paths) of bounded height (walks below one wall, or between two walls, for $\textit{any}$ finite set of jumps). Thus, for any lattice paths, we give the generating functions of bridges ("discrete'' Brownian bridges) and reflected bridges ("discrete'' reflected Brownian bridges) of a given height. It is a new success of the "kernel method'' that the generating functions of such walks have some nice expressions as symmetric functions in terms of the roots of the kernel. These formulae also lead to fast algorithms for computing the $n$-th Taylor coefficients of the corresponding generating functions. For a large class of walks, we give the discrete distribution of the height of bridges, and show the convergence to a Rayleigh limit law. For the family of walks consisting of a $-1$ jump and many positive jumps, we give more precise bounds for the speed of convergence. We end our article with a heuristic application to bioinformatics that has a high speed-up relative to previous work.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2792 https://doi.org/10.46298/dmtcs.2792 Banderier, C. Nicodème, P. Banderier, C. Nicodème, P. <![CDATA[This article tackles the enumeration and asymptotics of directed lattice paths (that are isomorphic to unidimensional paths) of bounded height (walks below one wall, or between two walls, for $\textit{any}$ finite set of jumps). Thus, for any lattice paths, we give the generating functions of bridges ("discrete'' Brownian bridges) and reflected bridges ("discrete'' reflected Brownian bridges) of a given height. It is a new success of the "kernel method'' that the generating functions of such walks have some nice expressions as symmetric functions in terms of the roots of the kernel. These formulae also lead to fast algorithms for computing the $n$-th Taylor coefficients of the corresponding generating functions. For a large class of walks, we give the discrete distribution of the height of bridges, and show the convergence to a Rayleigh limit law. For the family of walks consisting of a $-1$ jump and many positive jumps, we give more precise bounds for the speed of convergence. We end our article with a heuristic application to bioinformatics that has a high speed-up relative to previous work.]]> 0 <![CDATA[In this paper, we significantly improve a previous result by the same author showing the existence of a weakly universal cellular automaton with five states living in the hyperbolic $3D$-space. Here, we get such a cellular automaton with three states only.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2755 https://doi.org/10.46298/dmtcs.2755 Margenstern, Maurice Margenstern, Maurice <![CDATA[In this paper, we significantly improve a previous result by the same author showing the existence of a weakly universal cellular automaton with five states living in the hyperbolic $3D$-space. Here, we get such a cellular automaton with three states only.]]> 0 <![CDATA[We discuss a very close relation between minimal recurrent configurations of Chip Firing Games and Directed Acyclic Graphs and demonstrate the usefulness of this relation by giving a lower bound for the number of minimal recurrent configurations of the Abelian Sandpile Model as well as a lower bound for the number of firings which are caused by the addition of two recurrent configurations on particular graphs.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2756 https://doi.org/10.46298/dmtcs.2756 Schulz, Matthias Schulz, Matthias <![CDATA[We discuss a very close relation between minimal recurrent configurations of Chip Firing Games and Directed Acyclic Graphs and demonstrate the usefulness of this relation by giving a lower bound for the number of minimal recurrent configurations of the Abelian Sandpile Model as well as a lower bound for the number of firings which are caused by the addition of two recurrent configurations on particular graphs.]]> 0 <![CDATA[Infinite systems of equations appear naturally in combinatorial counting problems. Formally, we consider functional equations of the form $\mathbf{y} (x)=F(x,\mathbf{y} (x))$, where $F(x,\mathbf{y} ):\mathbb{C} \times \ell^p \to \ell^p$ is a positive and nonlinear function, and analyze the behavior of the solution $\mathbf{y} (x)$ at the boundary of the domain of convergence. In contrast to the finite dimensional case different types of singularities are possible. We show that if the Jacobian operator of the function $F$ is compact, then the occurring singularities are of square root type, as it is in the finite dimensional setting. This leads to asymptotic expansions of the Taylor coefficients of $\mathbf{y} (x)$.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2782 https://doi.org/10.46298/dmtcs.2782 Morgenbesser, Johannes F. Morgenbesser, Johannes F. <![CDATA[Infinite systems of equations appear naturally in combinatorial counting problems. Formally, we consider functional equations of the form $\mathbf{y} (x)=F(x,\mathbf{y} (x))$, where $F(x,\mathbf{y} ):\mathbb{C} \times \ell^p \to \ell^p$ is a positive and nonlinear function, and analyze the behavior of the solution $\mathbf{y} (x)$ at the boundary of the domain of convergence. In contrast to the finite dimensional case different types of singularities are possible. We show that if the Jacobian operator of the function $F$ is compact, then the occurring singularities are of square root type, as it is in the finite dimensional setting. This leads to asymptotic expansions of the Taylor coefficients of $\mathbf{y} (x)$.]]> 0 <![CDATA[We show that the diameter $D(G_n)$ of a random (unembedded) labelled connected planar graph with $n$ vertices is asymptotically almost surely of order $n^{1/4}$, in the sense that there exists a constant $c>0$ such that $P(D(G_n) \in (n^{1/4-\epsilon} ,n^{1/4+\epsilon})) \geq 1-\exp (-n^{c\epsilon})$ for $\epsilon$ small enough and $n$ large enough $(n \geq n_0(\epsilon))$. We prove similar statements for rooted $2$-connected and $3$-connected embedded (maps) and unembedded planar graphs.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2790 https://doi.org/10.46298/dmtcs.2790 Chapuy, Guillaume Fusy, Eric Gimenez, Omer Noy, Marc Chapuy, Guillaume Fusy, Eric Gimenez, Omer Noy, Marc <![CDATA[We show that the diameter $D(G_n)$ of a random (unembedded) labelled connected planar graph with $n$ vertices is asymptotically almost surely of order $n^{1/4}$, in the sense that there exists a constant $c>0$ such that $P(D(G_n) \in (n^{1/4-\epsilon} ,n^{1/4+\epsilon})) \geq 1-\exp (-n^{c\epsilon})$ for $\epsilon$ small enough and $n$ large enough $(n \geq n_0(\epsilon))$. We prove similar statements for rooted $2$-connected and $3$-connected embedded (maps) and unembedded planar graphs.]]> 0 <![CDATA[We state and prove new properties about Doeblin's ergodicity coefficient for finite Markov chains. We show that this coefficient satisfies a sub-multiplicative type inequality (analogous to the Markov-Dobrushin's ergodicity coefficient), and provide a novel but elementary proof of Doeblin's characterization of weak-ergodicity for non-homogeneous chains. Using Doeblin's coefficient, we illustrate how to approximate a homogeneous but possibly non-stationary Markov chain of duration $n$ by independent and short-lived realizations of an auxiliary chain of duration of order $\ln (n)$. This leads to approximations of occupancy distributions in homogeneous chains, which may be particularly useful when exact calculations via one-step methods or transfer matrices are impractical, and when asymptotic approximations may not be yet reliable. Our findings may find applications to pattern problems in Markovian and non-Markovian sequences that are treatable via embedding techniques.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2789 https://doi.org/10.46298/dmtcs.2789 Chestnut, Stephen Lladser, Manuel E. Chestnut, Stephen Lladser, Manuel E. <![CDATA[We state and prove new properties about Doeblin's ergodicity coefficient for finite Markov chains. We show that this coefficient satisfies a sub-multiplicative type inequality (analogous to the Markov-Dobrushin's ergodicity coefficient), and provide a novel but elementary proof of Doeblin's characterization of weak-ergodicity for non-homogeneous chains. Using Doeblin's coefficient, we illustrate how to approximate a homogeneous but possibly non-stationary Markov chain of duration $n$ by independent and short-lived realizations of an auxiliary chain of duration of order $\ln (n)$. This leads to approximations of occupancy distributions in homogeneous chains, which may be particularly useful when exact calculations via one-step methods or transfer matrices are impractical, and when asymptotic approximations may not be yet reliable. Our findings may find applications to pattern problems in Markovian and non-Markovian sequences that are treatable via embedding techniques.]]> 0 <![CDATA[We study depth properties of a general class of random recursive trees where each node $n$ attaches to the random node $\lfloor nX_n \rfloor$ and $X_0, \ldots , X_n$ is a sequence of i.i.d. random variables taking values in $[0,1)$. We call such trees scaled attachment random recursive trees (SARRT). We prove that the height $H_n$ of a SARRT is asymptotically given by $H_n \sim \alpha_{\max} \log n$ where $\alpha_{\max}$ is a constant depending only on the distribution of $X_0$ whenever $X_0$ has a bounded density. This gives a new elementary proof for the height of uniform random recursive trees $H_n \sim e \log n$ that does not use branching random walks.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2788 https://doi.org/10.46298/dmtcs.2788 Devroye, Luc Fawzi, Omar Fraiman, Nicolas Devroye, Luc Fawzi, Omar Fraiman, Nicolas <![CDATA[We study depth properties of a general class of random recursive trees where each node $n$ attaches to the random node $\lfloor nX_n \rfloor$ and $X_0, \ldots , X_n$ is a sequence of i.i.d. random variables taking values in $[0,1)$. We call such trees scaled attachment random recursive trees (SARRT). We prove that the height $H_n$ of a SARRT is asymptotically given by $H_n \sim \alpha_{\max} \log n$ where $\alpha_{\max}$ is a constant depending only on the distribution of $X_0$ whenever $X_0$ has a bounded density. This gives a new elementary proof for the height of uniform random recursive trees $H_n \sim e \log n$ that does not use branching random walks.]]> 0 <![CDATA[In this paper we obtain the expectation and variance of the number of Euler tours of a random $d$-in/$d$-out directed graph, for $d \geq 2$. We use this to obtain the asymptotic distribution and prove a concentration result. We are then able to show that a very simple approach for uniform sampling or approximately counting Euler tours yields algorithms running in expected polynomial time for almost every $d$-in/$d$-out graph. We make use of the BEST theorem of de Bruijn, van Aardenne-Ehrenfest, Smith and Tutte, which shows that the number of Euler tours of a $d$-in/$d$-out graph is the product of the number of arborescences and the term $[(d-1)!]^n/n$. Therefore most of our effort is towards estimating the asymptotic distribution of the number of arborescences of a random $d$-in/$d$-out graph.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2787 https://doi.org/10.46298/dmtcs.2787 Creed, Páidí Cryan, Mary Creed, Páidí Cryan, Mary <![CDATA[In this paper we obtain the expectation and variance of the number of Euler tours of a random $d$-in/$d$-out directed graph, for $d \geq 2$. We use this to obtain the asymptotic distribution and prove a concentration result. We are then able to show that a very simple approach for uniform sampling or approximately counting Euler tours yields algorithms running in expected polynomial time for almost every $d$-in/$d$-out graph. We make use of the BEST theorem of de Bruijn, van Aardenne-Ehrenfest, Smith and Tutte, which shows that the number of Euler tours of a $d$-in/$d$-out graph is the product of the number of arborescences and the term $[(d-1)!]^n/n$. Therefore most of our effort is towards estimating the asymptotic distribution of the number of arborescences of a random $d$-in/$d$-out graph.]]> 0 <![CDATA[In this paper we present a simple theory, based on the notion of group action on a set, which explains why processes of throwing random sets of points and throwing random lines are similar up to the second moment of counting functions connected with them. We also discuss other applications of this method and show how to calculate higher moments using the group acting on a set.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2786 https://doi.org/10.46298/dmtcs.2786 Cichoń, Jacek Klonowski, Marek Cichoń, Jacek Klonowski, Marek <![CDATA[In this paper we present a simple theory, based on the notion of group action on a set, which explains why processes of throwing random sets of points and throwing random lines are similar up to the second moment of counting functions connected with them. We also discuss other applications of this method and show how to calculate higher moments using the group acting on a set.]]> 0 <![CDATA[Consider the following weighted digital sum (WDS) variant: write integer $n$ as $n=2^{i_1} + 2^{i_2} + \cdots + 2^{i_k}$ with $i_1 > i_2 > \cdots > i_k \geq 0$ and set $W_M(n) := \sum_{t=1}^k t^M 2^{i_t}$. This type of weighted digital sum arises (when $M=1$) in the analysis of bottom-up mergesort but is not "smooth'' enough to permit a clean analysis. We therefore analyze its average $TW_M(n) := \frac{1}{n}\sum_{j \gt n} W_M(j)$. We show that $TW_M(n)$ has a solution of the form $n G_M(\lg n) + d_M \lg ^M n + \sum\limits_{d=0}^{M-1}(\lg ^d n)G_{M,d}(\lg n)$, where $d_M$ is a constant and $G_M(u), G_{M,d}(u)$'s are periodic functions with period one (given by absolutely convergent Fourier series).]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2785 https://doi.org/10.46298/dmtcs.2785 Cheung, Y. K. Golin, Mordecai Cheung, Y. K. Golin, Mordecai <![CDATA[Consider the following weighted digital sum (WDS) variant: write integer $n$ as $n=2^{i_1} + 2^{i_2} + \cdots + 2^{i_k}$ with $i_1 > i_2 > \cdots > i_k \geq 0$ and set $W_M(n) := \sum_{t=1}^k t^M 2^{i_t}$. This type of weighted digital sum arises (when $M=1$) in the analysis of bottom-up mergesort but is not "smooth'' enough to permit a clean analysis. We therefore analyze its average $TW_M(n) := \frac{1}{n}\sum_{j \gt n} W_M(j)$. We show that $TW_M(n)$ has a solution of the form $n G_M(\lg n) + d_M \lg ^M n + \sum\limits_{d=0}^{M-1}(\lg ^d n)G_{M,d}(\lg n)$, where $d_M$ is a constant and $G_M(u), G_{M,d}(u)$'s are periodic functions with period one (given by absolutely convergent Fourier series).]]> 0 <![CDATA[The Shannon effect states that "almost all'' Boolean functions have a complexity close to the maximal possible for the uniform probability distribution. In this paper we use some probability distributions on functions, induced by random expressions, and prove that this model does not exhibit the Shannon effect.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2784 https://doi.org/10.46298/dmtcs.2784 Genitrini, Antoine Gittenberger, Bernhard Genitrini, Antoine Gittenberger, Bernhard <![CDATA[The Shannon effect states that "almost all'' Boolean functions have a complexity close to the maximal possible for the uniform probability distribution. In this paper we use some probability distributions on functions, induced by random expressions, and prove that this model does not exhibit the Shannon effect.]]> 0 <![CDATA[We build and analyze in this paper Markov chains for the random sampling of some one-dimensional lattice paths with constraints, for various constraints. These chains are easy to implement, and sample an "almost" uniform path of length $n$ in $n^{3+\epsilon}$ steps. This bound makes use of a certain $\textit{contraction property}$ of the Markov chain, and is proved with an approach inspired by optimal transport.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2783 https://doi.org/10.46298/dmtcs.2783 Gerin, Lucas Gerin, Lucas <![CDATA[We build and analyze in this paper Markov chains for the random sampling of some one-dimensional lattice paths with constraints, for various constraints. These chains are easy to implement, and sample an "almost" uniform path of length $n$ in $n^{3+\epsilon}$ steps. This bound makes use of a certain $\textit{contraction property}$ of the Markov chain, and is proved with an approach inspired by optimal transport.]]> 0 <![CDATA[The analysis of pattern occurrences has numerous applications, in particular in biology. In this article, a symbolic method is proposed to compute the distribution associated to the number of occurences of a specific pattern in a random text generated by a stochastic 0L-system. To that purpose, a semiring structure is set for combinatorial classes composed of weighted words. This algebraic structure relies on new union and concatenation operators which, under some assumptions, are admissible constructions. Decomposing the combinatorial classes of interest by using these binary operators enables the direct translation of specifications into a set of functional equations relating generating functions thanks to transformation rules. The article ends with two examples. The first one deals with unary patterns and the connection with multitype branching process is established. The second one is about a pattern composed of two letters and underlines the importance of writing a proper specification.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2774 https://doi.org/10.46298/dmtcs.2774 Loi, Cedric Cournède, Paul-Henry Françon, Jean Loi, Cedric Cournède, Paul-Henry Françon, Jean <![CDATA[The analysis of pattern occurrences has numerous applications, in particular in biology. In this article, a symbolic method is proposed to compute the distribution associated to the number of occurences of a specific pattern in a random text generated by a stochastic 0L-system. To that purpose, a semiring structure is set for combinatorial classes composed of weighted words. This algebraic structure relies on new union and concatenation operators which, under some assumptions, are admissible constructions. Decomposing the combinatorial classes of interest by using these binary operators enables the direct translation of specifications into a set of functional equations relating generating functions thanks to transformation rules. The article ends with two examples. The first one deals with unary patterns and the connection with multitype branching process is established. The second one is about a pattern composed of two letters and underlines the importance of writing a proper specification.]]> 0 <![CDATA[Partial Quicksort sorts the $l$ smallest elements in a list of length $n$. We provide a complete running time analysis for this combination of Find and Quicksort. Further we give some optimal adapted versions, called Partition Quicksort, with an asymptotic running time $c_1l\ln l+c_2l+n+o(n)$. The constant $c_1$ can be as small as the information theoretic lower bound $\log_2 e$.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2781 https://doi.org/10.46298/dmtcs.2781 Martínez, Conrado Rösler, Uwe Martínez, Conrado Rösler, Uwe <![CDATA[Partial Quicksort sorts the $l$ smallest elements in a list of length $n$. We provide a complete running time analysis for this combination of Find and Quicksort. Further we give some optimal adapted versions, called Partition Quicksort, with an asymptotic running time $c_1l\ln l+c_2l+n+o(n)$. The constant $c_1$ can be as small as the information theoretic lower bound $\log_2 e$.]]> 0 <![CDATA[Building on the ideas of Flajolet and Martin (1985), Alon et al. (1987), Bar-Yossef et al. (2002), Giroire (2005), we develop a new algorithm for cardinality estimation, based on order statistics which, according to Chassaing and Gerin (2006), is optimal among similar algorithms. This algorithm has a remarkably simple analysis that allows us to take its $\textit{fine-tuning}$ and the $\textit{characterization of its properties}$ further than has been done until now. We prove that, asymptotically, it is $\textit{strictly unbiased}$ (contrarily to Probabilistic Counting, Loglog, Hyperloglog), we verify that its relative precision is about $1/\sqrt{m-2}$ when $m$ words of storage are used, and we fully characterize the limit law of the estimates it provides, in terms of gamma distribution―-this is the first such algorithm for which the limit law has been established. We also develop a Poisson analysis for the pre-asymptotic regime. In this way, we are able to devise a complete algorithm, covering all cardinalities ranges from $0$ to very large.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2780 https://doi.org/10.46298/dmtcs.2780 Lumbroso, Jérémie Lumbroso, Jérémie <![CDATA[Building on the ideas of Flajolet and Martin (1985), Alon et al. (1987), Bar-Yossef et al. (2002), Giroire (2005), we develop a new algorithm for cardinality estimation, based on order statistics which, according to Chassaing and Gerin (2006), is optimal among similar algorithms. This algorithm has a remarkably simple analysis that allows us to take its $\textit{fine-tuning}$ and the $\textit{characterization of its properties}$ further than has been done until now. We prove that, asymptotically, it is $\textit{strictly unbiased}$ (contrarily to Probabilistic Counting, Loglog, Hyperloglog), we verify that its relative precision is about $1/\sqrt{m-2}$ when $m$ words of storage are used, and we fully characterize the limit law of the estimates it provides, in terms of gamma distribution―-this is the first such algorithm for which the limit law has been established. We also develop a Poisson analysis for the pre-asymptotic regime. In this way, we are able to devise a complete algorithm, covering all cardinalities ranges from $0$ to very large.]]> 0 <![CDATA[We prove a limit theorem for the total Steiner $k$-distance of a random $b$-ary recursive tree with weighted edges. The total Steiner $k$-distance is the sum of all Steiner $k$-distances in a tree and it generalises the Wiener index. The limit theorem is obtained by using a limit theorem in the general setting of the contraction method. In order to use the contraction method we prove a recursion formula and determine the asymptotic expansion of the expectation using the so-called Master Theorem by Roura (2001). In a second step we prove a transformation of the total Steiner $k$-distance of $b$-ary trees with weighted edges to arbitrary recursive trees. This transformation yields the limit theorem for the total Steiner $k$-distance of the linear recursive trees when the parameter of these trees is a non-negative integer.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2779 https://doi.org/10.46298/dmtcs.2779 Munsonius, Götz Olaf Munsonius, Götz Olaf <![CDATA[We prove a limit theorem for the total Steiner $k$-distance of a random $b$-ary recursive tree with weighted edges. The total Steiner $k$-distance is the sum of all Steiner $k$-distances in a tree and it generalises the Wiener index. The limit theorem is obtained by using a limit theorem in the general setting of the contraction method. In order to use the contraction method we prove a recursion formula and determine the asymptotic expansion of the expectation using the so-called Master Theorem by Roura (2001). In a second step we prove a transformation of the total Steiner $k$-distance of $b$-ary trees with weighted edges to arbitrary recursive trees. This transformation yields the limit theorem for the total Steiner $k$-distance of the linear recursive trees when the parameter of these trees is a non-negative integer.]]> 0 <![CDATA[We introduce a random graph model based on $k$-trees, which can be generated by applying a probabilistic preferential attachment rule, but which also has a simple combinatorial description. We carry out a precise distributional analysis of important parameters for the network model such as the degree, the local clustering coefficient and the number of descendants of the nodes and root-to-node distances. We do not only obtain results for random nodes, but in particular we also get a precise description of the behaviour of parameters for the $j$-th inserted node in a random $k$-tree of size $n$, where $j=j(n)$ might grow with $n$. The approach presented is not restricted to this specific $k$-tree model, but can also be applied to other evolving $k$-tree models.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2778 https://doi.org/10.46298/dmtcs.2778 Panholzer, Alois Seitz, Georg Panholzer, Alois Seitz, Georg <![CDATA[We introduce a random graph model based on $k$-trees, which can be generated by applying a probabilistic preferential attachment rule, but which also has a simple combinatorial description. We carry out a precise distributional analysis of important parameters for the network model such as the degree, the local clustering coefficient and the number of descendants of the nodes and root-to-node distances. We do not only obtain results for random nodes, but in particular we also get a precise description of the behaviour of parameters for the $j$-th inserted node in a random $k$-tree of size $n$, where $j=j(n)$ might grow with $n$. The approach presented is not restricted to this specific $k$-tree model, but can also be applied to other evolving $k$-tree models.]]> 0 <![CDATA[The webpage of Herbert Wilf describes eight Unsolved Problems. Here, we completely resolve the third of these eight problems. The task seems innocent: find the first term of the asymptotic behavior of the coefficients of an ordinary generating function, whose coefficients naturally yield rational approximations to $\pi$. Upon closer examination, however, the analysis is fraught with difficulties. For instance, the function is the composition of three functions, but the innermost function has a non-zero constant term, so many standard techniques for analyzing function compositions will completely fail. Additionally, the signs of the coefficients are neither all positive, nor alternating in a regular manner. The generating function involves both a square root and an arctangent. The complex-valued square root and arctangent functions each rely on complex logarithms, which are multivalued and fundamentally depend on branch cuts. These multiple values and branch cuts make the function extremely tedious to visualize using Maple. We provide a complete asymptotic analysis of the coefficients of Wilf's generating function. The asymptotic expansion is naturally additive (not multiplicative); each term of the expansion contains oscillations, which we precisely characterize. The proofs rely on complex analysis, in particular, singularity analysis (which, in turn, rely on a Hankel contour and transfer theorems).]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2777 https://doi.org/10.46298/dmtcs.2777 Ward, Mark Daniel Ward, Mark Daniel <![CDATA[The webpage of Herbert Wilf describes eight Unsolved Problems. Here, we completely resolve the third of these eight problems. The task seems innocent: find the first term of the asymptotic behavior of the coefficients of an ordinary generating function, whose coefficients naturally yield rational approximations to $\pi$. Upon closer examination, however, the analysis is fraught with difficulties. For instance, the function is the composition of three functions, but the innermost function has a non-zero constant term, so many standard techniques for analyzing function compositions will completely fail. Additionally, the signs of the coefficients are neither all positive, nor alternating in a regular manner. The generating function involves both a square root and an arctangent. The complex-valued square root and arctangent functions each rely on complex logarithms, which are multivalued and fundamentally depend on branch cuts. These multiple values and branch cuts make the function extremely tedious to visualize using Maple. We provide a complete asymptotic analysis of the coefficients of Wilf's generating function. The asymptotic expansion is naturally additive (not multiplicative); each term of the expansion contains oscillations, which we precisely characterize. The proofs rely on complex analysis, in particular, singularity analysis (which, in turn, rely on a Hankel contour and transfer theorems).]]> 0 <![CDATA[The difference between ordinary tries and Patricia tries lies in the fact that all unary nodes are removed in the latter. Their average number is thus easily determined from earlier results on the size of tries/Patricia tries. In a well-known contention resolution algorithm, whose probabilistic model is essentially equivalent to tries, unary nodes correspond to repetitions, i.e., steps in the algorithm that do not resolve anything at all. In this paper, we take an individual's view on such repetitions: we consider the distribution of the number of repetitions a certain contender encounters in the course of the algorithm―-which is equivalent to the number of unary nodes on the path from the root to a random string in a trie. We encounter an example of a sequence of distributions that does not actually converge to a limit distribution, but rather oscillates around a (discrete) limit distribution.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2776 https://doi.org/10.46298/dmtcs.2776 Wagner, Stephan Wagner, Stephan <![CDATA[The difference between ordinary tries and Patricia tries lies in the fact that all unary nodes are removed in the latter. Their average number is thus easily determined from earlier results on the size of tries/Patricia tries. In a well-known contention resolution algorithm, whose probabilistic model is essentially equivalent to tries, unary nodes correspond to repetitions, i.e., steps in the algorithm that do not resolve anything at all. In this paper, we take an individual's view on such repetitions: we consider the distribution of the number of repetitions a certain contender encounters in the course of the algorithm―-which is equivalent to the number of unary nodes on the path from the root to a random string in a trie. We encounter an example of a sequence of distributions that does not actually converge to a limit distribution, but rather oscillates around a (discrete) limit distribution.]]> 0 <![CDATA[We consider a (random permutation model) binary search tree with $n$ nodes and give asymptotics on the $\log$ $\log$ scale for the height $H_n$ and saturation level $h_n$ of the tree as $n \to \infty$, both almost surely and in probability. We then consider the number $F_n$ of particles at level $H_n$ at time $n$, and show that $F_n$ is unbounded almost surely.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2775 https://doi.org/10.46298/dmtcs.2775 Roberts, Matthew, Roberts, Matthew, <![CDATA[We consider a (random permutation model) binary search tree with $n$ nodes and give asymptotics on the $\log$ $\log$ scale for the height $H_n$ and saturation level $h_n$ of the tree as $n \to \infty$, both almost surely and in probability. We then consider the number $F_n$ of particles at level $H_n$ at time $n$, and show that $F_n$ is unbounded almost surely.]]> 0 <![CDATA[We give a survey of a number of simple applications of renewal theory to problems on random strings, in particular to tries and Khodak and Tunstall codes.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2765 https://doi.org/10.46298/dmtcs.2765 Janson, Svante Janson, Svante <![CDATA[We give a survey of a number of simple applications of renewal theory to problems on random strings, in particular to tries and Khodak and Tunstall codes.]]> 0 <![CDATA[We study the random variable $X_n^k$, counting the number of vertices of degree $k$ in a randomly chosen $2$-connected graph of given families. We prove a central limit theorem for $X_n^k$ with expected value $\mathbb{E}X_n^k \sim \mu_kn$ and variance $\mathbb{V}X_n^k \sim \sigma_k^2n$, both asymptotically linear in $n$, for both rooted and unrooted unlabelled $2$-connected outerplanar or series-parallel graphs.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2773 https://doi.org/10.46298/dmtcs.2773 Kraus, Veronika Kraus, Veronika <![CDATA[We study the random variable $X_n^k$, counting the number of vertices of degree $k$ in a randomly chosen $2$-connected graph of given families. We prove a central limit theorem for $X_n^k$ with expected value $\mathbb{E}X_n^k \sim \mu_kn$ and variance $\mathbb{V}X_n^k \sim \sigma_k^2n$, both asymptotically linear in $n$, for both rooted and unrooted unlabelled $2$-connected outerplanar or series-parallel graphs.]]> 0 <![CDATA[Generalizing an idea used by Alonso to generate uniformly at random Motzkin words, we outline an approach to build efficient random generators using binomial distributions and rejection algorithms. As an application of this method, we present random generators, both efficient and easy to implement, for partial injections and colored unary-binary trees.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2772 https://doi.org/10.46298/dmtcs.2772 Gouyou-Beauchamps, Dominique Nicaud, Cyril Gouyou-Beauchamps, Dominique Nicaud, Cyril <![CDATA[Generalizing an idea used by Alonso to generate uniformly at random Motzkin words, we outline an approach to build efficient random generators using binomial distributions and rejection algorithms. As an application of this method, we present random generators, both efficient and easy to implement, for partial injections and colored unary-binary trees.]]> 0 <![CDATA[Using standard methods of analytic combinatorics we elaborate critical points (thresholds) of phase transitions from provability to unprovability of arithmetical well-partial-ordering assertions in several familiar theories occurring in the reverse mathematics program.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2771 https://doi.org/10.46298/dmtcs.2771 Gordeev, Lev Weiermann, Andreas Gordeev, Lev Weiermann, Andreas <![CDATA[Using standard methods of analytic combinatorics we elaborate critical points (thresholds) of phase transitions from provability to unprovability of arithmetical well-partial-ordering assertions in several familiar theories occurring in the reverse mathematics program.]]> 0 <![CDATA[The Bernoulli sieve is a version of the classical balls-in-boxes occupancy scheme, in which random frequencies of infinitely many boxes are produced by a multiplicative random walk, also known as the residual allocation model or stick-breaking. We give an overview of the limit theorems concerning the number of boxes occupied by some balls out of the first $n$ balls thrown, and present some new results concerning the number of empty boxes within the occupancy range.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2770 https://doi.org/10.46298/dmtcs.2770 Gnedin, Alexander Iksanov, Alexander Marynych, Alexander Gnedin, Alexander Iksanov, Alexander Marynych, Alexander <![CDATA[The Bernoulli sieve is a version of the classical balls-in-boxes occupancy scheme, in which random frequencies of infinitely many boxes are produced by a multiplicative random walk, also known as the residual allocation model or stick-breaking. We give an overview of the limit theorems concerning the number of boxes occupied by some balls out of the first $n$ balls thrown, and present some new results concerning the number of empty boxes within the occupancy range.]]> 0 <![CDATA[The analysis of probabilistic algorithms has proved to be very successful for finding asymptotic bounds on parameters of random regular graphs. In this paper, we show that similar ideas may be used to obtain deterministic bounds for one such parameter in the case of regular graphs with large girth. More precisely, we address the problem of finding a large induced forest in a graph $G$, by which we mean an acyclic induced subgraph of $G$ with a lot of vertices. For a fixed integer $r \geq 3$, we obtain new lower bounds on the size of a maximum induced forest in graphs with maximum degree $r$ and large girth. These bounds are derived from the solution of a system of differential equations that arises naturally in the analysis of an iterative probabilistic procedure to generate an induced forest in a graph. Numerical approximations suggest that these bounds improve substantially the best previous bounds. Moreover, they improve previous asymptotic lower bounds on the size of a maximum induced forest in a random regular graph.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2769 https://doi.org/10.46298/dmtcs.2769 Hoppen, Carlos Hoppen, Carlos <![CDATA[The analysis of probabilistic algorithms has proved to be very successful for finding asymptotic bounds on parameters of random regular graphs. In this paper, we show that similar ideas may be used to obtain deterministic bounds for one such parameter in the case of regular graphs with large girth. More precisely, we address the problem of finding a large induced forest in a graph $G$, by which we mean an acyclic induced subgraph of $G$ with a lot of vertices. For a fixed integer $r \geq 3$, we obtain new lower bounds on the size of a maximum induced forest in graphs with maximum degree $r$ and large girth. These bounds are derived from the solution of a system of differential equations that arises naturally in the analysis of an iterative probabilistic procedure to generate an induced forest in a graph. Numerical approximations suggest that these bounds improve substantially the best previous bounds. Moreover, they improve previous asymptotic lower bounds on the size of a maximum induced forest in a random regular graph.]]> 0 <![CDATA[The method of types is one of the most popular techniques in information theory and combinatorics. Two sequences of equal length have the same type if they have identical empirical distributions. In this paper, we focus on Markov types, that is, sequences generated by a Markov source (of order one). We note that sequences having the same Markov type share the same so called $\textit{balanced frequency matrix}$ that counts the number of distinct pairs of symbols. We enumerate the number of Markov types for sequences of length $n$ over an alphabet of size $m$. This turns out to coincide with the number of the balanced frequency matrices as well as with the number of special $\textit{linear diophantine equations}$, and also balanced directed multigraphs. For fixed $m$ we prove that the number of Markov types is asymptotically equal to $d(m) \frac{n^{m^{2-m}}}{(m^2-m)!}$, where $d(m)$ is a constant for which we give an integral representation. For $m \to \infty$ we conclude that asymptotically the number of types is equivalent to $\frac{\sqrt{2}m^{3m/2} e^{m^2}}{m^{2m^2} 2^m \pi^{m/2}} n^{m^2-m}$ provided that $m=o(n^{1/4})$ (however, our techniques work for $m=o(\sqrt{n})$). These findings are derived by analytical techniques ranging from multidimensional generating functions to the saddle point method.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2768 https://doi.org/10.46298/dmtcs.2768 Jacquet, Philippe Knessl, Charles Szpankowski, Wojciech Jacquet, Philippe Knessl, Charles Szpankowski, Wojciech <![CDATA[The method of types is one of the most popular techniques in information theory and combinatorics. Two sequences of equal length have the same type if they have identical empirical distributions. In this paper, we focus on Markov types, that is, sequences generated by a Markov source (of order one). We note that sequences having the same Markov type share the same so called $\textit{balanced frequency matrix}$ that counts the number of distinct pairs of symbols. We enumerate the number of Markov types for sequences of length $n$ over an alphabet of size $m$. This turns out to coincide with the number of the balanced frequency matrices as well as with the number of special $\textit{linear diophantine equations}$, and also balanced directed multigraphs. For fixed $m$ we prove that the number of Markov types is asymptotically equal to $d(m) \frac{n^{m^{2-m}}}{(m^2-m)!}$, where $d(m)$ is a constant for which we give an integral representation. For $m \to \infty$ we conclude that asymptotically the number of types is equivalent to $\frac{\sqrt{2}m^{3m/2} e^{m^2}}{m^{2m^2} 2^m \pi^{m/2}} n^{m^2-m}$ provided that $m=o(n^{1/4})$ (however, our techniques work for $m=o(\sqrt{n})$). These findings are derived by analytical techniques ranging from multidimensional generating functions to the saddle point method.]]> 0 <![CDATA[We propose a new strategy for universal best choice problem for partially ordered sets. We present its partial analysis which is sufficient to prove that the probability of success with this strategy is asymptotically strictly greater than 1/4, which is the value of the best universal strategy known so far.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2767 https://doi.org/10.46298/dmtcs.2767 Kozik, Jakub Kozik, Jakub <![CDATA[We propose a new strategy for universal best choice problem for partially ordered sets. We present its partial analysis which is sufficient to prove that the probability of success with this strategy is asymptotically strictly greater than 1/4, which is the value of the best universal strategy known so far.]]> 0 <![CDATA[We study the maximum of a Brownian motion with a parabolic drift; this is a random variable that often occurs as a limit of the maximum of discrete processes whose expectations have a maximum at an interior point. This has some applications in algorithmic and data structures analysis. We give series expansions and integral formulas for the distribution and the first two moments, together with numerical values to high precision.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2766 https://doi.org/10.46298/dmtcs.2766 Janson, Svante Louchard, Guy Martin-Löf, Anders Janson, Svante Louchard, Guy Martin-Löf, Anders <![CDATA[We study the maximum of a Brownian motion with a parabolic drift; this is a random variable that often occurs as a limit of the maximum of discrete processes whose expectations have a maximum at an interior point. This has some applications in algorithmic and data structures analysis. We give series expansions and integral formulas for the distribution and the first two moments, together with numerical values to high precision.]]> 0 <![CDATA[We study how hard is to determine some fundamental properties of dynamics of certain types of network automata. We address the computational complexity of determining how many different possible dynamic evolutions can arise from some structurally very simple, deterministic and sparsely connected network automata. In this as well as our prior, related work, we try to push the limits on the underlying simplicity of two structural aspects of such network automata: (i) the uniform sparseness of their topologies, and (ii) severely restricted local behaviors of the individual agents (that is, the local update rules of the network nodes). In this endeavor, we prove that counting the Fixed Point (FP) configurations and the predecessor and ancestor configurations in two classes of network automata, called Sequential and Synchronous Dynamical Systems (SDSs and SyDSs, respectively), are computationally intractable problems. Moreover, this intractability is shown to hold when each node in such a network is required to update according to (i) a monotone Boolean function, (ii) a symmetric Boolean function, or even (iii) a simple threshold function that is both monotone and symmetric. Furthermore, the hardness of the exact enumeration of FPs and other types of configurations of interest remains to hold even in some severely restricted cases with respect to both the network topology and the diversity (or lack thereof) of individual node's local update rules. Namely, we show that the counting problems of interest remain hard even when the nodes of an SDS or SyDS use at most two different update rules from a given restricted class, and, additionally, when the network topologies are constrained so that each node has only $c = O(1)$ neighbors for small values of constant $c$. Our results also have considerable implications for other discrete dynamical system models studied in applied mathematics, physics, biology and computer science, such as Hopfield networks and spin glasses. In particular, one corollary of our results is that determining the memory capacity of sparse discrete Hopfield networks (viewed as associative memories) remains computationally intractable even when the interconnection and dependence structure among the nodes of a Hopfield network is severely restricted.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2757 https://doi.org/10.46298/dmtcs.2757 Tošić, Predrag T. Tošić, Predrag T. <![CDATA[We study how hard is to determine some fundamental properties of dynamics of certain types of network automata. We address the computational complexity of determining how many different possible dynamic evolutions can arise from some structurally very simple, deterministic and sparsely connected network automata. In this as well as our prior, related work, we try to push the limits on the underlying simplicity of two structural aspects of such network automata: (i) the uniform sparseness of their topologies, and (ii) severely restricted local behaviors of the individual agents (that is, the local update rules of the network nodes). In this endeavor, we prove that counting the Fixed Point (FP) configurations and the predecessor and ancestor configurations in two classes of network automata, called Sequential and Synchronous Dynamical Systems (SDSs and SyDSs, respectively), are computationally intractable problems. Moreover, this intractability is shown to hold when each node in such a network is required to update according to (i) a monotone Boolean function, (ii) a symmetric Boolean function, or even (iii) a simple threshold function that is both monotone and symmetric. Furthermore, the hardness of the exact enumeration of FPs and other types of configurations of interest remains to hold even in some severely restricted cases with respect to both the network topology and the diversity (or lack thereof) of individual node's local update rules. Namely, we show that the counting problems of interest remain hard even when the nodes of an SDS or SyDS use at most two different update rules from a given restricted class, and, additionally, when the network topologies are constrained so that each node has only $c = O(1)$ neighbors for small values of constant $c$. Our results also have considerable implications for other discrete dynamical system models studied in applied mathematics, physics, biology and computer science, such as Hopfield networks and spin glasses. In particular, one corollary of our results is that determining the memory capacity of sparse discrete Hopfield networks (viewed as associative memories) remains computationally intractable even when the interconnection and dependence structure among the nodes of a Hopfield network is severely restricted.]]> 0 <![CDATA[We analyse the average number of buckets in a Linear Bucket tree created by $n$ points uniformly dispatched on an interval of length $y$. A new bucket is created when a point does not fall in an existing bucket. The bucket is the interval of length 2 centered on the point. We illustrate this concept by an interesting tale of how the moon's surface took on its present form. Thanks to an explicit Laplace transform of the Poissonized sequence, and the use of dePoissonization tools, we obtain the explicit asymptotic expansions of the average number of buckets in most of the asymptotic regimes relative to $n$ and $y$.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2764 https://doi.org/10.46298/dmtcs.2764 Jacquet, Philippe Muhlethaler, Paul Jacquet, Philippe Muhlethaler, Paul <![CDATA[We analyse the average number of buckets in a Linear Bucket tree created by $n$ points uniformly dispatched on an interval of length $y$. A new bucket is created when a point does not fall in an existing bucket. The bucket is the interval of length 2 centered on the point. We illustrate this concept by an interesting tale of how the moon's surface took on its present form. Thanks to an explicit Laplace transform of the Poissonized sequence, and the use of dePoissonization tools, we obtain the explicit asymptotic expansions of the average number of buckets in most of the asymptotic regimes relative to $n$ and $y$.]]> 0 <![CDATA[The biggest obstacle to the efficient discovery of conserved energy functions for cellular auotmata is the elimination of the trivial functions from the solution space. Once this is accomplished, the identification of nontrivial conserved functions can be accomplished computationally through appropriate linear algebra. As a means to this end, we introduce a general theory of trivial conserved functions. We consider the existence of nontrivial additive conserved energy functions ("nontrivials") for cellular automata in any number of dimensions, with any size of neighborhood, and with any number of cell states. We give the first known basis set for all trivial conserved functions in the general case, and use this to derive a number of optimizations for reducing time and memory for the discovery of nontrivials. We report that the Game of Life has no nontrivials with energy windows of size 13 or smaller. Other $2D$ automata, however, do have nontrivials. We give the complete list of those functions for binary outer-totalistic automata with energy windows of size 9 or smaller, and discuss patterns we have observed.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2763 https://doi.org/10.46298/dmtcs.2763 Baird, Leemon Fagin, Barry Baird, Leemon Fagin, Barry <![CDATA[The biggest obstacle to the efficient discovery of conserved energy functions for cellular auotmata is the elimination of the trivial functions from the solution space. Once this is accomplished, the identification of nontrivial conserved functions can be accomplished computationally through appropriate linear algebra. As a means to this end, we introduce a general theory of trivial conserved functions. We consider the existence of nontrivial additive conserved energy functions ("nontrivials") for cellular automata in any number of dimensions, with any size of neighborhood, and with any number of cell states. We give the first known basis set for all trivial conserved functions in the general case, and use this to derive a number of optimizations for reducing time and memory for the discovery of nontrivials. We report that the Game of Life has no nontrivials with energy windows of size 13 or smaller. Other $2D$ automata, however, do have nontrivials. We give the complete list of those functions for binary outer-totalistic automata with energy windows of size 9 or smaller, and discuss patterns we have observed.]]> 0 <![CDATA[Our work is set in the framework of complex dynamical systems and, more precisely, that of Boolean automata networks modeling regulation networks. We study how the choice of an update schedule impacts on the dynamics of such a network. To do this, we explain how studying the dynamics of any network updated with an arbitrary block-sequential update schedule can be reduced to the study of the dynamics of a different network updated in parallel. We give special attention to networks whose underlying structure is a circuit, that is, Boolean automata circuits. These particular and simple networks are known to serve as the "engines'' of the dynamics of arbitrary regulation networks containing them as sub-networks in that they are responsible for their variety of dynamical behaviours. We give both the number of attractors of period $p$, $\forall p\in \mathbb{N}$ and the total number of attractors in the dynamics of Boolean automata circuits updated with any block-sequential update schedule. We also detail the variety of dynamical behaviours that such networks may exhibit according to the update schedule.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2762 https://doi.org/10.46298/dmtcs.2762 Goles, Eric Noual, Mathilde Goles, Eric Noual, Mathilde <![CDATA[Our work is set in the framework of complex dynamical systems and, more precisely, that of Boolean automata networks modeling regulation networks. We study how the choice of an update schedule impacts on the dynamics of such a network. To do this, we explain how studying the dynamics of any network updated with an arbitrary block-sequential update schedule can be reduced to the study of the dynamics of a different network updated in parallel. We give special attention to networks whose underlying structure is a circuit, that is, Boolean automata circuits. These particular and simple networks are known to serve as the "engines'' of the dynamics of arbitrary regulation networks containing them as sub-networks in that they are responsible for their variety of dynamical behaviours. We give both the number of attractors of period $p$, $\forall p\in \mathbb{N}$ and the total number of attractors in the dynamics of Boolean automata circuits updated with any block-sequential update schedule. We also detail the variety of dynamical behaviours that such networks may exhibit according to the update schedule.]]> 0 <![CDATA[We present a method of solving of the probabilistic initial value problem for cellular automata (CA) using CA rule 172 as an example. For a disordered initial condition on an infinite lattice, we derive exact expressions for the density of ones at arbitrary time step. In order to do this, we analyze topological structure of preimage trees of finite strings of length 3. Level sets of these trees can be enumerated directly using classical combinatorial methods, yielding expressions for the number of $n$-step preimages of all strings of length 3, and, subsequently, probabilities of occurrence of these strings in a configuration obtained from the initial one after $n$ iterations of rule 172. The density of ones can be expressed in terms of Fibonacci numbers, while expressions for probabilities of other strings involve Lucas numbers. Applicability of this method to other CA rules is briefly discussed.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2761 https://doi.org/10.46298/dmtcs.2761 Fuks, Henryk Fuks, Henryk <![CDATA[We present a method of solving of the probabilistic initial value problem for cellular automata (CA) using CA rule 172 as an example. For a disordered initial condition on an infinite lattice, we derive exact expressions for the density of ones at arbitrary time step. In order to do this, we analyze topological structure of preimage trees of finite strings of length 3. Level sets of these trees can be enumerated directly using classical combinatorial methods, yielding expressions for the number of $n$-step preimages of all strings of length 3, and, subsequently, probabilities of occurrence of these strings in a configuration obtained from the initial one after $n$ iterations of rule 172. The density of ones can be expressed in terms of Fibonacci numbers, while expressions for probabilities of other strings involve Lucas numbers. Applicability of this method to other CA rules is briefly discussed.]]> 0 <![CDATA[Expander graphs are useful in the design and analysis of communication networks. Mukhopadhyay et al. introduced a method to generate a family of expander graphs based on nongroup two predecessor single attractor Cellular Automata(CA). In this paper we propose a method to generate a family of expander graphs based on 60/102 Null Boundary CA(NBCA) which is a group CA. The spectral gap generated by our method is maximal. Moreover, the spectral gap is larger than that of Mukhopadhyay et al.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2760 https://doi.org/10.46298/dmtcs.2760 Cho, Sung-Jin Choi, Un-Sook Kim, Han-Doo Hwang, Yoon-Hee Kim, Jin-Gyoung Cho, Sung-Jin Choi, Un-Sook Kim, Han-Doo Hwang, Yoon-Hee Kim, Jin-Gyoung <![CDATA[Expander graphs are useful in the design and analysis of communication networks. Mukhopadhyay et al. introduced a method to generate a family of expander graphs based on nongroup two predecessor single attractor Cellular Automata(CA). In this paper we propose a method to generate a family of expander graphs based on 60/102 Null Boundary CA(NBCA) which is a group CA. The spectral gap generated by our method is maximal. Moreover, the spectral gap is larger than that of Mukhopadhyay et al.]]> 0 <![CDATA[It is a well-known fact that the spacetime diagrams of some cellular automata have a fractal structure: for instance Pascal's triangle modulo $2$ generates a Sierpinski triangle. Explaining the fractal structure of the spacetime diagrams of cellular automata is a much explored topic, but virtually all of the results revolve around a special class of automata, whose main features include irreversibility, an alphabet with a ring structure and a rule respecting this structure, and a property known as being (weakly) $p$-Fermat. The class of automata that we study in this article fulfills none of these properties. Their cell structure is weaker and they are far from being $p$-Fermat, even weakly. However, they do produce fractal spacetime diagrams, and we will explain why and how. These automata emerge naturally from the field of quantum cellular automata, as they include the classical equivalent of the Clifford quantum cellular automata, which have been studied by the quantum community for several reasons. They are a basic building block of a universal model of quantum computation, and they can be used to generate highly entangled states, which are a primary resource for measurement-based models of quantum computing.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2759 https://doi.org/10.46298/dmtcs.2759 Gütschow, Johannes Nesme, Vincent Werner, Reinhard F. Gütschow, Johannes Nesme, Vincent Werner, Reinhard F. <![CDATA[It is a well-known fact that the spacetime diagrams of some cellular automata have a fractal structure: for instance Pascal's triangle modulo $2$ generates a Sierpinski triangle. Explaining the fractal structure of the spacetime diagrams of cellular automata is a much explored topic, but virtually all of the results revolve around a special class of automata, whose main features include irreversibility, an alphabet with a ring structure and a rule respecting this structure, and a property known as being (weakly) $p$-Fermat. The class of automata that we study in this article fulfills none of these properties. Their cell structure is weaker and they are far from being $p$-Fermat, even weakly. However, they do produce fractal spacetime diagrams, and we will explain why and how. These automata emerge naturally from the field of quantum cellular automata, as they include the classical equivalent of the Clifford quantum cellular automata, which have been studied by the quantum community for several reasons. They are a basic building block of a universal model of quantum computation, and they can be used to generate highly entangled states, which are a primary resource for measurement-based models of quantum computing.]]> 0 <![CDATA[We investigate the descriptional complexity of basic operations on real-time one-way cellular automata with an unbounded as well well as a fixed number of cells. The size of the automata is measured by their number of states. Most of the bounds shown are tight in the order of magnitude, that is, the sizes resulting from the effective constructions given are optimal with respect to worst case complexity. Conversely, these bounds also show the maximal savings of size that can be achieved when a given minimal real-time OCA is decomposed into smaller ones with respect to a given operation. From this point of view the natural problem of whether a decomposition can algorithmically be solved is studied. It turns out that all decomposition problems considered are algorithmically unsolvable. Therefore, a very restricted cellular model is studied in the second part of the paper, namely, real-time one-way cellular automata with a fixed number of cells. These devices are known to capture the regular languages and, thus, all the problems being undecidable for general one-way cellular automata become decidable. It is shown that these decision problems are $\textsf{NLOGSPACE}$-complete and thus share the attractive computational complexity of deterministic finite automata. Furthermore, the state complexity of basic operations for these devices is studied and upper and lower bounds are given.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2758 https://doi.org/10.46298/dmtcs.2758 Kutrib, Martin Lefèvre, Jonas Malcher, Andreas Kutrib, Martin Lefèvre, Jonas Malcher, Andreas <![CDATA[We investigate the descriptional complexity of basic operations on real-time one-way cellular automata with an unbounded as well well as a fixed number of cells. The size of the automata is measured by their number of states. Most of the bounds shown are tight in the order of magnitude, that is, the sizes resulting from the effective constructions given are optimal with respect to worst case complexity. Conversely, these bounds also show the maximal savings of size that can be achieved when a given minimal real-time OCA is decomposed into smaller ones with respect to a given operation. From this point of view the natural problem of whether a decomposition can algorithmically be solved is studied. It turns out that all decomposition problems considered are algorithmically unsolvable. Therefore, a very restricted cellular model is studied in the second part of the paper, namely, real-time one-way cellular automata with a fixed number of cells. These devices are known to capture the regular languages and, thus, all the problems being undecidable for general one-way cellular automata become decidable. It is shown that these decision problems are $\textsf{NLOGSPACE}$-complete and thus share the attractive computational complexity of deterministic finite automata. Furthermore, the state complexity of basic operations for these devices is studied and upper and lower bounds are given.]]> 0 <![CDATA[Motivated by the classical Frobenius problem, we introduce the Frobenius poset on the integers $\mathbb{Z}$, that is, for a sub-semigroup $\Lambda$ of the non-negative integers $(\mathbb{N},+)$, we define the order by $n \leq_{\Lambda} m$ if $m-n \in \Lambda$. When $\Lambda$ is generated by two relatively prime integers $a$ and $b$, we show that the order complex of an interval in the Frobenius poset is either contractible or homotopy equivalent to a sphere. We also show that when $\Lambda$ is generated by the integers $\{a,a+d,a+2d,\ldots,a+(a-1)d\}$, the order complex is homotopy equivalent to a wedge of spheres.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2816 https://doi.org/10.46298/dmtcs.2816 Clark, Eric Ehrenborg, Richard Clark, Eric Ehrenborg, Richard <![CDATA[Motivated by the classical Frobenius problem, we introduce the Frobenius poset on the integers $\mathbb{Z}$, that is, for a sub-semigroup $\Lambda$ of the non-negative integers $(\mathbb{N},+)$, we define the order by $n \leq_{\Lambda} m$ if $m-n \in \Lambda$. When $\Lambda$ is generated by two relatively prime integers $a$ and $b$, we show that the order complex of an interval in the Frobenius poset is either contractible or homotopy equivalent to a sphere. We also show that when $\Lambda$ is generated by the integers $\{a,a+d,a+2d,\ldots,a+(a-1)d\}$, the order complex is homotopy equivalent to a wedge of spheres.]]> 0 <![CDATA[The classical quadratically convergent Newton-Raphson iterative scheme for successive approximations of a root of an equation $f(t)=0$ has been extended in various ways by different authors, going from cubical convergence to convergence of arbitrary orders. We introduce two such extensions, using appropriate differential operators as well as combinatorial arguments. We conclude with some applications including special series expansions for functions of the root and enumeration of classes of tree-like structures according to their number of leaves.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2824 https://doi.org/10.46298/dmtcs.2824 Labelle, Gilbert Labelle, Gilbert <![CDATA[The classical quadratically convergent Newton-Raphson iterative scheme for successive approximations of a root of an equation $f(t)=0$ has been extended in various ways by different authors, going from cubical convergence to convergence of arbitrary orders. We introduce two such extensions, using appropriate differential operators as well as combinatorial arguments. We conclude with some applications including special series expansions for functions of the root and enumeration of classes of tree-like structures according to their number of leaves.]]> 0 <![CDATA[We present a simple technique to compute moments of derivatives of unitary characteristic polynomials. The first part of the technique relies on an idea of Bump and Gamburd: it uses orthonormality of Schur functions over unitary groups to compute matrix averages of characteristic polynomials. In order to consider derivatives of those polynomials, we here need the added strength of the Generalized Binomial Theorem of Okounkov and Olshanski. This result is very natural as it provides coefficients for the Taylor expansions of Schur functions, in terms of shifted Schur functions. The answer is finally given as a sum over partitions of functions of the contents. One can also obtain alternative expressions involving hypergeometric functions of matrix arguments.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2823 https://doi.org/10.46298/dmtcs.2823 Dehaye, Paul-Olivier Dehaye, Paul-Olivier <![CDATA[We present a simple technique to compute moments of derivatives of unitary characteristic polynomials. The first part of the technique relies on an idea of Bump and Gamburd: it uses orthonormality of Schur functions over unitary groups to compute matrix averages of characteristic polynomials. In order to consider derivatives of those polynomials, we here need the added strength of the Generalized Binomial Theorem of Okounkov and Olshanski. This result is very natural as it provides coefficients for the Taylor expansions of Schur functions, in terms of shifted Schur functions. The answer is finally given as a sum over partitions of functions of the contents. One can also obtain alternative expressions involving hypergeometric functions of matrix arguments.]]> 0 <![CDATA[We take a geometric point of view on the recent result by Brenti and Welker, who showed that the roots of the $f$-polynomials of successive barycentric subdivisions of a finite simplicial complex $X$ converge to fixed values depending only on the dimension of $X$. We show that these numbers are roots of a certain polynomial whose coefficients can be computed explicitly. We observe and prove an interesting symmetry of these roots about the real number $-2$. This symmetry can be seen via a nice realization of barycentric subdivision as a simple map on formal power series. We then examine how such a symmetry extends to more general types of subdivisions. The generalization is formulated in terms of an operator on the (formal) ring on the set of simplices of the complex.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2822 https://doi.org/10.46298/dmtcs.2822 Delucchi, Emanuele Pixton, Aaron Sabalka, Lucas Delucchi, Emanuele Pixton, Aaron Sabalka, Lucas <![CDATA[We take a geometric point of view on the recent result by Brenti and Welker, who showed that the roots of the $f$-polynomials of successive barycentric subdivisions of a finite simplicial complex $X$ converge to fixed values depending only on the dimension of $X$. We show that these numbers are roots of a certain polynomial whose coefficients can be computed explicitly. We observe and prove an interesting symmetry of these roots about the real number $-2$. This symmetry can be seen via a nice realization of barycentric subdivision as a simple map on formal power series. We then examine how such a symmetry extends to more general types of subdivisions. The generalization is formulated in terms of an operator on the (formal) ring on the set of simplices of the complex.]]> 0 <![CDATA[Building on the work of P.N. Norton, we give combinatorial formulae for two maximal decompositions of the identity into orthogonal idempotents in the $0$-Hecke algebra of the symmetric group, $\mathbb{C}H_0(S_N)$. This construction is compatible with the branching from $H_0(S_{N-1})$ to $H_0(S_N)$.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2821 https://doi.org/10.46298/dmtcs.2821 Denton, Tom Denton, Tom <![CDATA[Building on the work of P.N. Norton, we give combinatorial formulae for two maximal decompositions of the identity into orthogonal idempotents in the $0$-Hecke algebra of the symmetric group, $\mathbb{C}H_0(S_N)$. This construction is compatible with the branching from $H_0(S_{N-1})$ to $H_0(S_N)$.]]> 0 <![CDATA[We construct and study an embedded weighted balanced graph in $\mathbb{R}^{n+1}$ parametrized by a strictly increasing sequence of $n$ coprime numbers $\{ i_1, \ldots, i_n\}$, called the $\textit{tropical secant surface graph}$. We identify it with the tropicalization of a surface in $\mathbb{C}^{n+1}$ parametrized by binomials. Using this graph, we construct the tropicalization of the first secant variety of a monomial projective curve with exponent vector $(0, i_1, \ldots, i_n)$, which can be described by a balanced graph called the $\textit{tropical secant graph}$. The combinatorics involved in computing the degree of this classical secant variety is non-trivial. One earlier approach to this is due to K. Ranestad. Using techniques from tropical geometry, we give algorithms to effectively compute this degree (as well as its multidegree) and the Newton polytope of the first secant variety of any given monomial curve in $\mathbb{P}^4$.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2820 https://doi.org/10.46298/dmtcs.2820 Cueto, María Angélica Lin, Shaowei Cueto, María Angélica Lin, Shaowei <![CDATA[We construct and study an embedded weighted balanced graph in $\mathbb{R}^{n+1}$ parametrized by a strictly increasing sequence of $n$ coprime numbers $\{ i_1, \ldots, i_n\}$, called the $\textit{tropical secant surface graph}$. We identify it with the tropicalization of a surface in $\mathbb{C}^{n+1}$ parametrized by binomials. Using this graph, we construct the tropicalization of the first secant variety of a monomial projective curve with exponent vector $(0, i_1, \ldots, i_n)$, which can be described by a balanced graph called the $\textit{tropical secant graph}$. The combinatorics involved in computing the degree of this classical secant variety is non-trivial. One earlier approach to this is due to K. Ranestad. Using techniques from tropical geometry, we give algorithms to effectively compute this degree (as well as its multidegree) and the Newton polytope of the first secant variety of any given monomial curve in $\mathbb{P}^4$.]]> 0 <![CDATA[In this paper we study pattern avoidance for affine permutations. In particular, we show that for a given pattern $p$, there are only finitely many affine permutations in $\widetilde{S}_n$ that avoid $p$ if and only if $p$ avoids the pattern $321$. We then count the number of affine permutations that avoid a given pattern $p$ for each $p$ in $S_3$, as well as give some conjectures for the patterns in $S_4$. This paper is just an outline; the full version will appear elsewhere.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2819 https://doi.org/10.46298/dmtcs.2819 Crites, Andrew Crites, Andrew <![CDATA[In this paper we study pattern avoidance for affine permutations. In particular, we show that for a given pattern $p$, there are only finitely many affine permutations in $\widetilde{S}_n$ that avoid $p$ if and only if $p$ avoids the pattern $321$. We then count the number of affine permutations that avoid a given pattern $p$ for each $p$ in $S_3$, as well as give some conjectures for the patterns in $S_4$. This paper is just an outline; the full version will appear elsewhere.]]> 0 <![CDATA[Motivated by the concept of partial words, we introduce an analogous concept of partial permutations. A $\textit{partial permutation of length n with k holes}$ is a sequence of symbols $\pi = \pi_1 \pi_2 \cdots \pi_n$ in which each of the symbols from the set $\{1,2,\ldots,n-k\}$ appears exactly once, while the remaining $k$ symbols of $\pi$ are "holes''. We introduce pattern-avoidance in partial permutations and prove that most of the previous results on Wilf equivalence of permutation patterns can be extended to partial permutations with an arbitrary number of holes. We also show that Baxter permutations of a given length $k$ correspond to a Wilf-type equivalence class with respect to partial permutations with $(k-2)$ holes. Lastly, we enumerate the partial permutations of length $n$ with $k$ holes avoiding a given pattern of length at most four, for each $n \geq k \geq 1$.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2818 https://doi.org/10.46298/dmtcs.2818 Claesson, Anders Jelínek, Vít Jelínková, Eva Kitaev, Sergey Claesson, Anders Jelínek, Vít Jelínková, Eva Kitaev, Sergey <![CDATA[Motivated by the concept of partial words, we introduce an analogous concept of partial permutations. A $\textit{partial permutation of length n with k holes}$ is a sequence of symbols $\pi = \pi_1 \pi_2 \cdots \pi_n$ in which each of the symbols from the set $\{1,2,\ldots,n-k\}$ appears exactly once, while the remaining $k$ symbols of $\pi$ are "holes''. We introduce pattern-avoidance in partial permutations and prove that most of the previous results on Wilf equivalence of permutation patterns can be extended to partial permutations with an arbitrary number of holes. We also show that Baxter permutations of a given length $k$ correspond to a Wilf-type equivalence class with respect to partial permutations with $(k-2)$ holes. Lastly, we enumerate the partial permutations of length $n$ with $k$ holes avoiding a given pattern of length at most four, for each $n \geq k \geq 1$.]]> 0 <![CDATA[We show that there are $n!$ matchings on $2n$ points without, so called, left (neighbor) nestings. We also define a set of naturally labelled $(2+2)$-free posets, and show that there are $n!$ such posets on $n$ elements. Our work was inspired by Bousquet-Mélou, Claesson, Dukes and Kitaev [J. Combin. Theory Ser. A. 117 (2010) 884―909]. They gave bijections between four classes of combinatorial objects: matchings with no neighbor nestings (due to Stoimenow), unlabelled $(2+2)$-free posets, permutations avoiding a specific pattern, and so called ascent sequences. We believe that certain statistics on our matchings and posets could generalize the work of Bousquet-Mélou et al. and we make a conjecture to that effect. We also identify natural subsets of matchings and posets that are equinumerous to the class of unlabeled $(2+2)$-free posets. We give bijections that show the equivalence of (neighbor) restrictions on nesting arcs with (neighbor) restrictions on crossing arcs. These bijections are thought to be of independent interest. One of the bijections maps via certain upper-triangular integer matrices that have recently been studied by Dukes and Parviainen [Electron. J. Combin. 17 (2010) #R53].]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2817 https://doi.org/10.46298/dmtcs.2817 Claesson, Anders Linusson, Svante Claesson, Anders Linusson, Svante <![CDATA[We show that there are $n!$ matchings on $2n$ points without, so called, left (neighbor) nestings. We also define a set of naturally labelled $(2+2)$-free posets, and show that there are $n!$ such posets on $n$ elements. Our work was inspired by Bousquet-Mélou, Claesson, Dukes and Kitaev [J. Combin. Theory Ser. A. 117 (2010) 884―909]. They gave bijections between four classes of combinatorial objects: matchings with no neighbor nestings (due to Stoimenow), unlabelled $(2+2)$-free posets, permutations avoiding a specific pattern, and so called ascent sequences. We believe that certain statistics on our matchings and posets could generalize the work of Bousquet-Mélou et al. and we make a conjecture to that effect. We also identify natural subsets of matchings and posets that are equinumerous to the class of unlabeled $(2+2)$-free posets. We give bijections that show the equivalence of (neighbor) restrictions on nesting arcs with (neighbor) restrictions on crossing arcs. These bijections are thought to be of independent interest. One of the bijections maps via certain upper-triangular integer matrices that have recently been studied by Dukes and Parviainen [Electron. J. Combin. 17 (2010) #R53].]]> 0 <![CDATA[We introduce a new family of polyominos that are inscribed in a rectangle of given size for which we establish a number of exact formulas and generating functions. In particular, we study polyominos inscribed in a rectangle with minimum area and minimum area plus one. These results are then used for the enumeration of lattice trees inscribed in a rectangle with minimum area plus one.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2808 https://doi.org/10.46298/dmtcs.2808 Goupil, Alain Cloutier, Hugo Nouboud, Fathallah Goupil, Alain Cloutier, Hugo Nouboud, Fathallah <![CDATA[We introduce a new family of polyominos that are inscribed in a rectangle of given size for which we establish a number of exact formulas and generating functions. In particular, we study polyominos inscribed in a rectangle with minimum area and minimum area plus one. These results are then used for the enumeration of lattice trees inscribed in a rectangle with minimum area plus one.]]> 0 <![CDATA[We look at the number of permutations $\beta$ of $[N]$ with $m$ cycles such that $(1 2 \ldots N) \beta^{-1}$ is a long cycle. These numbers appear as coefficients of linear monomials in Kerov's and Stanley's character polynomials. D. Zagier, using algebraic methods, found an unexpected connection with Stirling numbers of size $N+1$. We present the first combinatorial proof of his result, introducing a new bijection between partitioned maps and thorn trees. Moreover, we obtain a finer result, which takes the type of the permutations into account.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2815 https://doi.org/10.46298/dmtcs.2815 Féray, Valentin Vassilieva, Ekaterina A. Féray, Valentin Vassilieva, Ekaterina A. <![CDATA[We look at the number of permutations $\beta$ of $[N]$ with $m$ cycles such that $(1 2 \ldots N) \beta^{-1}$ is a long cycle. These numbers appear as coefficients of linear monomials in Kerov's and Stanley's character polynomials. D. Zagier, using algebraic methods, found an unexpected connection with Stirling numbers of size $N+1$. We present the first combinatorial proof of his result, introducing a new bijection between partitioned maps and thorn trees. Moreover, we obtain a finer result, which takes the type of the permutations into account.]]> 0 <![CDATA[We show that the set of balanced binary trees is closed by interval in the Tamari lattice. We establish that the intervals $[T_0, T_1]$ where $T_0$ and $T_1$ are balanced trees are isomorphic as posets to a hypercube. We introduce tree patterns and synchronous grammars to get a functional equation of the generating series enumerating balanced tree intervals.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2814 https://doi.org/10.46298/dmtcs.2814 Giraudo, Samuele Giraudo, Samuele <![CDATA[We show that the set of balanced binary trees is closed by interval in the Tamari lattice. We establish that the intervals $[T_0, T_1]$ where $T_0$ and $T_1$ are balanced trees are isomorphic as posets to a hypercube. We introduce tree patterns and synchronous grammars to get a functional equation of the generating series enumerating balanced tree intervals.]]> 0 <![CDATA[We give combinatorial proofs of the formulas for the number of multichains in the $k-divisible$ noncrossing partitions of classical types with certain conditions on the rank and the block size due to Krattenthaler and Müller. We also prove Armstrong's conjecture on the zeta polynomial of the poset of k-divisible noncrossing partitions of type A invariant under the 180° rotation in the cyclic representation.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2813 https://doi.org/10.46298/dmtcs.2813 Kim, Jang Soo Kim, Jang Soo <![CDATA[We give combinatorial proofs of the formulas for the number of multichains in the $k-divisible$ noncrossing partitions of classical types with certain conditions on the rank and the block size due to Krattenthaler and Müller. We also prove Armstrong's conjecture on the zeta polynomial of the poset of k-divisible noncrossing partitions of type A invariant under the 180° rotation in the cyclic representation.]]> 0 <![CDATA[A poset is said to be (2+2)-free if it does not contain an induced subposet that is isomorphic to 2+2, the union of two disjoint 2-element chains. In a recent paper, Bousquet-Mélou et al. found, using so called ascent sequences, the generating function for the number of (2+2)-free posets: $P(t)=∑_n≥ 0 ∏_i=1^n ( 1-(1-t)^i)$. We extend this result by finding the generating function for (2+2)-free posets when four statistics are taken into account, one of which is the number of minimal elements in a poset. We also show that in a special case when only minimal elements are of interest, our rather involved generating function can be rewritten in the form $P(t,z)=∑_n,k ≥0 p_n,k t^n z^k = 1+ ∑_n ≥0\frac{zt}{(1-zt)^n+1}∏_i=1^n (1-(1-t)^i)$ where $p_n,k$ equals the number of (2+2)-free posets of size $n$ with $k$ minimal elements.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2812 https://doi.org/10.46298/dmtcs.2812 Kitaev, Sergey Remmel, Jeffrey Kitaev, Sergey Remmel, Jeffrey <![CDATA[A poset is said to be (2+2)-free if it does not contain an induced subposet that is isomorphic to 2+2, the union of two disjoint 2-element chains. In a recent paper, Bousquet-Mélou et al. found, using so called ascent sequences, the generating function for the number of (2+2)-free posets: $P(t)=∑_n≥ 0 ∏_i=1^n ( 1-(1-t)^i)$. We extend this result by finding the generating function for (2+2)-free posets when four statistics are taken into account, one of which is the number of minimal elements in a poset. We also show that in a special case when only minimal elements are of interest, our rather involved generating function can be rewritten in the form $P(t,z)=∑_n,k ≥0 p_n,k t^n z^k = 1+ ∑_n ≥0\frac{zt}{(1-zt)^n+1}∏_i=1^n (1-(1-t)^i)$ where $p_n,k$ equals the number of (2+2)-free posets of size $n$ with $k$ minimal elements.]]> 0 <![CDATA[We prove a closed character formula for the symmetric powers $S^N V(λ )$ of a fixed irreducible representation $V(λ )$ of a complex semi-simple Lie algebra $\mathfrak{g}$ by means of partial fraction decomposition. The formula involves rational functions in rank of $\mathfrak{g}$ many variables which are easier to determine than the weight multiplicities of $S^N V(λ )$ themselves. We compute those rational functions in some interesting cases. Furthermore, we introduce a residue-type generating function for the weight multiplicities of $S^N V(λ )$ and explain the connections between our character formula, vector partition functions and iterated partial fraction decomposition.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2811 https://doi.org/10.46298/dmtcs.2811 Kousidis, Stavros Kousidis, Stavros <![CDATA[We prove a closed character formula for the symmetric powers $S^N V(λ )$ of a fixed irreducible representation $V(λ )$ of a complex semi-simple Lie algebra $\mathfrak{g}$ by means of partial fraction decomposition. The formula involves rational functions in rank of $\mathfrak{g}$ many variables which are easier to determine than the weight multiplicities of $S^N V(λ )$ themselves. We compute those rational functions in some interesting cases. Furthermore, we introduce a residue-type generating function for the weight multiplicities of $S^N V(λ )$ and explain the connections between our character formula, vector partition functions and iterated partial fraction decomposition.]]> 0 <![CDATA[We provide formulas for the Weyl-Kac denominator and superdenominator of a basic classical Lie superalgebra for a distinguished set of positive roots. \par]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2810 https://doi.org/10.46298/dmtcs.2810 Kac, Victor Möseneder Frajria, Pierluigi Papi, Paolo Kac, Victor Möseneder Frajria, Pierluigi Papi, Paolo <![CDATA[We provide formulas for the Weyl-Kac denominator and superdenominator of a basic classical Lie superalgebra for a distinguished set of positive roots. \par]]> 0 <![CDATA[We provide the unique affine crystal structure for type $E_6^{(1)}$ Kirillov―Reshetikhin crystals corresponding to the multiples of fundamental weights $s\Lambda _1, s\Lambda _2$, and $s\Lambda _6$ for all $s≥ 1$ (in Bourbaki's labeling of the Dynkin nodes, where 2 is the adjoint node). Our methods introduce a generalized tableaux model for classical highest weight crystals of type $E$ and use the order three automorphism of the affine $E_6^{(1)}$ Dynkin diagram. In addition, we provide a conjecture for the affine crystal structure of type $E_7^{(1)}$ Kirillov―Reshetikhin crystals corresponding to the adjoint node.]]> Thu, 31 Dec 2009 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2809 https://doi.org/10.46298/dmtcs.2809 Jones, Brant Schilling, Anne Jones, Brant Schilling, Anne <![CDATA[We provide the unique affine crystal structure for type $E_6^{(1)}$ Kirillov―Reshetikhin crystals corresponding to the multiples of fundamental weights $s\Lambda _1, s\Lambda _2$, and $s\Lambda _6$ for all $s≥ 1$ (in Bourbaki's labeling of the Dynkin nodes, where 2 is the adjoint node). Our methods introduce a generalized tableaux model for classical highest weight crystals of type $E$ and use the order three automorphism of the affine $E_6^{(1)}$ Dynkin diagram. In addition, we provide a conjecture for the affine crystal structure of type $E_7^{(1)}$ Kirillov―Reshetikhin crystals corresponding to the adjoint node.]]> 0 <![CDATA[Several matrices can be associated to a graph such as the adjacency matrix or the Laplacian matrix. The spectrum of these matrices gives some informations about the structure of the graph and the question ''Which graphs are determined by their spectrum?'' remains a difficult problem in algebraic graph theory. In this article we enlarge the known families of graphs determined by their spectrum by considering some unicyclic graphs. An odd (resp. even) sun is a graph obtained by appending a pendant vertex to each vertex of an odd (resp. even) cycle. A broken sun is a graph obtained by deleting pendant vertices of a sun. In this paper we prove that a sun is determined by its Laplacian spectrum, an odd sun is determined by its adjacency spectrum (counter-examples are given for even suns) and we give some spectral characterizations of broken suns.]]> Tue, 01 Dec 2009 07:00:00 +0000 https://doi.org/10.46298/dmtcs.456 https://doi.org/10.46298/dmtcs.456 Boulet, Romain Boulet, Romain <![CDATA[Several matrices can be associated to a graph such as the adjacency matrix or the Laplacian matrix. The spectrum of these matrices gives some informations about the structure of the graph and the question ''Which graphs are determined by their spectrum?'' remains a difficult problem in algebraic graph theory. In this article we enlarge the known families of graphs determined by their spectrum by considering some unicyclic graphs. An odd (resp. even) sun is a graph obtained by appending a pendant vertex to each vertex of an odd (resp. even) cycle. A broken sun is a graph obtained by deleting pendant vertices of a sun. In this paper we prove that a sun is determined by its Laplacian spectrum, an odd sun is determined by its adjacency spectrum (counter-examples are given for even suns) and we give some spectral characterizations of broken suns.]]> 0 <![CDATA[We give a new expression for the expected number of inversions in the product of n random adjacent transpositions in the symmetric group S_{m+1}. We then derive from this expression the asymptotic behaviour of this number when n scales with m in various ways. Our starting point is an equivalence, due to Eriksson et al., with a problem of weighted walks confined to a triangular area of the plane.]]> Mon, 31 Aug 2009 06:00:00 +0000 https://doi.org/10.46298/dmtcs.478 https://doi.org/10.46298/dmtcs.478 Bousquet-Mélou, Mireille Bousquet-Mélou, Mireille <![CDATA[We give a new expression for the expected number of inversions in the product of n random adjacent transpositions in the symmetric group S_{m+1}. We then derive from this expression the asymptotic behaviour of this number when n scales with m in various ways. Our starting point is an equivalence, due to Eriksson et al., with a problem of weighted walks confined to a triangular area of the plane.]]> 0 <![CDATA[A partition of $x > 0$ of the form $x = \sum_i 2^{a_i}3^{b_i}$ with distinct parts is called a double-base expansion of $x$. Such a representation can be obtained using a greedy approach, assuming one can efficiently compute the largest \mbox{$\{2,3\}$-integer}, i.e., a number of the form $2^a3^b$, less than or equal to $x$. In order to solve this problem, we propose an algorithm based on continued fractions in the vein of the Ostrowski number system, we prove its correctness and we analyse its complexity. In a second part, we present some experimental results on the length of double-base expansions when only a few iterations of our algorithm are performed.]]> Sun, 01 Mar 2009 07:00:00 +0000 https://doi.org/10.46298/dmtcs.450 https://doi.org/10.46298/dmtcs.450 Berthe, Valerie Imbert, Laurent Berthe, Valerie Imbert, Laurent <![CDATA[A partition of $x > 0$ of the form $x = \sum_i 2^{a_i}3^{b_i}$ with distinct parts is called a double-base expansion of $x$. Such a representation can be obtained using a greedy approach, assuming one can efficiently compute the largest \mbox{$\{2,3\}$-integer}, i.e., a number of the form $2^a3^b$, less than or equal to $x$. In order to solve this problem, we propose an algorithm based on continued fractions in the vein of the Ostrowski number system, we prove its correctness and we analyse its complexity. In a second part, we present some experimental results on the length of double-base expansions when only a few iterations of our algorithm are performed.]]> 0 <![CDATA[Two-dimensional structures of various kinds can be viewed as generalizations of words. Codicity verification and the defect effect, important properties related to word codes, are studied also in this context. Unfortunately, both are lost in the case of two common structures, polyominoes and figures. We consider directed figures defined as labelled polyominoes with designated start and end points, equipped with catenation operation that uses a merging function to resolve possible conflicts. We prove that in this setting verification whether a given finite set of directed figures is a code is decidable and we give a constructive algorithm. We also clarify the status of the defect effect for directed figures.]]> Thu, 01 Jan 2009 07:00:00 +0000 https://doi.org/10.46298/dmtcs.455 https://doi.org/10.46298/dmtcs.455 Kolarz, Michal Moczurad, Wlodzimierz Kolarz, Michal Moczurad, Wlodzimierz <![CDATA[Two-dimensional structures of various kinds can be viewed as generalizations of words. Codicity verification and the defect effect, important properties related to word codes, are studied also in this context. Unfortunately, both are lost in the case of two common structures, polyominoes and figures. We consider directed figures defined as labelled polyominoes with designated start and end points, equipped with catenation operation that uses a merging function to resolve possible conflicts. We prove that in this setting verification whether a given finite set of directed figures is a code is decidable and we give a constructive algorithm. We also clarify the status of the defect effect for directed figures.]]> 0 <![CDATA[We study the Fox coloring invariants of rational knots. We express the propagation of the colors down the twists of these knots and ultimately the determinant of them with the help of finite increasing sequences whose terms of even order are even and whose terms of odd order are odd.]]> Thu, 01 Jan 2009 07:00:00 +0000 https://doi.org/10.46298/dmtcs.454 https://doi.org/10.46298/dmtcs.454 Kauffman, Louis H. Lopes, Pedro Kauffman, Louis H. Lopes, Pedro <![CDATA[We study the Fox coloring invariants of rational knots. We express the propagation of the colors down the twists of these knots and ultimately the determinant of them with the help of finite increasing sequences whose terms of even order are even and whose terms of odd order are odd.]]> 0 <![CDATA[A clique-transversal set in a graph is a subset of the vertices that meets all maximal complete subgraphs on at least two vertices. We prove that every connected graph of order n and maximum degree three has a clique-transversal set of size left perpendicular19n/30 + 2/15right perpendicular. This bound is tight, since 19n/30 - 1/15 is a lower bound for infinitely many values of n. We also prove that the vertex set of any connected claw-free graph of maximum degree at most four, other than an odd cycle longer than three, can be partitioned into two clique-transversal sets. The proofs of both results yield polynomial-time algorithms that find corresponding solutions.]]> Thu, 01 Jan 2009 07:00:00 +0000 https://doi.org/10.46298/dmtcs.453 https://doi.org/10.46298/dmtcs.453 Bacsó, Gábor Tuza, Zsolt Bacsó, Gábor Tuza, Zsolt <![CDATA[A clique-transversal set in a graph is a subset of the vertices that meets all maximal complete subgraphs on at least two vertices. We prove that every connected graph of order n and maximum degree three has a clique-transversal set of size left perpendicular19n/30 + 2/15right perpendicular. This bound is tight, since 19n/30 - 1/15 is a lower bound for infinitely many values of n. We also prove that the vertex set of any connected claw-free graph of maximum degree at most four, other than an odd cycle longer than three, can be partitioned into two clique-transversal sets. The proofs of both results yield polynomial-time algorithms that find corresponding solutions.]]> 0 <![CDATA[The following problem was solved by Woodall in 1972: for any pair of nonnegative integers n and k < n/2 - 1 find the minimum integer g(n, k) such that every graph with n vertices and at least g(n, k) edges contains a cycle of length n - k. Woodall proved even more: the size g(n, k), in fact, guarantees the existence of cycles C, for all 3 <= p <= n - k. In the paper an analogous problem for bipartite graphs is considered. It is proved that every bipartite graph with color classes of cardinalities m and n, m <= n, and size greater than n(m - k - 1) + k + 1 contains a cycle of length 2m - 2k, where m >= 1/2k(2) + 3/2k + 4, k is an element of N. The bound on the number of edges is best possible. Moreover, this size condition guarantees the existence of cycles of all even lengths up to 2m - 2k. We also characterize all extremal graphs for this problem. Finally, we conjecture that the condition on the order may be relaxed to m >= 2k + 2.]]> Thu, 01 Jan 2009 07:00:00 +0000 https://doi.org/10.46298/dmtcs.452 https://doi.org/10.46298/dmtcs.452 Adamus, Lech Adamus, Lech <![CDATA[The following problem was solved by Woodall in 1972: for any pair of nonnegative integers n and k < n/2 - 1 find the minimum integer g(n, k) such that every graph with n vertices and at least g(n, k) edges contains a cycle of length n - k. Woodall proved even more: the size g(n, k), in fact, guarantees the existence of cycles C, for all 3 <= p <= n - k. In the paper an analogous problem for bipartite graphs is considered. It is proved that every bipartite graph with color classes of cardinalities m and n, m <= n, and size greater than n(m - k - 1) + k + 1 contains a cycle of length 2m - 2k, where m >= 1/2k(2) + 3/2k + 4, k is an element of N. The bound on the number of edges is best possible. Moreover, this size condition guarantees the existence of cycles of all even lengths up to 2m - 2k. We also characterize all extremal graphs for this problem. Finally, we conjecture that the condition on the order may be relaxed to m >= 2k + 2.]]> 0 <![CDATA[In this paper we discuss the bounds of and relations among various kinds of intersection numbers of graphs. Especially, we address extremal graphs with respect to the established bounds. The uniqueness of the minimum-size intersection representations for some graphs is also studied. In the course of this work, we introduce a superclass of chordal graphs, defined in terms of a generalization of simplicial vertex and perfect elimination ordering.]]> Thu, 01 Jan 2009 07:00:00 +0000 https://doi.org/10.46298/dmtcs.451 https://doi.org/10.46298/dmtcs.451 Kong, Jing Wu, Yaokun Kong, Jing Wu, Yaokun <![CDATA[In this paper we discuss the bounds of and relations among various kinds of intersection numbers of graphs. Especially, we address extremal graphs with respect to the established bounds. The uniqueness of the minimum-size intersection representations for some graphs is also studied. In the course of this work, we introduce a superclass of chordal graphs, defined in terms of a generalization of simplicial vertex and perfect elimination ordering.]]> 0 <![CDATA[In the extensible bin packing problem we are asked to pack a set of items into a given number of bins, each with an original size. However, the original bin sizes can be extended if necessary. The goal is to minimize the total size of the bins. We consider the problem with unequal (original) bin sizes and give the complete analysis on a list scheduling algorithm (LS). Namely we present tight bounds of LS for every collection of original bin sizes and every number of bins. We further show better on-line algorithms for the two-bin case and the three-bin case. Interestingly, it is proved that the on-line algorithms have better competitive ratios for unequal bins than for equal bins. Some variants of the problem are also discussed.]]> Thu, 01 Jan 2009 07:00:00 +0000 https://doi.org/10.46298/dmtcs.472 https://doi.org/10.46298/dmtcs.472 Ye, Deshi Zhang, Guochuan Ye, Deshi Zhang, Guochuan <![CDATA[In the extensible bin packing problem we are asked to pack a set of items into a given number of bins, each with an original size. However, the original bin sizes can be extended if necessary. The goal is to minimize the total size of the bins. We consider the problem with unequal (original) bin sizes and give the complete analysis on a list scheduling algorithm (LS). Namely we present tight bounds of LS for every collection of original bin sizes and every number of bins. We further show better on-line algorithms for the two-bin case and the three-bin case. Interestingly, it is proved that the on-line algorithms have better competitive ratios for unequal bins than for equal bins. Some variants of the problem are also discussed.]]> 0 <![CDATA[We consider a general concept of composition and decomposition of objects, and discuss a few natural properties one may expect from a reasonable choice thereof. It will be demonstrated how this leads to multiplication and co-multiplication laws, thereby providing a generic scheme furnishing combinatorial classes with an algebraic structure. The paper is meant as a gentle introduction to the concepts of composition and decomposition with the emphasis on combinatorial origin of the ensuing algebraic constructions.]]> Thu, 01 Jan 2009 07:00:00 +0000 https://doi.org/10.46298/dmtcs.481 https://doi.org/10.46298/dmtcs.481 Blasiak, Pawel Blasiak, Pawel <![CDATA[We consider a general concept of composition and decomposition of objects, and discuss a few natural properties one may expect from a reasonable choice thereof. It will be demonstrated how this leads to multiplication and co-multiplication laws, thereby providing a generic scheme furnishing combinatorial classes with an algebraic structure. The paper is meant as a gentle introduction to the concepts of composition and decomposition with the emphasis on combinatorial origin of the ensuing algebraic constructions.]]> 0 <![CDATA[Motivated by the recent refutation of information loss paradox in black hole by Hawking, we investigate the new concept of {\it non unitary random walks}. In a non unitary random walk, we consider that the state 0, called the {\it black hole}, has a probability weight that decays exponentially in $e^{-\lambda t}$ for some $\lambda>0$. This decaying probabilities affect the probability weight of the other states, so that the the apparent transition probabilities are affected by a repulsion factor that depends on the factors $\lambda$ and black hole lifetime $t$. If $\lambda$ is large enough, then the resulting transition probabilities correspond to a neutral random walk. We generalize to {\it non unitary gravitational walks} where the transition probabilities are function of the distance to the black hole. We show the surprising result that the black hole remains attractive below a certain distance and becomes repulsive with an exactly reversed random walk beyond this distance. This effect has interesting analogy with so-called dark energy effect in astrophysics.]]> Thu, 01 Jan 2009 07:00:00 +0000 https://doi.org/10.46298/dmtcs.480 https://doi.org/10.46298/dmtcs.480 Jacquet, Philippe Jacquet, Philippe <![CDATA[Motivated by the recent refutation of information loss paradox in black hole by Hawking, we investigate the new concept of {\it non unitary random walks}. In a non unitary random walk, we consider that the state 0, called the {\it black hole}, has a probability weight that decays exponentially in $e^{-\lambda t}$ for some $\lambda>0$. This decaying probabilities affect the probability weight of the other states, so that the the apparent transition probabilities are affected by a repulsion factor that depends on the factors $\lambda$ and black hole lifetime $t$. If $\lambda$ is large enough, then the resulting transition probabilities correspond to a neutral random walk. We generalize to {\it non unitary gravitational walks} where the transition probabilities are function of the distance to the black hole. We show the surprising result that the black hole remains attractive below a certain distance and becomes repulsive with an exactly reversed random walk beyond this distance. This effect has interesting analogy with so-called dark energy effect in astrophysics.]]> 0 <![CDATA[We prove that on the class of (P6,diamond)-free graphs the Maximum-Weight Independent Set problem and the Minimum-Weight Independent Dominating Set problem can be solved in polynomial time.]]> Thu, 01 Jan 2009 07:00:00 +0000 https://doi.org/10.46298/dmtcs.473 https://doi.org/10.46298/dmtcs.473 Mosca, Raffaele Mosca, Raffaele <![CDATA[We prove that on the class of (P6,diamond)-free graphs the Maximum-Weight Independent Set problem and the Minimum-Weight Independent Dominating Set problem can be solved in polynomial time.]]> 0 <![CDATA[A (cyclic) n-bit Gray code is a (cyclic) ordering of all 2(n) binary strings of length n such that consecutive strings differ in a single bit. Equivalently, an n-bit Gray code can be viewed as a Hamiltonian path of the n-dimensional hypercube Q(n), and a cyclic Gray code as a Hamiltonian cycle of Q(n). In this paper we study (cyclic) Gray codes avoiding a given set of faulty edges that form a matching. Given a matching M and two vertices u, v of Q(n), n >= 4, our main result provides a necessary and sufficient condition, expressed in terms of forbidden configurations for M, for the existence of a Gray code between u and v that avoids M. As a corollary. we obtain a similar characterization for a cyclic Gray code avoiding M. In particular, in the case that M is a perfect matching, Q(n) has a (cyclic) Gray code that avoids M if and only if Q(n) - M is a connected graph. This complements a recent result of Fink, who proved that every perfect matching of Q(n) can be extended to a Hamiltonian cycle. Furthermore, our results imply that the problem of Hamilionicity of Q(n) with faulty edges, which is NP-complete in general, becomes polynomial for up to 2(n-1) edges provided they form a matching.]]> Thu, 01 Jan 2009 07:00:00 +0000 https://doi.org/10.46298/dmtcs.457 https://doi.org/10.46298/dmtcs.457 Dimitrov, Darko Dvořák, Tomáš Gregor, Petr Škrekovski, Riste Dimitrov, Darko Dvořák, Tomáš Gregor, Petr Škrekovski, Riste <![CDATA[A (cyclic) n-bit Gray code is a (cyclic) ordering of all 2(n) binary strings of length n such that consecutive strings differ in a single bit. Equivalently, an n-bit Gray code can be viewed as a Hamiltonian path of the n-dimensional hypercube Q(n), and a cyclic Gray code as a Hamiltonian cycle of Q(n). In this paper we study (cyclic) Gray codes avoiding a given set of faulty edges that form a matching. Given a matching M and two vertices u, v of Q(n), n >= 4, our main result provides a necessary and sufficient condition, expressed in terms of forbidden configurations for M, for the existence of a Gray code between u and v that avoids M. As a corollary. we obtain a similar characterization for a cyclic Gray code avoiding M. In particular, in the case that M is a perfect matching, Q(n) has a (cyclic) Gray code that avoids M if and only if Q(n) - M is a connected graph. This complements a recent result of Fink, who proved that every perfect matching of Q(n) can be extended to a Hamiltonian cycle. Furthermore, our results imply that the problem of Hamilionicity of Q(n) with faulty edges, which is NP-complete in general, becomes polynomial for up to 2(n-1) edges provided they form a matching.]]> 0 <![CDATA[We study two graph parameters, namely the number of spanning forests and the number of connected subgraphs, for self-similar graphs with exactly two boundary vertices. In both cases, we determine the general behavior for these and related auxiliary quantities by means of polynomial recurrences and a careful asymptotic analysis. It turns out that the so-called resistance scaling factor of a graph plays an essential role in both instances, a phenomenon that was previously observed for the number of spanning trees. Several explicit examples show that our findings are likely to hold in an even more general setting.]]> Thu, 01 Jan 2009 07:00:00 +0000 https://doi.org/10.46298/dmtcs.471 https://doi.org/10.46298/dmtcs.471 Teufl, Elmar Wagner, Stephan Teufl, Elmar Wagner, Stephan <![CDATA[We study two graph parameters, namely the number of spanning forests and the number of connected subgraphs, for self-similar graphs with exactly two boundary vertices. In both cases, we determine the general behavior for these and related auxiliary quantities by means of polynomial recurrences and a careful asymptotic analysis. It turns out that the so-called resistance scaling factor of a graph plays an essential role in both instances, a phenomenon that was previously observed for the number of spanning trees. Several explicit examples show that our findings are likely to hold in an even more general setting.]]> 0 <![CDATA[We study the number of connected spanning subgraphs f(d,b) (n) on the generalized Sierpinski gasket SG(d,b) (n) at stage n with dimension d equal to two, three and four for b = 2, and layer b equal to three and four for d = 2. The upper and lower bounds for the asymptotic growth constant, defined as zSG(d,b) = lim(v ->infinity) ln f(d,b)(n)/v where v is the number of vertices, on SG(2,b) (n) with b = 2, 3, 4 are derived in terms of the results at a certain stage. The numerical values of zSG(d,b) are obtained.]]> Thu, 01 Jan 2009 07:00:00 +0000 https://doi.org/10.46298/dmtcs.470 https://doi.org/10.46298/dmtcs.470 Chang, Shu-Chiuan Chen, Lung-Chi Chang, Shu-Chiuan Chen, Lung-Chi <![CDATA[We study the number of connected spanning subgraphs f(d,b) (n) on the generalized Sierpinski gasket SG(d,b) (n) at stage n with dimension d equal to two, three and four for b = 2, and layer b equal to three and four for d = 2. The upper and lower bounds for the asymptotic growth constant, defined as zSG(d,b) = lim(v ->infinity) ln f(d,b)(n)/v where v is the number of vertices, on SG(2,b) (n) with b = 2, 3, 4 are derived in terms of the results at a certain stage. The numerical values of zSG(d,b) are obtained.]]> 0 <![CDATA[We present an extensive study of the Eulerian distribution on the set of centrosymmetric involutions, namely, involutions in S(n) satisfying the property sigma(i) + sigma(n + 1 - i) = n + 1 for every i = 1 ... n. We find some combinatorial properties for the generating polynomial of such distribution, together with an explicit formula for its coefficients. Afterwards, we carry out an analogous study for the subset of centrosymmetric involutions without fixed points.]]> Thu, 01 Jan 2009 07:00:00 +0000 https://doi.org/10.46298/dmtcs.469 https://doi.org/10.46298/dmtcs.469 Barnabei, Marilena Bonetti, Flavio Silimbani, Matteo Barnabei, Marilena Bonetti, Flavio Silimbani, Matteo <![CDATA[We present an extensive study of the Eulerian distribution on the set of centrosymmetric involutions, namely, involutions in S(n) satisfying the property sigma(i) + sigma(n + 1 - i) = n + 1 for every i = 1 ... n. We find some combinatorial properties for the generating polynomial of such distribution, together with an explicit formula for its coefficients. Afterwards, we carry out an analogous study for the subset of centrosymmetric involutions without fixed points.]]> 0 <![CDATA[A k-uniform hypergraph H = ( V; E) is said to be self-complementary whenever it is isomorphic with its complement (H) over bar = ( V; ((V)(k)) - E). Every permutation sigma of the set V such that sigma(e) is an edge of (H) over bar if and only if e is an element of E is called self-complementing. 2-self-comlementary hypergraphs are exactly self complementary graphs introduced independently by Ringel ( 1963) and Sachs ( 1962). For any positive integer n we denote by lambda(n) the unique integer such that n = 2(lambda(n)) c, where c is odd. In the paper we prove that a permutation sigma of [1, n] with orbits O-1,..., O-m O m is a self-complementing permutation of a k-uniform hypergraph of order n if and only if there is an integer l >= 0 such that k = a2(l) + s, a is odd, 0 <= s <= 2(l) and the following two conditions hold: (i)n = b2(l+1) + r,r is an element of {0,..., 2(l) - 1 + s}, and (ii) Sigma(i:lambda(vertical bar Oi vertical bar)<= l) vertical bar O-i vertical bar <= r. For k = 2 this result is the very well known characterization of self-complementing permutation of graphs given by Ringel and Sachs.]]> Thu, 01 Jan 2009 07:00:00 +0000 https://doi.org/10.46298/dmtcs.468 https://doi.org/10.46298/dmtcs.468 Szymański, Artur Wojda, Adam Pawel Szymański, Artur Wojda, Adam Pawel <![CDATA[A k-uniform hypergraph H = ( V; E) is said to be self-complementary whenever it is isomorphic with its complement (H) over bar = ( V; ((V)(k)) - E). Every permutation sigma of the set V such that sigma(e) is an edge of (H) over bar if and only if e is an element of E is called self-complementing. 2-self-comlementary hypergraphs are exactly self complementary graphs introduced independently by Ringel ( 1963) and Sachs ( 1962). For any positive integer n we denote by lambda(n) the unique integer such that n = 2(lambda(n)) c, where c is odd. In the paper we prove that a permutation sigma of [1, n] with orbits O-1,..., O-m O m is a self-complementing permutation of a k-uniform hypergraph of order n if and only if there is an integer l >= 0 such that k = a2(l) + s, a is odd, 0 <= s <= 2(l) and the following two conditions hold: (i)n = b2(l+1) + r,r is an element of {0,..., 2(l) - 1 + s}, and (ii) Sigma(i:lambda(vertical bar Oi vertical bar)<= l) vertical bar O-i vertical bar <= r. For k = 2 this result is the very well known characterization of self-complementing permutation of graphs given by Ringel and Sachs.]]> 0 <![CDATA[We consider the sum u of differences between adjacent letters of a word of n letters, chosen uniformly at random from a given alphabet. This paper obtains the enumerating generating function for the number of such words with respect to the sum u, as well as explicit formulas for the mean and variance of u.]]> Thu, 01 Jan 2009 07:00:00 +0000 https://doi.org/10.46298/dmtcs.467 https://doi.org/10.46298/dmtcs.467 Mansour, Toufik Mansour, Toufik <![CDATA[We consider the sum u of differences between adjacent letters of a word of n letters, chosen uniformly at random from a given alphabet. This paper obtains the enumerating generating function for the number of such words with respect to the sum u, as well as explicit formulas for the mean and variance of u.]]> 0 <![CDATA[In this paper, we study long cycles in induced subgraphs of hypercubes obtained by removing a given set of faulty vertices such that every two faults are distant. First, we show that every induced subgraph of Q(n) with minimum degree n - 1 contains a cycle of length at least 2(n) - 2(f) where f is the number of removed vertices. This length is the best possible when all removed vertices are from the same bipartite class of Q(n). Next, we prove that every induced subgraph of Q(n) obtained by removing vertices of some given set M of edges of Q(n) contains a Hamiltonian cycle if every two edges of M are at distance at least 3. The last result shows that the shell of every linear code with odd minimum distance at least 3 contains a Hamiltonian cycle. In all these results we obtain significantly more tolerable faulty vertices than in the previously known results. We also conjecture that every induced subgraph of Q(n) obtained by removing a balanced set of vertices with minimum distance at least 3 contains a Hamiltonian cycle.]]> Thu, 01 Jan 2009 07:00:00 +0000 https://doi.org/10.46298/dmtcs.466 https://doi.org/10.46298/dmtcs.466 Gregor, Petr Škrekovski, Riste Gregor, Petr Škrekovski, Riste <![CDATA[In this paper, we study long cycles in induced subgraphs of hypercubes obtained by removing a given set of faulty vertices such that every two faults are distant. First, we show that every induced subgraph of Q(n) with minimum degree n - 1 contains a cycle of length at least 2(n) - 2(f) where f is the number of removed vertices. This length is the best possible when all removed vertices are from the same bipartite class of Q(n). Next, we prove that every induced subgraph of Q(n) obtained by removing vertices of some given set M of edges of Q(n) contains a Hamiltonian cycle if every two edges of M are at distance at least 3. The last result shows that the shell of every linear code with odd minimum distance at least 3 contains a Hamiltonian cycle. In all these results we obtain significantly more tolerable faulty vertices than in the previously known results. We also conjecture that every induced subgraph of Q(n) obtained by removing a balanced set of vertices with minimum distance at least 3 contains a Hamiltonian cycle.]]> 0 <![CDATA[The problem of string pattern avoidance in generalized non-crossing trees is studied. The generating functions for generalized non-crossing trees avoiding string patterns of length one and two are obtained. The Lagrange inversion formula is used to obtain the explicit formulas for some special cases. A bijection is also established between generalized non-crossing trees with special string pattern avoidance and little Schr ̈oder paths.]]> Thu, 01 Jan 2009 07:00:00 +0000 https://doi.org/10.46298/dmtcs.465 https://doi.org/10.46298/dmtcs.465 Sun, Yidong Wang, Zhiping Sun, Yidong Wang, Zhiping <![CDATA[The problem of string pattern avoidance in generalized non-crossing trees is studied. The generating functions for generalized non-crossing trees avoiding string patterns of length one and two are obtained. The Lagrange inversion formula is used to obtain the explicit formulas for some special cases. A bijection is also established between generalized non-crossing trees with special string pattern avoidance and little Schr ̈oder paths.]]> 0 <![CDATA[The Centerpoint Theorem states that, for any set S of n points in R(d), there exists a point p in R(d) such that every closed halfspace containing p contains at least [n/(d + 1)] points of S. We consider generalizations of the Centerpoint Theorem in which halfspaces are replaced with wedges (cones) of angle alpha. In R(2), we give bounds that are tight for all values of ff and give an O(n) time algorithm to find a point satisfying these bounds. We also give partial results for R(3) and, more generally, R(d).]]> Thu, 01 Jan 2009 07:00:00 +0000 https://doi.org/10.46298/dmtcs.464 https://doi.org/10.46298/dmtcs.464 Erickson, Jeff Hurtado, Ferran Morin, Pat Erickson, Jeff Hurtado, Ferran Morin, Pat <![CDATA[The Centerpoint Theorem states that, for any set S of n points in R(d), there exists a point p in R(d) such that every closed halfspace containing p contains at least [n/(d + 1)] points of S. We consider generalizations of the Centerpoint Theorem in which halfspaces are replaced with wedges (cones) of angle alpha. In R(2), we give bounds that are tight for all values of ff and give an O(n) time algorithm to find a point satisfying these bounds. We also give partial results for R(3) and, more generally, R(d).]]> 0 <![CDATA[Given a word w over a finite alphabet Sigma and a finite deterministic automaton A = < Q,Sigma,delta >, the inequality vertical bar delta(Q,w)vertical bar <= vertical bar Q vertical bar - k means that under the natural action of the word w the image of the state set Q is reduced by at least k states. The word w is k-collapsing (k-synchronizing) if this inequality holds for any deterministic finite automaton ( with k + 1 states) that satisfies such an inequality for at least one word. We prove that for each alphabet Sigma there is a 2-collapsing word whose length is vertical bar Sigma vertical bar(3)+6 vertical bar Sigma vertical bar(2)+5 vertical bar Sigma vertical bar/2. Then we produce shorter 2-collapsing and 2-synchronizing words over alphabets of 4 and 5 letters.]]> Thu, 01 Jan 2009 07:00:00 +0000 https://doi.org/10.46298/dmtcs.463 https://doi.org/10.46298/dmtcs.463 Cherubini, Alessandra Kisielewicz, Andrzej Piochi, Brunetto Cherubini, Alessandra Kisielewicz, Andrzej Piochi, Brunetto <![CDATA[Given a word w over a finite alphabet Sigma and a finite deterministic automaton A = < Q,Sigma,delta >, the inequality vertical bar delta(Q,w)vertical bar <= vertical bar Q vertical bar - k means that under the natural action of the word w the image of the state set Q is reduced by at least k states. The word w is k-collapsing (k-synchronizing) if this inequality holds for any deterministic finite automaton ( with k + 1 states) that satisfies such an inequality for at least one word. We prove that for each alphabet Sigma there is a 2-collapsing word whose length is vertical bar Sigma vertical bar(3)+6 vertical bar Sigma vertical bar(2)+5 vertical bar Sigma vertical bar/2. Then we produce shorter 2-collapsing and 2-synchronizing words over alphabets of 4 and 5 letters.]]> 0 <![CDATA[A composition of a positive integer n is a finite sequence of positive integers a(1), a(2), ..., a(k) such that a(1) + a(2) + ... + a(k) = n. Let d be a fixed nonnegative integer. We say that we have an ascent of size d or more if a(i+1) >= a(i) + d. We determine the mean, variance and limiting distribution of the number of ascents of size d or more in the set of compositions of n. We also study the average size of the greatest ascent over all compositions of n.]]> Thu, 01 Jan 2009 07:00:00 +0000 https://doi.org/10.46298/dmtcs.462 https://doi.org/10.46298/dmtcs.462 Brennan, Charlotte Knopfmacher, Arnold Brennan, Charlotte Knopfmacher, Arnold <![CDATA[A composition of a positive integer n is a finite sequence of positive integers a(1), a(2), ..., a(k) such that a(1) + a(2) + ... + a(k) = n. Let d be a fixed nonnegative integer. We say that we have an ascent of size d or more if a(i+1) >= a(i) + d. We determine the mean, variance and limiting distribution of the number of ascents of size d or more in the set of compositions of n. We also study the average size of the greatest ascent over all compositions of n.]]> 0 <![CDATA[Let q be a prime a power and k an integer such that 3 ≤ k ≤ q. In this paper we present a method using Latin squares to construct adjacency matrices of k-regular bipartite graphs of girth 8 on 2(kq2 -- q) vertices. Some of these graphs have the smallest number of vertices among the known regular graphs with girth 8.]]> Thu, 01 Jan 2009 07:00:00 +0000 https://doi.org/10.46298/dmtcs.461 https://doi.org/10.46298/dmtcs.461 Balbuena, Camino Balbuena, Camino <![CDATA[Let q be a prime a power and k an integer such that 3 ≤ k ≤ q. In this paper we present a method using Latin squares to construct adjacency matrices of k-regular bipartite graphs of girth 8 on 2(kq2 -- q) vertices. Some of these graphs have the smallest number of vertices among the known regular graphs with girth 8.]]> 0 <![CDATA[This paper studies the chromatic number of the following four flip graphs (under suitable definitions of a flip): the flip graph of perfect matchings of a complete graph of even order, the flip graph of triangulations of a convex polygon (the associahedron), the flip graph of non-crossing Hamiltonian paths of a set of points in convex position, and the flip graph of triangles in a convex point set. We give tight bounds for the latter two cases and upper bounds for the first two.]]> Thu, 01 Jan 2009 07:00:00 +0000 https://doi.org/10.46298/dmtcs.460 https://doi.org/10.46298/dmtcs.460 Fabila-Monroy, Ruy Flores-Peñaloza, David Huemer, Clemens Hurtado, Ferran Urrutia, Jorge Wood, David R. Fabila-Monroy, Ruy Flores-Peñaloza, David Huemer, Clemens Hurtado, Ferran Urrutia, Jorge Wood, David R. <![CDATA[This paper studies the chromatic number of the following four flip graphs (under suitable definitions of a flip): the flip graph of perfect matchings of a complete graph of even order, the flip graph of triangulations of a convex polygon (the associahedron), the flip graph of non-crossing Hamiltonian paths of a set of points in convex position, and the flip graph of triangles in a convex point set. We give tight bounds for the latter two cases and upper bounds for the first two.]]> 0 <![CDATA[We conjecture Ore and Erdős type criteria for a balanced bipartite graph of order 2n to contain a long cycle C(2n-2k), where 0 <= k < n/2. For k = 0, these are the classical hamiltonicity criteria of Moon and Moser. The main two results of the paper assert that our conjectures hold for k = 1 as well.]]> Thu, 01 Jan 2009 07:00:00 +0000 https://doi.org/10.46298/dmtcs.459 https://doi.org/10.46298/dmtcs.459 Adamus, Janusz Adamus, Lech Adamus, Janusz Adamus, Lech <![CDATA[We conjecture Ore and Erdős type criteria for a balanced bipartite graph of order 2n to contain a long cycle C(2n-2k), where 0 <= k < n/2. For k = 0, these are the classical hamiltonicity criteria of Moon and Moser. The main two results of the paper assert that our conjectures hold for k = 1 as well.]]> 0 <![CDATA[A family T of digraphs is a complete set of obstructions for a digraph H if for an arbitrary digraph G the existence of a homomorphism from G to H is equivalent to the non-existence of a homomorphism from any member of T to G. A digraph H is said to have tree duality if there exists a complete set of obstructions T consisting of orientations of trees. We show that if H has tree duality, then its arc graph delta H also has tree duality, and we derive a family of tree obstructions for delta H from the obstructions for H. Furthermore we generalise our result to right adjoint functors on categories of relational structures. We show that these functors always preserve tree duality, as well as polynomial CSPs and the existence of near-unanimity functions.]]> Thu, 01 Jan 2009 07:00:00 +0000 https://doi.org/10.46298/dmtcs.458 https://doi.org/10.46298/dmtcs.458 Foniok, Jan Tardif, Claude Foniok, Jan Tardif, Claude <![CDATA[A family T of digraphs is a complete set of obstructions for a digraph H if for an arbitrary digraph G the existence of a homomorphism from G to H is equivalent to the non-existence of a homomorphism from any member of T to G. A digraph H is said to have tree duality if there exists a complete set of obstructions T consisting of orientations of trees. We show that if H has tree duality, then its arc graph delta H also has tree duality, and we derive a family of tree obstructions for delta H from the obstructions for H. Furthermore we generalise our result to right adjoint functors on categories of relational structures. We show that these functors always preserve tree duality, as well as polynomial CSPs and the existence of near-unanimity functions.]]> 0 <![CDATA[We construct unital extensions of the higher order peak algebras defined by Krob and the third author in [Ann. Comb. 9 (2005), 411―430], and show that they can be obtained as homomorphic images of certain subalgebras of the Mantaci-Reutenauer algebras of type $B$. This generalizes a result of Bergeron, Nyman and the first author [Trans. AMS 356 (2004), 2781―2824].]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2729 https://doi.org/10.46298/dmtcs.2729 Aguiar, Marcelo Novelli, Jean-Christophe Thibon, Jean-Yves Aguiar, Marcelo Novelli, Jean-Christophe Thibon, Jean-Yves <![CDATA[We construct unital extensions of the higher order peak algebras defined by Krob and the third author in [Ann. Comb. 9 (2005), 411―430], and show that they can be obtained as homomorphic images of certain subalgebras of the Mantaci-Reutenauer algebras of type $B$. This generalizes a result of Bergeron, Nyman and the first author [Trans. AMS 356 (2004), 2781―2824].]]> 0 <![CDATA[A new class of functions is studied. We define the Brauer-Schur functions $B^{(p)}_{\lambda}$ for a prime number $p$, and investigate their properties. We construct a basis for the space of symmetric functions, which consists of products of $p$-Brauer-Schur functions and Schur functions. We will see that the transition matrix from the natural Schur function basis has some interesting numerical properties.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2730 https://doi.org/10.46298/dmtcs.2730 Aokage, Kazuya Aokage, Kazuya <![CDATA[A new class of functions is studied. We define the Brauer-Schur functions $B^{(p)}_{\lambda}$ for a prime number $p$, and investigate their properties. We construct a basis for the space of symmetric functions, which consists of products of $p$-Brauer-Schur functions and Schur functions. We will see that the transition matrix from the natural Schur function basis has some interesting numerical properties.]]> 0 <![CDATA[The $\textit{hiring problem}$ has been recently introduced by Broder et al. in last year's ACM-SIAM Symp. on Discrete Algorithms (SODA 2008), as a simple model for decision making under uncertainty. Candidates are interviewed in a sequential fashion, each one endowed with a quality score, and decisions to hire or discard them must be taken on the fly. The goal is to maintain a good rate of hiring while improving the "average'' quality of the hired staff. We provide here an alternative formulation of the hiring problem in combinatorial terms. This combinatorial model allows us the systematic use of techniques from combinatorial analysis, e. g., generating functions, to study the problem. Consider a permutation $\sigma :[1,\ldots, n] \to [1,\ldots, n]$. We process this permutation in a sequential fashion, so that at step $i$, we see the score or quality of candidate $i$, which is actually her face value $\sigma (i)$. Thus $\sigma (i)$ is the rank of candidate $i$; the best candidate among the $n$ gets rank $n$, while the worst one gets rank $1$. We define $\textit{rank-based}$ strategies, those that take their decisions using only the relative rank of the current candidate compared to the score of the previous candidates. For these strategies we can prove general theorems about the number of hired candidates in a permutation of length $n$, the time of the last hiring, and the average quality of the last hired candidate, using techniques from the area of analytic combinatorics. We apply these general results to specific strategies like hiring above the best, hiring above the median or hiring above the $m$th best; some of our results provide a complementary view to those of Broder et al., but on the other hand, our general results apply to a large family of hiring strategies, not just to specific cases.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2731 https://doi.org/10.46298/dmtcs.2731 Archibald, Margaret Martínez, Conrado Archibald, Margaret Martínez, Conrado <![CDATA[The $\textit{hiring problem}$ has been recently introduced by Broder et al. in last year's ACM-SIAM Symp. on Discrete Algorithms (SODA 2008), as a simple model for decision making under uncertainty. Candidates are interviewed in a sequential fashion, each one endowed with a quality score, and decisions to hire or discard them must be taken on the fly. The goal is to maintain a good rate of hiring while improving the "average'' quality of the hired staff. We provide here an alternative formulation of the hiring problem in combinatorial terms. This combinatorial model allows us the systematic use of techniques from combinatorial analysis, e. g., generating functions, to study the problem. Consider a permutation $\sigma :[1,\ldots, n] \to [1,\ldots, n]$. We process this permutation in a sequential fashion, so that at step $i$, we see the score or quality of candidate $i$, which is actually her face value $\sigma (i)$. Thus $\sigma (i)$ is the rank of candidate $i$; the best candidate among the $n$ gets rank $n$, while the worst one gets rank $1$. We define $\textit{rank-based}$ strategies, those that take their decisions using only the relative rank of the current candidate compared to the score of the previous candidates. For these strategies we can prove general theorems about the number of hired candidates in a permutation of length $n$, the time of the last hiring, and the average quality of the last hired candidate, using techniques from the area of analytic combinatorics. We apply these general results to specific strategies like hiring above the best, hiring above the median or hiring above the $m$th best; some of our results provide a complementary view to those of Broder et al., but on the other hand, our general results apply to a large family of hiring strategies, not just to specific cases.]]> 0 <![CDATA[In the early 1990s, Garsia and Haiman conjectured that the dimension of the Garsia-Haiman module $R_{\mu}$ is $n!$, and they showed that the resolution of this conjecture implies the Macdonald Positivity Conjecture. Haiman proved these conjectures in 2001 using algebraic geometry, but the question remains to find an explicit basis for $R_{\mu}$ which would give a simple proof of the dimension. Using the theory of Orbit Harmonics developed by Garsia and Haiman, we present a "kicking basis" for $R_{\mu}$ when $\mu$ has two columns.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2732 https://doi.org/10.46298/dmtcs.2732 Assaf, Sami Garsia, Adriano Assaf, Sami Garsia, Adriano <![CDATA[In the early 1990s, Garsia and Haiman conjectured that the dimension of the Garsia-Haiman module $R_{\mu}$ is $n!$, and they showed that the resolution of this conjecture implies the Macdonald Positivity Conjecture. Haiman proved these conjectures in 2001 using algebraic geometry, but the question remains to find an explicit basis for $R_{\mu}$ which would give a simple proof of the dimension. Using the theory of Orbit Harmonics developed by Garsia and Haiman, we present a "kicking basis" for $R_{\mu}$ when $\mu$ has two columns.]]> 0 <![CDATA[We study the Gilbert-Shannon-Reeds model for riffle shuffles and ask 'How many times must a deck of cards be shuffled for the deck to be in close to random order?'. In 1992, Bayer and Diaconis gave a solution which gives exact and asymptotic results for all decks of practical interest, e.g. a deck of 52 cards. But what if one only cares about the colors of the cards or disregards the suits focusing solely on the ranks? More generally, how does the rate of convergence of a Markov chain change if we are interested in only certain features? Our exploration of this problem takes us through random walks on groups and their cosets, discovering along the way exact formulas leading to interesting combinatorics, an 'amazing matrix', and new analytic methods which produce a completely general asymptotic solution that is remarkable accurate.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2733 https://doi.org/10.46298/dmtcs.2733 Assaf, Sami Diaconis, Persi Soundararajan, K. Assaf, Sami Diaconis, Persi Soundararajan, K. <![CDATA[We study the Gilbert-Shannon-Reeds model for riffle shuffles and ask 'How many times must a deck of cards be shuffled for the deck to be in close to random order?'. In 1992, Bayer and Diaconis gave a solution which gives exact and asymptotic results for all decks of practical interest, e.g. a deck of 52 cards. But what if one only cares about the colors of the cards or disregards the suits focusing solely on the ranks? More generally, how does the rate of convergence of a Markov chain change if we are interested in only certain features? Our exploration of this problem takes us through random walks on groups and their cosets, discovering along the way exact formulas leading to interesting combinatorics, an 'amazing matrix', and new analytic methods which produce a completely general asymptotic solution that is remarkable accurate.]]> 0 <![CDATA[We express the matroid polytope $P_M$ of a matroid $M$ as a signed Minkowski sum of simplices, and obtain a formula for the volume of $P_M$. This gives a combinatorial expression for the degree of an arbitrary torus orbit closure in the Grassmannian $Gr_{k,n}$. We then derive analogous results for the independent set polytope and the associated flag matroid polytope of $M$. Our proofs are based on a natural extension of Postnikov's theory of generalized permutohedra.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2734 https://doi.org/10.46298/dmtcs.2734 Ardila, Federico Benedetti, Carolina Doker, Jeffrey Ardila, Federico Benedetti, Carolina Doker, Jeffrey <![CDATA[We express the matroid polytope $P_M$ of a matroid $M$ as a signed Minkowski sum of simplices, and obtain a formula for the volume of $P_M$. This gives a combinatorial expression for the degree of an arbitrary torus orbit closure in the Grassmannian $Gr_{k,n}$. We then derive analogous results for the independent set polytope and the associated flag matroid polytope of $M$. Our proofs are based on a natural extension of Postnikov's theory of generalized permutohedra.]]> 0 <![CDATA[We show that the Kronecker coefficients indexed by two two―row shapes are given by quadratic quasipolynomial formulas whose domains are the maximal cells of a fan. Simple calculations provide explicitly the quasipolynomial formulas and a description of the associated fan. These new formulas are obtained from analogous formulas for the corresponding reduced Kronecker coefficients and a formula recovering the Kronecker coefficients from the reduced Kronecker coefficients. As an application, we characterize all the Kronecker coefficients indexed by two two-row shapes that are equal to zero. This allowed us to disprove a conjecture of Mulmuley about the behavior of the stretching functions attached to the Kronecker coefficients.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2735 https://doi.org/10.46298/dmtcs.2735 Briand, Emmanuel Orellana, Rosa Rosas, Mercedes Briand, Emmanuel Orellana, Rosa Rosas, Mercedes <![CDATA[We show that the Kronecker coefficients indexed by two two―row shapes are given by quadratic quasipolynomial formulas whose domains are the maximal cells of a fan. Simple calculations provide explicitly the quasipolynomial formulas and a description of the associated fan. These new formulas are obtained from analogous formulas for the corresponding reduced Kronecker coefficients and a formula recovering the Kronecker coefficients from the reduced Kronecker coefficients. As an application, we characterize all the Kronecker coefficients indexed by two two-row shapes that are equal to zero. This allowed us to disprove a conjecture of Mulmuley about the behavior of the stretching functions attached to the Kronecker coefficients.]]> 0 <![CDATA[We use the polynomial ring $\mathbb{C}[x_{1,1},\ldots,x_{n,n}]$ to modify the Kazhdan-Lusztig construction of irreducible $S_n$-modules. This modified construction produces exactly the same matrices as the original construction in [$\textit{Invent. Math}$ $\mathbf{53}$ (1979)], but does not employ the Kazhdan-Lusztig preorders. We also show that our modules are related by unitriangular transition matrices to those constructed by Clausen in [$\textit{J. Symbolic Comput.}$ $\textbf{11}$ (1991)]. This provides a $\mathbb{C}[x_{1,1},\ldots,x_{n,n}]$-analog of results of Garsia-McLarnan in [$\textit{Adv. Math.}$ $\textbf{69}$ (1988)].]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2736 https://doi.org/10.46298/dmtcs.2736 Buehrle, Charles Skandera, Mark Buehrle, Charles Skandera, Mark <![CDATA[We use the polynomial ring $\mathbb{C}[x_{1,1},\ldots,x_{n,n}]$ to modify the Kazhdan-Lusztig construction of irreducible $S_n$-modules. This modified construction produces exactly the same matrices as the original construction in [$\textit{Invent. Math}$ $\mathbf{53}$ (1979)], but does not employ the Kazhdan-Lusztig preorders. We also show that our modules are related by unitriangular transition matrices to those constructed by Clausen in [$\textit{J. Symbolic Comput.}$ $\textbf{11}$ (1991)]. This provides a $\mathbb{C}[x_{1,1},\ldots,x_{n,n}]$-analog of results of Garsia-McLarnan in [$\textit{Adv. Math.}$ $\textbf{69}$ (1988)].]]> 0 <![CDATA[We present $\textit{type preserving}$ bijections between noncrossing and nonnesting partitions for all classical reflection groups, answering a question of Athanasiadis and Reiner. The bijections for the abstract Coxeter types $B$, $C$ and $D$ are new in the literature. To find them we define, for every type, sets of statistics that are in bijection with noncrossing and nonnesting partitions, and this correspondence is established by means of elementary methods in all cases. The statistics can be then seen to be counted by the generalized Catalan numbers Cat$(W)$ when $W$ is a classical reflection group. In particular, the statistics of type $A$ appear as a new explicit example of objects that are counted by the classical Catalan numbers.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2737 https://doi.org/10.46298/dmtcs.2737 Fink, Alex Giraldo, Benjamin Iriarte Fink, Alex Giraldo, Benjamin Iriarte <![CDATA[We present $\textit{type preserving}$ bijections between noncrossing and nonnesting partitions for all classical reflection groups, answering a question of Athanasiadis and Reiner. The bijections for the abstract Coxeter types $B$, $C$ and $D$ are new in the literature. To find them we define, for every type, sets of statistics that are in bijection with noncrossing and nonnesting partitions, and this correspondence is established by means of elementary methods in all cases. The statistics can be then seen to be counted by the generalized Catalan numbers Cat$(W)$ when $W$ is a classical reflection group. In particular, the statistics of type $A$ appear as a new explicit example of objects that are counted by the classical Catalan numbers.]]> 0 <![CDATA[We enumerate derangements with descents in prescribed positions. A generating function was given by Guo-Niu Han and Guoce Xin in 2007. We give a combinatorial proof of this result, and derive several explicit formulas. To this end, we consider fixed point $\lambda$-coloured permutations, which are easily enumerated. Several formulae regarding these numbers are given, as well as a generalisation of Euler's difference tables. We also prove that except in a trivial special case, if a permutation $\pi$ is chosen uniformly among all permutations on $n$ elements, the events that $\pi$ has descents in a set $S$ of positions, and that $\pi$ is a derangement, are positively correlated.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2738 https://doi.org/10.46298/dmtcs.2738 Eriksen, Niklas Freij, Ragnar Wästlund, Johan Eriksen, Niklas Freij, Ragnar Wästlund, Johan <![CDATA[We enumerate derangements with descents in prescribed positions. A generating function was given by Guo-Niu Han and Guoce Xin in 2007. We give a combinatorial proof of this result, and derive several explicit formulas. To this end, we consider fixed point $\lambda$-coloured permutations, which are easily enumerated. Several formulae regarding these numbers are given, as well as a generalisation of Euler's difference tables. We also prove that except in a trivial special case, if a permutation $\pi$ is chosen uniformly among all permutations on $n$ elements, the events that $\pi$ has descents in a set $S$ of positions, and that $\pi$ is a derangement, are positively correlated.]]> 0 <![CDATA[The median problem seeks a permutation whose total distance to a given set of permutations (the base set) is minimal. This is an important problem in comparative genomics and has been studied for several distance measures such as reversals. The transposition distance is less relevant biologically, but it has been shown that it behaves similarly to the most important biological distances, and can thus give important information on their properties. We have derived an algorithm which solves the transposition median problem, giving all transposition medians (the median cloud). We show that our algorithm can be modified to accept median clouds as elements in the base set and briefly discuss the new concept of median iterates (medians of medians) and limit medians, that is the limit of this iterate.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2739 https://doi.org/10.46298/dmtcs.2739 Eriksen, Niklas Eriksen, Niklas <![CDATA[The median problem seeks a permutation whose total distance to a given set of permutations (the base set) is minimal. This is an important problem in comparative genomics and has been studied for several distance measures such as reversals. The transposition distance is less relevant biologically, but it has been shown that it behaves similarly to the most important biological distances, and can thus give important information on their properties. We have derived an algorithm which solves the transposition median problem, giving all transposition medians (the median cloud). We show that our algorithm can be modified to accept median clouds as elements in the base set and briefly discuss the new concept of median iterates (medians of medians) and limit medians, that is the limit of this iterate.]]> 0 <![CDATA[The multiplihedra $\mathcal{M}_{\bullet} = (\mathcal{M}_n)_{n \geq 1}$ form a family of polytopes originating in the study of higher categories and homotopy theory. While the multiplihedra may be unfamiliar to the algebraic combinatorics community, it is nestled between two families of polytopes that certainly are not: the permutahedra $\mathfrak{S}_{\bullet}$ and associahedra $\mathcal{Y}_{\bullet}$. The maps $\mathfrak{S}_{\bullet} \twoheadrightarrow \mathcal{M}_{\bullet} \twoheadrightarrow \mathcal{Y}_{\bullet}$ reveal several new Hopf structures on tree-like objects nestled between the Hopf algebras $\mathfrak{S}Sym$ and $\mathcal{Y}Sym$. We begin their study here, showing that $\mathcal{M}Sym$ is a module over $\mathfrak{S}Sym$ and a Hopf module over $\mathcal{Y}Sym$. An elegant description of the coinvariants for $\mathcal{M}Sym$ over $\mathcal{Y}Sym$ is uncovered via a change of basis-using Möbius inversion in posets built on the $1$-skeleta of $\mathcal{M}_{\bullet}$. Our analysis uses the notion of an $\textit{interval retract}$ that should be of independent interest in poset combinatorics. It also reveals new families of polytopes, and even a new factorization of a known projection from the associahedra to hypercubes.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2740 https://doi.org/10.46298/dmtcs.2740 Forcey, Stefan Lauve, Aaron Sottile, Frank Forcey, Stefan Lauve, Aaron Sottile, Frank <![CDATA[The multiplihedra $\mathcal{M}_{\bullet} = (\mathcal{M}_n)_{n \geq 1}$ form a family of polytopes originating in the study of higher categories and homotopy theory. While the multiplihedra may be unfamiliar to the algebraic combinatorics community, it is nestled between two families of polytopes that certainly are not: the permutahedra $\mathfrak{S}_{\bullet}$ and associahedra $\mathcal{Y}_{\bullet}$. The maps $\mathfrak{S}_{\bullet} \twoheadrightarrow \mathcal{M}_{\bullet} \twoheadrightarrow \mathcal{Y}_{\bullet}$ reveal several new Hopf structures on tree-like objects nestled between the Hopf algebras $\mathfrak{S}Sym$ and $\mathcal{Y}Sym$. We begin their study here, showing that $\mathcal{M}Sym$ is a module over $\mathfrak{S}Sym$ and a Hopf module over $\mathcal{Y}Sym$. An elegant description of the coinvariants for $\mathcal{M}Sym$ over $\mathcal{Y}Sym$ is uncovered via a change of basis-using Möbius inversion in posets built on the $1$-skeleta of $\mathcal{M}_{\bullet}$. Our analysis uses the notion of an $\textit{interval retract}$ that should be of independent interest in poset combinatorics. It also reveals new families of polytopes, and even a new factorization of a known projection from the associahedra to hypercubes.]]> 0 <![CDATA[For nonexceptional types, we prove a conjecture of Hatayama et al. about the prefectness of Kirillov―Reshetikhin crystals.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2741 https://doi.org/10.46298/dmtcs.2741 Fourier, Ghislain Okado, Masato Schilling, Anne Fourier, Ghislain Okado, Masato Schilling, Anne <![CDATA[For nonexceptional types, we prove a conjecture of Hatayama et al. about the prefectness of Kirillov―Reshetikhin crystals.]]> 0 <![CDATA[Let $\Gamma$ be a quiver on $n$ vertices $v_1, v_2, \ldots , v_n$ with $g_{ij}$ edges between $v_i$ and $v_j$, and let $\boldsymbol{\alpha} \in \mathbb{N}^n$. Hua gave a formula for $A_{\Gamma}(\boldsymbol{\alpha}, q)$, the number of isomorphism classes of absolutely indecomposable representations of $\Gamma$ over the finite field $\mathbb{F}_q$ with dimension vector $\boldsymbol{\alpha}$. We use Hua's formula to show that the derivatives of $A_{\Gamma}(\boldsymbol{\alpha}, q)$ with respect to $q$, when evaluated at $q = 1$, are polynomials in the variables $g_{ij}$, and we can compute the highest degree terms in these polynomials. The formulas for these coefficients depend on the enumeration of certain families of connected graphs. This note simply gives an overview of these results; a complete account of this research is available on the arXiv and has been submitted for publication.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2742 https://doi.org/10.46298/dmtcs.2742 Helleloid, Geir Rodriguez-Villegas, Fernando Helleloid, Geir Rodriguez-Villegas, Fernando <![CDATA[Let $\Gamma$ be a quiver on $n$ vertices $v_1, v_2, \ldots , v_n$ with $g_{ij}$ edges between $v_i$ and $v_j$, and let $\boldsymbol{\alpha} \in \mathbb{N}^n$. Hua gave a formula for $A_{\Gamma}(\boldsymbol{\alpha}, q)$, the number of isomorphism classes of absolutely indecomposable representations of $\Gamma$ over the finite field $\mathbb{F}_q$ with dimension vector $\boldsymbol{\alpha}$. We use Hua's formula to show that the derivatives of $A_{\Gamma}(\boldsymbol{\alpha}, q)$ with respect to $q$, when evaluated at $q = 1$, are polynomials in the variables $g_{ij}$, and we can compute the highest degree terms in these polynomials. The formulas for these coefficients depend on the enumeration of certain families of connected graphs. This note simply gives an overview of these results; a complete account of this research is available on the arXiv and has been submitted for publication.]]> 0 <![CDATA[We discuss some recent progress on the Monotone Column Permanent (MCP) conjecture. We use a general method for proving that a univariate polynomial has real roots only, namely by showing that a corresponding multivariate polynomial is stable. Recent connections between stability of polynomials and the strong Rayleigh property revealed by Brändén allows for a computationally feasible check of stability for multi-affine polynomials. Using this method we obtain a simpler proof for the $n=3$ case of the MCP conjecture, and a new proof for the $n=4$ case. We also show a multivariate version of the stability of Eulerian polynomials for $n \leq 5$ which arises as a special case of the multivariate MCP conjecture.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2743 https://doi.org/10.46298/dmtcs.2743 Haglund, James Visontai, Mirkó Haglund, James Visontai, Mirkó <![CDATA[We discuss some recent progress on the Monotone Column Permanent (MCP) conjecture. We use a general method for proving that a univariate polynomial has real roots only, namely by showing that a corresponding multivariate polynomial is stable. Recent connections between stability of polynomials and the strong Rayleigh property revealed by Brändén allows for a computationally feasible check of stability for multi-affine polynomials. Using this method we obtain a simpler proof for the $n=3$ case of the MCP conjecture, and a new proof for the $n=4$ case. We also show a multivariate version of the stability of Eulerian polynomials for $n \leq 5$ which arises as a special case of the multivariate MCP conjecture.]]> 0 <![CDATA[We refine the classical Littlewood-Richardson rule in several different settings. We begin with a combinatorial rule for the product of a Demazure atom and a Schur function. Building on this, we also describe the product of a quasisymmetric Schur function and a Schur function as a positive sum of quasisymmetric Schur functions. Finally, we provide a combinatorial formula for the product of a Demazure character and a Schur function as a positive sum of Demazure characters. This last rule implies the classical Littlewood-Richardson rule for the multiplication of two Schur functions.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2744 https://doi.org/10.46298/dmtcs.2744 Haglund, J. Luoto, K. Mason, S. van Willigenburg, S. Haglund, J. Luoto, K. Mason, S. van Willigenburg, S. <![CDATA[We refine the classical Littlewood-Richardson rule in several different settings. We begin with a combinatorial rule for the product of a Demazure atom and a Schur function. Building on this, we also describe the product of a quasisymmetric Schur function and a Schur function as a positive sum of quasisymmetric Schur functions. Finally, we provide a combinatorial formula for the product of a Demazure character and a Schur function as a positive sum of Demazure characters. This last rule implies the classical Littlewood-Richardson rule for the multiplication of two Schur functions.]]> 0 <![CDATA[A permutation $\pi$ is realized by the shift on $N$ symbols if there is an infinite word on an $N$-letter alphabet whose successive left shifts by one position are lexicographically in the same relative order as $\pi$. The set of realized permutations is closed under consecutive pattern containment. Permutations that cannot be realized are called forbidden patterns. It was shown in [J.M. Amigó, S. Elizalde and M. Kennel, $\textit{J. Combin. Theory Ser. A}$ 115 (2008), 485―504] that the shortest forbidden patterns of the shift on $N$ symbols have length $N+2$. In this paper we give a characterization of the set of permutations that are realized by the shift on $N$ symbols, and we enumerate them with respect to their length.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2745 https://doi.org/10.46298/dmtcs.2745 Elizalde, Sergi Elizalde, Sergi <![CDATA[A permutation $\pi$ is realized by the shift on $N$ symbols if there is an infinite word on an $N$-letter alphabet whose successive left shifts by one position are lexicographically in the same relative order as $\pi$. The set of realized permutations is closed under consecutive pattern containment. Permutations that cannot be realized are called forbidden patterns. It was shown in [J.M. Amigó, S. Elizalde and M. Kennel, $\textit{J. Combin. Theory Ser. A}$ 115 (2008), 485―504] that the shortest forbidden patterns of the shift on $N$ symbols have length $N+2$. In this paper we give a characterization of the set of permutations that are realized by the shift on $N$ symbols, and we enumerate them with respect to their length.]]> 0 <![CDATA[We define and consider $k$-distant crossings and nestings for matchings and set partitions, which are a variation of crossings and nestings in which the distance between vertices is important. By modifying an involution of Kasraoui and Zeng (Electronic J. Combinatorics 2006, research paper 33), we show that the joint distribution of $k$-distant crossings and nestings is symmetric. We also study the numbers of $k$-distant noncrossing matchings and partitions for small $k$, which are counted by well-known sequences, as well as the orthogonal polynomials related to $k$-distant noncrossing matchings and partitions. We extend Chen et al.'s $r$-crossings and enhanced $r$-crossings.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2746 https://doi.org/10.46298/dmtcs.2746 Drake, Dan Kim, Jang Soo Drake, Dan Kim, Jang Soo <![CDATA[We define and consider $k$-distant crossings and nestings for matchings and set partitions, which are a variation of crossings and nestings in which the distance between vertices is important. By modifying an involution of Kasraoui and Zeng (Electronic J. Combinatorics 2006, research paper 33), we show that the joint distribution of $k$-distant crossings and nestings is symmetric. We also study the numbers of $k$-distant noncrossing matchings and partitions for small $k$, which are counted by well-known sequences, as well as the orthogonal polynomials related to $k$-distant noncrossing matchings and partitions. We extend Chen et al.'s $r$-crossings and enhanced $r$-crossings.]]> 0 <![CDATA[We give a bijective operation that relates unicellular maps of given genus to unicellular maps of lower genus, with distinguished vertices. This gives a new combinatorial identity relating the number $\epsilon_g(n)$ of unicellular maps of size $n$ and genus $g$ to the numbers $\epsilon _j(n)$'s, for $j \lt g$. In particular for each $g$ this enables to compute the closed-form formula for $\epsilon_g(n)$ much more easily than with other known identities, like the Harer-Zagier formula. From the combinatorial point of view, we give an explanation to the fact that $\epsilon_g(n)=R_g(n) \mathrm{Cat}(n)$, where $\mathrm{Cat}(n$) is the $n$-th Catalan number and $R_g$ is a polynomial of degree $3g$, with explicit interpretation.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2747 https://doi.org/10.46298/dmtcs.2747 Chapuy, Guillaume Chapuy, Guillaume <![CDATA[We give a bijective operation that relates unicellular maps of given genus to unicellular maps of lower genus, with distinguished vertices. This gives a new combinatorial identity relating the number $\epsilon_g(n)$ of unicellular maps of size $n$ and genus $g$ to the numbers $\epsilon _j(n)$'s, for $j \lt g$. In particular for each $g$ this enables to compute the closed-form formula for $\epsilon_g(n)$ much more easily than with other known identities, like the Harer-Zagier formula. From the combinatorial point of view, we give an explanation to the fact that $\epsilon_g(n)=R_g(n) \mathrm{Cat}(n)$, where $\mathrm{Cat}(n$) is the $n$-th Catalan number and $R_g$ is a polynomial of degree $3g$, with explicit interpretation.]]> 0 <![CDATA[We introduce the class of projective reflection groups which includes all complex reflection groups. We show that several aspects involving the combinatorics and the representation theory of complex reflection groups find a natural description in this wider setting.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2748 https://doi.org/10.46298/dmtcs.2748 Caselli, Fabrizio Caselli, Fabrizio <![CDATA[We introduce the class of projective reflection groups which includes all complex reflection groups. We show that several aspects involving the combinatorics and the representation theory of complex reflection groups find a natural description in this wider setting.]]> 0 <![CDATA[Littlewood-Richardson coefficients are the multiplicities in the tensor product decomposition of two irreducible representations of the general linear group $\mathrm{GL}(n,\mathbb{C})$. They have a wide variety of interpretations in combinatorics, representation theory and geometry. Mulmuley and Sohoni pointed out that it is possible to decide the positivity of Littlewood-Richardson coefficients in polynomial time. This follows by combining the saturation property of Littlewood-Richardson coefficients (shown by Knutson and Tao 1999) with the well-known fact that linear optimization is solvable in polynomial time. We design an explicit $\textit{combinatorial}$ polynomial time algorithm for deciding the positivity of Littlewood-Richardson coefficients. This algorithm is highly adapted to the problem and it is based on ideas from the theory of optimizing flows in networks.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2749 https://doi.org/10.46298/dmtcs.2749 Bürgisser, Peter Ikenmeyer, Christian Bürgisser, Peter Ikenmeyer, Christian <![CDATA[Littlewood-Richardson coefficients are the multiplicities in the tensor product decomposition of two irreducible representations of the general linear group $\mathrm{GL}(n,\mathbb{C})$. They have a wide variety of interpretations in combinatorics, representation theory and geometry. Mulmuley and Sohoni pointed out that it is possible to decide the positivity of Littlewood-Richardson coefficients in polynomial time. This follows by combining the saturation property of Littlewood-Richardson coefficients (shown by Knutson and Tao 1999) with the well-known fact that linear optimization is solvable in polynomial time. We design an explicit $\textit{combinatorial}$ polynomial time algorithm for deciding the positivity of Littlewood-Richardson coefficients. This algorithm is highly adapted to the problem and it is based on ideas from the theory of optimizing flows in networks.]]> 0 <![CDATA[A permutation $a_1a_2 \ldots a_n$ is $\textit{indecomposable}$ if there does not exist $p \lt n$ such that $a_1a_2 \ldots a_p$ is a permutation of $\{ 1,2, \ldots ,p\}$. We compute the asymptotic probability that a permutation of $\mathbb{S}_n$ with $m$ cycles is indecomposable as $n$ goes to infinity with $m/n$ fixed. The error term is $O(\frac{\log(n-m)}{ n-m})$. The asymptotic probability is monotone in $m/n$, and there is no threshold phenomenon: it degrades gracefully from $1$ to $0$. When $n=2m$, a slight majority ($51.1 \ldots$ percent) of the permutations are indecomposable. We also consider indecomposable fixed point free involutions which are in bijection with maps of arbitrary genus on orientable surfaces, for these involutions with $m$ left-to-right maxima we obtain a lower bound for the probability of being indecomposable.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2750 https://doi.org/10.46298/dmtcs.2750 Cori, Robert Mathieu, Claire Cori, Robert Mathieu, Claire <![CDATA[A permutation $a_1a_2 \ldots a_n$ is $\textit{indecomposable}$ if there does not exist $p \lt n$ such that $a_1a_2 \ldots a_p$ is a permutation of $\{ 1,2, \ldots ,p\}$. We compute the asymptotic probability that a permutation of $\mathbb{S}_n$ with $m$ cycles is indecomposable as $n$ goes to infinity with $m/n$ fixed. The error term is $O(\frac{\log(n-m)}{ n-m})$. The asymptotic probability is monotone in $m/n$, and there is no threshold phenomenon: it degrades gracefully from $1$ to $0$. When $n=2m$, a slight majority ($51.1 \ldots$ percent) of the permutations are indecomposable. We also consider indecomposable fixed point free involutions which are in bijection with maps of arbitrary genus on orientable surfaces, for these involutions with $m$ left-to-right maxima we obtain a lower bound for the probability of being indecomposable.]]> 0 <![CDATA[We give two combinatorial interpretations of the Matrix Ansatz of the PASEP in terms of lattice paths and rook placements. This gives two (mostly) combinatorial proofs of a new enumeration formula for the partition function of the PASEP. Besides other interpretations, this formula gives the generating function for permutations of a given size with respect to the number of ascents and occurrences of the pattern $13-2$, the generating function according to weak exceedances and crossings, and the $n^{\mathrm{th}}$ moment of certain $q$-Laguerre polynomials.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2751 https://doi.org/10.46298/dmtcs.2751 Corteel, S. Josuat-Vergès, M. Prellberg, T. Rubey, M. Corteel, S. Josuat-Vergès, M. Prellberg, T. Rubey, M. <![CDATA[We give two combinatorial interpretations of the Matrix Ansatz of the PASEP in terms of lattice paths and rook placements. This gives two (mostly) combinatorial proofs of a new enumeration formula for the partition function of the PASEP. Besides other interpretations, this formula gives the generating function for permutations of a given size with respect to the number of ascents and occurrences of the pattern $13-2$, the generating function according to weak exceedances and crossings, and the $n^{\mathrm{th}}$ moment of certain $q$-Laguerre polynomials.]]> 0 <![CDATA[Free cumulants are nice and useful functionals of the shape of a Young diagram, in particular they give the asymptotics of normalized characters of symmetric groups $\mathfrak{S}(n)$ in the limit $n \to \infty$. We give an explicit combinatorial formula for normalized characters of the symmetric groups in terms of free cumulants. We also express characters in terms of Frobenius coordinates. Our formulas involve counting certain factorizations of a given permutation. The main tool are Stanley polynomials which give values of characters on multirectangular Young diagrams.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2752 https://doi.org/10.46298/dmtcs.2752 Dolega, Maciej Féray, Valentin Sniady, Piotr Dolega, Maciej Féray, Valentin Sniady, Piotr <![CDATA[Free cumulants are nice and useful functionals of the shape of a Young diagram, in particular they give the asymptotics of normalized characters of symmetric groups $\mathfrak{S}(n)$ in the limit $n \to \infty$. We give an explicit combinatorial formula for normalized characters of the symmetric groups in terms of free cumulants. We also express characters in terms of Frobenius coordinates. Our formulas involve counting certain factorizations of a given permutation. The main tool are Stanley polynomials which give values of characters on multirectangular Young diagrams.]]> 0 <![CDATA[We consider the class of bases $B$ of tropical Plücker functions on the Boolean $n$-cube such that $B$ can be obtained by a series of flips from the basis formed by the intervals of the ordered set of $n$ elements. We show that these bases are representable by special wiring diagrams and by certain arrangements generalizing rhombus tilings on a zonogon.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2753 https://doi.org/10.46298/dmtcs.2753 Danilov, Vladimir I. Karzanov, Alexander V. Koshevoy, Gleb A. Danilov, Vladimir I. Karzanov, Alexander V. Koshevoy, Gleb A. <![CDATA[We consider the class of bases $B$ of tropical Plücker functions on the Boolean $n$-cube such that $B$ can be obtained by a series of flips from the basis formed by the intervals of the ordered set of $n$ elements. We show that these bases are representable by special wiring diagrams and by certain arrangements generalizing rhombus tilings on a zonogon.]]> 0 <![CDATA[In this article, we propose a generalization of the notion of chordal graphs to signed graphs, which is based on the existence of a perfect elimination ordering for a chordal graph. We give a special kind of filtrations of the generalized chordal graphs, and show a characterization of those graphs. Moreover, we also describe a relation between signed graphs and a certain class of multiarrangements of hyperplanes, and show a characterization of free multiarrangements in that class in terms of the generalized chordal graphs, which generalizes a well-known result by Stanley on free hyperplane arrangements. Finally, we give a remark on a relation of our results with a recent conjecture by Athanasiadis on freeness characterization for another class of hyperplane arrangements.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2754 https://doi.org/10.46298/dmtcs.2754 Abe, Takuro Nuida, Koji Numata, Yasuhide Abe, Takuro Nuida, Koji Numata, Yasuhide <![CDATA[In this article, we propose a generalization of the notion of chordal graphs to signed graphs, which is based on the existence of a perfect elimination ordering for a chordal graph. We give a special kind of filtrations of the generalized chordal graphs, and show a characterization of those graphs. Moreover, we also describe a relation between signed graphs and a certain class of multiarrangements of hyperplanes, and show a characterization of free multiarrangements in that class in terms of the generalized chordal graphs, which generalizes a well-known result by Stanley on free hyperplane arrangements. Finally, we give a remark on a relation of our results with a recent conjecture by Athanasiadis on freeness characterization for another class of hyperplane arrangements.]]> 0 <![CDATA[We derive a new formula for the number of factorizations of a full cycle into an ordered product of two permutations of given cycle types. For the first time, a purely combinatorial argument involving a bijective description of bicolored maps of specified vertex degree distribution is used. All the previous results in the field rely either partially or totally on a character theoretic approach. The combinatorial proof relies on a new bijection extending the one in [G. Schaeffer and E. Vassilieva. $\textit{J. Comb. Theory Ser. A}$, 115(6):903―924, 2008] that focused only on the number of cycles. As a salient ingredient, we introduce the notion of thorn trees of given vertex degree distribution which are recursive planar objects allowing simple description of maps of arbitrary genus. \par]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2682 https://doi.org/10.46298/dmtcs.2682 Morales, Alejandro Vassilieva, Ekaterina Morales, Alejandro Vassilieva, Ekaterina <![CDATA[We derive a new formula for the number of factorizations of a full cycle into an ordered product of two permutations of given cycle types. For the first time, a purely combinatorial argument involving a bijective description of bicolored maps of specified vertex degree distribution is used. All the previous results in the field rely either partially or totally on a character theoretic approach. The combinatorial proof relies on a new bijection extending the one in [G. Schaeffer and E. Vassilieva. $\textit{J. Comb. Theory Ser. A}$, 115(6):903―924, 2008] that focused only on the number of cycles. As a salient ingredient, we introduce the notion of thorn trees of given vertex degree distribution which are recursive planar objects allowing simple description of maps of arbitrary genus. \par]]> 0 <![CDATA[A $k$-triangulation of the $n$-gon is a maximal set of diagonals of the $n$-gon containing no subset of $k+1$ mutually crossing diagonals. The number of $k$-triangulations of the $n$-gon, determined by Jakob Jonsson, is equal to a $k \times k$ Hankel determinant of Catalan numbers. This determinant is also equal to the number of $k$ non-crossing Dyck paths of semi-length $n-2k$. This brings up the problem of finding a combinatorial bijection between these two sets. In FPSAC 2007, Elizalde presented such a bijection for the case $k=2$. We construct another bijection for this case that is stronger and simpler that Elizalde's. The bijection preserves two sets of parameters, degrees and generalized returns. As a corollary, we generalize Jonsson's formula for $k=2$ by counting the number of $2$-triangulations of the $n$-gon with a given degree at a fixed vertex.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2683 https://doi.org/10.46298/dmtcs.2683 Nicolás, Carlos M. Nicolás, Carlos M. <![CDATA[A $k$-triangulation of the $n$-gon is a maximal set of diagonals of the $n$-gon containing no subset of $k+1$ mutually crossing diagonals. The number of $k$-triangulations of the $n$-gon, determined by Jakob Jonsson, is equal to a $k \times k$ Hankel determinant of Catalan numbers. This determinant is also equal to the number of $k$ non-crossing Dyck paths of semi-length $n-2k$. This brings up the problem of finding a combinatorial bijection between these two sets. In FPSAC 2007, Elizalde presented such a bijection for the case $k=2$. We construct another bijection for this case that is stronger and simpler that Elizalde's. The bijection preserves two sets of parameters, degrees and generalized returns. As a corollary, we generalize Jonsson's formula for $k=2$ by counting the number of $2$-triangulations of the $n$-gon with a given degree at a fixed vertex.]]> 0 <![CDATA[The purpose of this paper is to present the $q$-hook formula of Gansner type for a generalized Young diagram in the sense of D. Peterson and R. A. Proctor. This gives a far-reaching generalization of a hook length formula due to J. S. Frame, G. de B. Robinson, and R. M. Thrall. Furthurmore, we give a generalization of P. MacMahon's identity as an application of the $q$-hook formula.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2684 https://doi.org/10.46298/dmtcs.2684 Nakada, Kento Nakada, Kento <![CDATA[The purpose of this paper is to present the $q$-hook formula of Gansner type for a generalized Young diagram in the sense of D. Peterson and R. A. Proctor. This gives a far-reaching generalization of a hook length formula due to J. S. Frame, G. de B. Robinson, and R. M. Thrall. Furthurmore, we give a generalization of P. MacMahon's identity as an application of the $q$-hook formula.]]> 0 <![CDATA[We study cluster algebras with principal coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of perfect matchings of a certain graph $G_{T,\gamma}$ that is constructed from the surface by recursive glueing of elementary pieces that we call tiles. We also give a second formula for these Laurent polynomial expansions in terms of subgraphs of the graph $G_{T,\gamma}$ .]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2685 https://doi.org/10.46298/dmtcs.2685 Musiker, Gregg Schiffler, Ralf Musiker, Gregg Schiffler, Ralf <![CDATA[We study cluster algebras with principal coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of perfect matchings of a certain graph $G_{T,\gamma}$ that is constructed from the surface by recursive glueing of elementary pieces that we call tiles. We also give a second formula for these Laurent polynomial expansions in terms of subgraphs of the graph $G_{T,\gamma}$ .]]> 0 <![CDATA[The total number of noncrossing partitions of type $\Psi$ is the $n$th Catalan number $\frac{1}{ n+1} \binom{2n}{n}$ when $\Psi =A_{n-1}$, and the binomial coefficient $\binom{2n}{n}$ when $\Psi =B_n$, and these numbers coincide with the correspondent number of nonnesting partitions. For type $A$, there are several bijective proofs of this equality; in particular, the intuitive map, which locally converts each crossing to a nesting, is one of them. In this paper we present a bijection between nonnesting and noncrossing partitions of types $A$ and $B$ that generalizes the type $A$ bijection that locally converts each crossing to a nesting.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2686 https://doi.org/10.46298/dmtcs.2686 Mamede, Ricardo Mamede, Ricardo <![CDATA[The total number of noncrossing partitions of type $\Psi$ is the $n$th Catalan number $\frac{1}{ n+1} \binom{2n}{n}$ when $\Psi =A_{n-1}$, and the binomial coefficient $\binom{2n}{n}$ when $\Psi =B_n$, and these numbers coincide with the correspondent number of nonnesting partitions. For type $A$, there are several bijective proofs of this equality; in particular, the intuitive map, which locally converts each crossing to a nesting, is one of them. In this paper we present a bijection between nonnesting and noncrossing partitions of types $A$ and $B$ that generalizes the type $A$ bijection that locally converts each crossing to a nesting.]]> 0 <![CDATA[We investigate the homogeneous symmetric Macdonald polynomials $P_{\lambda} (\mathbb{X} ;q,t)$ for the specialization $t=q^k$. We show an identity relying the polynomials $P_{\lambda} (\mathbb{X} ;q,q^k)$ and $P_{\lambda} (\frac{1-q}{1-q^k}\mathbb{X} ;q,q^k)$. As a consequence, we describe an operator whose eigenvalues characterize the polynomials $P_{\lambda} (\mathbb{X} ;q,q^k)$.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2687 https://doi.org/10.46298/dmtcs.2687 Luque, Jean-Gabriel Luque, Jean-Gabriel <![CDATA[We investigate the homogeneous symmetric Macdonald polynomials $P_{\lambda} (\mathbb{X} ;q,t)$ for the specialization $t=q^k$. We show an identity relying the polynomials $P_{\lambda} (\mathbb{X} ;q,q^k)$ and $P_{\lambda} (\frac{1-q}{1-q^k}\mathbb{X} ;q,q^k)$. As a consequence, we describe an operator whose eigenvalues characterize the polynomials $P_{\lambda} (\mathbb{X} ;q,q^k)$.]]> 0 <![CDATA[Let $P$ be a partially ordered set and consider the free monoid $P^{\ast}$ of all words over $P$. If $w,w' \in P^{\ast}$ then $w'$ is a factor of $w$ if there are words $u,v$ with $w=uw'v$. Define generalized factor order on $P^{\ast}$ by letting $u \leq w$ if there is a factor $w'$ of $w$ having the same length as $u$ such that $u \leq w'$, where the comparison of $u$ and $w'$ is done componentwise using the partial order in $P$. One obtains ordinary factor order by insisting that $u=w'$ or, equivalently, by taking $P$ to be an antichain. Given $u \in P^{\ast}$, we prove that the language $\mathcal{F}(u)=\{w : w \geq u\}$ is accepted by a finite state automaton. If $P$ is finite then it follows that the generating function $F(u)=\sum_{w \geq u} w$ is rational. This is an analogue of a theorem of Björner and Sagan for generalized subword order. We also consider $P=\mathbb{P}$, the positive integers with the usual total order, so that $\mathbb{P}^{\ast}$ is the set of compositions. In this case one obtains a weight generating function $F(u;t,x)$ by substituting $tx^n$ each time $n \in \mathbb{P}$ appears in $F(u)$. We show that this generating function is also rational by using the transfer-matrix method. Words $u,v$ are said to be Wilf equivalent if $F(u;t,x)=F(v;t,x)$ and we can prove various Wilf equivalences combinatorially. Björner found a recursive formula for the Möbius function of ordinary factor order on $P^{\ast}$. It follows that one always has $\mu (u,w)=0, \pm 1$. Using the Pumping Lemma we show that the generating function $M(u)= \sum_{w \geq u} | \mu (u,w) | w$ can be irrational.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2688 https://doi.org/10.46298/dmtcs.2688 Kitaev, Sergey Liese, Jeffrey Remmel, Jeffrey Sagan, Bruce Kitaev, Sergey Liese, Jeffrey Remmel, Jeffrey Sagan, Bruce <![CDATA[Let $P$ be a partially ordered set and consider the free monoid $P^{\ast}$ of all words over $P$. If $w,w' \in P^{\ast}$ then $w'$ is a factor of $w$ if there are words $u,v$ with $w=uw'v$. Define generalized factor order on $P^{\ast}$ by letting $u \leq w$ if there is a factor $w'$ of $w$ having the same length as $u$ such that $u \leq w'$, where the comparison of $u$ and $w'$ is done componentwise using the partial order in $P$. One obtains ordinary factor order by insisting that $u=w'$ or, equivalently, by taking $P$ to be an antichain. Given $u \in P^{\ast}$, we prove that the language $\mathcal{F}(u)=\{w : w \geq u\}$ is accepted by a finite state automaton. If $P$ is finite then it follows that the generating function $F(u)=\sum_{w \geq u} w$ is rational. This is an analogue of a theorem of Björner and Sagan for generalized subword order. We also consider $P=\mathbb{P}$, the positive integers with the usual total order, so that $\mathbb{P}^{\ast}$ is the set of compositions. In this case one obtains a weight generating function $F(u;t,x)$ by substituting $tx^n$ each time $n \in \mathbb{P}$ appears in $F(u)$. We show that this generating function is also rational by using the transfer-matrix method. Words $u,v$ are said to be Wilf equivalent if $F(u;t,x)=F(v;t,x)$ and we can prove various Wilf equivalences combinatorially. Björner found a recursive formula for the Möbius function of ordinary factor order on $P^{\ast}$. It follows that one always has $\mu (u,w)=0, \pm 1$. Using the Pumping Lemma we show that the generating function $M(u)= \sum_{w \geq u} | \mu (u,w) | w$ can be irrational.]]> 0 <![CDATA[The absolute order on the hyperoctahedral group $B_n$ is investigated. It is shown that every closed interval in this order is shellable, those closed intervals which are lattices are characterized and their zeta polynomials are computed. Moreover, using the notion of strong constructibility, it is proved that the order ideal generated by the Coxeter elements of $B_n$ is homotopy Cohen-Macaulay and the Euler characteristic of the order complex of the proper part of this ideal is computed. Finally, an example of a non Cohen-Macaulay closed interval in the absolute order on the group $D_4$ is given and the closed intervals of $D_n$ which are lattices are characterized.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2689 https://doi.org/10.46298/dmtcs.2689 Kallipoliti, Myrto Kallipoliti, Myrto <![CDATA[The absolute order on the hyperoctahedral group $B_n$ is investigated. It is shown that every closed interval in this order is shellable, those closed intervals which are lattices are characterized and their zeta polynomials are computed. Moreover, using the notion of strong constructibility, it is proved that the order ideal generated by the Coxeter elements of $B_n$ is homotopy Cohen-Macaulay and the Euler characteristic of the order complex of the proper part of this ideal is computed. Finally, an example of a non Cohen-Macaulay closed interval in the absolute order on the group $D_4$ is given and the closed intervals of $D_n$ which are lattices are characterized.]]> 0 <![CDATA[The $\mathtt{polymake}$ software system deals with convex polytopes and related objects from geometric combinatorics. This note reports on a new implementation of a subclass for lattice polytopes. The features displayed are enabled by recent changes to the $\mathtt{polymake}$ core, which will be discussed briefly.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2690 https://doi.org/10.46298/dmtcs.2690 Joswig, Michael Müller, Benjamin Paffenholz, Andreas Joswig, Michael Müller, Benjamin Paffenholz, Andreas <![CDATA[The $\mathtt{polymake}$ software system deals with convex polytopes and related objects from geometric combinatorics. This note reports on a new implementation of a subclass for lattice polytopes. The features displayed are enabled by recent changes to the $\mathtt{polymake}$ core, which will be discussed briefly.]]> 0 <![CDATA[A $\textit{composition}$ $\sigma =a_1 a_2 \ldots a_m$ of $n$ is an ordered collection of positive integers whose sum is $n$. An element $a_i$ in $\sigma$ is a strong (weak) $\textit{record}$ if $a_i> a_j (a_i \geq a_j)$ for all $j=1,2,\ldots,i-1$. Furthermore, the position of this record is $i$. We derive generating functions for the total number of strong (weak) records in all compositions of $n$, as well as for the sum of the positions of the records in all compositions of $n$, where the parts $a_i$ belong to a fixed subset $A$ of the natural numbers. In particular when $A=\mathbb{N}$, we find the asymptotic mean values for the number, and for the sum of positions, of records in compositions of $n$.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2691 https://doi.org/10.46298/dmtcs.2691 Knopfmacher, Arnold Mansour, Toufik Knopfmacher, Arnold Mansour, Toufik <![CDATA[A $\textit{composition}$ $\sigma =a_1 a_2 \ldots a_m$ of $n$ is an ordered collection of positive integers whose sum is $n$. An element $a_i$ in $\sigma$ is a strong (weak) $\textit{record}$ if $a_i> a_j (a_i \geq a_j)$ for all $j=1,2,\ldots,i-1$. Furthermore, the position of this record is $i$. We derive generating functions for the total number of strong (weak) records in all compositions of $n$, as well as for the sum of the positions of the records in all compositions of $n$, where the parts $a_i$ belong to a fixed subset $A$ of the natural numbers. In particular when $A=\mathbb{N}$, we find the asymptotic mean values for the number, and for the sum of positions, of records in compositions of $n$.]]> 0 <![CDATA[In this paper we analyze O'Hara's partition bijection. We present three type of results. First, we see that O'Hara's bijection can be viewed geometrically as a certain scissor congruence type result. Second, we present a number of new complexity bounds, proving that O'Hara's bijection is efficient in most cases and mildly exponential in general. Finally, we see that for identities with finite support, the map of the O'Hara's bijection can be computed in polynomial time, i.e. much more efficiently than by O'Hara's construction.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2692 https://doi.org/10.46298/dmtcs.2692 Konvalinka, Matjaž Pak, Igor Konvalinka, Matjaž Pak, Igor <![CDATA[In this paper we analyze O'Hara's partition bijection. We present three type of results. First, we see that O'Hara's bijection can be viewed geometrically as a certain scissor congruence type result. Second, we present a number of new complexity bounds, proving that O'Hara's bijection is efficient in most cases and mildly exponential in general. Finally, we see that for identities with finite support, the map of the O'Hara's bijection can be computed in polynomial time, i.e. much more efficiently than by O'Hara's construction.]]> 0 <![CDATA[The devil's staircase ― a continuous function on the unit interval $[0,1]$ which is not constant, yet is locally constant on an open dense set ― is the sort of exotic creature a combinatorialist might never expect to encounter in "real life.'' We show how a devil's staircase arises from the combinatorial problem of parallel chip-firing on the complete graph. This staircase helps explain a previously observed "mode locking'' phenomenon, as well as the surprising tendency of parallel chip-firing to find periodic states of small period.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2693 https://doi.org/10.46298/dmtcs.2693 Levine, Lionel Levine, Lionel <![CDATA[The devil's staircase ― a continuous function on the unit interval $[0,1]$ which is not constant, yet is locally constant on an open dense set ― is the sort of exotic creature a combinatorialist might never expect to encounter in "real life.'' We show how a devil's staircase arises from the combinatorial problem of parallel chip-firing on the complete graph. This staircase helps explain a previously observed "mode locking'' phenomenon, as well as the surprising tendency of parallel chip-firing to find periodic states of small period.]]> 0 <![CDATA[Let $n$ and $k$ be positive integers, $d(k)$ and $\nu_2(k)$ denote the number of ones in the binary representation of $k$ and the highest power of two dividing $k$, respectively. De Wannemacker recently proved for the Stirling numbers of the second kind that $\nu_2(S(2^n,k))=d(k)-1, 1\leq k \leq 2^n$. Here we prove that $\nu_2(S(c2^n,k))=d(k)-1, 1\leq k \leq 2^n$, for any positive integer $c$. We improve and extend this statement in some special cases. For the difference, we obtain lower bounds on $\nu_2(S(c2^{n+1}+u,k)-S(c2^n+u,k))$ for any nonnegative integer $u$, make a conjecture on the exact order and, for $u=0$, prove part of it when $k \leq 6$, or $k \geq 5$ and $d(k) \leq 2$. The proofs rely on congruential identities for power series and polynomials related to the Stirling numbers and Bell polynomials, and some divisibility properties.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2694 https://doi.org/10.46298/dmtcs.2694 Lengyel, Tamás Lengyel, Tamás <![CDATA[Let $n$ and $k$ be positive integers, $d(k)$ and $\nu_2(k)$ denote the number of ones in the binary representation of $k$ and the highest power of two dividing $k$, respectively. De Wannemacker recently proved for the Stirling numbers of the second kind that $\nu_2(S(2^n,k))=d(k)-1, 1\leq k \leq 2^n$. Here we prove that $\nu_2(S(c2^n,k))=d(k)-1, 1\leq k \leq 2^n$, for any positive integer $c$. We improve and extend this statement in some special cases. For the difference, we obtain lower bounds on $\nu_2(S(c2^{n+1}+u,k)-S(c2^n+u,k))$ for any nonnegative integer $u$, make a conjecture on the exact order and, for $u=0$, prove part of it when $k \leq 6$, or $k \geq 5$ and $d(k) \leq 2$. The proofs rely on congruential identities for power series and polynomials related to the Stirling numbers and Bell polynomials, and some divisibility properties.]]> 0 <![CDATA[A breakthrough in the theory of (type $A$) Macdonald polynomials is due to Haglund, Haiman and Loehr, who exhibited a combinatorial formula for these polynomials in terms of fillings of Young diagrams. Recently, Ram and Yip gave a formula for the Macdonald polynomials of arbitrary type in terms of the corresponding affine Weyl group. In this paper, we show that a Haglund-Haiman-Loehr type formula follows naturally from the more general Ram-Yip formula, via compression. Then we extend this approach to the Hall-Littlewood polynomials of type $C$, which are specializations of the corresponding Macdonald polynomials at $q=0$. We note that no analog of the Haglund-Haiman-Loehr formula exists beyond type $A$, so our work is a first step towards finding such a formula.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2695 https://doi.org/10.46298/dmtcs.2695 Lenart, Cristian Lenart, Cristian <![CDATA[A breakthrough in the theory of (type $A$) Macdonald polynomials is due to Haglund, Haiman and Loehr, who exhibited a combinatorial formula for these polynomials in terms of fillings of Young diagrams. Recently, Ram and Yip gave a formula for the Macdonald polynomials of arbitrary type in terms of the corresponding affine Weyl group. In this paper, we show that a Haglund-Haiman-Loehr type formula follows naturally from the more general Ram-Yip formula, via compression. Then we extend this approach to the Hall-Littlewood polynomials of type $C$, which are specializations of the corresponding Macdonald polynomials at $q=0$. We note that no analog of the Haglund-Haiman-Loehr formula exists beyond type $A$, so our work is a first step towards finding such a formula.]]> 0 <![CDATA[In this article, we investigate the asymptotic occurrence rates of specific subwords in any infinite binary word. We prove that the asymptotic occurrence rate for the subwords is upper- and lower-bounded in the same way for every infinite binary word, in terms of the asymptotic occurrence rate of the zeros. We also show that both of the bounds are best-possible by constructing, for each bound, a concrete infinite binary word such that the bound is reached. Moreover, we apply the result to analyses of recently-proposed pseudorandom number generators that are based on integer-valued variants of logistic maps.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2696 https://doi.org/10.46298/dmtcs.2696 Nuida, Koji Nuida, Koji <![CDATA[In this article, we investigate the asymptotic occurrence rates of specific subwords in any infinite binary word. We prove that the asymptotic occurrence rate for the subwords is upper- and lower-bounded in the same way for every infinite binary word, in terms of the asymptotic occurrence rate of the zeros. We also show that both of the bounds are best-possible by constructing, for each bound, a concrete infinite binary word such that the bound is reached. Moreover, we apply the result to analyses of recently-proposed pseudorandom number generators that are based on integer-valued variants of logistic maps.]]> 0 <![CDATA[Given a sequence $(a_k)=a_0,a_1,a_2,\ldots$ of real numbers, define a new sequence $\mathcal{L}(a_k)=(b_k)$ where $b_k=a_k^2-a_{k-1}a_{k+1}$. So $(a_k)$ is log-concave if and only if $(b_k)$ is a nonnegative sequence. Call $(a_k)$ $\textit{infinitely log-concave}$ if $\mathcal{L}^i(a_k)$ is nonnegative for all $i \geq 1$. Boros and Moll conjectured that the rows of Pascal's triangle are infinitely log-concave. Using a computer and a stronger version of log-concavity, we prove their conjecture for the $n$th row for all $n \leq 1450$. We can also use our methods to give a simple proof of a recent result of Uminsky and Yeats about regions of infinite log-concavity. We investigate related questions about the columns of Pascal's triangle, $q$-analogues, symmetric functions, real-rooted polynomials, and Toeplitz matrices. In addition, we offer several conjectures.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2678 https://doi.org/10.46298/dmtcs.2678 McNamara, Peter R. W. Sagan, Bruce E. McNamara, Peter R. W. Sagan, Bruce E. <![CDATA[Given a sequence $(a_k)=a_0,a_1,a_2,\ldots$ of real numbers, define a new sequence $\mathcal{L}(a_k)=(b_k)$ where $b_k=a_k^2-a_{k-1}a_{k+1}$. So $(a_k)$ is log-concave if and only if $(b_k)$ is a nonnegative sequence. Call $(a_k)$ $\textit{infinitely log-concave}$ if $\mathcal{L}^i(a_k)$ is nonnegative for all $i \geq 1$. Boros and Moll conjectured that the rows of Pascal's triangle are infinitely log-concave. Using a computer and a stronger version of log-concavity, we prove their conjecture for the $n$th row for all $n \leq 1450$. We can also use our methods to give a simple proof of a recent result of Uminsky and Yeats about regions of infinite log-concavity. We investigate related questions about the columns of Pascal's triangle, $q$-analogues, symmetric functions, real-rooted polynomials, and Toeplitz matrices. In addition, we offer several conjectures.]]> 0 <![CDATA[This document is an extended abstract of the paper `Hopf algebras and the logarithm of the S-transform in free probability' in which we introduce a Hopf algebraic approach to the study of the operation $\boxtimes$ (free multiplicative convolution) from free probability.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2679 https://doi.org/10.46298/dmtcs.2679 Mastnak, Mitja Nica, Alexandru Mastnak, Mitja Nica, Alexandru <![CDATA[This document is an extended abstract of the paper `Hopf algebras and the logarithm of the S-transform in free probability' in which we introduce a Hopf algebraic approach to the study of the operation $\boxtimes$ (free multiplicative convolution) from free probability.]]> 0 <![CDATA[The election is a classical problem in distributed algorithmic. It aims to design and to analyze a distributed algorithm choosing a node in a graph, here, in a tree. In this paper, a class of randomized algorithms for the election is studied. The election amounts to removing leaves one by one until the tree is reduced to a unique node which is then elected. The algorithm assigns to each leaf a probability distribution (that may depends on the information transmitted by the eliminated nodes) used by the leaf to generate its remaining random lifetime. In the general case, the probability of each node to be elected is given. For two categories of algorithms, close formulas are provided.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2680 https://doi.org/10.46298/dmtcs.2680 Marckert, Jean-François Saheb-Djahromi, Nasser Zemmari, Akka Marckert, Jean-François Saheb-Djahromi, Nasser Zemmari, Akka <![CDATA[The election is a classical problem in distributed algorithmic. It aims to design and to analyze a distributed algorithm choosing a node in a graph, here, in a tree. In this paper, a class of randomized algorithms for the election is studied. The election amounts to removing leaves one by one until the tree is reduced to a unique node which is then elected. The algorithm assigns to each leaf a probability distribution (that may depends on the information transmitted by the eliminated nodes) used by the leaf to generate its remaining random lifetime. In the general case, the probability of each node to be elected is given. For two categories of algorithms, close formulas are provided.]]> 0 <![CDATA[The type $A_n$ root polytope $\mathcal{P}(A_n^+)$ is the convex hull in $\mathbb{R}^{n+1}$ of the origin and the points $e_i-e_j$ for $1 \leq i < j \leq n+1$. Given a tree $T$ on vertex set $[n+1]$, the associated root polytope $\mathcal{P}(T)$ is the intersection of $\mathcal{P}(A_n^+)$ with the cone generated by the vectors $e_i-e_j$, where $(i, j) \in E(T)$, $i < j$. The reduced forms of a certain monomial $m[T]$ in commuting variables $x_{ij}$ under the reduction $x_{ij} x_{jk} \to x_{ik} x_{ij} + x_{jk} x_{ik} + \beta x_{ik}$, can be interpreted as triangulations of $\mathcal{P}(T)$. If we allow variables $x_{ij}$ and$x_{kl}$ to commute only when $i, j, k, l$ are distinct, then the reduced form of $m[T]$ is unique and yields a canonical triangulation of $\mathcal{P}(T)$ in which each simplex corresponds to a noncrossing alternating forest.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2681 https://doi.org/10.46298/dmtcs.2681 Mészáros, Karola Mészáros, Karola <![CDATA[The type $A_n$ root polytope $\mathcal{P}(A_n^+)$ is the convex hull in $\mathbb{R}^{n+1}$ of the origin and the points $e_i-e_j$ for $1 \leq i < j \leq n+1$. Given a tree $T$ on vertex set $[n+1]$, the associated root polytope $\mathcal{P}(T)$ is the intersection of $\mathcal{P}(A_n^+)$ with the cone generated by the vectors $e_i-e_j$, where $(i, j) \in E(T)$, $i < j$. The reduced forms of a certain monomial $m[T]$ in commuting variables $x_{ij}$ under the reduction $x_{ij} x_{jk} \to x_{ik} x_{ij} + x_{jk} x_{ik} + \beta x_{ik}$, can be interpreted as triangulations of $\mathcal{P}(T)$. If we allow variables $x_{ij}$ and$x_{kl}$ to commute only when $i, j, k, l$ are distinct, then the reduced form of $m[T]$ is unique and yields a canonical triangulation of $\mathcal{P}(T)$ in which each simplex corresponds to a noncrossing alternating forest.]]> 0 <![CDATA[Using resolutions of singularities introduced by Cortez and a method for calculating Kazhdan-Lusztig polynomials due to Polo, we prove the conjecture of Billey and Braden characterizing permutations w with Kazhdan-Lusztig polynomial$ P_id,w(q)=1+q^h$ for some $h$.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2705 https://doi.org/10.46298/dmtcs.2705 Woo, Alexander Woo, Alexander <![CDATA[Using resolutions of singularities introduced by Cortez and a method for calculating Kazhdan-Lusztig polynomials due to Polo, we prove the conjecture of Billey and Braden characterizing permutations w with Kazhdan-Lusztig polynomial$ P_id,w(q)=1+q^h$ for some $h$.]]> 0 <![CDATA[We consider Buch's rule for K-theory of the Grassmannian, in the Schur multiplicity-free cases classified by Stembridge. Using a result of Knutson, one sees that Buch's coefficients are related to Möbius inversion. We give a direct combinatorial proof of this by considering the product expansion for Grassmannian Grothendieck polynomials. We end with an extension to the multiplicity-free cases of Thomas and Yong.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2713 https://doi.org/10.46298/dmtcs.2713 Snider, Michelle Snider, Michelle <![CDATA[We consider Buch's rule for K-theory of the Grassmannian, in the Schur multiplicity-free cases classified by Stembridge. Using a result of Knutson, one sees that Buch's coefficients are related to Möbius inversion. We give a direct combinatorial proof of this by considering the product expansion for Grassmannian Grothendieck polynomials. We end with an extension to the multiplicity-free cases of Thomas and Yong.]]> 0 <![CDATA[The associahedron is an object that has been well studied and has numerous applications, particularly in the theory of operads, the study of non-crossing partitions, lattice theory and more recently in the study of cluster algebras. We approach the associahedron from the point of view of discrete homotopy theory, that is we consider 5-cycles in the 1-skeleton of the associahedron to be combinatorial holes, but 4-cycles to be contractible. We give a simple description of the equivalence classes of 5-cycles in the 1-skeleton and then identify a set of 5-cycles from which we may produce all other cycles. This set of 5-cycle equivalence classes turns out to be the generating set for the abelianization of the discrete fundamental group of the associahedron. In this paper we provide presentations for the discrete fundamental group and the abelianization of the discrete fundamental group. We also discuss applications to cluster algebras as well as generalizations to type B and D associahedra. \par]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2712 https://doi.org/10.46298/dmtcs.2712 Severs, Christopher White, Jacob Severs, Christopher White, Jacob <![CDATA[The associahedron is an object that has been well studied and has numerous applications, particularly in the theory of operads, the study of non-crossing partitions, lattice theory and more recently in the study of cluster algebras. We approach the associahedron from the point of view of discrete homotopy theory, that is we consider 5-cycles in the 1-skeleton of the associahedron to be combinatorial holes, but 4-cycles to be contractible. We give a simple description of the equivalence classes of 5-cycles in the 1-skeleton and then identify a set of 5-cycles from which we may produce all other cycles. This set of 5-cycle equivalence classes turns out to be the generating set for the abelianization of the discrete fundamental group of the associahedron. In this paper we provide presentations for the discrete fundamental group and the abelianization of the discrete fundamental group. We also discuss applications to cluster algebras as well as generalizations to type B and D associahedra. \par]]> 0 <![CDATA[In this paper, we study k-parabolic arrangements, a generalization of the k-equal arrangement for any finite real reflection group. When k=2, these arrangements correspond to the well-studied Coxeter arrangements. Brieskorn (1971) showed that the fundamental group of the complement of the type W Coxeter arrangement (over $\mathbb{C}$) is isomorphic to the pure Artin group of type W. Khovanov (1996) gave an algebraic description for the fundamental group of the complement of the 3-equal arrangement (over $\mathbb{R}$). We generalize Khovanov's result to obtain an algebraic description of the fundamental group of the complement of the 3-parabolic arrangement for arbitrary finite reflection group. Our description is a real analogue to Brieskorn's description.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2711 https://doi.org/10.46298/dmtcs.2711 Severs, Christopher White, Jacob Severs, Christopher White, Jacob <![CDATA[In this paper, we study k-parabolic arrangements, a generalization of the k-equal arrangement for any finite real reflection group. When k=2, these arrangements correspond to the well-studied Coxeter arrangements. Brieskorn (1971) showed that the fundamental group of the complement of the type W Coxeter arrangement (over $\mathbb{C}$) is isomorphic to the pure Artin group of type W. Khovanov (1996) gave an algebraic description for the fundamental group of the complement of the 3-equal arrangement (over $\mathbb{R}$). We generalize Khovanov's result to obtain an algebraic description of the fundamental group of the complement of the 3-parabolic arrangement for arbitrary finite reflection group. Our description is a real analogue to Brieskorn's description.]]> 0 <![CDATA[This work is devoted to the study of typical properties of random graphs from classes with structural constraints, like for example planar graphs, with the additional restriction that the average degree is fixed. More precisely, within a general analytic framework, we provide sharp concentration results for the number of blocks (maximal biconnected subgraphs) in a random graph from the class in question. Among other results, we discover that essentially such a random graph belongs with high probability to only one of two possible types: it either has blocks of at most logarithmic size, or there is a \emphgiant block that contains linearly many vertices, and all other blocks are significantly smaller. This extends and generalizes the results in the previous work [K. Panagiotou and A. Steger. Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '09), pp. 432-440, 2009], where similar statements were shown without the restriction on the average degree.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2710 https://doi.org/10.46298/dmtcs.2710 Panagiotou, Konstantinos Panagiotou, Konstantinos <![CDATA[This work is devoted to the study of typical properties of random graphs from classes with structural constraints, like for example planar graphs, with the additional restriction that the average degree is fixed. More precisely, within a general analytic framework, we provide sharp concentration results for the number of blocks (maximal biconnected subgraphs) in a random graph from the class in question. Among other results, we discover that essentially such a random graph belongs with high probability to only one of two possible types: it either has blocks of at most logarithmic size, or there is a \emphgiant block that contains linearly many vertices, and all other blocks are significantly smaller. This extends and generalizes the results in the previous work [K. Panagiotou and A. Steger. Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '09), pp. 432-440, 2009], where similar statements were shown without the restriction on the average degree.]]> 0 <![CDATA[We define a new lattice structure (W,\preceq ) on the elements of a finite Coxeter group W. This lattice, called the \emphshard intersection order, is weaker than the weak order and has the noncrossing partition lattice \NC (W) as a sublattice. The new construction of \NC (W) yields a new proof that \NC (W) is a lattice. The shard intersection order is graded and its rank generating function is the W-Eulerian polynomial. Many order-theoretic properties of (W,\preceq ), like Möbius number, number of maximal chains, etc., are exactly analogous to the corresponding properties of \NC (W). There is a natural dimension-preserving bijection between simplices in the order complex of (W,\preceq ) (i.e. chains in (W,\preceq )) and simplices in a certain pulling triangulation of the W-permutohedron. Restricting the bijection to the order complex of \NC (W) yields a bijection to simplices in a pulling triangulation of the W-associahedron. The lattice (W,\preceq ) is defined indirectly via the polyhedral geometry of the reflecting hyperplanes of W\!. Indeed, most of the results of the paper are proven in the more general setting of simplicial hyperplane arrangements.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2709 https://doi.org/10.46298/dmtcs.2709 Reading, Nathan Reading, Nathan <![CDATA[We define a new lattice structure (W,\preceq ) on the elements of a finite Coxeter group W. This lattice, called the \emphshard intersection order, is weaker than the weak order and has the noncrossing partition lattice \NC (W) as a sublattice. The new construction of \NC (W) yields a new proof that \NC (W) is a lattice. The shard intersection order is graded and its rank generating function is the W-Eulerian polynomial. Many order-theoretic properties of (W,\preceq ), like Möbius number, number of maximal chains, etc., are exactly analogous to the corresponding properties of \NC (W). There is a natural dimension-preserving bijection between simplices in the order complex of (W,\preceq ) (i.e. chains in (W,\preceq )) and simplices in a certain pulling triangulation of the W-permutohedron. Restricting the bijection to the order complex of \NC (W) yields a bijection to simplices in a pulling triangulation of the W-associahedron. The lattice (W,\preceq ) is defined indirectly via the polyhedral geometry of the reflecting hyperplanes of W\!. Indeed, most of the results of the paper are proven in the more general setting of simplicial hyperplane arrangements.]]> 0 <![CDATA[We introduce a shifted analog of the plactic monoid of Lascoux and Schützenberger, the \emphshifted plactic monoid. It can be defined in two different ways: via the \emphshifted Knuth relations, or using Haiman's mixed insertion. Applications include: a new combinatorial derivation (and a new version of) the shifted Littlewood-Richardson Rule; similar results for the coefficients in the Schur expansion of a Schur P-function; a shifted counterpart of the Lascoux-Schützenberger theory of noncommutative Schur functions in plactic variables; a characterization of shifted tableau words; and more.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2708 https://doi.org/10.46298/dmtcs.2708 Serrano, Luis Serrano, Luis <![CDATA[We introduce a shifted analog of the plactic monoid of Lascoux and Schützenberger, the \emphshifted plactic monoid. It can be defined in two different ways: via the \emphshifted Knuth relations, or using Haiman's mixed insertion. Applications include: a new combinatorial derivation (and a new version of) the shifted Littlewood-Richardson Rule; similar results for the coefficients in the Schur expansion of a Schur P-function; a shifted counterpart of the Lascoux-Schützenberger theory of noncommutative Schur functions in plactic variables; a characterization of shifted tableau words; and more.]]> 0 <![CDATA[Alternating sign matrices (ASMs) are square matrices with entries 0, 1, or -1 whose rows and columns sum to 1 and whose nonzero entries alternate in sign. We put ASMs into a larger context by studying the order ideals of subposets of a certain poset, proving that they are in bijection with a variety of interesting combinatorial objects, including ASMs, totally symmetric self―complementary plane partitions (TSSCPPs), Catalan objects, tournaments, semistandard Young tableaux, and totally symmetric plane partitions. We use this perspective to prove an expansion of the tournament generating function as a sum over TSSCPPs which is analogous to a known formula involving ASMs.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2707 https://doi.org/10.46298/dmtcs.2707 Striker, Jessica Striker, Jessica <![CDATA[Alternating sign matrices (ASMs) are square matrices with entries 0, 1, or -1 whose rows and columns sum to 1 and whose nonzero entries alternate in sign. We put ASMs into a larger context by studying the order ideals of subposets of a certain poset, proving that they are in bijection with a variety of interesting combinatorial objects, including ASMs, totally symmetric self―complementary plane partitions (TSSCPPs), Catalan objects, tournaments, semistandard Young tableaux, and totally symmetric plane partitions. We use this perspective to prove an expansion of the tournament generating function as a sum over TSSCPPs which is analogous to a known formula involving ASMs.]]> 0 <![CDATA[Postnikov constructed a decomposition of a totally nonnegative Grassmannian $(Gr _{kn})_≥0$ into positroid cells. We provide combinatorial formulas that allow one to decide which cell a given point in $(Gr _{kn})_≥0$ belongs to and to determine affine coordinates of the point within this cell. This simplifies Postnikov's description of the inverse boundary measurement map and generalizes formulas for the top cell given by Speyer and Williams. In addition, we identify a particular subset of Plücker coordinates as a totally positive base for the set of non-vanishing Plücker coordinates for a given positroid cell.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2706 https://doi.org/10.46298/dmtcs.2706 Talaska, Kelli Talaska, Kelli <![CDATA[Postnikov constructed a decomposition of a totally nonnegative Grassmannian $(Gr _{kn})_≥0$ into positroid cells. We provide combinatorial formulas that allow one to decide which cell a given point in $(Gr _{kn})_≥0$ belongs to and to determine affine coordinates of the point within this cell. This simplifies Postnikov's description of the inverse boundary measurement map and generalizes formulas for the top cell given by Speyer and Williams. In addition, we identify a particular subset of Plücker coordinates as a totally positive base for the set of non-vanishing Plücker coordinates for a given positroid cell.]]> 0 <![CDATA[Recently Postnikov gave a combinatorial description of the cells in a totally-nonnegative Grassmannian. These cells correspond to a special class of matroids called positroids. There are many interesting combinatorial objects associated to a positroid. We introduce some recent results, including the generalization and proof of the purity conjecture by Leclerc and Zelevinsky on weakly separated sets.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2697 https://doi.org/10.46298/dmtcs.2697 Oh, Suho Oh, Suho <![CDATA[Recently Postnikov gave a combinatorial description of the cells in a totally-nonnegative Grassmannian. These cells correspond to a special class of matroids called positroids. There are many interesting combinatorial objects associated to a positroid. We introduce some recent results, including the generalization and proof of the purity conjecture by Leclerc and Zelevinsky on weakly separated sets.]]> 0 <![CDATA[We introduce a combinatorial way of calculating the Hilbert series of bigraded $S_n$-modules as a weighted sum over standard Young tableaux in the hook shape case. This method is based on Macdonald formula for Hall-Littlewood polynomial and extends the result of $A$. Garsia and $C$. Procesi for the Hilbert series when $q=0$. Moreover, we give the way of associating the fillings giving the monomial terms of Macdonald polynomials to the standard Young tableaux.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2704 https://doi.org/10.46298/dmtcs.2704 Yoo, Meesue Yoo, Meesue <![CDATA[We introduce a combinatorial way of calculating the Hilbert series of bigraded $S_n$-modules as a weighted sum over standard Young tableaux in the hook shape case. This method is based on Macdonald formula for Hall-Littlewood polynomial and extends the result of $A$. Garsia and $C$. Procesi for the Hilbert series when $q=0$. Moreover, we give the way of associating the fillings giving the monomial terms of Macdonald polynomials to the standard Young tableaux.]]> 0 <![CDATA[We show that dual canonical basis elements of the quantum polynomial ring in $n^2$ variables can be expressed as specializations of dual canonical basis elements of $0$-weight spaces of other quantum polynomial rings. Our results rely upon the natural appearance in the quantum polynomial ring of Kazhdan-Lusztig polynomials, $R$-polynomials, and certain single and double parabolic generalizations of these.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2703 https://doi.org/10.46298/dmtcs.2703 Skandera, Mark Lambright, Justin Skandera, Mark Lambright, Justin <![CDATA[We show that dual canonical basis elements of the quantum polynomial ring in $n^2$ variables can be expressed as specializations of dual canonical basis elements of $0$-weight spaces of other quantum polynomial rings. Our results rely upon the natural appearance in the quantum polynomial ring of Kazhdan-Lusztig polynomials, $R$-polynomials, and certain single and double parabolic generalizations of these.]]> 0 <![CDATA[In a recent paper, Schilling proposed an operator $\overline{\mathrm{pr}}$ on unrestricted rigged configurations $RC$, and conjectured it to be the promotion operator of the type $A$ crystal formed by $RC$. In this paper we announce a proof for this conjecture.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2702 https://doi.org/10.46298/dmtcs.2702 Wang, Qiang Wang, Qiang <![CDATA[In a recent paper, Schilling proposed an operator $\overline{\mathrm{pr}}$ on unrestricted rigged configurations $RC$, and conjectured it to be the promotion operator of the type $A$ crystal formed by $RC$. In this paper we announce a proof for this conjecture.]]> 0 <![CDATA[We show that if the cardinality of a subset of the $(2k-1)$-dimensional vector space over a finite field with $q$ elements is $\gg q^{2k-1-\frac{1}{ 2k}}$, then it contains a positive proportional of all $k$-simplexes up to congruence.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2701 https://doi.org/10.46298/dmtcs.2701 Vinh, Le Anh Vinh, Le Anh <![CDATA[We show that if the cardinality of a subset of the $(2k-1)$-dimensional vector space over a finite field with $q$ elements is $\gg q^{2k-1-\frac{1}{ 2k}}$, then it contains a positive proportional of all $k$-simplexes up to congruence.]]> 0 <![CDATA[The recent work of Bonnafé et al. (2007) shows through two conjectures that $r$-domino tableaux have an important role in Kazhdan-Lusztig theory of type $B$ with unequal parameters. In this paper we provide plactic relations on signed permutations which determine whether given two signed permutations have the same insertion $r$-domino tableaux in Garfinkle's algorithm (1990). Moreover, we show that a particular extension of these relations can describe Garfinkle's equivalence relation on $r$-domino tableaux which is given through the notion of open cycles. With these results we enunciate the conjectures of Bonnafé et al. and provide necessary tool for their proofs.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2700 https://doi.org/10.46298/dmtcs.2700 Taşkın, Müge Taşkın, Müge <![CDATA[The recent work of Bonnafé et al. (2007) shows through two conjectures that $r$-domino tableaux have an important role in Kazhdan-Lusztig theory of type $B$ with unequal parameters. In this paper we provide plactic relations on signed permutations which determine whether given two signed permutations have the same insertion $r$-domino tableaux in Garfinkle's algorithm (1990). Moreover, we show that a particular extension of these relations can describe Garfinkle's equivalence relation on $r$-domino tableaux which is given through the notion of open cycles. With these results we enunciate the conjectures of Bonnafé et al. and provide necessary tool for their proofs.]]> 0 <![CDATA[We aim to generalize a theorem on the number of rooted spanning forests of a highly symmetric graph to the case of asymmetric graphs. We show that this can be achieved by means of an identity between the minor determinants of a Laplace matrix, for which we provide two different (combinatorial as well as algebraic) proofs in the simplest case. Furthermore, we discuss the connections to electrical networks and the enumeration of spanning trees in sequences of self-similar graphs.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2699 https://doi.org/10.46298/dmtcs.2699 Teufl, Elmar Wagner, Stephan Teufl, Elmar Wagner, Stephan <![CDATA[We aim to generalize a theorem on the number of rooted spanning forests of a highly symmetric graph to the case of asymmetric graphs. We show that this can be achieved by means of an identity between the minor determinants of a Laplace matrix, for which we provide two different (combinatorial as well as algebraic) proofs in the simplest case. Furthermore, we discuss the connections to electrical networks and the enumeration of spanning trees in sequences of self-similar graphs.]]> 0 <![CDATA[It is becoming increasingly clear that the supercharacter theory of the finite group of unipotent upper-triangular matrices has a rich combinatorial structure built on set-partitions that is analogous to the partition combinatorics of the classical representation theory of the symmetric group. This paper begins by exploring a connection to the ring of symmetric functions in non-commuting variables that mirrors the symmetric group's relationship with the ring of symmetric functions. It then also investigates some of the representation theoretic structure constants arising from the restriction, tensor products and superinduction of supercharacters.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2698 https://doi.org/10.46298/dmtcs.2698 Thiem, Nathaniel Thiem, Nathaniel <![CDATA[It is becoming increasingly clear that the supercharacter theory of the finite group of unipotent upper-triangular matrices has a rich combinatorial structure built on set-partitions that is analogous to the partition combinatorics of the classical representation theory of the symmetric group. This paper begins by exploring a connection to the ring of symmetric functions in non-commuting variables that mirrors the symmetric group's relationship with the ring of symmetric functions. It then also investigates some of the representation theoretic structure constants arising from the restriction, tensor products and superinduction of supercharacters.]]> 0 <![CDATA[For a fixed sequence of $n$ positive integers $(a,\bar{b}) := (a, b, b,\ldots, b)$, an $(a,\bar{b})$-parking function of length $n$ is a sequence $(p_1, p_2, \ldots, p_n)$ of positive integers whose nondecreasing rearrangement $q_1 \leq q_2 \leq \cdots \leq q_n$ satisfies $q_i \leq a+(i-1)b$ for any $i=1,\ldots, n$. A $(a,\bar{b})$-forest on $n$-set is a rooted vertex-colored forests on $n$-set whose roots are colored with the colors $0, 1, \ldots, a-1$ and the other vertices are colored with the colors $0, 1, \ldots, b-1$. In this paper, we construct a bijection between $(bc,\bar{b})$-parking functions of length $n$ and $(bc,\bar{b})$-forests on $n$-set with some interesting properties. As applications, we obtain a generalization of Gessel and Seo's result about $(c,\bar{1})$-parking functions [Ira M. Gessel and Seunghyun Seo, Electron. J. Combin. $\textbf{11}$(2)R27, 2004] and a refinement of Yan's identity [Catherine H. Yan, Adv. Appl. Math. $\textbf{27}$(2―3):641―670, 2001] between an inversion enumerator for $(bc,\bar{b})$-forests and a complement enumerator for $(bc,\bar{b})$-parking functions.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2714 https://doi.org/10.46298/dmtcs.2714 Shin, Heesung Zeng, Jiang Shin, Heesung Zeng, Jiang <![CDATA[For a fixed sequence of $n$ positive integers $(a,\bar{b}) := (a, b, b,\ldots, b)$, an $(a,\bar{b})$-parking function of length $n$ is a sequence $(p_1, p_2, \ldots, p_n)$ of positive integers whose nondecreasing rearrangement $q_1 \leq q_2 \leq \cdots \leq q_n$ satisfies $q_i \leq a+(i-1)b$ for any $i=1,\ldots, n$. A $(a,\bar{b})$-forest on $n$-set is a rooted vertex-colored forests on $n$-set whose roots are colored with the colors $0, 1, \ldots, a-1$ and the other vertices are colored with the colors $0, 1, \ldots, b-1$. In this paper, we construct a bijection between $(bc,\bar{b})$-parking functions of length $n$ and $(bc,\bar{b})$-forests on $n$-set with some interesting properties. As applications, we obtain a generalization of Gessel and Seo's result about $(c,\bar{1})$-parking functions [Ira M. Gessel and Seunghyun Seo, Electron. J. Combin. $\textbf{11}$(2)R27, 2004] and a refinement of Yan's identity [Catherine H. Yan, Adv. Appl. Math. $\textbf{27}$(2―3):641―670, 2001] between an inversion enumerator for $(bc,\bar{b})$-forests and a complement enumerator for $(bc,\bar{b})$-parking functions.]]> 0 <![CDATA[Surveying the results of three recent papers and some currently ongoing research, we show how a generalization of Brylawski's tensor product formula to colored graphs may be used to compute the Jones polynomial of some fairly complicated knots and, in the future, even virtual knots.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2715 https://doi.org/10.46298/dmtcs.2715 Hetyei, Gábor Diao, Yuanan Hinson, Kenneth Hetyei, Gábor Diao, Yuanan Hinson, Kenneth <![CDATA[Surveying the results of three recent papers and some currently ongoing research, we show how a generalization of Brylawski's tensor product formula to colored graphs may be used to compute the Jones polynomial of some fairly complicated knots and, in the future, even virtual knots.]]> 0 <![CDATA[Algebraic complexes whose "faces'' are indexed by partitions and plane partitions are introduced, and their homology is proven to be concentrated in even dimensions with homology basis indexed by fixed points of an involution, thereby explaining topologically two quite important instances of Stembridge's $q=-1$ phenomenon. A more general framework of invariant and coinvariant complexes with coefficients taken $\mod 2$ is developed, and as a part of this story an analogous topological result for necklaces is conjectured.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2716 https://doi.org/10.46298/dmtcs.2716 Hersh, P. Shareshian, J. Stanton, D. Hersh, P. Shareshian, J. Stanton, D. <![CDATA[Algebraic complexes whose "faces'' are indexed by partitions and plane partitions are introduced, and their homology is proven to be concentrated in even dimensions with homology basis indexed by fixed points of an involution, thereby explaining topologically two quite important instances of Stembridge's $q=-1$ phenomenon. A more general framework of invariant and coinvariant complexes with coefficients taken $\mod 2$ is developed, and as a part of this story an analogous topological result for necklaces is conjectured.]]> 0 <![CDATA[In this paper, we introduce a new model of the crystal $B(\Lambda _0)$ of $\widehat{\mathfrak{sl}_{\ell}}$. We briefly describe some of the properties of this crystal and compare it to the combinatorial model of Misra and Miwa.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2717 https://doi.org/10.46298/dmtcs.2717 Berg, Chris Berg, Chris <![CDATA[In this paper, we introduce a new model of the crystal $B(\Lambda _0)$ of $\widehat{\mathfrak{sl}_{\ell}}$. We briefly describe some of the properties of this crystal and compare it to the combinatorial model of Misra and Miwa.]]> 0 <![CDATA[A shuffle of two words is a word obtained by concatenating the two original words in either order and then sliding any letters from the second word back past letters of the first word, in such a way that the letters of each original word remain spelled out in their original relative order. Examples of shuffles of the words $1234$ and $5678$ are, for instance, $15236784$ and $51236748$. In this paper, we enumerate the distinct shuffles of two permutations of any two lengths, where the permutations are written as words in the letters $1,2,3,\ldots ,m$ and $1,2,3,\ldots ,n$, respectively.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2718 https://doi.org/10.46298/dmtcs.2718 Smith Barnes, Camillia Smith Barnes, Camillia <![CDATA[A shuffle of two words is a word obtained by concatenating the two original words in either order and then sliding any letters from the second word back past letters of the first word, in such a way that the letters of each original word remain spelled out in their original relative order. Examples of shuffles of the words $1234$ and $5678$ are, for instance, $15236784$ and $51236748$. In this paper, we enumerate the distinct shuffles of two permutations of any two lengths, where the permutations are written as words in the letters $1,2,3,\ldots ,m$ and $1,2,3,\ldots ,n$, respectively.]]> 0 <![CDATA[Let $W$ be a finite crystallographic reflection group, with root system $\Phi$. Associated to $W$ there is a positive integer, the generalized Catalan number, which counts the clusters in the associated cluster algebra, the noncrossing partitions for $W$, and several other interesting sets. Bijections have been found between the clusters and the noncrossing partitions by Reading and Athanasiadis et al. There is a further generalization of the generalized Catalan number, sometimes called the Fuss-Catalan number for $W$, which we will denote $C_m(W)$. Here $m$ is a positive integer, and $C_1(W)$ is the usual generalized Catalan number. $C_m(W)$ counts the $m$-noncrossing partitions for $W$ and the $m$-clusters for $\Phi$. In this abstract, we will give an explicit description of a bijection between these two sets. The proof depends on a representation-theoretic reinterpretation of the problem, in terms of exceptional sequences of representations of quivers.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2719 https://doi.org/10.46298/dmtcs.2719 Bakke Buan, Aslak Reiten, Idun Thomas, Hugh Bakke Buan, Aslak Reiten, Idun Thomas, Hugh <![CDATA[Let $W$ be a finite crystallographic reflection group, with root system $\Phi$. Associated to $W$ there is a positive integer, the generalized Catalan number, which counts the clusters in the associated cluster algebra, the noncrossing partitions for $W$, and several other interesting sets. Bijections have been found between the clusters and the noncrossing partitions by Reading and Athanasiadis et al. There is a further generalization of the generalized Catalan number, sometimes called the Fuss-Catalan number for $W$, which we will denote $C_m(W)$. Here $m$ is a positive integer, and $C_1(W)$ is the usual generalized Catalan number. $C_m(W)$ counts the $m$-noncrossing partitions for $W$ and the $m$-clusters for $\Phi$. In this abstract, we will give an explicit description of a bijection between these two sets. The proof depends on a representation-theoretic reinterpretation of the problem, in terms of exceptional sequences of representations of quivers.]]> 0 <![CDATA[Let $V$ be a complex vector space with basis $\{x_1,x_2,\ldots,x_n\}$ and $G$ be a finite subgroup of $GL(V)$. The tensor algebra $T(V)$ over the complex is isomorphic to the polynomials in the non-commutative variables $x_1, x_2, \ldots, x_n$ with complex coefficients. We want to give a combinatorial interpretation for the decomposition of $T(V)$ into simple $G$-modules. In particular, we want to study the graded space of invariants in $T(V)$ with respect to the action of $G$. We give a general method for decomposing the space $T(V)$ into simple $G$-module in terms of words in a particular Cayley graph of $G$. To apply the method to a particular group, we require a surjective homomorphism from a subalgebra of the group algebra into the character algebra. In the case of the symmetric group, we give an example of this homomorphism from the descent algebra. When $G$ is the dihedral group, we have a realization of the character algebra as a subalgebra of the group algebra. In those two cases, we have an interpretation for the graded dimensions of the invariant space in term of those words.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2720 https://doi.org/10.46298/dmtcs.2720 Bergeron-Brlek, Anouk Hohlweg, Christophe Zabrocki, Mike Bergeron-Brlek, Anouk Hohlweg, Christophe Zabrocki, Mike <![CDATA[Let $V$ be a complex vector space with basis $\{x_1,x_2,\ldots,x_n\}$ and $G$ be a finite subgroup of $GL(V)$. The tensor algebra $T(V)$ over the complex is isomorphic to the polynomials in the non-commutative variables $x_1, x_2, \ldots, x_n$ with complex coefficients. We want to give a combinatorial interpretation for the decomposition of $T(V)$ into simple $G$-modules. In particular, we want to study the graded space of invariants in $T(V)$ with respect to the action of $G$. We give a general method for decomposing the space $T(V)$ into simple $G$-module in terms of words in a particular Cayley graph of $G$. To apply the method to a particular group, we require a surjective homomorphism from a subalgebra of the group algebra into the character algebra. In the case of the symmetric group, we give an example of this homomorphism from the descent algebra. When $G$ is the dihedral group, we have a realization of the character algebra as a subalgebra of the group algebra. In those two cases, we have an interpretation for the graded dimensions of the invariant space in term of those words.]]> 0 <![CDATA[We define a poset using the shortest paths in the Bruhat graph of a finite Coxeter group $W$ from the identity to the longest word in $W, w_0$. We show that this poset is the union of Boolean posets of rank absolute length of $w_0$; that is, any shortest path labeled by reflections $t_1,\ldots,t_m$ is fully commutative. This allows us to give a combinatorial interpretation to the lowest-degree terms in the complete $\textbf{cd}$-index of $W$.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2721 https://doi.org/10.46298/dmtcs.2721 Blanco, Saúl A. Blanco, Saúl A. <![CDATA[We define a poset using the shortest paths in the Bruhat graph of a finite Coxeter group $W$ from the identity to the longest word in $W, w_0$. We show that this poset is the union of Boolean posets of rank absolute length of $w_0$; that is, any shortest path labeled by reflections $t_1,\ldots,t_m$ is fully commutative. This allows us to give a combinatorial interpretation to the lowest-degree terms in the complete $\textbf{cd}$-index of $W$.]]> 0 <![CDATA[To a word $w$, we associate the rational function $\Psi_w = \prod (x_{w_i} - x_{w_{i+1}})^{-1}$. The main object, introduced by C. Greene to generalize identities linked to Murnaghan-Nakayama rule, is a sum of its images by certain permutations of the variables. The sets of permutations that we consider are the linear extensions of oriented graphs. We explain how to compute this rational function, using the combinatorics of the graph $G$. We also establish a link between an algebraic property of the rational function (the factorization of the numerator) and a combinatorial property of the graph (the existence of a disconnecting chain).]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2722 https://doi.org/10.46298/dmtcs.2722 Boussicault, Adrien Féray, Valentin Boussicault, Adrien Féray, Valentin <![CDATA[To a word $w$, we associate the rational function $\Psi_w = \prod (x_{w_i} - x_{w_{i+1}})^{-1}$. The main object, introduced by C. Greene to generalize identities linked to Murnaghan-Nakayama rule, is a sum of its images by certain permutations of the variables. The sets of permutations that we consider are the linear extensions of oriented graphs. We explain how to compute this rational function, using the combinatorics of the graph $G$. We also establish a link between an algebraic property of the rational function (the factorization of the numerator) and a combinatorial property of the graph (the existence of a disconnecting chain).]]> 0 <![CDATA[We present statistic-preserving bijections between four classes of combinatorial objects. Two of them, the class of unlabeled $(\textrm{2+2})$-free posets and a certain class of chord diagrams (or involutions), already appeared in the literature, but were apparently not known to be equinumerous. The third one is a new class of pattern avoiding permutations, and the fourth one consists of certain integer sequences called $\textit{ascent sequences}$. We also determine the generating function of these classes of objects, thus recovering a non-D-finite series obtained by Zagier for chord diagrams. Finally, we characterize the ascent sequences that correspond to permutations avoiding the barred pattern $3\bar{1}52\bar{4}$, and enumerate those permutations, thus settling a conjecture of Pudwell.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2723 https://doi.org/10.46298/dmtcs.2723 Bousquet-Mélou, Mireille Claesson, Anders Dukes, Mark Kitaev, Sergey Bousquet-Mélou, Mireille Claesson, Anders Dukes, Mark Kitaev, Sergey <![CDATA[We present statistic-preserving bijections between four classes of combinatorial objects. Two of them, the class of unlabeled $(\textrm{2+2})$-free posets and a certain class of chord diagrams (or involutions), already appeared in the literature, but were apparently not known to be equinumerous. The third one is a new class of pattern avoiding permutations, and the fourth one consists of certain integer sequences called $\textit{ascent sequences}$. We also determine the generating function of these classes of objects, thus recovering a non-D-finite series obtained by Zagier for chord diagrams. Finally, we characterize the ascent sequences that correspond to permutations avoiding the barred pattern $3\bar{1}52\bar{4}$, and enumerate those permutations, thus settling a conjecture of Pudwell.]]> 0 <![CDATA[We propose an $\textit{experimental mathematics approach}$ leading to the computer-driven $\textit{discovery}$ of various conjectures about structural properties of generating functions coming from enumeration of restricted lattice walks in 2D and in 3D.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2724 https://doi.org/10.46298/dmtcs.2724 Bostan, Alin Kauers, Manuel Bostan, Alin Kauers, Manuel <![CDATA[We propose an $\textit{experimental mathematics approach}$ leading to the computer-driven $\textit{discovery}$ of various conjectures about structural properties of generating functions coming from enumeration of restricted lattice walks in 2D and in 3D.]]> 0 <![CDATA[Benkart, Sottile, and Stroomer have completely characterized by Knuth and dual Knuth equivalence a bijective proof of the Littlewood―Richardson coefficient conjugation symmetry, i.e. $c_{\mu, \nu}^{\lambda} =c_{\mu^t,\nu^t}^{\lambda ^t}$. Tableau―switching provides an algorithm to produce such a bijective proof. Fulton has shown that the White and the Hanlon―Sundaram maps are versions of that bijection. In this paper one exhibits explicitly the Yamanouchi word produced by that conjugation symmetry map which on its turn leads to a new and very natural version of the same map already considered independently. A consequence of this latter construction is that using notions of Relative Computational Complexity we are allowed to show that this conjugation symmetry map is linear time reducible to the Schützenberger involution and reciprocally. Thus the Benkart―Sottile―Stroomer conjugation symmetry map with the two mentioned versions, the three versions of the commutative symmetry map, and Schützenberger involution, are linear time reducible to each other. This answers a question posed by Pak and Vallejo.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2725 https://doi.org/10.46298/dmtcs.2725 Azenhas, Olga Conflitti, Alessandro Mamede, Ricardo Azenhas, Olga Conflitti, Alessandro Mamede, Ricardo <![CDATA[Benkart, Sottile, and Stroomer have completely characterized by Knuth and dual Knuth equivalence a bijective proof of the Littlewood―Richardson coefficient conjugation symmetry, i.e. $c_{\mu, \nu}^{\lambda} =c_{\mu^t,\nu^t}^{\lambda ^t}$. Tableau―switching provides an algorithm to produce such a bijective proof. Fulton has shown that the White and the Hanlon―Sundaram maps are versions of that bijection. In this paper one exhibits explicitly the Yamanouchi word produced by that conjugation symmetry map which on its turn leads to a new and very natural version of the same map already considered independently. A consequence of this latter construction is that using notions of Relative Computational Complexity we are allowed to show that this conjugation symmetry map is linear time reducible to the Schützenberger involution and reciprocally. Thus the Benkart―Sottile―Stroomer conjugation symmetry map with the two mentioned versions, the three versions of the commutative symmetry map, and Schützenberger involution, are linear time reducible to each other. This answers a question posed by Pak and Vallejo.]]> 0 <![CDATA[The aim of this work is to enumerate alternating sign matrices (ASM) that are quasi-invariant under a quarter-turn. The enumeration formula (conjectured by Duchon) involves, as a product of three terms, the number of unrestrited ASm's and the number of half-turn symmetric ASM's.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2726 https://doi.org/10.46298/dmtcs.2726 Aval, Jean-Christophe Duchon, Philippe Aval, Jean-Christophe Duchon, Philippe <![CDATA[The aim of this work is to enumerate alternating sign matrices (ASM) that are quasi-invariant under a quarter-turn. The enumeration formula (conjectured by Duchon) involves, as a product of three terms, the number of unrestrited ASm's and the number of half-turn symmetric ASM's.]]> 0 <![CDATA[We define a universal cycle for a class of $n$-permutations as a cyclic word in which each element of the class occurs exactly once as an $n$-factor. We give a general result for cyclically closed classes, and then survey the situation when the class is defined as the avoidance class of a set of permutations of length $3$, or of a set of permutations of mixed lengths $3$ and $4$.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2727 https://doi.org/10.46298/dmtcs.2727 Albert, Michael West, Julian Albert, Michael West, Julian <![CDATA[We define a universal cycle for a class of $n$-permutations as a cyclic word in which each element of the class occurs exactly once as an $n$-factor. We give a general result for cyclically closed classes, and then survey the situation when the class is defined as the avoidance class of a set of permutations of length $3$, or of a set of permutations of mixed lengths $3$ and $4$.]]> 0 <![CDATA[In this article we study a class of monoids that includes Garside monoids, and give a simple combinatorial proof of a formula for the formal sum of all elements of the monoid. This leads to a formula for the growth function of the monoid in the homogeneous case, and can also be lifted to a resolution of the monoid algebra. These results are then applied to known monoids related to Coxeter systems: we give the growth function of the Artin-Tits monoids, and do the same for the dual braid monoids. In this last case we show that the monoid algebras of the dual braid monoids of type A and B are Koszul algebras.]]> Wed, 31 Dec 2008 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2728 https://doi.org/10.46298/dmtcs.2728 Albenque, Marie Nadeau, Philippe Albenque, Marie Nadeau, Philippe <![CDATA[In this article we study a class of monoids that includes Garside monoids, and give a simple combinatorial proof of a formula for the formal sum of all elements of the monoid. This leads to a formula for the growth function of the monoid in the homogeneous case, and can also be lifted to a resolution of the monoid algebra. These results are then applied to known monoids related to Coxeter systems: we give the growth function of the Artin-Tits monoids, and do the same for the dual braid monoids. In this last case we show that the monoid algebras of the dual braid monoids of type A and B are Koszul algebras.]]> 0 <![CDATA[Let M be an additive abelian group. An M-strong-oriented coloring of an oriented graph G is a mapping f from V(G) to M such that f(u) <> j(v) whenever uv is an arc in G and f(v)−f(u) <> −(f(t)−f(z)) whenever uv and zt are two arcs in G. The strong oriented chromatic number of an oriented graph is the minimal order of a group M such that G has an M-strong-oriented coloring. This notion was introduced by Nesetril and Raspaud [Ann. Inst. Fourier, 49(3):1037-1056, 1999]. We prove that the strong oriented chromatic number of oriented planar graphs without cycles of lengths 4 to 12 (resp. 4 or 6) is at most 7 (resp. 19). Moreover, for all i ≥ 4, we construct outerplanar graphs without cycles of lengths 4 to i whose oriented chromatic number is 7.]]> Thu, 10 Jan 2008 07:00:00 +0000 https://doi.org/10.46298/dmtcs.418 https://doi.org/10.46298/dmtcs.418 Montassier, Mickael Ochem, Pascal Pinlou, Alexandre Montassier, Mickael Ochem, Pascal Pinlou, Alexandre <![CDATA[Let M be an additive abelian group. An M-strong-oriented coloring of an oriented graph G is a mapping f from V(G) to M such that f(u) <> j(v) whenever uv is an arc in G and f(v)−f(u) <> −(f(t)−f(z)) whenever uv and zt are two arcs in G. The strong oriented chromatic number of an oriented graph is the minimal order of a group M such that G has an M-strong-oriented coloring. This notion was introduced by Nesetril and Raspaud [Ann. Inst. Fourier, 49(3):1037-1056, 1999]. We prove that the strong oriented chromatic number of oriented planar graphs without cycles of lengths 4 to 12 (resp. 4 or 6) is at most 7 (resp. 19). Moreover, for all i ≥ 4, we construct outerplanar graphs without cycles of lengths 4 to i whose oriented chromatic number is 7.]]> 0 <![CDATA[For Dyck paths (nonnegative symmetric) random walks, the location of the first maximum within the first sojourn is studied. Generating functions and explicit resp. asymptotic expressions for the average are derived. Related parameters are also discussed.]]> Tue, 01 Jan 2008 07:00:00 +0000 https://doi.org/10.46298/dmtcs.426 https://doi.org/10.46298/dmtcs.426 Prodinger, Helmut Prodinger, Helmut <![CDATA[For Dyck paths (nonnegative symmetric) random walks, the location of the first maximum within the first sojourn is studied. Generating functions and explicit resp. asymptotic expressions for the average are derived. Related parameters are also discussed.]]> 0 <![CDATA[We consider subshifts of the full shift of all binary bi-infinite sequences. On the one hand, the topological entropy of any subshift with computably co-enumerable language is a right-computable real number between 0 and 1. We show that, on the other hand, any right-computable real number between 0 and 1, whether computable or not, is the entropy of some subshift with even polynomial time decidable language. In addition, we show that computability of the entropy of a subshift does not imply any kind of computability of the language of the subshift]]> Tue, 01 Jan 2008 07:00:00 +0000 https://doi.org/10.46298/dmtcs.425 https://doi.org/10.46298/dmtcs.425 Hertling, Peter Spandl, Christoph Hertling, Peter Spandl, Christoph <![CDATA[We consider subshifts of the full shift of all binary bi-infinite sequences. On the one hand, the topological entropy of any subshift with computably co-enumerable language is a right-computable real number between 0 and 1. We show that, on the other hand, any right-computable real number between 0 and 1, whether computable or not, is the entropy of some subshift with even polynomial time decidable language. In addition, we show that computability of the entropy of a subshift does not imply any kind of computability of the language of the subshift]]> 0 <![CDATA[By application of the Markov-WZ method, we prove a more general form of a bivariate generating function identity containing, as particular cases, Koecher's and Almkvist-Granville's Apéry-like formulae for odd zeta values. As a consequence, we get a new identity producing Apéry-like series for all ζ(2n+4m+3),n,m ≥ 0, convergent at the geometric rate with ratio 2−10.]]> Tue, 01 Jan 2008 07:00:00 +0000 https://doi.org/10.46298/dmtcs.424 https://doi.org/10.46298/dmtcs.424 Pilehrood, Khodabakhsh Hessami Pilehrood, Tatiana Hessami Pilehrood, Khodabakhsh Hessami Pilehrood, Tatiana Hessami <![CDATA[By application of the Markov-WZ method, we prove a more general form of a bivariate generating function identity containing, as particular cases, Koecher's and Almkvist-Granville's Apéry-like formulae for odd zeta values. As a consequence, we get a new identity producing Apéry-like series for all ζ(2n+4m+3),n,m ≥ 0, convergent at the geometric rate with ratio 2−10.]]> 0 <![CDATA[A planar k-restricted structure is a simple graph whose blocks are planar and each has at most k vertices. Planar k-restricted structures are used by approximation algorithms for Maximum Weight Planar Subgraph, which motivates this work. The planar k-restricted ratio is the infimum, over simple planar graphs H, of the ratio of the number of edges in a maximum k-restricted structure subgraph of H to the number edges of H. We prove that, as k tends to infinity, the planar k-restricted ratio tends to 1 = 2. The same result holds for the weighted version. Our results are based on analyzing the analogous ratios for outerplanar and weighted outerplanar graphs. Here both ratios tend to 1 as k goes to infinity, and we provide good estimates of the rates of convergence, showing that they differ in the weighted from the unweighted case.]]> Tue, 01 Jan 2008 07:00:00 +0000 https://doi.org/10.46298/dmtcs.423 https://doi.org/10.46298/dmtcs.423 Călinescu, Gruia Fernandes, Cristina G. Călinescu, Gruia Fernandes, Cristina G. <![CDATA[A planar k-restricted structure is a simple graph whose blocks are planar and each has at most k vertices. Planar k-restricted structures are used by approximation algorithms for Maximum Weight Planar Subgraph, which motivates this work. The planar k-restricted ratio is the infimum, over simple planar graphs H, of the ratio of the number of edges in a maximum k-restricted structure subgraph of H to the number edges of H. We prove that, as k tends to infinity, the planar k-restricted ratio tends to 1 = 2. The same result holds for the weighted version. Our results are based on analyzing the analogous ratios for outerplanar and weighted outerplanar graphs. Here both ratios tend to 1 as k goes to infinity, and we provide good estimates of the rates of convergence, showing that they differ in the weighted from the unweighted case.]]> 0 <![CDATA[Let D ∈ D(n, p) denote a simple random digraph obtained by choosing each of the (n 2) undirected edges independently with probability 2p and then orienting each chosen edge independently in one of the two directions with equal probability 1/2. Let mas(D) denote the maximum size of an induced acyclic subgraph in D. We obtain tight concentration results on the size of mas(D). Precisely, we show that $mas(D) \le \frac{2}{ln(1-p)^-1} (ln np + 3e)$ almost surely, provided p ≥ W/n for some fixed constant W. This combined with known and new lower bounds shows that (for p satisfying p = ω(1/n) and p ≤ 0.5) $mas(D) = \frac{2(ln np)}{ln(1-p)^-1} (1± o(1))$. This proves a conjecture stated by Subramanian in 2003 for those p such that p = ω(1/n). Our results are also valid for the random digraph obtained by choosing each of the n(n − 1) directed edges independently with probability p.]]> Tue, 01 Jan 2008 07:00:00 +0000 https://doi.org/10.46298/dmtcs.422 https://doi.org/10.46298/dmtcs.422 Spencer, Joel Subramanian, C.R. Spencer, Joel Subramanian, C.R. <![CDATA[Let D ∈ D(n, p) denote a simple random digraph obtained by choosing each of the (n 2) undirected edges independently with probability 2p and then orienting each chosen edge independently in one of the two directions with equal probability 1/2. Let mas(D) denote the maximum size of an induced acyclic subgraph in D. We obtain tight concentration results on the size of mas(D). Precisely, we show that $mas(D) \le \frac{2}{ln(1-p)^-1} (ln np + 3e)$ almost surely, provided p ≥ W/n for some fixed constant W. This combined with known and new lower bounds shows that (for p satisfying p = ω(1/n) and p ≤ 0.5) $mas(D) = \frac{2(ln np)}{ln(1-p)^-1} (1± o(1))$. This proves a conjecture stated by Subramanian in 2003 for those p such that p = ω(1/n). Our results are also valid for the random digraph obtained by choosing each of the n(n − 1) directed edges independently with probability p.]]> 0 <![CDATA[We show that a determinant of Stirling cycle numbers counts unlabeled acyclic single-source automata. The proof involves a bijection from these automata to certain marked lattice paths and a sign-reversing involution to evaluate the determinant. We also give a formula for the number of acyclic automata with a given set of sources.]]> Tue, 01 Jan 2008 07:00:00 +0000 https://doi.org/10.46298/dmtcs.421 https://doi.org/10.46298/dmtcs.421 Callan, David Callan, David <![CDATA[We show that a determinant of Stirling cycle numbers counts unlabeled acyclic single-source automata. The proof involves a bijection from these automata to certain marked lattice paths and a sign-reversing involution to evaluate the determinant. We also give a formula for the number of acyclic automata with a given set of sources.]]> 0 <![CDATA[Let (w_n)0 < n be the sequence known as Integer Sequence A047749 In this paper, we show that the integer w_n enumerates various kinds of symmetric structures of order two. We first consider ternary trees having a reflexive symmetry and we relate all symmetric combinatorial objects by means of bijection. We then generalize the symmetric structures and correspondences to an infinite family of symmetric objects.]]> Tue, 01 Jan 2008 07:00:00 +0000 https://doi.org/10.46298/dmtcs.420 https://doi.org/10.46298/dmtcs.420 Bousquet, Michel Lamathe, Cédric Bousquet, Michel Lamathe, Cédric <![CDATA[Let (w_n)0 < n be the sequence known as Integer Sequence A047749 In this paper, we show that the integer w_n enumerates various kinds of symmetric structures of order two. We first consider ternary trees having a reflexive symmetry and we relate all symmetric combinatorial objects by means of bijection. We then generalize the symmetric structures and correspondences to an infinite family of symmetric objects.]]> 0 <![CDATA[We consider relational periods where the relation is a compatibility relation on words induced by a relation on letters. We introduce three types of periods, namely global, external and local relational periods, and we compare their properties by proving variants of the theorem of Fine and Wilf for these periods.]]> Tue, 01 Jan 2008 07:00:00 +0000 https://doi.org/10.46298/dmtcs.419 https://doi.org/10.46298/dmtcs.419 Halava, Vesa Harju, Tero Kärki, Tomi Halava, Vesa Harju, Tero Kärki, Tomi <![CDATA[We consider relational periods where the relation is a compatibility relation on words induced by a relation on letters. We introduce three types of periods, namely global, external and local relational periods, and we compare their properties by proving variants of the theorem of Fine and Wilf for these periods.]]> 0 <![CDATA[For a set D ⊂ Zn, the distance graph Pn(D) has Zn as its vertex set and the edges are between vertices i and j with |i − j| ∈ D. The circulant graph Cn(D) is defined analogously by considering operations modulo n. The minimum feedback vertex set problem consists in finding the smallest number of vertices to be removed in order to cut all cycles in the graph. This paper studies the minimum feedback vertex set problem for some families of distance graphs and circulant graphs depending on the value of D.]]> Tue, 01 Jan 2008 07:00:00 +0000 https://doi.org/10.46298/dmtcs.417 https://doi.org/10.46298/dmtcs.417 Kheddouci, Hamamache Togni, Olivier Kheddouci, Hamamache Togni, Olivier <![CDATA[For a set D ⊂ Zn, the distance graph Pn(D) has Zn as its vertex set and the edges are between vertices i and j with |i − j| ∈ D. The circulant graph Cn(D) is defined analogously by considering operations modulo n. The minimum feedback vertex set problem consists in finding the smallest number of vertices to be removed in order to cut all cycles in the graph. This paper studies the minimum feedback vertex set problem for some families of distance graphs and circulant graphs depending on the value of D.]]> 0 <![CDATA[Fekete's lemma is a well-known combinatorial result on number sequences: we extend it to functions defined on d-tuples of integers. As an application of the new variant, we show that nonsurjective d-dimensional cellular automata are characterized by loss of arbitrarily much information on finite supports, at a growth rate greater than that of the support's boundary determined by the automaton's neighbourhood index.]]> Tue, 01 Jan 2008 07:00:00 +0000 https://doi.org/10.46298/dmtcs.442 https://doi.org/10.46298/dmtcs.442 Capobianco, Silvio Capobianco, Silvio <![CDATA[Fekete's lemma is a well-known combinatorial result on number sequences: we extend it to functions defined on d-tuples of integers. As an application of the new variant, we show that nonsurjective d-dimensional cellular automata are characterized by loss of arbitrarily much information on finite supports, at a growth rate greater than that of the support's boundary determined by the automaton's neighbourhood index.]]> 0 <![CDATA[In this paper we consider the distribution of a pattern of interest in a binary random (d; k)-sequence generated by a Markov source. Such constrained sequences are frequently encountered in communication systems. Unlike the previous approach based on generating function we have chosen here to use Markov chain embedding techniques. By doing so, we get both previous results (sequence constrained up to the rth occurrence), and new ones (sequence constrained up to its end). We also provide in both cases efficient algorithms using basic linear algebra only. We then compare our numerical results to previous ones and finally propose several interesting extensions of our method which further illustrate the usefulness of our approach. That is to say higher order Markov chains, renewal occurrences rather than overlapping ones, heterogeneous models, more complex patterns of interest, and multistate trial sequences.]]> Tue, 01 Jan 2008 07:00:00 +0000 https://doi.org/10.46298/dmtcs.449 https://doi.org/10.46298/dmtcs.449 Nuel, Gregory Nuel, Gregory <![CDATA[In this paper we consider the distribution of a pattern of interest in a binary random (d; k)-sequence generated by a Markov source. Such constrained sequences are frequently encountered in communication systems. Unlike the previous approach based on generating function we have chosen here to use Markov chain embedding techniques. By doing so, we get both previous results (sequence constrained up to the rth occurrence), and new ones (sequence constrained up to its end). We also provide in both cases efficient algorithms using basic linear algebra only. We then compare our numerical results to previous ones and finally propose several interesting extensions of our method which further illustrate the usefulness of our approach. That is to say higher order Markov chains, renewal occurrences rather than overlapping ones, heterogeneous models, more complex patterns of interest, and multistate trial sequences.]]> 0 <![CDATA[For any class of binary functions on [n]={1, ..., n} a classical result by Sauer states a sufficient condition for its VC-dimension to be at least d: its cardinality should be at least O(nd-1). A necessary condition is that its cardinality be at least 2d (which is O(1) with respect to n). How does the size of a 'typical' class of VC-dimension d compare to these two extreme thresholds ? To answer this, we consider classes generated randomly by two methods, repeated biased coin flips on the n-dimensional hypercube or uniform sampling over the space of all possible classes of cardinality k on [n]. As it turns out, the typical behavior of such classes is much more similar to the necessary condition; the cardinality k need only be larger than a threshold of 2d for its VC-dimension to be at least d with high probability. If its expected size is greater than a threshold of O(&log;n) (which is still significantly smaller than the sufficient size of O(nd-1)) then it shatters every set of size d with high probability. The behavior in the neighborhood of these thresholds is described by the asymptotic probability distribution of the VC-dimension and of the largest d such that all sets of size d are shattered.]]> Tue, 01 Jan 2008 07:00:00 +0000 https://doi.org/10.46298/dmtcs.448 https://doi.org/10.46298/dmtcs.448 Ycart, Bernard Ratsaby, Joel Ycart, Bernard Ratsaby, Joel <![CDATA[For any class of binary functions on [n]={1, ..., n} a classical result by Sauer states a sufficient condition for its VC-dimension to be at least d: its cardinality should be at least O(nd-1). A necessary condition is that its cardinality be at least 2d (which is O(1) with respect to n). How does the size of a 'typical' class of VC-dimension d compare to these two extreme thresholds ? To answer this, we consider classes generated randomly by two methods, repeated biased coin flips on the n-dimensional hypercube or uniform sampling over the space of all possible classes of cardinality k on [n]. As it turns out, the typical behavior of such classes is much more similar to the necessary condition; the cardinality k need only be larger than a threshold of 2d for its VC-dimension to be at least d with high probability. If its expected size is greater than a threshold of O(&log;n) (which is still significantly smaller than the sufficient size of O(nd-1)) then it shatters every set of size d with high probability. The behavior in the neighborhood of these thresholds is described by the asymptotic probability distribution of the VC-dimension and of the largest d such that all sets of size d are shattered.]]> 0 <![CDATA[Priority trees are a certain data structure used for priority queue administration. Under the model that all permutations of the numbers 1, . . . , n are equally likely to construct a priority tree of size n we study the following parameters in size-n trees: depth of a random node, number of right edges to a random node, and number of descendants of a random node. For all parameters studied we give limiting distribution results.]]> Tue, 01 Jan 2008 07:00:00 +0000 https://doi.org/10.46298/dmtcs.447 https://doi.org/10.46298/dmtcs.447 Panholzer, Alois Panholzer, Alois <![CDATA[Priority trees are a certain data structure used for priority queue administration. Under the model that all permutations of the numbers 1, . . . , n are equally likely to construct a priority tree of size n we study the following parameters in size-n trees: depth of a random node, number of right edges to a random node, and number of descendants of a random node. For all parameters studied we give limiting distribution results.]]> 0 <![CDATA[In 1973, V. Virkkunen proved that propagating scattered context grammars which use leftmost derivations are as powerful as context-sensitive grammars. This paper brings a significantly simplified proof of this result.]]> Tue, 01 Jan 2008 07:00:00 +0000 https://doi.org/10.46298/dmtcs.446 https://doi.org/10.46298/dmtcs.446 Masopust, Tomáš Techet, Jiří Masopust, Tomáš Techet, Jiří <![CDATA[In 1973, V. Virkkunen proved that propagating scattered context grammars which use leftmost derivations are as powerful as context-sensitive grammars. This paper brings a significantly simplified proof of this result.]]> 0 <![CDATA[A dominating set D of vertices in a graph G is called an efficient dominating set if the distance between any two vertices in D is at least three. A tree T of order n is called maximum if T has the largest number of efficient dominating sets among all n-vertex trees. A constructive characterization of all maximum trees is given. Their structure has recurring aspects with period 7. Moreover, the number of efficient dominating sets in maximum n-vertex trees is determined and is exponential. Also the number of maximum n-vertex trees is shown to be bounded below by an increasing exponential function in n.]]> Tue, 01 Jan 2008 07:00:00 +0000 https://doi.org/10.46298/dmtcs.445 https://doi.org/10.46298/dmtcs.445 Bród, Dorota Skupień, Zdzisław Bród, Dorota Skupień, Zdzisław <![CDATA[A dominating set D of vertices in a graph G is called an efficient dominating set if the distance between any two vertices in D is at least three. A tree T of order n is called maximum if T has the largest number of efficient dominating sets among all n-vertex trees. A constructive characterization of all maximum trees is given. Their structure has recurring aspects with period 7. Moreover, the number of efficient dominating sets in maximum n-vertex trees is determined and is exponential. Also the number of maximum n-vertex trees is shown to be bounded below by an increasing exponential function in n.]]> 0 <![CDATA[For a set D ⊂ Zn, the distance graph Pn(D) has Zn as its vertex set and the edges are between vertices i and j with |i − j| ∈ D. The circulant graph Cn(D) is defined analogously by considering operations modulo n. The minimum feedback vertex set problem consists in finding the smallest number of vertices to be removed in order to cut all cycles in the graph. This paper studies the minimum feedback vertex set problem for some families of distance graphs and circulant graphs depending on the value of D.]]> Tue, 01 Jan 2008 07:00:00 +0000 https://doi.org/10.46298/dmtcs.444 https://doi.org/10.46298/dmtcs.444 Kheddouci, Hamamache Togni, Olivier Kheddouci, Hamamache Togni, Olivier <![CDATA[For a set D ⊂ Zn, the distance graph Pn(D) has Zn as its vertex set and the edges are between vertices i and j with |i − j| ∈ D. The circulant graph Cn(D) is defined analogously by considering operations modulo n. The minimum feedback vertex set problem consists in finding the smallest number of vertices to be removed in order to cut all cycles in the graph. This paper studies the minimum feedback vertex set problem for some families of distance graphs and circulant graphs depending on the value of D.]]> 0 <![CDATA[In the permutation routing problem, each processor is the origin of at most one packet and the destination of no more than one packet. The goal is to minimize the number of time steps required to route all packets to their respective destinations, under the constraint that each link can be crossed simultaneously by no more than one packet. We study this problem in a hexagonal network, i.e. a finite subgraph of a triangular grid, which is a widely used network in practical applications. We present an optimal distributed permutation routing algorithm on full-duplex hexagonal networks, using the addressing scheme described by F.G. Nocetti, I. Stojmenovic and J. Zhang (IEEE TPDS 13(9): 962-971, 2002). Furthermore, we prove that this algorithm is oblivious and translation invariant.]]> Tue, 01 Jan 2008 07:00:00 +0000 https://doi.org/10.46298/dmtcs.443 https://doi.org/10.46298/dmtcs.443 Sau, Ignasi Žerovnik, Janez Sau, Ignasi Žerovnik, Janez <![CDATA[In the permutation routing problem, each processor is the origin of at most one packet and the destination of no more than one packet. The goal is to minimize the number of time steps required to route all packets to their respective destinations, under the constraint that each link can be crossed simultaneously by no more than one packet. We study this problem in a hexagonal network, i.e. a finite subgraph of a triangular grid, which is a widely used network in practical applications. We present an optimal distributed permutation routing algorithm on full-duplex hexagonal networks, using the addressing scheme described by F.G. Nocetti, I. Stojmenovic and J. Zhang (IEEE TPDS 13(9): 962-971, 2002). Furthermore, we prove that this algorithm is oblivious and translation invariant.]]> 0 <![CDATA[We study on-line version of size-Ramsey numbers of graphs defined via a game played between Builder and Painter: in one round Builder joins two vertices by an edge and Painter paints it red or blue. The goal of Builder is to force Painter to create a monochromatic copy of a fixed graph H in as few rounds as possible. The minimum number of rounds (assuming both players play perfectly) is the on-line Ramsey number r(H) of the graph H. We determine exact values of r(H) for a few short paths and obtain a general upper bound r(Pn) ≤ 4n −7. We also study asymmetric version of this parameter when one of the target graphs is a star Sn with n edges. We prove that r(Sn, H) ≤ n*e(H) when H is any tree, cycle or clique]]> Tue, 01 Jan 2008 07:00:00 +0000 https://doi.org/10.46298/dmtcs.427 https://doi.org/10.46298/dmtcs.427 Grytczuk, J. A. Kierstead, H. A. Prałat, P. Grytczuk, J. A. Kierstead, H. A. Prałat, P. <![CDATA[We study on-line version of size-Ramsey numbers of graphs defined via a game played between Builder and Painter: in one round Builder joins two vertices by an edge and Painter paints it red or blue. The goal of Builder is to force Painter to create a monochromatic copy of a fixed graph H in as few rounds as possible. The minimum number of rounds (assuming both players play perfectly) is the on-line Ramsey number r(H) of the graph H. We determine exact values of r(H) for a few short paths and obtain a general upper bound r(Pn) ≤ 4n −7. We also study asymmetric version of this parameter when one of the target graphs is a star Sn with n edges. We prove that r(Sn, H) ≤ n*e(H) when H is any tree, cycle or clique]]> 0 <![CDATA[We study the number of spanning forests on the Sierpinski gasket SGd(n) at stage n with dimension d equal to two, three and four, and determine the asymptotic behaviors. The corresponding results on the generalized Sierpinski gasket SGd;b(n) with d = 2 and b = 3 ; 4 are obtained. We also derive upper bounds for the asymptotic growth constants for both SGd and SG2,b.]]> Tue, 01 Jan 2008 07:00:00 +0000 https://doi.org/10.46298/dmtcs.441 https://doi.org/10.46298/dmtcs.441 Chang, Shu-Chiuan Chen, Lung-Chi Chang, Shu-Chiuan Chen, Lung-Chi <![CDATA[We study the number of spanning forests on the Sierpinski gasket SGd(n) at stage n with dimension d equal to two, three and four, and determine the asymptotic behaviors. The corresponding results on the generalized Sierpinski gasket SGd;b(n) with d = 2 and b = 3 ; 4 are obtained. We also derive upper bounds for the asymptotic growth constants for both SGd and SG2,b.]]> 0 <![CDATA[In graph theory, the Helly property has been applied to families of sets, such as cliques, disks, bicliques, and neighbourhoods, leading to the classes of clique-Helly, disk-Helly, biclique-Helly, neighbourhood-Helly graphs, respectively. A natural question is to determine for which graphs the corresponding Helly property holds, for every induced subgraph. This leads to the corresponding classes of hereditary clique-Helly, hereditary disk-Helly, hereditary biclique-Helly and hereditary neighbourhood-Helly graphs. In this paper, we describe characterizations in terms of families of forbidden subgraphs, for the classes of hereditary biclique-Helly and hereditary neighbourhood-Helly graphs. We consider both open and closed neighbourhoods. The forbidden subgraphs are all of fixed size, implying polynomial time recognition for these classes.]]> Tue, 01 Jan 2008 07:00:00 +0000 https://doi.org/10.46298/dmtcs.440 https://doi.org/10.46298/dmtcs.440 Groshaus, Marina Szwarcfiter, Jayme Luiz Groshaus, Marina Szwarcfiter, Jayme Luiz <![CDATA[In graph theory, the Helly property has been applied to families of sets, such as cliques, disks, bicliques, and neighbourhoods, leading to the classes of clique-Helly, disk-Helly, biclique-Helly, neighbourhood-Helly graphs, respectively. A natural question is to determine for which graphs the corresponding Helly property holds, for every induced subgraph. This leads to the corresponding classes of hereditary clique-Helly, hereditary disk-Helly, hereditary biclique-Helly and hereditary neighbourhood-Helly graphs. In this paper, we describe characterizations in terms of families of forbidden subgraphs, for the classes of hereditary biclique-Helly and hereditary neighbourhood-Helly graphs. We consider both open and closed neighbourhoods. The forbidden subgraphs are all of fixed size, implying polynomial time recognition for these classes.]]> 0 <![CDATA[The Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the second smallest eigenvalue of the Laplacian matrix of the graph. In this paper, we show that the star is the unique tree with maximal Laplacian spread among all trees of given order, and the path is the unique one with minimal Laplacian spread among all trees of given order.]]> Tue, 01 Jan 2008 07:00:00 +0000 https://doi.org/10.46298/dmtcs.439 https://doi.org/10.46298/dmtcs.439 Fan, Yi-Zheng Xu, Jing Wang, Yi Liang, Dong Fan, Yi-Zheng Xu, Jing Wang, Yi Liang, Dong <![CDATA[The Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the second smallest eigenvalue of the Laplacian matrix of the graph. In this paper, we show that the star is the unique tree with maximal Laplacian spread among all trees of given order, and the path is the unique one with minimal Laplacian spread among all trees of given order.]]> 0 <![CDATA[We start with a set of $n$ players. With some probability $P(n,k)$, we kill $n-k$ players; the other ones stay alive, and we repeat with them. What is the distribution of the number $X_n$ of \emph{phases} (or rounds) before getting only one player? We present a probabilistic analysis of this algorithm under some conditions on the probability distributions $P(n,k)$, including stochastic monotonicity and the assumption that roughly a fixed proportion $\al$ of the players survive in each round. We prove a kind of convergence in distribution for $X_n - \log_{1/\!\alpha}(n)$; as in many other similar problems there are oscillations and no true limit distribution, but suitable subsequences converge, and there is an absolutely continuous random variable $Z$ such that $d\l(X_n, \lceil Z + \log_{1/\!\alpha} (n)\rceil\r) \to 0$, where $d$ is either the total variation distance or the Wasserstein distance. Applications of the general result include the leader election algorithm where players are eliminated by independent coin tosses and a variation of the leader election algorithm proposed by W.R. Franklin. We study the latter algorithm further, including numerical results.]]> Tue, 01 Jan 2008 07:00:00 +0000 https://doi.org/10.46298/dmtcs.437 https://doi.org/10.46298/dmtcs.437 Janson, Svante Lavault, Christian Louchard, Guy Janson, Svante Lavault, Christian Louchard, Guy <![CDATA[We start with a set of $n$ players. With some probability $P(n,k)$, we kill $n-k$ players; the other ones stay alive, and we repeat with them. What is the distribution of the number $X_n$ of \emph{phases} (or rounds) before getting only one player? We present a probabilistic analysis of this algorithm under some conditions on the probability distributions $P(n,k)$, including stochastic monotonicity and the assumption that roughly a fixed proportion $\al$ of the players survive in each round. We prove a kind of convergence in distribution for $X_n - \log_{1/\!\alpha}(n)$; as in many other similar problems there are oscillations and no true limit distribution, but suitable subsequences converge, and there is an absolutely continuous random variable $Z$ such that $d\l(X_n, \lceil Z + \log_{1/\!\alpha} (n)\rceil\r) \to 0$, where $d$ is either the total variation distance or the Wasserstein distance. Applications of the general result include the leader election algorithm where players are eliminated by independent coin tosses and a variation of the leader election algorithm proposed by W.R. Franklin. We study the latter algorithm further, including numerical results.]]> 0 <![CDATA[In this paper we show an information-theoretic lower bound of kn - o(kn) on the minimum number of bits to represent an unlabeled simple connected n-node graph of pagenumber k. This has to be compared with the efficient encoding scheme of Munro and Raman of 2kn + 2m + o(kn+m) bits (m the number of edges), that is 4kn + 2n + o(kn) bits in the worst-case. For m-edge graphs of pagenumber k (with multi-edges and loops), we propose a 2mlog2k + O(m) bits encoding improving the best previous upper bound of Munro and Raman whenever m ≤ 1 / 2kn/log2 k. Actually our scheme applies to k-page embedding containing multi-edge and loops. Moreover, with an auxiliary table of o(m log k) bits, our coding supports (1) the computation of the degree of a node in constant time, (2) adjacency queries with O(logk) queries of type rank, select and match, that is in O(logk *minlogk / loglogm, loglogk) time and (3) the access to δ neighbors in O(δ) runs of select, rank or match;.]]> Tue, 01 Jan 2008 07:00:00 +0000 https://doi.org/10.46298/dmtcs.436 https://doi.org/10.46298/dmtcs.436 Gavoille, Cyril Hanusse, Nicolas Gavoille, Cyril Hanusse, Nicolas <![CDATA[In this paper we show an information-theoretic lower bound of kn - o(kn) on the minimum number of bits to represent an unlabeled simple connected n-node graph of pagenumber k. This has to be compared with the efficient encoding scheme of Munro and Raman of 2kn + 2m + o(kn+m) bits (m the number of edges), that is 4kn + 2n + o(kn) bits in the worst-case. For m-edge graphs of pagenumber k (with multi-edges and loops), we propose a 2mlog2k + O(m) bits encoding improving the best previous upper bound of Munro and Raman whenever m ≤ 1 / 2kn/log2 k. Actually our scheme applies to k-page embedding containing multi-edge and loops. Moreover, with an auxiliary table of o(m log k) bits, our coding supports (1) the computation of the degree of a node in constant time, (2) adjacency queries with O(logk) queries of type rank, select and match, that is in O(logk *minlogk / loglogm, loglogk) time and (3) the access to δ neighbors in O(δ) runs of select, rank or match;.]]> 0 <![CDATA[In this paper new exact values of the Zarankiewicz function z(m,n;s,t) are obtained assuming certain requirements on the parameters. Moreover, all the corresponding extremal graphs are characterized. Finally, an extension of this problem to 3-partite graphs is studied.]]> Tue, 01 Jan 2008 07:00:00 +0000 https://doi.org/10.46298/dmtcs.435 https://doi.org/10.46298/dmtcs.435 Balbuena, Camino García-Vázquez, P. Marcote, Xavier Valenzuela, J. C. Balbuena, Camino García-Vázquez, P. Marcote, Xavier Valenzuela, J. C. <![CDATA[In this paper new exact values of the Zarankiewicz function z(m,n;s,t) are obtained assuming certain requirements on the parameters. Moreover, all the corresponding extremal graphs are characterized. Finally, an extension of this problem to 3-partite graphs is studied.]]> 0 <![CDATA[It is proved that there exist graphs of bounded degree with arbitrarily large queue-number. In particular, for all \Delta ≥ 3 and for all sufficiently large n, there is a simple \Delta-regular n-vertex graph with queue-number at least c√\Delta_n^{1/2-1/\Delta} for some absolute constant c.]]> Tue, 01 Jan 2008 07:00:00 +0000 https://doi.org/10.46298/dmtcs.434 https://doi.org/10.46298/dmtcs.434 Wood, David R. Wood, David R. <![CDATA[It is proved that there exist graphs of bounded degree with arbitrarily large queue-number. In particular, for all \Delta ≥ 3 and for all sufficiently large n, there is a simple \Delta-regular n-vertex graph with queue-number at least c√\Delta_n^{1/2-1/\Delta} for some absolute constant c.]]> 0 <![CDATA[A graph G is Kr-covered if each vertex of G is contained in a Kr-clique. Let $\gamma_t(G)$ denote the total domination number of G. It has been conjectured that every Kr-covered graph of order n with no Kr-component satisfies $\gamma_t(G) \le \frac{2n}{r+1}$. We prove that this conjecture is true for r = 5 and 6.]]> Tue, 01 Jan 2008 07:00:00 +0000 https://doi.org/10.46298/dmtcs.433 https://doi.org/10.46298/dmtcs.433 Favaron, Odile Karami, H. Sheikholeslami, S. M. Favaron, Odile Karami, H. Sheikholeslami, S. M. <![CDATA[A graph G is Kr-covered if each vertex of G is contained in a Kr-clique. Let $\gamma_t(G)$ denote the total domination number of G. It has been conjectured that every Kr-covered graph of order n with no Kr-component satisfies $\gamma_t(G) \le \frac{2n}{r+1}$. We prove that this conjecture is true for r = 5 and 6.]]> 0 <![CDATA[Recently, Kitaev and Remmel refined the well-known permutation statistic "descent" by fixing parity of one of the descent's numbers which was extended and generalized in several ways in the literature. In this paper, we shall fix a set partition of the natural numbers N,(N1, ..., Ns), and we study the distribution of descents, levels, and rises according to whether the first letter of the descent, rise, or level lies in Ni over the set of words over the alphabet [k] = 1, ..., k. In particular, we refine and generalize some of the results by Burstein and Mansour]]> Tue, 01 Jan 2008 07:00:00 +0000 https://doi.org/10.46298/dmtcs.432 https://doi.org/10.46298/dmtcs.432 Kitaev, Sergey Mansour, Toufik Remmel, Jeff Kitaev, Sergey Mansour, Toufik Remmel, Jeff <![CDATA[Recently, Kitaev and Remmel refined the well-known permutation statistic "descent" by fixing parity of one of the descent's numbers which was extended and generalized in several ways in the literature. In this paper, we shall fix a set partition of the natural numbers N,(N1, ..., Ns), and we study the distribution of descents, levels, and rises according to whether the first letter of the descent, rise, or level lies in Ni over the set of words over the alphabet [k] = 1, ..., k. In particular, we refine and generalize some of the results by Burstein and Mansour]]> 0 <![CDATA[We study two variants of edge-coloring of edge-weighted graphs, namely compact edge-coloring and circular compact edge-coloring. First, we discuss relations between these two coloring models. We prove that every outerplanar bipartite graph admits a compact edge-coloring and that the decision problem of the existence of compact circular edge-coloring is NP-complete in general. Then we provide a polynomial time 1:5-approximation algorithm and pseudo-polynomial exact algorithm for compact circular coloring of odd cycles and prove that it is NP-hard to optimally color these graphs. Finally, we prove that if a path P2 is joined by an edge to an odd cycle then the problem of the existence of a compact circular coloring becomes NP-complete.]]> Tue, 01 Jan 2008 07:00:00 +0000 https://doi.org/10.46298/dmtcs.431 https://doi.org/10.46298/dmtcs.431 Dereniowski, Dariusz Nadolski, Adam Dereniowski, Dariusz Nadolski, Adam <![CDATA[We study two variants of edge-coloring of edge-weighted graphs, namely compact edge-coloring and circular compact edge-coloring. First, we discuss relations between these two coloring models. We prove that every outerplanar bipartite graph admits a compact edge-coloring and that the decision problem of the existence of compact circular edge-coloring is NP-complete in general. Then we provide a polynomial time 1:5-approximation algorithm and pseudo-polynomial exact algorithm for compact circular coloring of odd cycles and prove that it is NP-hard to optimally color these graphs. Finally, we prove that if a path P2 is joined by an edge to an odd cycle then the problem of the existence of a compact circular coloring becomes NP-complete.]]> 0 <![CDATA[If F(x) = e^G(x), where F(x) = \Sum f(n)x^n and G(x) = \Sum g(n)x^n, with 0 ≤ g(n) = O(n^θn/n!), θ ∈ (0,1), and gcd(n : g(n) > 0) = 1, then f(n) = o(f(n − 1)). This gives an answer to Compton's request in Question 8.3 [Compton 1987] for an "easily verifiable sufficient condition" to show that an adequate class of structures has a labelled first-order 0-1 law, namely it suffices to show that the labelled component count function is O(n^θn) for some θ ∈ (0,1). It also provides the means to recursively construct an adequate class of structures with a labelled 0-1 law but not an unlabelled 0-1 law, answering Compton's Question 8.4.]]> Tue, 01 Jan 2008 07:00:00 +0000 https://doi.org/10.46298/dmtcs.430 https://doi.org/10.46298/dmtcs.430 Burris, Stanley N. Yeats, Karen A. Burris, Stanley N. Yeats, Karen A. <![CDATA[If F(x) = e^G(x), where F(x) = \Sum f(n)x^n and G(x) = \Sum g(n)x^n, with 0 ≤ g(n) = O(n^θn/n!), θ ∈ (0,1), and gcd(n : g(n) > 0) = 1, then f(n) = o(f(n − 1)). This gives an answer to Compton's request in Question 8.3 [Compton 1987] for an "easily verifiable sufficient condition" to show that an adequate class of structures has a labelled first-order 0-1 law, namely it suffices to show that the labelled component count function is O(n^θn) for some θ ∈ (0,1). It also provides the means to recursively construct an adequate class of structures with a labelled 0-1 law but not an unlabelled 0-1 law, answering Compton's Question 8.4.]]> 0 <![CDATA[For an odd prime p and each non-empty subset S ⊂ GF(p), consider the hyperelliptic curve X_S defined by y^2 = f_s(x), where f_s(x) = \P_{a2S} (x-a). Using a connection between binary quadratic residue codes and hyperelliptic curves over GF(p), this paper investigates how coding theory bounds give rise to bounds such as the following example: for all sufficiently large primes p there exists a subset S ⊂ GF(p) for which the bound |X_S(GF(p))| > 1.39p holds. We also use the quasi-quadratic residue codes defined below to construct an example of a formally self-dual optimal code whose zeta function does not satisfy the "Riemann hypothesis."]]> Tue, 01 Jan 2008 07:00:00 +0000 https://doi.org/10.46298/dmtcs.429 https://doi.org/10.46298/dmtcs.429 Joyner, David Joyner, David <![CDATA[For an odd prime p and each non-empty subset S ⊂ GF(p), consider the hyperelliptic curve X_S defined by y^2 = f_s(x), where f_s(x) = \P_{a2S} (x-a). Using a connection between binary quadratic residue codes and hyperelliptic curves over GF(p), this paper investigates how coding theory bounds give rise to bounds such as the following example: for all sufficiently large primes p there exists a subset S ⊂ GF(p) for which the bound |X_S(GF(p))| > 1.39p holds. We also use the quasi-quadratic residue codes defined below to construct an example of a formally self-dual optimal code whose zeta function does not satisfy the "Riemann hypothesis."]]> 0 <![CDATA[A digraph is k-traceable if each of its induced subdigraphs of order k is traceable. The Traceability Conjecture is that for k ≥ 2 every k-traceable oriented graph of order at least 2k − 1 is traceable. The conjecture has been proved for k ≤ 5. We prove that it also holds for k = 6.]]> Tue, 01 Jan 2008 07:00:00 +0000 https://doi.org/10.46298/dmtcs.428 https://doi.org/10.46298/dmtcs.428 Frick, Marietjie Katrenič, Peter Frick, Marietjie Katrenič, Peter <![CDATA[A digraph is k-traceable if each of its induced subdigraphs of order k is traceable. The Traceability Conjecture is that for k ≥ 2 every k-traceable oriented graph of order at least 2k − 1 is traceable. The conjecture has been proved for k ≤ 5. We prove that it also holds for k = 6.]]> 0 <![CDATA[Two processes of random fragmentation of an interval are investigated. For each of them, there is a splitting probability at each step of the fragmentation process whose overall effect is to stabilize the global number of splitting events. More precisely, we consider two models. In the first model, the fragmentation stops which a probability $p$ witch can not depend on the fragment size. The number of stable fragments with sizes less than a given $t \geq 0$, denoted by $K(t)$, is introduced and studied. In the second one the probability to split a fragment of size $x$ is $p(x)=1-e^{-x}$. For this model we utilize the contraction method to show that the distribution of a suitably normalized version of the number of stable fragments converges in law. It's shown that the limit is the fixed-point solution (in the Wasserstein space) to a distributional equation. An explicit solution to the fixed-point equation is easily verified to be Gaussian.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3588 https://doi.org/10.46298/dmtcs.3588 Aguech, Rafik Aguech, Rafik <![CDATA[Two processes of random fragmentation of an interval are investigated. For each of them, there is a splitting probability at each step of the fragmentation process whose overall effect is to stabilize the global number of splitting events. More precisely, we consider two models. In the first model, the fragmentation stops which a probability $p$ witch can not depend on the fragment size. The number of stable fragments with sizes less than a given $t \geq 0$, denoted by $K(t)$, is introduced and studied. In the second one the probability to split a fragment of size $x$ is $p(x)=1-e^{-x}$. For this model we utilize the contraction method to show that the distribution of a suitably normalized version of the number of stable fragments converges in law. It's shown that the limit is the fixed-point solution (in the Wasserstein space) to a distributional equation. An explicit solution to the fixed-point equation is easily verified to be Gaussian.]]> 0 <![CDATA[We asymptotically analyse the volume random variables of general, symmetric and cyclically symmetric plane partitions fitting inside a box. We consider the respective symmetry class equipped with the uniform distribution. We also prove area limit laws for two ensembles of Ferrers diagrams. Most limit laws are Gaussian.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3589 https://doi.org/10.46298/dmtcs.3589 Schwerdtfeger, Uwe Schwerdtfeger, Uwe <![CDATA[We asymptotically analyse the volume random variables of general, symmetric and cyclically symmetric plane partitions fitting inside a box. We consider the respective symmetry class equipped with the uniform distribution. We also prove area limit laws for two ensembles of Ferrers diagrams. Most limit laws are Gaussian.]]> 0 <![CDATA[We exploit a bijection between plane recursive trees and Stirling permutations; this yields the equivalence of some results previously proven separately by different methods for the two types of objects as well as some new results. We also prove results on the joint distribution of the numbers of ascents, descents and plateaux in a random Stirling permutation. The proof uses an interesting generalized Pólya urn.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3590 https://doi.org/10.46298/dmtcs.3590 Janson, Svante Janson, Svante <![CDATA[We exploit a bijection between plane recursive trees and Stirling permutations; this yields the equivalence of some results previously proven separately by different methods for the two types of objects as well as some new results. We also prove results on the joint distribution of the numbers of ascents, descents and plateaux in a random Stirling permutation. The proof uses an interesting generalized Pólya urn.]]> 0 <![CDATA[The Mabinogion urn is a simple model of the spread of influences amongst versatile populations. It corresponds to a non-standard urn with balls of two colours: each time a ball is drawn, it causes a ball of the other kind to switch its colour. The process stops once unanimity has been reached. This note provides analytic expressions describing the evolution of the Mabinogion urn, based on a time-reversal transformation applied to the classical Ehrenfest urn. Consequences include a precise asymptotic analysis of the stopping-time distribution―it is asymptotically normal in the "unfair'' case and akin to an extreme-value (double exponential) distribution in the "fair'' case―as well as a characterization of the exponentially small probability of reversing a majority.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3591 https://doi.org/10.46298/dmtcs.3591 Flajolet, Philippe Huillet, Thierry Flajolet, Philippe Huillet, Thierry <![CDATA[The Mabinogion urn is a simple model of the spread of influences amongst versatile populations. It corresponds to a non-standard urn with balls of two colours: each time a ball is drawn, it causes a ball of the other kind to switch its colour. The process stops once unanimity has been reached. This note provides analytic expressions describing the evolution of the Mabinogion urn, based on a time-reversal transformation applied to the classical Ehrenfest urn. Consequences include a precise asymptotic analysis of the stopping-time distribution―it is asymptotically normal in the "unfair'' case and akin to an extreme-value (double exponential) distribution in the "fair'' case―as well as a characterization of the exponentially small probability of reversing a majority.]]> 0 <![CDATA[The factorization theorem by King, Tollu and Toumazet gives four different reduction formulae of Littlewood-Richardson coefficients. One of them is the classical reduction formula of the first type while others are new. Moreover, the classical reduction formula of the second type is not a special case of KTT theorem. We give combinatorial proofs of reduction formulae in terms of tableaux or hives. The proofs for the cases $r=1, 2, n-2$ in terms of tableaux and the proof for the classical reduction formula of the second type in terms of hives are new.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3592 https://doi.org/10.46298/dmtcs.3592 Cho, Soojin Jung, Eun-Kyoung Moon, Dongho Cho, Soojin Jung, Eun-Kyoung Moon, Dongho <![CDATA[The factorization theorem by King, Tollu and Toumazet gives four different reduction formulae of Littlewood-Richardson coefficients. One of them is the classical reduction formula of the first type while others are new. Moreover, the classical reduction formula of the second type is not a special case of KTT theorem. We give combinatorial proofs of reduction formulae in terms of tableaux or hives. The proofs for the cases $r=1, 2, n-2$ in terms of tableaux and the proof for the classical reduction formula of the second type in terms of hives are new.]]> 0 <![CDATA[We give a combinatorial proof of the factorization formula of modified Macdonald polynomials $\widetilde{H}_{\lambda} (X;q,t)$ when $t$ is specialized at a primitive root of unity. Our proof is restricted to the special case where $\lambda$ is a two columns partition. We mainly use the combinatorial interpretation of Haiman, Haglund and Loehr giving the expansion of $\widetilde{H}_{\lambda} (X;q,t)$ on the monomial basis.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3593 https://doi.org/10.46298/dmtcs.3593 Descouens, F. Morita, H. Numata, Y. Descouens, F. Morita, H. Numata, Y. <![CDATA[We give a combinatorial proof of the factorization formula of modified Macdonald polynomials $\widetilde{H}_{\lambda} (X;q,t)$ when $t$ is specialized at a primitive root of unity. Our proof is restricted to the special case where $\lambda$ is a two columns partition. We mainly use the combinatorial interpretation of Haiman, Haglund and Loehr giving the expansion of $\widetilde{H}_{\lambda} (X;q,t)$ on the monomial basis.]]> 0 <![CDATA[It is well-known, and was first established by Knuth in 1969, that the number of 321-avoiding permutations is equal to that of 132-avoiding permutations. In the literature one can find many subsequent bijective proofs confirming this fact. It turns out that some of the published bijections can easily be obtained from others. In this paper we describe all bijections we were able to find in the literature and we show how they are related to each other (via "trivial'' bijections). Thus, we give a comprehensive survey and a systematic analysis of these bijections. We also analyze how many permutation statistics (from a fixed, but large, set of statistics) each of the known bijections preserves, obtaining substantial extensions of known results. We also give a recursive description of the algorithmic bijection given by Richards in 1988 (combined with a bijection by Knuth from 1969). This bijection is equivalent to the celebrated bijection of Simion and Schmidt (1985), as well as to the bijection given by Krattenthaler in 2001, and it respects 11 statistics (the largest number of statistics any of the bijections respect).]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3594 https://doi.org/10.46298/dmtcs.3594 Claesson, Anders Kitaev, Sergey Claesson, Anders Kitaev, Sergey <![CDATA[It is well-known, and was first established by Knuth in 1969, that the number of 321-avoiding permutations is equal to that of 132-avoiding permutations. In the literature one can find many subsequent bijective proofs confirming this fact. It turns out that some of the published bijections can easily be obtained from others. In this paper we describe all bijections we were able to find in the literature and we show how they are related to each other (via "trivial'' bijections). Thus, we give a comprehensive survey and a systematic analysis of these bijections. We also analyze how many permutation statistics (from a fixed, but large, set of statistics) each of the known bijections preserves, obtaining substantial extensions of known results. We also give a recursive description of the algorithmic bijection given by Richards in 1988 (combined with a bijection by Knuth from 1969). This bijection is equivalent to the celebrated bijection of Simion and Schmidt (1985), as well as to the bijection given by Krattenthaler in 2001, and it respects 11 statistics (the largest number of statistics any of the bijections respect).]]> 0 <![CDATA[We give a compact expression for the number of factorizations of any permutation into a minimal number of transpositions of the form $(1 i)$. Our result generalizes earlier work of Pak ($\textit{Reduced decompositions of permutations in terms of star transpositions, generalized catalan numbers and k-ary trees}$, Discrete Math. $\textbf{204}$:329―335, 1999) in which substantial restrictions were placed on the permutation being factored.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3595 https://doi.org/10.46298/dmtcs.3595 Irving, J. Rattan, A. Irving, J. Rattan, A. <![CDATA[We give a compact expression for the number of factorizations of any permutation into a minimal number of transpositions of the form $(1 i)$. Our result generalizes earlier work of Pak ($\textit{Reduced decompositions of permutations in terms of star transpositions, generalized catalan numbers and k-ary trees}$, Discrete Math. $\textbf{204}$:329―335, 1999) in which substantial restrictions were placed on the permutation being factored.]]> 0 <![CDATA[Pak and Vallejo have defined fundamental symmetry map as any Young tableau bijection for the commutativity of the Littlewood-Richardson coefficients $c_{\mu,\nu}^{\lambda}=c_{\nu, \mu}^{\lambda}$. They have considered four fundamental symmetry maps and conjectured that they are all equivalent (2004). The three first ones are based on standard operations in Young tableau theory and, in this case, the conjecture was proved by Danilov and Koshevoy (2005). The fourth fundamental symmetry, given by the author in (1999;2000) and reformulated by Pak and Vallejo, is defined by nonstandard operations in Young tableau theory and will be shown to be equivalent to the first one defined by the involution property of the Benkart-Sottile-Stroomer tableau switching. The proof of this equivalence provides, in the case the first tableau is Yamanouchi, a variation of the tableau switching algorithm which shows $\textit{switching}$ as an operation that takes two tableaux sharing a common border and moves them trough each other by decomposing the first tableau into a sequence of tableaux whose sequence of partition shapes defines a Gelfand-Tsetlin pattern. This property leads to a $\textit{jeu de taquin-chain sliding}$ on Littlewood-Richardson tableaux.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3596 https://doi.org/10.46298/dmtcs.3596 Azenhas, Olga Azenhas, Olga <![CDATA[Pak and Vallejo have defined fundamental symmetry map as any Young tableau bijection for the commutativity of the Littlewood-Richardson coefficients $c_{\mu,\nu}^{\lambda}=c_{\nu, \mu}^{\lambda}$. They have considered four fundamental symmetry maps and conjectured that they are all equivalent (2004). The three first ones are based on standard operations in Young tableau theory and, in this case, the conjecture was proved by Danilov and Koshevoy (2005). The fourth fundamental symmetry, given by the author in (1999;2000) and reformulated by Pak and Vallejo, is defined by nonstandard operations in Young tableau theory and will be shown to be equivalent to the first one defined by the involution property of the Benkart-Sottile-Stroomer tableau switching. The proof of this equivalence provides, in the case the first tableau is Yamanouchi, a variation of the tableau switching algorithm which shows $\textit{switching}$ as an operation that takes two tableaux sharing a common border and moves them trough each other by decomposing the first tableau into a sequence of tableaux whose sequence of partition shapes defines a Gelfand-Tsetlin pattern. This property leads to a $\textit{jeu de taquin-chain sliding}$ on Littlewood-Richardson tableaux.]]> 0 <![CDATA[We introduce non-commutative analogs of $k$-Schur functions and prove that their images by the non-commutative nabla operator $\blacktriangledown$ is ribbon Schur positive, up to a global sign. Inspired by these results, we define new filtrations of the usual $(q,t)$-Catalan polynomials by computing the image of certain commutative $k$-Schur functions by the commutative nabla operator $\nabla$. In some particular cases, we give a combinatorial interpretation of these polynomials in terms of nested quantum Dick paths.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3597 https://doi.org/10.46298/dmtcs.3597 Bergeron, N. Descouens, F. Zabrocki, M. Bergeron, N. Descouens, F. Zabrocki, M. <![CDATA[We introduce non-commutative analogs of $k$-Schur functions and prove that their images by the non-commutative nabla operator $\blacktriangledown$ is ribbon Schur positive, up to a global sign. Inspired by these results, we define new filtrations of the usual $(q,t)$-Catalan polynomials by computing the image of certain commutative $k$-Schur functions by the commutative nabla operator $\nabla$. In some particular cases, we give a combinatorial interpretation of these polynomials in terms of nested quantum Dick paths.]]> 0 <![CDATA[We show that the set of cluster monomials for the cluster algebra of type $D_4$ contains a basis of the $\mathbb{Z}$-module $\mathbb{Z}[x_{1,1},\ldots ,x_{3,3}]$. We also show that the transition matrices relating this cluster basis to the natural and the dual canonical bases are unitriangular and nonnegative. These results support a conjecture of Fomin and Zelevinsky on the equality of the cluster and dual canonical bases. In the event that this conjectured equality is true, our results also imply an explicit factorization of each dual canonical basis element as a product of cluster variables.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3598 https://doi.org/10.46298/dmtcs.3598 Skandera, Mark Skandera, Mark <![CDATA[We show that the set of cluster monomials for the cluster algebra of type $D_4$ contains a basis of the $\mathbb{Z}$-module $\mathbb{Z}[x_{1,1},\ldots ,x_{3,3}]$. We also show that the transition matrices relating this cluster basis to the natural and the dual canonical bases are unitriangular and nonnegative. These results support a conjecture of Fomin and Zelevinsky on the equality of the cluster and dual canonical bases. In the event that this conjectured equality is true, our results also imply an explicit factorization of each dual canonical basis element as a product of cluster variables.]]> 0 <![CDATA[We construct an $n$-dimensional polytope whose boundary complex is compressed and whose face numbers for any pulling triangulation are the coefficients of the powers of $(x-1)/2$ in the $n$-th Legendre polynomial. We show that the non-central Delannoy numbers count all faces in the lexicographic pulling triangulation that contain a point in a given open quadrant. We thus provide a geometric interpretation of a relation between the central Delannoy numbers and Legendre polynomials, observed over 50 years ago. The polytopes we construct are closely related to the root polytopes introduced by Gelfand, Graev, and Postnikov. \par]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3599 https://doi.org/10.46298/dmtcs.3599 Hetyei, Gábor Hetyei, Gábor <![CDATA[We construct an $n$-dimensional polytope whose boundary complex is compressed and whose face numbers for any pulling triangulation are the coefficients of the powers of $(x-1)/2$ in the $n$-th Legendre polynomial. We show that the non-central Delannoy numbers count all faces in the lexicographic pulling triangulation that contain a point in a given open quadrant. We thus provide a geometric interpretation of a relation between the central Delannoy numbers and Legendre polynomials, observed over 50 years ago. The polytopes we construct are closely related to the root polytopes introduced by Gelfand, Graev, and Postnikov. \par]]> 0 <![CDATA[We prove a collection of conjectures due to Abuzzahab-Korson-Li-Meyer, Reiner, and White regarding the cyclic sieving phenomenon as it applies to jeu-de-taquin promotion on rectangular tableaux. To do this, we use Kazhdan-Lusztig theory and a characterization of the dual canonical basis of $\mathbb{C}[x_{11}, \ldots , x_{nn}]$ due to Skandera. Afterwards, we extend our results to analyzing the fixed points of a dihedral action on rectangular tableaux generated by promotion and evacuation, suggesting a possible sieving phenomenon for dihedral groups. Finally, we give applications of this theory to cyclic sieving phenomena involving reduced words for the long elements of hyperoctohedral groups, handshake patterns, and noncrossing partitions.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3600 https://doi.org/10.46298/dmtcs.3600 Rhoades, Brendon Rhoades, Brendon <![CDATA[We prove a collection of conjectures due to Abuzzahab-Korson-Li-Meyer, Reiner, and White regarding the cyclic sieving phenomenon as it applies to jeu-de-taquin promotion on rectangular tableaux. To do this, we use Kazhdan-Lusztig theory and a characterization of the dual canonical basis of $\mathbb{C}[x_{11}, \ldots , x_{nn}]$ due to Skandera. Afterwards, we extend our results to analyzing the fixed points of a dihedral action on rectangular tableaux generated by promotion and evacuation, suggesting a possible sieving phenomenon for dihedral groups. Finally, we give applications of this theory to cyclic sieving phenomena involving reduced words for the long elements of hyperoctohedral groups, handshake patterns, and noncrossing partitions.]]> 0 <![CDATA[We prove that a $q$-deformation $\mathfrak{D}_k(\mathbb{X};q)$ of the powers of the discriminant is equal, up to a normalization, to a specialization of a Macdonald polynomial indexed by a staircase partition. We investigate the expansion of $\mathfrak{D}_k(\mathbb{X};q)$ on different bases of symmetric functions. In particular, we show that its expansion on the monomial basis can be explicitly described in terms of standard tableaux and we generalize a result of King-Toumazet-Wybourne about the expansion of the $q$-discriminant on the Schur basis.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3601 https://doi.org/10.46298/dmtcs.3601 Boussicault, Adrien Luque, Jean-Gabriel Boussicault, Adrien Luque, Jean-Gabriel <![CDATA[We prove that a $q$-deformation $\mathfrak{D}_k(\mathbb{X};q)$ of the powers of the discriminant is equal, up to a normalization, to a specialization of a Macdonald polynomial indexed by a staircase partition. We investigate the expansion of $\mathfrak{D}_k(\mathbb{X};q)$ on different bases of symmetric functions. In particular, we show that its expansion on the monomial basis can be explicitly described in terms of standard tableaux and we generalize a result of King-Toumazet-Wybourne about the expansion of the $q$-discriminant on the Schur basis.]]> 0 <![CDATA[Let $(W,S)$ be an arbitrary Coxeter system. For each sequence $\omega =(\omega_1,\omega_2,\ldots) \in S^{\ast}$ in the generators we define a partial order― called the $\omega \mathsf{-sorting order}$ ―on the set of group elements $W_{\omega} \subseteq W$ that occur as finite subwords of $\omega$ . We show that the $\omega$-sorting order is a supersolvable join-distributive lattice and that it is strictly between the weak and strong Bruhat orders on the group. Moreover, the $\omega$-sorting order is a "maximal lattice'' in the sense that the addition of any collection of edges from the Bruhat order results in a nonlattice. Along the way we define a class of structures called $\mathsf{supersolvable}$ $\mathsf{antimatroids}$ and we show that these are equivalent to the class of supersolvable join-distributive lattices.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3602 https://doi.org/10.46298/dmtcs.3602 Armstrong, Drew Armstrong, Drew <![CDATA[Let $(W,S)$ be an arbitrary Coxeter system. For each sequence $\omega =(\omega_1,\omega_2,\ldots) \in S^{\ast}$ in the generators we define a partial order― called the $\omega \mathsf{-sorting order}$ ―on the set of group elements $W_{\omega} \subseteq W$ that occur as finite subwords of $\omega$ . We show that the $\omega$-sorting order is a supersolvable join-distributive lattice and that it is strictly between the weak and strong Bruhat orders on the group. Moreover, the $\omega$-sorting order is a "maximal lattice'' in the sense that the addition of any collection of edges from the Bruhat order results in a nonlattice. Along the way we define a class of structures called $\mathsf{supersolvable}$ $\mathsf{antimatroids}$ and we show that these are equivalent to the class of supersolvable join-distributive lattices.]]> 0 <![CDATA[We consider Markovian models on graphs with local dynamics. We show that, under suitable conditions, such Markov chains exhibit both rapid convergence to equilibrium and strong concentration of measure in the stationary distribution. We illustrate our results with applications to some known chains from computer science and statistical mechanics.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3558 https://doi.org/10.46298/dmtcs.3558 Luczak, Malwina, Luczak, Malwina, <![CDATA[We consider Markovian models on graphs with local dynamics. We show that, under suitable conditions, such Markov chains exhibit both rapid convergence to equilibrium and strong concentration of measure in the stationary distribution. We illustrate our results with applications to some known chains from computer science and statistical mechanics.]]> 0 <![CDATA[This extended abstract is dedicated to the analysis of the height of non-plane unlabelled rooted binary trees. The height of such a tree chosen uniformly among those of size $n$ is proved to have a limiting theta distribution, both in a central and local sense. Moderate as well as large deviations estimates are also derived. The proofs rely on the analysis (in the complex plane) of generating functions associated with trees of bounded height.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3559 https://doi.org/10.46298/dmtcs.3559 Broutin, Nicolas Flajolet, Philippe Broutin, Nicolas Flajolet, Philippe <![CDATA[This extended abstract is dedicated to the analysis of the height of non-plane unlabelled rooted binary trees. The height of such a tree chosen uniformly among those of size $n$ is proved to have a limiting theta distribution, both in a central and local sense. Moderate as well as large deviations estimates are also derived. The proofs rely on the analysis (in the complex plane) of generating functions associated with trees of bounded height.]]> 0 <![CDATA[The register function for binary trees is the minimal number of extra registers required to evaluate the tree. This concept is also known as Horton-Strahler numbers. We extend this definition to lattice paths, built from steps $\pm 1$, without positivity restriction. Exact expressions are derived for appropriate generating functions. A procedure is presented how to get asymptotics of all moments, in an almost automatic way; this is based on an earlier paper of the authors.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3560 https://doi.org/10.46298/dmtcs.3560 Louchard, Guy Prodinger, Helmut Louchard, Guy Prodinger, Helmut <![CDATA[The register function for binary trees is the minimal number of extra registers required to evaluate the tree. This concept is also known as Horton-Strahler numbers. We extend this definition to lattice paths, built from steps $\pm 1$, without positivity restriction. Exact expressions are derived for appropriate generating functions. A procedure is presented how to get asymptotics of all moments, in an almost automatic way; this is based on an earlier paper of the authors.]]> 0 <![CDATA[We develop a combinatorial structure to serve as model of random real world networks. Starting with plane oriented recursive trees we substitute the nodes by more complex graphs. In such a way we obtain graphs having a global tree-like structure while locally looking clustered. This fits with observations obtained from real-world networks. In particular we show that the resulting graphs are scale-free, that is, the degree distribution has an asymptotic power law.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3561 https://doi.org/10.46298/dmtcs.3561 Drmota, Michael Gittenberger, Bernhard Panholzer, Alois Drmota, Michael Gittenberger, Bernhard Panholzer, Alois <![CDATA[We develop a combinatorial structure to serve as model of random real world networks. Starting with plane oriented recursive trees we substitute the nodes by more complex graphs. In such a way we obtain graphs having a global tree-like structure while locally looking clustered. This fits with observations obtained from real-world networks. In particular we show that the resulting graphs are scale-free, that is, the degree distribution has an asymptotic power law.]]> 0 <![CDATA[We prove that for each $k \geq 0$, the probability that a root vertex in a random planar graph has degree $k$ tends to a computable constant $d_k$, and moreover that $\sum_k d_k =1$. The proof uses the tools developed by Gimènez and Noy in their solution to the problem of the asymptotic enumeration of planar graphs, and is based on a detailed analysis of the generating functions involved in counting planar graphs. However, in order to keep track of the degree of the root, new technical difficulties arise. We obtain explicit, although quite involved expressions, for the coefficients in the singular expansions of interest, which allow us to use transfer theorems in order to get an explicit expression for the probability generating function $p(w)=\sum_k d_k w^k$. From the explicit expression for $p(w)$ we can compute the $d_k$ to any degree of accuracy, and derive asymptotic estimates for large values of $k$.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3562 https://doi.org/10.46298/dmtcs.3562 Drmota, Michael Gimenez, Omer Noy, Marc Drmota, Michael Gimenez, Omer Noy, Marc <![CDATA[We prove that for each $k \geq 0$, the probability that a root vertex in a random planar graph has degree $k$ tends to a computable constant $d_k$, and moreover that $\sum_k d_k =1$. The proof uses the tools developed by Gimènez and Noy in their solution to the problem of the asymptotic enumeration of planar graphs, and is based on a detailed analysis of the generating functions involved in counting planar graphs. However, in order to keep track of the degree of the root, new technical difficulties arise. We obtain explicit, although quite involved expressions, for the coefficients in the singular expansions of interest, which allow us to use transfer theorems in order to get an explicit expression for the probability generating function $p(w)=\sum_k d_k w^k$. From the explicit expression for $p(w)$ we can compute the $d_k$ to any degree of accuracy, and derive asymptotic estimates for large values of $k$.]]> 0 <![CDATA[We consider a component of the word statistics known as clump; starting from a finite set of words, clumps are maximal overlapping sets of these occurrences. This object has first been studied by Schbath with the aim of counting the number of occurrences of words in random texts. Later work with similar probabilistic approach used the Chen-Stein approximation for a compound Poisson distribution, where the number of clumps follows a law close to Poisson. Presently there is no combinatorial counterpart to this approach, and we fill the gap here. We also provide a construction for the yet unsolved problem of clumps of an arbitrary finite set of words. In contrast with the probabilistic approach which only provides asymptotic results, the combinatorial method provides exact results that are useful when considering short sequences.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3563 https://doi.org/10.46298/dmtcs.3563 Bassino, Frédérique Clément, Julien Fayolle, Julien Nicodème, Pierre Bassino, Frédérique Clément, Julien Fayolle, Julien Nicodème, Pierre <![CDATA[We consider a component of the word statistics known as clump; starting from a finite set of words, clumps are maximal overlapping sets of these occurrences. This object has first been studied by Schbath with the aim of counting the number of occurrences of words in random texts. Later work with similar probabilistic approach used the Chen-Stein approximation for a compound Poisson distribution, where the number of clumps follows a law close to Poisson. Presently there is no combinatorial counterpart to this approach, and we fill the gap here. We also provide a construction for the yet unsolved problem of clumps of an arbitrary finite set of words. In contrast with the probabilistic approach which only provides asymptotic results, the combinatorial method provides exact results that are useful when considering short sequences.]]> 0 <![CDATA[We study the number of encryptions necessary to revoke a set of users in the complete subtree scheme (CST) and the subset-difference scheme (SD). These are well-known tree based broadcast encryption schemes. Park and Blake in: Journal of Discrete Algorithms, vol. 4, 2006, pp. 215―238, give the mean number of encryptions for these schemes. We continue their analysis and show that the limiting distribution of the number of encryptions for these schemes is normal. This implies that the mean numbers of Park and Blake are good estimates for the number of necessary encryptions used by these schemes.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3564 https://doi.org/10.46298/dmtcs.3564 Eagle, Christopher Gao, Zhicheng Omar, Mohamed Panario, Daniel Richmond, Bruce Eagle, Christopher Gao, Zhicheng Omar, Mohamed Panario, Daniel Richmond, Bruce <![CDATA[We study the number of encryptions necessary to revoke a set of users in the complete subtree scheme (CST) and the subset-difference scheme (SD). These are well-known tree based broadcast encryption schemes. Park and Blake in: Journal of Discrete Algorithms, vol. 4, 2006, pp. 215―238, give the mean number of encryptions for these schemes. We continue their analysis and show that the limiting distribution of the number of encryptions for these schemes is normal. This implies that the mean numbers of Park and Blake are good estimates for the number of necessary encryptions used by these schemes.]]> 0 <![CDATA[Severini and Mansour introduced $\textit{square polygons}$, as graphical representations of $\textit{square permutations}$, that is, permutations such that all entries are records (left or right, minimum or maximum), and they obtained a nice formula for their number. In this paper we give a recursive construction for this class of permutations, that allows to simplify the derivation of their formula and to enumerate the subclass of square permutations with a simple record polygon. We also show that the generating function of these permutations with respect to the number of records of each type is algebraic, answering a question of Wilf in a particular case.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3565 https://doi.org/10.46298/dmtcs.3565 Duchi, Enrica Poulalhon, Dominique Duchi, Enrica Poulalhon, Dominique <![CDATA[Severini and Mansour introduced $\textit{square polygons}$, as graphical representations of $\textit{square permutations}$, that is, permutations such that all entries are records (left or right, minimum or maximum), and they obtained a nice formula for their number. In this paper we give a recursive construction for this class of permutations, that allows to simplify the derivation of their formula and to enumerate the subclass of square permutations with a simple record polygon. We also show that the generating function of these permutations with respect to the number of records of each type is algebraic, answering a question of Wilf in a particular case.]]> 0 <![CDATA[Polynomial bounds and tail estimates are derived for additive random recursive sequences, which typically arise as functionals of recursive structures, of random trees, or in recursive algorithms. In particular they arise as parameters of divide and conquer type algorithms. We mainly focuss on polynomial tails that arise due to heavy tail bounds of the toll term and the starting distributions. Besides estimating the tail probability directly we use a modified version of a theorem from regular variation theory. This theorem states that upper bounds on the asymptotic tail probability can be derived from upper bounds of the Laplace―Stieltjes transforms near zero.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3566 https://doi.org/10.46298/dmtcs.3566 Schopp, Eva-Maria Schopp, Eva-Maria <![CDATA[Polynomial bounds and tail estimates are derived for additive random recursive sequences, which typically arise as functionals of recursive structures, of random trees, or in recursive algorithms. In particular they arise as parameters of divide and conquer type algorithms. We mainly focuss on polynomial tails that arise due to heavy tail bounds of the toll term and the starting distributions. Besides estimating the tail probability directly we use a modified version of a theorem from regular variation theory. This theorem states that upper bounds on the asymptotic tail probability can be derived from upper bounds of the Laplace―Stieltjes transforms near zero.]]> 0 <![CDATA[Sampling from a random discrete distribution induced by a 'stick-breaking' process is considered. Under a moment condition, it is shown that the asymptotics of the sequence of occupancy numbers, and of the small-parts counts (singletons, doubletons, etc) can be read off from a limiting model involving a unit Poisson point process and a self-similar renewal process on the half-line.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3567 https://doi.org/10.46298/dmtcs.3567 Gnedin, Alexander Iksanov, Alex Roesler, Uwe Gnedin, Alexander Iksanov, Alex Roesler, Uwe <![CDATA[Sampling from a random discrete distribution induced by a 'stick-breaking' process is considered. Under a moment condition, it is shown that the asymptotics of the sequence of occupancy numbers, and of the small-parts counts (singletons, doubletons, etc) can be read off from a limiting model involving a unit Poisson point process and a self-similar renewal process on the half-line.]]> 0 <![CDATA[Continuing the line of research initiated in Iksanov and Möhle (2008) and Negadajlov (2008) we investigate the asymptotic (as $n \to \infty$) behaviour of $V_n$ the number of zero increments before the absorption in a random walk with the barrier $n$. In particular, when the step of the unrestricted random walk has a finite mean, we prove that the number of zero increments converges in distribution. We also establish a weak law of large numbers for $V_n$ under a regular variation assumption.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3568 https://doi.org/10.46298/dmtcs.3568 Iksanov, Alex Negadajlov, Pavlo Iksanov, Alex Negadajlov, Pavlo <![CDATA[Continuing the line of research initiated in Iksanov and Möhle (2008) and Negadajlov (2008) we investigate the asymptotic (as $n \to \infty$) behaviour of $V_n$ the number of zero increments before the absorption in a random walk with the barrier $n$. In particular, when the step of the unrestricted random walk has a finite mean, we prove that the number of zero increments converges in distribution. We also establish a weak law of large numbers for $V_n$ under a regular variation assumption.]]> 0 <![CDATA[This paper makes use of the recently introduced technique of $\gamma$-operators to evaluate the Hankel determinant with binomial coefficient entries $a_k = (3 k)! / (2k)! k!$. We actually evaluate the determinant of a class of polynomials $a_k(x)$ having this binomial coefficient as constant term. The evaluation in the polynomial case is as an almost product, i.e. as a sum of a small number of products. The $\gamma$-operator technique to find the explicit form of the almost product relies on differential-convolution equations and establishes a second order differential equation for the determinant. In addition to $x=0$, product form evaluations for $x = \frac{3}{5}, \frac{3}{4}, \frac{3}{2}, 3$ are also presented. At $x=1$, we obtain another almost product evaluation for the Hankel determinant with $a_k = ( 3 k+1) ! / (2k+1)!k!$.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3569 https://doi.org/10.46298/dmtcs.3569 Eugeciouglu, Ömer Redmond, Timothy Ryavec, Charles Eugeciouglu, Ömer Redmond, Timothy Ryavec, Charles <![CDATA[This paper makes use of the recently introduced technique of $\gamma$-operators to evaluate the Hankel determinant with binomial coefficient entries $a_k = (3 k)! / (2k)! k!$. We actually evaluate the determinant of a class of polynomials $a_k(x)$ having this binomial coefficient as constant term. The evaluation in the polynomial case is as an almost product, i.e. as a sum of a small number of products. The $\gamma$-operator technique to find the explicit form of the almost product relies on differential-convolution equations and establishes a second order differential equation for the determinant. In addition to $x=0$, product form evaluations for $x = \frac{3}{5}, \frac{3}{4}, \frac{3}{2}, 3$ are also presented. At $x=1$, we obtain another almost product evaluation for the Hankel determinant with $a_k = ( 3 k+1) ! / (2k+1)!k!$.]]> 0 <![CDATA[We study the number of records in random split trees on $n$ randomly labelled vertices. Equivalently the number of random cuttings required to eliminate an arbitrary random split tree can be studied. After normalization the distributions are shown to be asymptotically $1$-stable. This work is a generalization of our earlier results for the random binary search tree which is one specific case of split trees. Other important examples of split trees include $m$-ary search trees, quadtrees, median of $(2k+1)$-trees, simplex trees, tries and digital search trees.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3570 https://doi.org/10.46298/dmtcs.3570 Holmgren, Cecilia Holmgren, Cecilia <![CDATA[We study the number of records in random split trees on $n$ randomly labelled vertices. Equivalently the number of random cuttings required to eliminate an arbitrary random split tree can be studied. After normalization the distributions are shown to be asymptotically $1$-stable. This work is a generalization of our earlier results for the random binary search tree which is one specific case of split trees. Other important examples of split trees include $m$-ary search trees, quadtrees, median of $(2k+1)$-trees, simplex trees, tries and digital search trees.]]> 0 <![CDATA[The Recoil Growth algorithm, proposed in 1999 by Consta $\textit{et al.}$, is one of the most efficient algorithm available in the literature to sample from a multi-polymer system. Such problems are closely related to the generation of self-avoiding paths. In this paper, we study a variant of the original Recoil Growth algorithm, where we constrain the generation of a new polymer to take place on a specific class of graphs. This makes it possible to make a fine trade-off between computational cost and success rate. We moreover give a simple proof for a lower bound on the irreducibility of this new algorithm, which applies to the original algorithm as well.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3571 https://doi.org/10.46298/dmtcs.3571 Simatos, Florian Simatos, Florian <![CDATA[The Recoil Growth algorithm, proposed in 1999 by Consta $\textit{et al.}$, is one of the most efficient algorithm available in the literature to sample from a multi-polymer system. Such problems are closely related to the generation of self-avoiding paths. In this paper, we study a variant of the original Recoil Growth algorithm, where we constrain the generation of a new polymer to take place on a specific class of graphs. This makes it possible to make a fine trade-off between computational cost and success rate. We moreover give a simple proof for a lower bound on the irreducibility of this new algorithm, which applies to the original algorithm as well.]]> 0 <![CDATA[The paper deals with the problem of catching the elephants in the Internet traffic. The aim is to investigate an algorithm proposed by Azzana based on a multistage Bloom filter, with a refreshment mechanism (called $\textit{shift}$ in the present paper), able to treat on-line a huge amount of flows with high traffic variations. An analysis of a simplified model estimates the number of false positives. Limit theorems for the Markov chain that describes the algorithm for large filters are rigorously obtained. The asymptotic behavior of the stochastic model is here deterministic. The limit has a nice formulation in terms of a $M/G/1/C$ queue, which is analytically tractable and which allows to tune the algorithm optimally.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3572 https://doi.org/10.46298/dmtcs.3572 Chabchoub, Yousra Fricker, Christine Meunier, Frédéric Tibi, Danielle Chabchoub, Yousra Fricker, Christine Meunier, Frédéric Tibi, Danielle <![CDATA[The paper deals with the problem of catching the elephants in the Internet traffic. The aim is to investigate an algorithm proposed by Azzana based on a multistage Bloom filter, with a refreshment mechanism (called $\textit{shift}$ in the present paper), able to treat on-line a huge amount of flows with high traffic variations. An analysis of a simplified model estimates the number of false positives. Limit theorems for the Markov chain that describes the algorithm for large filters are rigorously obtained. The asymptotic behavior of the stochastic model is here deterministic. The limit has a nice formulation in terms of a $M/G/1/C$ queue, which is analytically tractable and which allows to tune the algorithm optimally.]]> 0 <![CDATA[We discuss scaling limits of random planar maps chosen uniformly over the set of all $2p$-angulations with $n$ faces. This leads to a limiting space called the Brownian map, which is viewed as a random compact metric space. Although we are not able to prove the uniqueness of the distribution of the Brownian map, many of its properties can be investigated in detail. In particular, we obtain a complete description of the geodesics starting from the distinguished point called the root. We give applications to various properties of large random planar maps.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3554 https://doi.org/10.46298/dmtcs.3554 Le Gall, Jean-François Le Gall, Jean-François <![CDATA[We discuss scaling limits of random planar maps chosen uniformly over the set of all $2p$-angulations with $n$ faces. This leads to a limiting space called the Brownian map, which is viewed as a random compact metric space. Although we are not able to prove the uniqueness of the distribution of the Brownian map, many of its properties can be investigated in detail. In particular, we obtain a complete description of the geodesics starting from the distinguished point called the root. We give applications to various properties of large random planar maps.]]> 0 <![CDATA[Analytic information theory aims at studying problems of information theory using analytic techniques of computer science and combinatorics. Following Hadamard's precept, these problems are tackled by complex analysis methods such as generating functions, Mellin transform, Fourier series, saddle point method, analytic poissonization and depoissonization, and singularity analysis. This approach lies at the crossroad of computer science and information theory. In this survey we concentrate on one facet of information theory (i.e., source coding better known as data compression), namely the $\textit{redundancy rate}$ problem. The redundancy rate problem determines by how much the actual code length exceeds the optimal code length. We further restrict our interest to the $\textit{average}$ redundancy for $\textit{known}$ sources, that is, when statistics of information sources are known. We present precise analyses of three types of lossless data compression schemes, namely fixed-to-variable (FV) length codes, variable-to-fixed (VF) length codes, and variable-to-variable (VV) length codes. In particular, we investigate average redundancy of Huffman, Tunstall, and Khodak codes. These codes have succinct representations as $\textit{trees}$, either as coding or parsing trees, and we analyze here some of their parameters (e.g., the average path from the root to a leaf).]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3555 https://doi.org/10.46298/dmtcs.3555 Szpankowski, Wojciech Szpankowski, Wojciech <![CDATA[Analytic information theory aims at studying problems of information theory using analytic techniques of computer science and combinatorics. Following Hadamard's precept, these problems are tackled by complex analysis methods such as generating functions, Mellin transform, Fourier series, saddle point method, analytic poissonization and depoissonization, and singularity analysis. This approach lies at the crossroad of computer science and information theory. In this survey we concentrate on one facet of information theory (i.e., source coding better known as data compression), namely the $\textit{redundancy rate}$ problem. The redundancy rate problem determines by how much the actual code length exceeds the optimal code length. We further restrict our interest to the $\textit{average}$ redundancy for $\textit{known}$ sources, that is, when statistics of information sources are known. We present precise analyses of three types of lossless data compression schemes, namely fixed-to-variable (FV) length codes, variable-to-fixed (VF) length codes, and variable-to-variable (VV) length codes. In particular, we investigate average redundancy of Huffman, Tunstall, and Khodak codes. These codes have succinct representations as $\textit{trees}$, either as coding or parsing trees, and we analyze here some of their parameters (e.g., the average path from the root to a leaf).]]> 0 <![CDATA[We provide an overview of stabilization methods for point processes and apply these methods to deduce a central limit theorem for statistical estimators of dimension.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3556 https://doi.org/10.46298/dmtcs.3556 Yukich, J. E. Yukich, J. E. <![CDATA[We provide an overview of stabilization methods for point processes and apply these methods to deduce a central limit theorem for statistical estimators of dimension.]]> 0 <![CDATA[We provide normal approximation error bounds for sums of the form $\sum_x \xi_x$, indexed by the points $x$ of a Poisson process (not necessarily homogeneous) in the unit $d$-cube, with each term $\xi_x$ determined by the configuration of Poisson points near to $x$ in some sense. We consider geometric graphs and coverage processes as examples of our general results.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3557 https://doi.org/10.46298/dmtcs.3557 Penrose, Mathew, Rosoman, Tom Penrose, Mathew, Rosoman, Tom <![CDATA[We provide normal approximation error bounds for sums of the form $\sum_x \xi_x$, indexed by the points $x$ of a Poisson process (not necessarily homogeneous) in the unit $d$-cube, with each term $\xi_x$ determined by the configuration of Poisson points near to $x$ in some sense. We consider geometric graphs and coverage processes as examples of our general results.]]> 0 <![CDATA[For the class of haploid exchangeable population models with non-overlapping generations and population size $N$ it is shown that, as $N$ tends to infinity, convergence of the time-scaled ancestral process to Kingman's coalescent and convergence in distribution of the scaled times back to the most recent common ancestor (MRCA) to the corresponding times back to the MRCA of the Kingman coalescent are equivalent. Extensions of this equivalence are derived for exchangeable population models being in the domain of attraction of a coalescent process with multiple collisions. The proofs are based on the property that the total rates of a coalescent with multiple collisions already determine the distribution of the coalescent. It is finally shown that similar results cannot be obtained for the full class of exchangeable coalescents allowing for simultaneous multiple collisions of ancestral lineages, essentially because the total rates do not determine the distribution of a general exchangeable coalescent.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3573 https://doi.org/10.46298/dmtcs.3573 Möhle, Martin Möhle, Martin <![CDATA[For the class of haploid exchangeable population models with non-overlapping generations and population size $N$ it is shown that, as $N$ tends to infinity, convergence of the time-scaled ancestral process to Kingman's coalescent and convergence in distribution of the scaled times back to the most recent common ancestor (MRCA) to the corresponding times back to the MRCA of the Kingman coalescent are equivalent. Extensions of this equivalence are derived for exchangeable population models being in the domain of attraction of a coalescent process with multiple collisions. The proofs are based on the property that the total rates of a coalescent with multiple collisions already determine the distribution of the coalescent. It is finally shown that similar results cannot be obtained for the full class of exchangeable coalescents allowing for simultaneous multiple collisions of ancestral lineages, essentially because the total rates do not determine the distribution of a general exchangeable coalescent.]]> 0 <![CDATA[If the interest of stochastic L-systems for plant growth simulation and visualization is broadly acknowledged, their full mathematical potential has not been taken advantage of. In this article, we show how to link stochastic L-systems to multitype branching processes, in order to characterize the probability distributions and moments of the numbers of organs in plant structure. Plant architectural development can be seen as the combination of two subprocesses driving the bud population dynamics, branching and differentiation. By writing the stochastic L-system associated to each subprocess, we get the generating function associated to the whole system by compounding the associated generating functions. The modelling of stochastic branching is classical, but to model differentiation, we introduce a new framework based on multivariate phase-type random vectors.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3574 https://doi.org/10.46298/dmtcs.3574 Loi, Cedric Cournède, Paul Henry Loi, Cedric Cournède, Paul Henry <![CDATA[If the interest of stochastic L-systems for plant growth simulation and visualization is broadly acknowledged, their full mathematical potential has not been taken advantage of. In this article, we show how to link stochastic L-systems to multitype branching processes, in order to characterize the probability distributions and moments of the numbers of organs in plant structure. Plant architectural development can be seen as the combination of two subprocesses driving the bud population dynamics, branching and differentiation. By writing the stochastic L-system associated to each subprocess, we get the generating function associated to the whole system by compounding the associated generating functions. The modelling of stochastic branching is classical, but to model differentiation, we introduce a new framework based on multivariate phase-type random vectors.]]> 0 <![CDATA[We give a functional limit law for the normalized profile of random plane-oriented recursive trees. The proof uses martingale convergence theorems in discrete and continuous-time. This complements results of Hwang (2007).]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3575 https://doi.org/10.46298/dmtcs.3575 Sulzbach, Henning Sulzbach, Henning <![CDATA[We give a functional limit law for the normalized profile of random plane-oriented recursive trees. The proof uses martingale convergence theorems in discrete and continuous-time. This complements results of Hwang (2007).]]> 0 <![CDATA[This paper is concerned with the analysis of the worst case behavior of Hopcroft's algorithm for minimizing deterministic finite state automata. We extend a result of Castiglione, Restivo and Sciortino. They show that Hopcroft's algorithm has a worst case behavior for the automata recognizing Fibonacci words. We prove that the same holds for all standard Sturmian words having an ultimately periodic directive sequence (the directive sequence for Fibonacci words is $(1,1,\ldots)$).]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3576 https://doi.org/10.46298/dmtcs.3576 Berstel, Jean Boasson, Luc Carton, Olivier Berstel, Jean Boasson, Luc Carton, Olivier <![CDATA[This paper is concerned with the analysis of the worst case behavior of Hopcroft's algorithm for minimizing deterministic finite state automata. We extend a result of Castiglione, Restivo and Sciortino. They show that Hopcroft's algorithm has a worst case behavior for the automata recognizing Fibonacci words. We prove that the same holds for all standard Sturmian words having an ultimately periodic directive sequence (the directive sequence for Fibonacci words is $(1,1,\ldots)$).]]> 0 <![CDATA[It is well known that a planar map is bipartite if and only if all its faces have even degree (what we call an even map). In this paper, we show that rooted even maps of positive genus $g$ chosen uniformly at random are bipartite with probability tending to $4^{−g}$ when their size goes to infinity. Loosely speaking, we show that each of the $2g$ fundamental cycles of the surface of genus $g$ contributes a factor $\frac{1}{2}$ to this probability.We actually do more than that: we obtain the explicit asymptotic behaviour of the number of even maps and bipartite maps of given genus with any finite set of allowed face degrees. This uses a generalisation of the Bouttier-Di Francesco-Guitter bijection to the case of positive genus, a decomposition inspired by previous works of Marcus, Schaeffer and the author, and some involved manipulations of generating series counting paths. A special case of our results implies former conjectures of Gao.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3577 https://doi.org/10.46298/dmtcs.3577 Chapuy, Guillaume Chapuy, Guillaume <![CDATA[It is well known that a planar map is bipartite if and only if all its faces have even degree (what we call an even map). In this paper, we show that rooted even maps of positive genus $g$ chosen uniformly at random are bipartite with probability tending to $4^{−g}$ when their size goes to infinity. Loosely speaking, we show that each of the $2g$ fundamental cycles of the surface of genus $g$ contributes a factor $\frac{1}{2}$ to this probability.We actually do more than that: we obtain the explicit asymptotic behaviour of the number of even maps and bipartite maps of given genus with any finite set of allowed face degrees. This uses a generalisation of the Bouttier-Di Francesco-Guitter bijection to the case of positive genus, a decomposition inspired by previous works of Marcus, Schaeffer and the author, and some involved manipulations of generating series counting paths. A special case of our results implies former conjectures of Gao.]]> 0 <![CDATA[Let $Z_n,n=0,1,\ldots,$ be a branching process evolving in the random environment generated by a sequence of iid generating functions $f_0(s),f_1(s),\ldots,$ and let $S_0=0$, $S_k=X_1+ \ldots +X_k,k \geq 1$, be the associated random walk with $X_i=\log f_{i-1}^{\prime}(1), \tau (m,n)$ be the left-most point of minimum of $\{S_k,k \geq 0 \}$ on the interval $[m,n]$, and $T=\min \{ k:Z_k=0\}$. Assuming that the associated random walk satisfies the Doney condition $P(S_n > 0) \to \rho \in (0,1), n \to \infty$, we prove (under the quenched approach) conditional limit theorems, as $n \to \infty$, for the distribution of $Z_{nt}, Z_{\tau (0,nt)}$, and $Z_{\tau (nt,n)}, t \in (0,1)$, given $T=n$. It is shown that the form of the limit distributions essentially depends on the location of $\tau (0,n)$ with respect to the point $nt$.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3578 https://doi.org/10.46298/dmtcs.3578 Vatutin, Vladimir, Kyprianou, Andreas, Vatutin, Vladimir, Kyprianou, Andreas, <![CDATA[Let $Z_n,n=0,1,\ldots,$ be a branching process evolving in the random environment generated by a sequence of iid generating functions $f_0(s),f_1(s),\ldots,$ and let $S_0=0$, $S_k=X_1+ \ldots +X_k,k \geq 1$, be the associated random walk with $X_i=\log f_{i-1}^{\prime}(1), \tau (m,n)$ be the left-most point of minimum of $\{S_k,k \geq 0 \}$ on the interval $[m,n]$, and $T=\min \{ k:Z_k=0\}$. Assuming that the associated random walk satisfies the Doney condition $P(S_n > 0) \to \rho \in (0,1), n \to \infty$, we prove (under the quenched approach) conditional limit theorems, as $n \to \infty$, for the distribution of $Z_{nt}, Z_{\tau (0,nt)}$, and $Z_{\tau (nt,n)}, t \in (0,1)$, given $T=n$. It is shown that the form of the limit distributions essentially depends on the location of $\tau (0,n)$ with respect to the point $nt$.]]> 0 <![CDATA[We investigate a multi-type Galton-Watson process in a random environment generated by a sequence of independent identically distributed random variables. Suppose that the associated random walk constructed by the logarithms of the Perron roots of the reproduction mean matrices has negative mean and assuming some additional conditions, we find the asymptotics of the survival probability at time $n$ as $n \to \infty$.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3579 https://doi.org/10.46298/dmtcs.3579 Dyakonova, Elena Dyakonova, Elena <![CDATA[We investigate a multi-type Galton-Watson process in a random environment generated by a sequence of independent identically distributed random variables. Suppose that the associated random walk constructed by the logarithms of the Perron roots of the reproduction mean matrices has negative mean and assuming some additional conditions, we find the asymptotics of the survival probability at time $n$ as $n \to \infty$.]]> 0 <![CDATA[We consider random walks on the set of all words over a finite alphabet such that in each step only the last two letters of the current word may be modified and only one letter may be adjoined or deleted. We assume that the transition probabilities depend only on the last two letters of the current word. Furthermore, we consider also the special case of random walks on free products by amalgamation of finite groups which arise in a natural way from random walks on the single factors. The aim of this paper is to compute several equivalent formulas for the rate of escape with respect to natural length functions for these random walks using different techniques.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3580 https://doi.org/10.46298/dmtcs.3580 Gilch, Lorenz A. Gilch, Lorenz A. <![CDATA[We consider random walks on the set of all words over a finite alphabet such that in each step only the last two letters of the current word may be modified and only one letter may be adjoined or deleted. We assume that the transition probabilities depend only on the last two letters of the current word. Furthermore, we consider also the special case of random walks on free products by amalgamation of finite groups which arise in a natural way from random walks on the single factors. The aim of this paper is to compute several equivalent formulas for the rate of escape with respect to natural length functions for these random walks using different techniques.]]> 0 <![CDATA[Gantert and Müller (2006) proved that a critical branching random walk (BRW) on the integer lattice is transient by analyzing this problem within the more general framework of branching Markov chains and making use of Lyapunov functions. The main purpose of this note is to show how the same result can be derived quite elegantly and even extended to the nonlattice case within the theory of weighted branching processes. This is done by an analysis of certain associated random weighted location measures which, upon taking expectations, provide a useful connection to the well established theory of ordinary random walks with i.i.d. increments. A brief discussion of the asymptotic behavior of the left- and rightmost particles in a critical BRW as time goes to infinity is provided in the final section by drawing on recent work by Hu and Shi (2008).]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3581 https://doi.org/10.46298/dmtcs.3581 Alsmeyer, Gerold Meiners, Matthias Alsmeyer, Gerold Meiners, Matthias <![CDATA[Gantert and Müller (2006) proved that a critical branching random walk (BRW) on the integer lattice is transient by analyzing this problem within the more general framework of branching Markov chains and making use of Lyapunov functions. The main purpose of this note is to show how the same result can be derived quite elegantly and even extended to the nonlattice case within the theory of weighted branching processes. This is done by an analysis of certain associated random weighted location measures which, upon taking expectations, provide a useful connection to the well established theory of ordinary random walks with i.i.d. increments. A brief discussion of the asymptotic behavior of the left- and rightmost particles in a critical BRW as time goes to infinity is provided in the final section by drawing on recent work by Hu and Shi (2008).]]> 0 <![CDATA[Let $P_k(f)$ denote the density of and/or trees defining a boolean function $f$ within the set of and/or trees with fixed number of variables $k$. We prove that there exists constant $B_f$ such that $P_k(f) \sim B_f \cdot k^{-L(f)-1}$ when $k \to \infty$, where $L(f)$ denote the complexity of $f$ (i.e. the size of a minimal and/or tree defining $f$). This theorem has been conjectured by Danièle Gardy and Alan Woods together with its counterpart for distribution $\pi$ defined by some critical Galton-Watson process. Methods presented in this paper can be also applied to prove the analogous property for $\pi$.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3582 https://doi.org/10.46298/dmtcs.3582 Kozik, Jakub Kozik, Jakub <![CDATA[Let $P_k(f)$ denote the density of and/or trees defining a boolean function $f$ within the set of and/or trees with fixed number of variables $k$. We prove that there exists constant $B_f$ such that $P_k(f) \sim B_f \cdot k^{-L(f)-1}$ when $k \to \infty$, where $L(f)$ denote the complexity of $f$ (i.e. the size of a minimal and/or tree defining $f$). This theorem has been conjectured by Danièle Gardy and Alan Woods together with its counterpart for distribution $\pi$ defined by some critical Galton-Watson process. Methods presented in this paper can be also applied to prove the analogous property for $\pi$.]]> 0 <![CDATA[In this paper we focus on the intuitionistic propositional logic with one propositional variable. More precisely we consider the standard fragment $\{ \to ,\vee ,\bot \}$ of this logic and compute the proportion of tautologies among all formulas. It turns out that this proportion is different from the analog one in the classical logic case.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3583 https://doi.org/10.46298/dmtcs.3583 Kostrzycka, Zofia Kostrzycka, Zofia <![CDATA[In this paper we focus on the intuitionistic propositional logic with one propositional variable. More precisely we consider the standard fragment $\{ \to ,\vee ,\bot \}$ of this logic and compute the proportion of tautologies among all formulas. It turns out that this proportion is different from the analog one in the classical logic case.]]> 0 <![CDATA[Within the language of propositional formulae built on implication and a finite number of variables $k$, we analyze the set of formulae which are classical tautologies but not intuitionistic (we call such formulae - Peirce's formulae). We construct the large family of so called simple Peirce's formulae, whose sequence of densities for different $k$ is asymptotically equivalent to the sequence $\frac{1}{ 2 k^2}$. We prove that the densities of the sets of remaining Peirce's formulae are asymptotically bounded from above by $\frac{c}{ k^3}$ for some constant $c \in \mathbb{R}$. The result justifies the statement that in the considered language almost all Peirce's formulae are simple. The result gives a partial answer to the question stated in the recent paper by H. Fournier, D. Gardy, A. Genitrini and M. Zaionc - although we have not proved the existence of the densities for Peirce's formulae, our result gives lower and upper bound for it (if it exists) and both bounds are asymptotically equivalent to $\frac{1}{ 2 k^2}$.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3584 https://doi.org/10.46298/dmtcs.3584 Genitrini, Antoine Kozik, Jakub Matecki, Grzegorz Genitrini, Antoine Kozik, Jakub Matecki, Grzegorz <![CDATA[Within the language of propositional formulae built on implication and a finite number of variables $k$, we analyze the set of formulae which are classical tautologies but not intuitionistic (we call such formulae - Peirce's formulae). We construct the large family of so called simple Peirce's formulae, whose sequence of densities for different $k$ is asymptotically equivalent to the sequence $\frac{1}{ 2 k^2}$. We prove that the densities of the sets of remaining Peirce's formulae are asymptotically bounded from above by $\frac{c}{ k^3}$ for some constant $c \in \mathbb{R}$. The result justifies the statement that in the considered language almost all Peirce's formulae are simple. The result gives a partial answer to the question stated in the recent paper by H. Fournier, D. Gardy, A. Genitrini and M. Zaionc - although we have not proved the existence of the densities for Peirce's formulae, our result gives lower and upper bound for it (if it exists) and both bounds are asymptotically equivalent to $\frac{1}{ 2 k^2}$.]]> 0 <![CDATA[Boltzmann random generation applies to well-defined systems of recursive combinatorial equations. It relies on oracles giving values of the enumeration generating series inside their disk of convergence. We show that the combinatorial systems translate into numerical iteration schemes that provide such oracles. In particular, we give a fast oracle based on Newton iteration.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3585 https://doi.org/10.46298/dmtcs.3585 Pivoteau, Carine Salvy, Bruno Soria, Michèle Pivoteau, Carine Salvy, Bruno Soria, Michèle <![CDATA[Boltzmann random generation applies to well-defined systems of recursive combinatorial equations. It relies on oracles giving values of the enumeration generating series inside their disk of convergence. We show that the combinatorial systems translate into numerical iteration schemes that provide such oracles. In particular, we give a fast oracle based on Newton iteration.]]> 0 <![CDATA[We introduce a recursive algorithm generating random trees, which we identify as skeletons of a continuous, stable tree. We deduce a representation of a fragmentation process on these trees.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3586 https://doi.org/10.46298/dmtcs.3586 Marchal, Philippe Marchal, Philippe <![CDATA[We introduce a recursive algorithm generating random trees, which we identify as skeletons of a continuous, stable tree. We deduce a representation of a fragmentation process on these trees.]]> 0 <![CDATA[According to the by now established theory developed in order to define a Laplacian or ― equivalently ― a Brownian motion on a nested fractal, one has to solve certain renormalization problems. In this paper, we present a Markov chain algorithm solving the problem for certain classes of simple fractals $K$ provided that there exists a unique Brownian motion and hence, a unique Laplacian on $K$.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3587 https://doi.org/10.46298/dmtcs.3587 Freiberg, Uta Thäle, Christoph Freiberg, Uta Thäle, Christoph <![CDATA[According to the by now established theory developed in order to define a Laplacian or ― equivalently ― a Brownian motion on a nested fractal, one has to solve certain renormalization problems. In this paper, we present a Markov chain algorithm solving the problem for certain classes of simple fractals $K$ provided that there exists a unique Brownian motion and hence, a unique Laplacian on $K$.]]> 0 <![CDATA[In this paper we solve the known V.A. Liskovets problem (1996) on the enumeration of orientable coverings over a non-orientable manifold with an arbitrary finitely generated fundamental group. As an application we obtain general formulas for the number of chiral and reflexible coverings over the manifold. As a further application, we count the chiral and reflexible maps and hypermaps on a closed orientable surface by the number of edges. Also, by this method the number of self-dual and Petri-dual maps can be determined. This will be done in forthcoming papers by authors.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3647 https://doi.org/10.46298/dmtcs.3647 Kwak, Jin Ho Mednykh, Alexander Nedela, Roman Kwak, Jin Ho Mednykh, Alexander Nedela, Roman <![CDATA[In this paper we solve the known V.A. Liskovets problem (1996) on the enumeration of orientable coverings over a non-orientable manifold with an arbitrary finitely generated fundamental group. As an application we obtain general formulas for the number of chiral and reflexible coverings over the manifold. As a further application, we count the chiral and reflexible maps and hypermaps on a closed orientable surface by the number of edges. Also, by this method the number of self-dual and Petri-dual maps can be determined. This will be done in forthcoming papers by authors.]]> 0 <![CDATA[Let $\Gamma$ be a simplicial complex with $n$ vertices, and let $\Delta (\Gamma)$ be either its exterior algebraic shifted complex or its symmetric algebraic shifted complex. If $\Gamma$ is a simplicial sphere, then it is known that (a) $\Delta (\Gamma)$ is pure and (b) $h$-vector of $\Gamma$ is symmetric. Kalai and Sarkaria conjectured that if $\Gamma$ is a simplicial sphere then its algebraic shifting also satisfies (c) $\Delta (\Gamma) \subset \Delta (C(n,d))$, where $C(n,d)$ is the boundary complex of the cyclic $d$-polytope with $n$ vertices. We show this conjecture for strongly edge decomposable spheres introduced by Nevo. We also show that any shifted simplicial complex satisfying (a), (b) and (c) is the algebraic shifted complex of some simplicial sphere.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3652 https://doi.org/10.46298/dmtcs.3652 Murai, Satoshi Murai, Satoshi <![CDATA[Let $\Gamma$ be a simplicial complex with $n$ vertices, and let $\Delta (\Gamma)$ be either its exterior algebraic shifted complex or its symmetric algebraic shifted complex. If $\Gamma$ is a simplicial sphere, then it is known that (a) $\Delta (\Gamma)$ is pure and (b) $h$-vector of $\Gamma$ is symmetric. Kalai and Sarkaria conjectured that if $\Gamma$ is a simplicial sphere then its algebraic shifting also satisfies (c) $\Delta (\Gamma) \subset \Delta (C(n,d))$, where $C(n,d)$ is the boundary complex of the cyclic $d$-polytope with $n$ vertices. We show this conjecture for strongly edge decomposable spheres introduced by Nevo. We also show that any shifted simplicial complex satisfying (a), (b) and (c) is the algebraic shifted complex of some simplicial sphere.]]> 0 <![CDATA[Since singletons are the connected sets, the species $X$ of singletons can be considered as the combinatorial logarithm of the species $E(X)$ of finite sets. In a previous work, we introduced the (rational) species $\widehat{X}$ of pseudo-singletons as the analytical logarithm of the species of finite sets. It follows that $E(X) = \exp (\widehat{X})$ in the context of rational species, where $\exp (T)$ denotes the classical analytical power series for the exponential function in the variable $T$. In the present work, we use the species $\widehat{X}$ to create new efficient recursive schemes for the computation of molecular expansions of species of rooted trees, of species of assemblies of structures, of the combinatorial logarithm species, of species of connected structures, and of species of structures with weighted connected components.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3651 https://doi.org/10.46298/dmtcs.3651 Labelle, Gilbert Labelle, Gilbert <![CDATA[Since singletons are the connected sets, the species $X$ of singletons can be considered as the combinatorial logarithm of the species $E(X)$ of finite sets. In a previous work, we introduced the (rational) species $\widehat{X}$ of pseudo-singletons as the analytical logarithm of the species of finite sets. It follows that $E(X) = \exp (\widehat{X})$ in the context of rational species, where $\exp (T)$ denotes the classical analytical power series for the exponential function in the variable $T$. In the present work, we use the species $\widehat{X}$ to create new efficient recursive schemes for the computation of molecular expansions of species of rooted trees, of species of assemblies of structures, of the combinatorial logarithm species, of species of connected structures, and of species of structures with weighted connected components.]]> 0 <![CDATA[In this paper we give an alternate combinatorial description of the "$(\ell,0)$-Carter partitions''. Our main theorem is the equivalence of our combinatoric and the one introduced by James and Mathas ($\textit{A q-analogue of the Jantzen-Schaper theorem}$). The condition of being an $(\ell,0)$-Carter partition is fundamentally related to the hook lengths of the partition. The representation-theoretic significance of their combinatoric on an $\ell$-regular partition is that it indicates the irreducibility of the corresponding Specht module over the finite Hecke algebra. We use our result to find a generating series which counts the number of such partitions, with respect to the statistic of a partition's first part. We then apply our description of these partitions to the crystal graph $B(\Lambda_0)$ of the basic representation of $\widehat{\mathfrak{sl}_{\ell}}$, whose nodes are labeled by $\ell$-regular partitions. Here we give a fairly simple crystal-theoretic rule which generates all $(\ell,0)$-Carter partitions in the graph of $B(\Lambda_0)$.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3650 https://doi.org/10.46298/dmtcs.3650 Berg, Chris Vazirani, Monica Berg, Chris Vazirani, Monica <![CDATA[In this paper we give an alternate combinatorial description of the "$(\ell,0)$-Carter partitions''. Our main theorem is the equivalence of our combinatoric and the one introduced by James and Mathas ($\textit{A q-analogue of the Jantzen-Schaper theorem}$). The condition of being an $(\ell,0)$-Carter partition is fundamentally related to the hook lengths of the partition. The representation-theoretic significance of their combinatoric on an $\ell$-regular partition is that it indicates the irreducibility of the corresponding Specht module over the finite Hecke algebra. We use our result to find a generating series which counts the number of such partitions, with respect to the statistic of a partition's first part. We then apply our description of these partitions to the crystal graph $B(\Lambda_0)$ of the basic representation of $\widehat{\mathfrak{sl}_{\ell}}$, whose nodes are labeled by $\ell$-regular partitions. Here we give a fairly simple crystal-theoretic rule which generates all $(\ell,0)$-Carter partitions in the graph of $B(\Lambda_0)$.]]> 0 <![CDATA[Motivated by the theory of operads, we introduce new combinatorial objects, called shrubs, that generalize forests of rooted trees. We show that the species of shrubs is isomorphic to the species of series-parallel posets.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3649 https://doi.org/10.46298/dmtcs.3649 Chapoton, Frédéric Chapoton, Frédéric <![CDATA[Motivated by the theory of operads, we introduce new combinatorial objects, called shrubs, that generalize forests of rooted trees. We show that the species of shrubs is isomorphic to the species of series-parallel posets.]]> 0 <![CDATA[We prove the conjecture of A. Postnikov that ($\mathrm{A}$) the number of regions in the inversion hyperplane arrangement associated with a permutation $w \in \mathfrak{S}_n$ is at most the number of elements below $w$ in the Bruhat order, and ($\mathrm{B}$) that equality holds if and only if $w$ avoids the patterns $4231$, $35142$, $42513$ and $351624$. Furthermore, assertion ($\mathrm{A}$) is extended to all finite reflection groups.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3648 https://doi.org/10.46298/dmtcs.3648 Hultman, Axel Linusson, Svante Shareshian, John Sjöstrand, Jonas Hultman, Axel Linusson, Svante Shareshian, John Sjöstrand, Jonas <![CDATA[We prove the conjecture of A. Postnikov that ($\mathrm{A}$) the number of regions in the inversion hyperplane arrangement associated with a permutation $w \in \mathfrak{S}_n$ is at most the number of elements below $w$ in the Bruhat order, and ($\mathrm{B}$) that equality holds if and only if $w$ avoids the patterns $4231$, $35142$, $42513$ and $351624$. Furthermore, assertion ($\mathrm{A}$) is extended to all finite reflection groups.]]> 0 <![CDATA[We show that the category of representations of the Euclidean group $E(2)$ is equivalent to the category of representations of the preprojective algebra of the quiver of type $A_{\infty}$. Furthermore, we consider the moduli space of $E(2)$-modules along with a set of generators. We show that these moduli spaces are quiver varieties of the type considered by Nakajima. These identifications allow us to draw on known results about preprojective algebras and quiver varieties to prove various statements about representations of $E(2)$. In particular, we show that $E(2)$ has wild representation type but that if we impose certain combinatorial restrictions on the weight decompositions of a representation, we obtain only a finite number of indecomposable representations.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3603 https://doi.org/10.46298/dmtcs.3603 Savage, Alistair Savage, Alistair <![CDATA[We show that the category of representations of the Euclidean group $E(2)$ is equivalent to the category of representations of the preprojective algebra of the quiver of type $A_{\infty}$. Furthermore, we consider the moduli space of $E(2)$-modules along with a set of generators. We show that these moduli spaces are quiver varieties of the type considered by Nakajima. These identifications allow us to draw on known results about preprojective algebras and quiver varieties to prove various statements about representations of $E(2)$. In particular, we show that $E(2)$ has wild representation type but that if we impose certain combinatorial restrictions on the weight decompositions of a representation, we obtain only a finite number of indecomposable representations.]]> 0 <![CDATA[In this paper we give a graph theoretic combinatorial interpretation for the cluster variables that arise in most cluster algebras of finite type. In particular, we provide a family of graphs such that a weighted enumeration of their perfect matchings encodes the numerator of the associated Laurent polynomial while decompositions of the graphs correspond to the denominator. This complements recent work by Schiffler and Carroll-Price for a cluster expansion formula for the $A_n$ case while providing a novel interpretation for the $B_n$, $C_n$, and $D_n$ cases.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3604 https://doi.org/10.46298/dmtcs.3604 Musiker, Gregg Musiker, Gregg <![CDATA[In this paper we give a graph theoretic combinatorial interpretation for the cluster variables that arise in most cluster algebras of finite type. In particular, we provide a family of graphs such that a weighted enumeration of their perfect matchings encodes the numerator of the associated Laurent polynomial while decompositions of the graphs correspond to the denominator. This complements recent work by Schiffler and Carroll-Price for a cluster expansion formula for the $A_n$ case while providing a novel interpretation for the $B_n$, $C_n$, and $D_n$ cases.]]> 0 <![CDATA[We introduce a new basis for the algebra of quasisymmetric functions that naturally partitions Schur functions, called quasisymmetric Schur functions. We describe their expansion in terms of fundamental quasisymmetric functions and determine when a quasisymmetric Schur function is equal to a fundamental quasisymmetric function. We conclude by describing a Pieri rule for quasisymmetric Schur functions that naturally generalizes the Pieri rule for Schur functions.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3605 https://doi.org/10.46298/dmtcs.3605 Haglund, James Mason, Sarah Luoto, Kurt van Willigenburg, Steph Haglund, James Mason, Sarah Luoto, Kurt van Willigenburg, Steph <![CDATA[We introduce a new basis for the algebra of quasisymmetric functions that naturally partitions Schur functions, called quasisymmetric Schur functions. We describe their expansion in terms of fundamental quasisymmetric functions and determine when a quasisymmetric Schur function is equal to a fundamental quasisymmetric function. We conclude by describing a Pieri rule for quasisymmetric Schur functions that naturally generalizes the Pieri rule for Schur functions.]]> 0 <![CDATA[We analyze the structure of the algebra $\mathbb{K}\langle \mathbf{x}\rangle^{\mathfrak{S}_n}$ of symmetric polynomials in non-commuting variables in so far as it relates to $\mathbb{K}[\mathbf{x}]^{\mathfrak{S}_n}$, its commutative counterpart. Using the "place-action'' of the symmetric group, we are able to realize the latter as the invariant polynomials inside the former. We discover a tensor product decomposition of $\mathbb{K}\langle \mathbf{x}\rangle^{\mathfrak{S}_n}$ analogous to the classical theorems of Chevalley, Shephard-Todd on finite reflection groups. In the case $|\mathbf{x}|= \infty$, our techniques simplify to a form readily generalized to many other familiar pairs of combinatorial Hopf algebras.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3606 https://doi.org/10.46298/dmtcs.3606 Bergeron, François Lauve, Aaron Bergeron, François Lauve, Aaron <![CDATA[We analyze the structure of the algebra $\mathbb{K}\langle \mathbf{x}\rangle^{\mathfrak{S}_n}$ of symmetric polynomials in non-commuting variables in so far as it relates to $\mathbb{K}[\mathbf{x}]^{\mathfrak{S}_n}$, its commutative counterpart. Using the "place-action'' of the symmetric group, we are able to realize the latter as the invariant polynomials inside the former. We discover a tensor product decomposition of $\mathbb{K}\langle \mathbf{x}\rangle^{\mathfrak{S}_n}$ analogous to the classical theorems of Chevalley, Shephard-Todd on finite reflection groups. In the case $|\mathbf{x}|= \infty$, our techniques simplify to a form readily generalized to many other familiar pairs of combinatorial Hopf algebras.]]> 0 <![CDATA[We describe a combinatorial model for the $q$-analogs of the generalized Stirling numbers in terms of bugs and colonies. Using both algebraic and combinatorial methods, we derive explicit formulas, recursions and generating functions for these $q$-analogs. We give a weight preserving bijective correspondence between our combinatorial model and rook placements on Ferrer boards. We outline a direct application of our theory to the theory of dual graded graphs developed by Fomin. Lastly we define a natural $p,q$-analog of these generalized Stirling numbers.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3607 https://doi.org/10.46298/dmtcs.3607 Méndez, Miguel Rodríguez, Adolfo Méndez, Miguel Rodríguez, Adolfo <![CDATA[We describe a combinatorial model for the $q$-analogs of the generalized Stirling numbers in terms of bugs and colonies. Using both algebraic and combinatorial methods, we derive explicit formulas, recursions and generating functions for these $q$-analogs. We give a weight preserving bijective correspondence between our combinatorial model and rook placements on Ferrer boards. We outline a direct application of our theory to the theory of dual graded graphs developed by Fomin. Lastly we define a natural $p,q$-analog of these generalized Stirling numbers.]]> 0 <![CDATA[For each infinite series of the classical Lie groups of type $B$, $C$ or $D$, we introduce a family of polynomials parametrized by the elements of the corresponding Weyl group of infinite rank. These polynomials represent the Schubert classes in the equivariant cohomology of the corresponding flag variety. They satisfy a stability property, and are a natural extension of the (single) Schubert polynomials of Billey and Haiman, which represent non-equivariant Schubert classes. When indexed by maximal Grassmannian elements of the Weyl group, these polynomials are equal to the factorial analogues of Schur $Q$- or $P$-functions defined earlier by Ivanov.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3608 https://doi.org/10.46298/dmtcs.3608 Ikeda, Takeshi Mihalcea, Leonardo Naruse, Hiroshi Ikeda, Takeshi Mihalcea, Leonardo Naruse, Hiroshi <![CDATA[For each infinite series of the classical Lie groups of type $B$, $C$ or $D$, we introduce a family of polynomials parametrized by the elements of the corresponding Weyl group of infinite rank. These polynomials represent the Schubert classes in the equivariant cohomology of the corresponding flag variety. They satisfy a stability property, and are a natural extension of the (single) Schubert polynomials of Billey and Haiman, which represent non-equivariant Schubert classes. When indexed by maximal Grassmannian elements of the Weyl group, these polynomials are equal to the factorial analogues of Schur $Q$- or $P$-functions defined earlier by Ivanov.]]> 0 <![CDATA[We consider the graded Hopf algebra $NCSym$ of symmetric functions with non-commutative variables, which is analogous to the algebra $Sym$ of the ordinary symmetric functions in commutative variables. We give formulaes for the product and coproduct on some of the analogues of the $Sym$ bases and expressions for a shuffle product on $NCSym$. We also consider the invariants of the hyperoctahedral group in the non-commutative case and a state a few results.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3609 https://doi.org/10.46298/dmtcs.3609 Bergeron-Brlek, Anouk Bergeron-Brlek, Anouk <![CDATA[We consider the graded Hopf algebra $NCSym$ of symmetric functions with non-commutative variables, which is analogous to the algebra $Sym$ of the ordinary symmetric functions in commutative variables. We give formulaes for the product and coproduct on some of the analogues of the $Sym$ bases and expressions for a shuffle product on $NCSym$. We also consider the invariants of the hyperoctahedral group in the non-commutative case and a state a few results.]]> 0 <![CDATA[A $\textit{grid shape}$ is a set of boxes chosen from a square grid; any Young diagram is an example. This paper considers a notion of pattern-avoidance for $0-1$ fillings of grid shapes, which generalizes permutation pattern-avoidance. A filling avoids some patterns if none of its sub-shapes equal any of the patterns. We focus on patterns that are $\textit{pairs}$ of $2 \times 2$ fillings. For some shapes, fillings that avoid specific $2 \times 2$ pairs are in bijection with totally nonnegative Grassmann cells, or with acyclic orientations of bipartite graphs. We prove a number of results analogous to Wilf-equivalence for these objects ―- that is, we show that for certain classes of shapes, some pattern-avoiding fillings are equinumerous with others.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3610 https://doi.org/10.46298/dmtcs.3610 Spiridonov, Alexey Spiridonov, Alexey <![CDATA[A $\textit{grid shape}$ is a set of boxes chosen from a square grid; any Young diagram is an example. This paper considers a notion of pattern-avoidance for $0-1$ fillings of grid shapes, which generalizes permutation pattern-avoidance. A filling avoids some patterns if none of its sub-shapes equal any of the patterns. We focus on patterns that are $\textit{pairs}$ of $2 \times 2$ fillings. For some shapes, fillings that avoid specific $2 \times 2$ pairs are in bijection with totally nonnegative Grassmann cells, or with acyclic orientations of bipartite graphs. We prove a number of results analogous to Wilf-equivalence for these objects ―- that is, we show that for certain classes of shapes, some pattern-avoiding fillings are equinumerous with others.]]> 0 <![CDATA[We present a simple bijective proof of the fact that matchings of $[2n]$ with N nestings are equinumerous to $\textit{partially directed self avoiding walks}$ confined to the symmetric wedge defined by $y= \pm x$, with $n$ east steps and $N$ north steps. A very similar construction connects permutations with $N$ nestings and $\textit{PDSAWs}$ remaining below the $x$-axis, again with $N$ north steps. Furthermore, both bijections transport several combinatorially meaningful parameters.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3611 https://doi.org/10.46298/dmtcs.3611 Rubey, Martin Rubey, Martin <![CDATA[We present a simple bijective proof of the fact that matchings of $[2n]$ with N nestings are equinumerous to $\textit{partially directed self avoiding walks}$ confined to the symmetric wedge defined by $y= \pm x$, with $n$ east steps and $N$ north steps. A very similar construction connects permutations with $N$ nestings and $\textit{PDSAWs}$ remaining below the $x$-axis, again with $N$ north steps. Furthermore, both bijections transport several combinatorially meaningful parameters.]]> 0 <![CDATA[A combinatorial construction of Gelfand models for the symmetric group, for its Iwahori-Hecke algebra and for the hyperoctahedral group is presented.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3612 https://doi.org/10.46298/dmtcs.3612 Adin, Ron M. Postnikov, Alex Roichman, Yuval Adin, Ron M. Postnikov, Alex Roichman, Yuval <![CDATA[A combinatorial construction of Gelfand models for the symmetric group, for its Iwahori-Hecke algebra and for the hyperoctahedral group is presented.]]> 0 <![CDATA[Schützenberger's theorem for the ordinary RSK correspondence naturally extends to Chen et. al's correspondence for matchings and partitions. Thus the counting of bilaterally symmetric $k$-noncrossing partitions naturally arises as an analogue for involutions. In obtaining the analogous result for $3$-noncrossing partitions, we use a different technique to develop a $\mathsf{MAPLE}$ package for $2$-dimensional vacillating lattice walk enumeration problems. As an application, we find an interesting relation between two special bilaterally symmetric partitions.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3613 https://doi.org/10.46298/dmtcs.3613 Xin, Guoce Zhang, Terence Y. J. Xin, Guoce Zhang, Terence Y. J. <![CDATA[Schützenberger's theorem for the ordinary RSK correspondence naturally extends to Chen et. al's correspondence for matchings and partitions. Thus the counting of bilaterally symmetric $k$-noncrossing partitions naturally arises as an analogue for involutions. In obtaining the analogous result for $3$-noncrossing partitions, we use a different technique to develop a $\mathsf{MAPLE}$ package for $2$-dimensional vacillating lattice walk enumeration problems. As an application, we find an interesting relation between two special bilaterally symmetric partitions.]]> 0 <![CDATA[In this paper we construct a bijection for partitioned 3-cacti that gives raise to a new formula for enumeration of factorizations of the long cycle into three permutations with given number of cycles.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3614 https://doi.org/10.46298/dmtcs.3614 Schaeffer, Gilles Vassilieva, Ekaterina Schaeffer, Gilles Vassilieva, Ekaterina <![CDATA[In this paper we construct a bijection for partitioned 3-cacti that gives raise to a new formula for enumeration of factorizations of the long cycle into three permutations with given number of cycles.]]> 0 <![CDATA[We give another construction of a permutation tableau from its corresponding permutation and construct a permutation-preserving bijection between $1$-hinge and $0$-hinge tableaux. We also consider certain alignment and crossing statistics on permutation tableaux that have previously been shown to be equidistributed by mapping them to patterns in related permutations. We give two direct maps on tableaux that prove the equidistribution of those statistics by exchanging some statistics and preserving the rest. Finally, we enumerate some sets of permutations that are restricted both by pattern avoidance and by certain parameters of their associated permutation tableaux.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3615 https://doi.org/10.46298/dmtcs.3615 Burstein, Alexander Eriksen, Niklas Burstein, Alexander Eriksen, Niklas <![CDATA[We give another construction of a permutation tableau from its corresponding permutation and construct a permutation-preserving bijection between $1$-hinge and $0$-hinge tableaux. We also consider certain alignment and crossing statistics on permutation tableaux that have previously been shown to be equidistributed by mapping them to patterns in related permutations. We give two direct maps on tableaux that prove the equidistribution of those statistics by exchanging some statistics and preserving the rest. Finally, we enumerate some sets of permutations that are restricted both by pattern avoidance and by certain parameters of their associated permutation tableaux.]]> 0 <![CDATA[Let $G$ be a perfectly oriented planar graph. Postnikov's boundary measurement construction provides a rational map from the set of positive weight functions on the edges of $G$ onto the appropriate totally nonnegative Grassmann cell. We establish an explicit combinatorial formula for Postnikov's map by expressing each Plücker coordinate of the image as a ratio of two polynomials in the edge weights, with positive integer coefficients. These polynomials are weight generating functions for certain subsets of edges in $G$.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3616 https://doi.org/10.46298/dmtcs.3616 Talaska, Kelli Talaska, Kelli <![CDATA[Let $G$ be a perfectly oriented planar graph. Postnikov's boundary measurement construction provides a rational map from the set of positive weight functions on the edges of $G$ onto the appropriate totally nonnegative Grassmann cell. We establish an explicit combinatorial formula for Postnikov's map by expressing each Plücker coordinate of the image as a ratio of two polynomials in the edge weights, with positive integer coefficients. These polynomials are weight generating functions for certain subsets of edges in $G$.]]> 0 <![CDATA[We study enumerative and homological properties of the Rees product of the cubical lattice with the chain. We give several explicit formulas for the Möbius function. The last formula is expressed in terms of the permanent of a matrix and is given by a bijective proof.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3617 https://doi.org/10.46298/dmtcs.3617 Muldoon, Patricia Readdy, Margaret A. Muldoon, Patricia Readdy, Margaret A. <![CDATA[We study enumerative and homological properties of the Rees product of the cubical lattice with the chain. We give several explicit formulas for the Möbius function. The last formula is expressed in terms of the permanent of a matrix and is given by a bijective proof.]]> 0 <![CDATA[The sandpile group of a graph $G$ is an abelian group whose order is the number of spanning trees of $G$. We find the decomposition of the sandpile group into cyclic subgroups when $G$ is a regular tree with the leaves are collapsed to a single vertex. This result can be used to understand the behavior of the rotor-router model, a deterministic analogue of random walk studied first by physicists and more recently rediscovered by combinatorialists. Several years ago, Jim Propp simulated a simple process called rotor-router aggregation and found that it produces a near perfect disk in the integer lattice $\mathbb{Z}^2$. We prove that this shape is close to circular, although it remains a challenge to explain the near perfect circularity produced by simulations. In the regular tree, we use the sandpile group to prove that rotor-router aggregation started from an acyclic initial condition yields a perfect ball.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3618 https://doi.org/10.46298/dmtcs.3618 Landau, Itamar Levine, Lionel Peres, Yuval Landau, Itamar Levine, Lionel Peres, Yuval <![CDATA[The sandpile group of a graph $G$ is an abelian group whose order is the number of spanning trees of $G$. We find the decomposition of the sandpile group into cyclic subgroups when $G$ is a regular tree with the leaves are collapsed to a single vertex. This result can be used to understand the behavior of the rotor-router model, a deterministic analogue of random walk studied first by physicists and more recently rediscovered by combinatorialists. Several years ago, Jim Propp simulated a simple process called rotor-router aggregation and found that it produces a near perfect disk in the integer lattice $\mathbb{Z}^2$. We prove that this shape is close to circular, although it remains a challenge to explain the near perfect circularity produced by simulations. In the regular tree, we use the sandpile group to prove that rotor-router aggregation started from an acyclic initial condition yields a perfect ball.]]> 0 <![CDATA[For $m$ a non-negative integer and $G$ a Coxeter group, we denote by $\mathbf{QI_m}(G)$ the ring of $m$-quasiinvariants of $G$, as defined by Chalykh, Feigin, and Veselov. These form a nested series of rings, with $\mathbf{QI_0}(G)$ the whole polynomial ring, and the limit $\mathbf{QI}_{\infty}(G)$ the usual ring of invariants. Remarkably, the ring $\mathbf{QI_m}(G)$ is freely generated over the ideal generated by the invariants of $G$ without constant term, and the quotient is isomorphic to the left regular representation of $G$. However, even in the case of the symmetric group, no basis for $\mathbf{QI_m}(G)$ is known. We provide a new description of $\mathbf{QI_m}(S_n)$, and use this to give a basis for the isotypic component of $\mathbf{QI_m}(S_n)$ indexed by the shape $[n-1,1]$.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3619 https://doi.org/10.46298/dmtcs.3619 Bandlow, Jason Musiker, Gregg Bandlow, Jason Musiker, Gregg <![CDATA[For $m$ a non-negative integer and $G$ a Coxeter group, we denote by $\mathbf{QI_m}(G)$ the ring of $m$-quasiinvariants of $G$, as defined by Chalykh, Feigin, and Veselov. These form a nested series of rings, with $\mathbf{QI_0}(G)$ the whole polynomial ring, and the limit $\mathbf{QI}_{\infty}(G)$ the usual ring of invariants. Remarkably, the ring $\mathbf{QI_m}(G)$ is freely generated over the ideal generated by the invariants of $G$ without constant term, and the quotient is isomorphic to the left regular representation of $G$. However, even in the case of the symmetric group, no basis for $\mathbf{QI_m}(G)$ is known. We provide a new description of $\mathbf{QI_m}(S_n)$, and use this to give a basis for the isotypic component of $\mathbf{QI_m}(S_n)$ indexed by the shape $[n-1,1]$.]]> 0 <![CDATA[The Hecke group algebra $\operatorname{H} \mathring{W}$ of a finite Coxeter group $\mathring{W}$, as introduced by the first and last author, is obtained from $\mathring{W}$ by gluing appropriately its $0$-Hecke algebra and its group algebra. In this paper, we give an equivalent alternative construction in the case when $\mathring{W}$ is the classical Weyl group associated to an affine Weyl group $W$. Namely, we prove that, for $q$ not a root of unity, $\operatorname{H} \mathring{W}$ is the natural quotient of the affine Hecke algebra $\operatorname{H}(W)(q)$ through its level $0$ representation. The proof relies on the following core combinatorial result: at level $0$ the $0$-Hecke algebra acts transitively on $\mathring{W}$. Equivalently, in type $A$, a word written on a circle can be both sorted and antisorted by elementary bubble sort operators. We further show that the level $0$ representation is a calibrated principal series representation $M(t)$ for a suitable choice of character $t$, so that the quotient factors (non trivially) through the principal central specialization. This explains in particular the similarities between the representation theory of the classical $0$-Hecke algebra and that of the affine Hecke algebra at this specialization.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3620 https://doi.org/10.46298/dmtcs.3620 Hivert, Florent Schilling, Anne Thiéry, Nicolas M. Hivert, Florent Schilling, Anne Thiéry, Nicolas M. <![CDATA[The Hecke group algebra $\operatorname{H} \mathring{W}$ of a finite Coxeter group $\mathring{W}$, as introduced by the first and last author, is obtained from $\mathring{W}$ by gluing appropriately its $0$-Hecke algebra and its group algebra. In this paper, we give an equivalent alternative construction in the case when $\mathring{W}$ is the classical Weyl group associated to an affine Weyl group $W$. Namely, we prove that, for $q$ not a root of unity, $\operatorname{H} \mathring{W}$ is the natural quotient of the affine Hecke algebra $\operatorname{H}(W)(q)$ through its level $0$ representation. The proof relies on the following core combinatorial result: at level $0$ the $0$-Hecke algebra acts transitively on $\mathring{W}$. Equivalently, in type $A$, a word written on a circle can be both sorted and antisorted by elementary bubble sort operators. We further show that the level $0$ representation is a calibrated principal series representation $M(t)$ for a suitable choice of character $t$, so that the quotient factors (non trivially) through the principal central specialization. This explains in particular the similarities between the representation theory of the classical $0$-Hecke algebra and that of the affine Hecke algebra at this specialization.]]> 0 <![CDATA[A theorem of Goulden and Jackson which gives interesting formulae for character immanants also implies MacMahon's Master Theorem. We quantize Goulden and Jackson's theorem to give formulae for quantum character immanants in such a way as to obtain a known quantization of MacMahon's Master Theorem due to Garoufalidis-Lê-Zeilberger. In doing so, we also quantize formulae of Littlewood, Merris and Watkins concerning induced character immanants.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3621 https://doi.org/10.46298/dmtcs.3621 Konvalinka, Matjaž Skandera, Mark Konvalinka, Matjaž Skandera, Mark <![CDATA[A theorem of Goulden and Jackson which gives interesting formulae for character immanants also implies MacMahon's Master Theorem. We quantize Goulden and Jackson's theorem to give formulae for quantum character immanants in such a way as to obtain a known quantization of MacMahon's Master Theorem due to Garoufalidis-Lê-Zeilberger. In doing so, we also quantize formulae of Littlewood, Merris and Watkins concerning induced character immanants.]]> 0 <![CDATA[Kronecker coefficients are the multiplicities in the tensor product decomposition of two irreducible representations of the symmetric group $S_n$. They can also be interpreted as the coefficients of the expansion of the internal product of two Schur polynomials in the basis of Schur polynomials. We show that the problem $\mathrm{KRONCOEFF}$ of computing Kronecker coefficients is very difficult. More specifically, we prove that $\mathrm{KRONCOEFF}$ is #$\mathrm{P}$-hard and contained in the complexity class $\mathrm{GapP}$. Formally, this means that the existence of a polynomial time algorithm for $\mathrm{KRONCOEFF}$ is equivalent to the existence of a polynomial time algorithm for evaluating permanents.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3622 https://doi.org/10.46298/dmtcs.3622 Bürgisser, Peter Ikenmeyer, Christian Bürgisser, Peter Ikenmeyer, Christian <![CDATA[Kronecker coefficients are the multiplicities in the tensor product decomposition of two irreducible representations of the symmetric group $S_n$. They can also be interpreted as the coefficients of the expansion of the internal product of two Schur polynomials in the basis of Schur polynomials. We show that the problem $\mathrm{KRONCOEFF}$ of computing Kronecker coefficients is very difficult. More specifically, we prove that $\mathrm{KRONCOEFF}$ is #$\mathrm{P}$-hard and contained in the complexity class $\mathrm{GapP}$. Formally, this means that the existence of a polynomial time algorithm for $\mathrm{KRONCOEFF}$ is equivalent to the existence of a polynomial time algorithm for evaluating permanents.]]> 0 <![CDATA[Using growth diagrams, we define a skew domino Schensted algorithm which is a domino analogue of the "Robinson-Schensted algorithm for skew tableaux'' due to Sagan and Stanley. The color-to-spin property of Shimozono and White is extended. As an application, we give a simple generating function for a weighted sum of skew domino tableaux whose special case is a generalization of Stanley's sign-imbalance formula. The generating function gives a method to calculate the generalized sign-imbalance formula.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3623 https://doi.org/10.46298/dmtcs.3623 Kim, Jang Soo Kim, Jang Soo <![CDATA[Using growth diagrams, we define a skew domino Schensted algorithm which is a domino analogue of the "Robinson-Schensted algorithm for skew tableaux'' due to Sagan and Stanley. The color-to-spin property of Shimozono and White is extended. As an application, we give a simple generating function for a weighted sum of skew domino tableaux whose special case is a generalization of Stanley's sign-imbalance formula. The generating function gives a method to calculate the generalized sign-imbalance formula.]]> 0 <![CDATA[We study two enumeration problems for $\textit{up-down alternating trees}$, i.e., rooted labelled trees $T$, where the labels $ v_1, v_2, v_3, \ldots$ on every path starting at the root of $T$ satisfy $v_1 < v_2 > v_3 < v_4 > \cdots$. First we consider various tree families of interest in combinatorics (such as unordered, ordered, $d$-ary and Motzkin trees) and study the number $T_n$ of different up-down alternating labelled trees of size $n$. We obtain for all tree families considered an implicit characterization of the exponential generating function $T(z)$ leading to asymptotic results of the coefficients $T_n$ for various tree families. Second we consider the particular family of up-down alternating labelled ordered trees and study the influence of such an alternating labelling to the average shape of the trees by analyzing the parameters $\textit{label of the root node}$, $\textit{degree of the root node}$ and $\textit{depth of a random node}$ in a random tree of size $n$. This leads to exact enumeration results and limiting distribution results.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3624 https://doi.org/10.46298/dmtcs.3624 Kuba, Markus Panholzer, Alois Kuba, Markus Panholzer, Alois <![CDATA[We study two enumeration problems for $\textit{up-down alternating trees}$, i.e., rooted labelled trees $T$, where the labels $ v_1, v_2, v_3, \ldots$ on every path starting at the root of $T$ satisfy $v_1 < v_2 > v_3 < v_4 > \cdots$. First we consider various tree families of interest in combinatorics (such as unordered, ordered, $d$-ary and Motzkin trees) and study the number $T_n$ of different up-down alternating labelled trees of size $n$. We obtain for all tree families considered an implicit characterization of the exponential generating function $T(z)$ leading to asymptotic results of the coefficients $T_n$ for various tree families. Second we consider the particular family of up-down alternating labelled ordered trees and study the influence of such an alternating labelling to the average shape of the trees by analyzing the parameters $\textit{label of the root node}$, $\textit{degree of the root node}$ and $\textit{depth of a random node}$ in a random tree of size $n$. This leads to exact enumeration results and limiting distribution results.]]> 0 <![CDATA[We extend the Billera―Ehrenborg―Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For toric arrangements, we also generalize Zaslavsky's fundamental results on the number of regions.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3625 https://doi.org/10.46298/dmtcs.3625 Ehrenborg, Richard Readdy, Margaret Slone, Michael Ehrenborg, Richard Readdy, Margaret Slone, Michael <![CDATA[We extend the Billera―Ehrenborg―Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For toric arrangements, we also generalize Zaslavsky's fundamental results on the number of regions.]]> 0 <![CDATA[For any polynomial representation of the special linear group, the nodes of the corresponding crystal may be indexed by semi-standard Young tableaux. Under certain conditions, the standard Young tableaux occur, and do so with weight $0$. Standard Young tableaux also parametrize the vertices of dual equivalence graphs. Motivated by the underlying representation theory, in this paper, we explain this connection by giving a combinatorial manifestation of Schur-Weyl duality. In particular, we put a dual equivalence graph structure on the $0$-weight space of certain crystal graphs, producing edges combinatorially from the crystal edges. The construction can be expressed in terms of the local characterizations given by Stembridge for crystal graphs and the author for dual equivalence graphs.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3626 https://doi.org/10.46298/dmtcs.3626 Assaf, S. Assaf, S. <![CDATA[For any polynomial representation of the special linear group, the nodes of the corresponding crystal may be indexed by semi-standard Young tableaux. Under certain conditions, the standard Young tableaux occur, and do so with weight $0$. Standard Young tableaux also parametrize the vertices of dual equivalence graphs. Motivated by the underlying representation theory, in this paper, we explain this connection by giving a combinatorial manifestation of Schur-Weyl duality. In particular, we put a dual equivalence graph structure on the $0$-weight space of certain crystal graphs, producing edges combinatorially from the crystal edges. The construction can be expressed in terms of the local characterizations given by Stembridge for crystal graphs and the author for dual equivalence graphs.]]> 0 <![CDATA[A self-avoiding walk on the square lattice is $\textit{prudent}$, if it never takes a step towards a vertex it has already visited. Préa was the first to address the enumeration of these walks, in 1997. For 4 natural classes of prudent walks, he wrote a system of recurrence relations, involving the length of the walks and some additional "catalytic'' parameters. The generating function of the first class is easily seen to be rational. The second class was proved to have an algebraic (quadratic) generating function by Duchi (FPSAC'05). Here, we solve exactly the third class, which turns out to be much more complex: its generating function is not algebraic, nor even $D$-finite. The fourth class ―- general prudent walks ―- still defeats us. However, we design an isotropic family of prudent walks on the triangular lattice, which we count exactly. Again, the generating function is proved to be non-$D$-finite. We also study the end-to-end distance of these walks and provide random generation procedures.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3627 https://doi.org/10.46298/dmtcs.3627 Bousquet-Mélou, Mireille Bousquet-Mélou, Mireille <![CDATA[A self-avoiding walk on the square lattice is $\textit{prudent}$, if it never takes a step towards a vertex it has already visited. Préa was the first to address the enumeration of these walks, in 1997. For 4 natural classes of prudent walks, he wrote a system of recurrence relations, involving the length of the walks and some additional "catalytic'' parameters. The generating function of the first class is easily seen to be rational. The second class was proved to have an algebraic (quadratic) generating function by Duchi (FPSAC'05). Here, we solve exactly the third class, which turns out to be much more complex: its generating function is not algebraic, nor even $D$-finite. The fourth class ―- general prudent walks ―- still defeats us. However, we design an isotropic family of prudent walks on the triangular lattice, which we count exactly. Again, the generating function is proved to be non-$D$-finite. We also study the end-to-end distance of these walks and provide random generation procedures.]]> 0 <![CDATA[This article describes new bijective links on planar maps, which are of incremental complexity and present original features. The first two bijections $\Phi _{1,2}$ are correspondences on oriented planar maps. They can be considered as variations on the classical edge-poset construction for bipolar orientations on graphs, suitably adapted so as to operate only on the embeddings in a simple local way. In turn, $\Phi_{1,2}$ yield two new bijections $F_{1,2}$ between families of (rooted) maps. (i) By identifying maps with specific constrained orientations, $\Phi_2 \circ \Phi_1$ specialises to a bijection $F_1$ between 2-connected maps and irreducible triangulations; (ii) $F_1$ gives rise to a bijection $F_2$ between loopless maps and triangulations, observing that these decompose respectively into 2-connected maps and into irreducible triangulations in a parallel way.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3628 https://doi.org/10.46298/dmtcs.3628 Fusy, Eric Fusy, Eric <![CDATA[This article describes new bijective links on planar maps, which are of incremental complexity and present original features. The first two bijections $\Phi _{1,2}$ are correspondences on oriented planar maps. They can be considered as variations on the classical edge-poset construction for bipolar orientations on graphs, suitably adapted so as to operate only on the embeddings in a simple local way. In turn, $\Phi_{1,2}$ yield two new bijections $F_{1,2}$ between families of (rooted) maps. (i) By identifying maps with specific constrained orientations, $\Phi_2 \circ \Phi_1$ specialises to a bijection $F_1$ between 2-connected maps and irreducible triangulations; (ii) $F_1$ gives rise to a bijection $F_2$ between loopless maps and triangulations, observing that these decompose respectively into 2-connected maps and into irreducible triangulations in a parallel way.]]> 0 <![CDATA[Kerov's polynomials give irreducible character values of the symmetric group in term of the free cumulants of the associated Young diagram. Using a combinatorial approach with maps, we prove in this article a positivity result on their coefficients, which extends a conjecture of S. Kerov.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3629 https://doi.org/10.46298/dmtcs.3629 Féray, Valentin Féray, Valentin <![CDATA[Kerov's polynomials give irreducible character values of the symmetric group in term of the free cumulants of the associated Young diagram. Using a combinatorial approach with maps, we prove in this article a positivity result on their coefficients, which extends a conjecture of S. Kerov.]]> 0 <![CDATA[We present a generalisation of the famous Selberg integral. This confirms the $\mathfrak{g}=A_n$ case of a conjecture by Mukhin and Varchenko concerning the existence of a Selberg integral for each simple Lie algebra $\mathfrak{g}$.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3630 https://doi.org/10.46298/dmtcs.3630 Warnaar, S. Ole Warnaar, S. Ole <![CDATA[We present a generalisation of the famous Selberg integral. This confirms the $\mathfrak{g}=A_n$ case of a conjecture by Mukhin and Varchenko concerning the existence of a Selberg integral for each simple Lie algebra $\mathfrak{g}$.]]> 0 <![CDATA[Let $W \ltimes L$ be an irreducible affine Weyl group with Coxeter complex $\Sigma$, where $W$ denotes the associated finite Weyl group and $L$ the translation subgroup. The Steinberg torus is the Boolean cell complex obtained by taking the quotient of $\Sigma$ by the lattice $L$. We show that the ordinary and flag $h$-polynomials of the Steinberg torus (with the empty face deleted) are generating functions over $W$ for a descent-like statistic first studied by Cellini. We also show that the ordinary $h$-polynomial has a nonnegative $\gamma$-vector, and hence, symmetric and unimodal coefficients. In the classical cases, we also provide expansions, identities, and generating functions for the $h$-polynomials of Steinberg tori.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3631 https://doi.org/10.46298/dmtcs.3631 Dilks, Kevin Petersen, T. Kyle Stembridge, John R. Dilks, Kevin Petersen, T. Kyle Stembridge, John R. <![CDATA[Let $W \ltimes L$ be an irreducible affine Weyl group with Coxeter complex $\Sigma$, where $W$ denotes the associated finite Weyl group and $L$ the translation subgroup. The Steinberg torus is the Boolean cell complex obtained by taking the quotient of $\Sigma$ by the lattice $L$. We show that the ordinary and flag $h$-polynomials of the Steinberg torus (with the empty face deleted) are generating functions over $W$ for a descent-like statistic first studied by Cellini. We also show that the ordinary $h$-polynomial has a nonnegative $\gamma$-vector, and hence, symmetric and unimodal coefficients. In the classical cases, we also provide expansions, identities, and generating functions for the $h$-polynomials of Steinberg tori.]]> 0 <![CDATA[In this paper we propose a new bijection between permutation tableaux and permutations. This bijection shows how natural statistics on the tableaux are equidistributed to classical statistics on permutations: descents, RL-minima and pattern enumerations. We then use the bijection, and a related encoding of tableaux by words, to prove results about the enumeration of permutations with a fixed number of 31-2 patterns, and to define subclasses of permutation tableaux that are in bijection with set partitions. An extended version of this work is available in [6].]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3632 https://doi.org/10.46298/dmtcs.3632 Corteel, Sylvie Nadeau, Philippe Corteel, Sylvie Nadeau, Philippe <![CDATA[In this paper we propose a new bijection between permutation tableaux and permutations. This bijection shows how natural statistics on the tableaux are equidistributed to classical statistics on permutations: descents, RL-minima and pattern enumerations. We then use the bijection, and a related encoding of tableaux by words, to prove results about the enumeration of permutations with a fixed number of 31-2 patterns, and to define subclasses of permutation tableaux that are in bijection with set partitions. An extended version of this work is available in [6].]]> 0 <![CDATA[In this paper we explore the combinatorics of the non-negative part $(G/P)_{\geq 0}$ of a cominuscule Grassmannian. For each such Grassmannian we define Le-diagrams ― certain fillings of generalized Young diagrams which are in bijection with the cells of $(G/P)_{\geq 0}$. In the classical cases, we describe Le-diagrams explicitly in terms of pattern avoidance. We also define a game on diagrams, by which one can reduce an arbitrary diagram to a Le-diagram. We give enumerative results and relate our Le-diagrams to other combinatorial objects. Surprisingly, the totally non-negative cells in the open Schubert cell of the odd and even orthogonal Grassmannians are (essentially) in bijection with preference functions and atomic preference functions respectively.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3633 https://doi.org/10.46298/dmtcs.3633 Lam, Thomas Williams, Lauren Lam, Thomas Williams, Lauren <![CDATA[In this paper we explore the combinatorics of the non-negative part $(G/P)_{\geq 0}$ of a cominuscule Grassmannian. For each such Grassmannian we define Le-diagrams ― certain fillings of generalized Young diagrams which are in bijection with the cells of $(G/P)_{\geq 0}$. In the classical cases, we describe Le-diagrams explicitly in terms of pattern avoidance. We also define a game on diagrams, by which one can reduce an arbitrary diagram to a Le-diagram. We give enumerative results and relate our Le-diagrams to other combinatorial objects. Surprisingly, the totally non-negative cells in the open Schubert cell of the odd and even orthogonal Grassmannians are (essentially) in bijection with preference functions and atomic preference functions respectively.]]> 0 <![CDATA[Bergeron and Li have introduced a set of axioms which guarantee that the Grothendieck groups of a tower of algebras $\bigoplus_{n \geq 0}A_n$ can be endowed with the structure of graded dual Hopf algebras. Hivert and Nzeutzhap, and independently Lam and Shimozono constructed dual graded graphs from primitive elements in Hopf algebras. In this paper we apply the composition of these constructions to towers of algebras. We show that if a tower $\bigoplus_{n \geq 0}A_n$ gives rise to graded dual Hopf algebras then we must have $\dim (A_n)=r^nn!$ where $r = \dim (A_1)$.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3634 https://doi.org/10.46298/dmtcs.3634 Bergeron, Nantel Lam, Thomas Li, Huilan Bergeron, Nantel Lam, Thomas Li, Huilan <![CDATA[Bergeron and Li have introduced a set of axioms which guarantee that the Grothendieck groups of a tower of algebras $\bigoplus_{n \geq 0}A_n$ can be endowed with the structure of graded dual Hopf algebras. Hivert and Nzeutzhap, and independently Lam and Shimozono constructed dual graded graphs from primitive elements in Hopf algebras. In this paper we apply the composition of these constructions to towers of algebras. We show that if a tower $\bigoplus_{n \geq 0}A_n$ gives rise to graded dual Hopf algebras then we must have $\dim (A_n)=r^nn!$ where $r = \dim (A_1)$.]]> 0 <![CDATA[Orbits generated by discrete-time dynamical systems have some interesting combinatorial properties. In this paper we address the existence of forbidden order patterns when the dynamics is generated by piecewise monotone maps on one-dimensional closed intervals. This means that the points belonging to a sufficiently long orbit cannot appear in any arbitrary order. The admissible patterns are then (the inverses of) those permutations avoiding the so-called forbidden root patterns in consecutive positions. The last part of the paper studies and enumerates forbidden order patterns in shift systems, which are universal models in information theory, dynamical systems and stochastic processes. In spite of their simple structure, shift systems exhibit all important features of low-dimensional chaos, allowing to export the results to other dynamical systems via order-isomorphisms. This paper summarizes some results from [1] and [11].]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3635 https://doi.org/10.46298/dmtcs.3635 Amigó, José María Elizalde, Sergi Kennel, Matthew B. Amigó, José María Elizalde, Sergi Kennel, Matthew B. <![CDATA[Orbits generated by discrete-time dynamical systems have some interesting combinatorial properties. In this paper we address the existence of forbidden order patterns when the dynamics is generated by piecewise monotone maps on one-dimensional closed intervals. This means that the points belonging to a sufficiently long orbit cannot appear in any arbitrary order. The admissible patterns are then (the inverses of) those permutations avoiding the so-called forbidden root patterns in consecutive positions. The last part of the paper studies and enumerates forbidden order patterns in shift systems, which are universal models in information theory, dynamical systems and stochastic processes. In spite of their simple structure, shift systems exhibit all important features of low-dimensional chaos, allowing to export the results to other dynamical systems via order-isomorphisms. This paper summarizes some results from [1] and [11].]]> 0 <![CDATA[In this paper we study the tangent spaces of the smooth nested Hilbert scheme $\mathrm{Hilb}^{n,n-1}(\mathbb{A}^2)$ of points in the plane, and give a general formula for computing the Euler characteristic of a $\mathbb{T}^2$-equivariant locally free sheaf on $\mathrm{Hilb}^{n,n-1}(\mathbb{A}^2)$. Applying our result to a particular sheaf, we conjecture that the result is a polynomial in the variables $q$ and $t$ with non-negative integer coefficients. We call this conjecturally positive polynomial as the "nested $q,t$-Catalan series,'' for it has many conjectural properties similar to that of the $q,t$-Catalan series.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3636 https://doi.org/10.46298/dmtcs.3636 Can, Mahir Bilen Can, Mahir Bilen <![CDATA[In this paper we study the tangent spaces of the smooth nested Hilbert scheme $\mathrm{Hilb}^{n,n-1}(\mathbb{A}^2)$ of points in the plane, and give a general formula for computing the Euler characteristic of a $\mathbb{T}^2$-equivariant locally free sheaf on $\mathrm{Hilb}^{n,n-1}(\mathbb{A}^2)$. Applying our result to a particular sheaf, we conjecture that the result is a polynomial in the variables $q$ and $t$ with non-negative integer coefficients. We call this conjecturally positive polynomial as the "nested $q,t$-Catalan series,'' for it has many conjectural properties similar to that of the $q,t$-Catalan series.]]> 0 <![CDATA[We complete the Wilf classification of signed patterns of length 5 for both signed permutations and signed involutions. New general equivalences of patterns are given which prove Jaggard's conjectures concerning involutions in the symmetric group avoiding certain patterns of length 5 and 6. In this way, we also complete the Wilf classification of $S_5$, $S_6$, and $S_7$ for both permutations and involutions.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3637 https://doi.org/10.46298/dmtcs.3637 Dukes, Mark Jelínek, Vít Mansour, Toufik Reifegerste, Astrid Dukes, Mark Jelínek, Vít Mansour, Toufik Reifegerste, Astrid <![CDATA[We complete the Wilf classification of signed patterns of length 5 for both signed permutations and signed involutions. New general equivalences of patterns are given which prove Jaggard's conjectures concerning involutions in the symmetric group avoiding certain patterns of length 5 and 6. In this way, we also complete the Wilf classification of $S_5$, $S_6$, and $S_7$ for both permutations and involutions.]]> 0 <![CDATA[A permutomino of size n is a polyomino determined by particular pairs $(\pi_1, \pi_2)$ of permutations of length $n$, such that $\pi_1(i) \neq \pi_2(i)$, for $1 \leq i \leq n$. In this paper we consider the class of convex permutominoes which are symmetric with respect to the diagonal $x = y$. We determine the number of these permutominoes according to the dimension and we characterize the class of permutations associated to these objects as particular involutions of length $n$.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3638 https://doi.org/10.46298/dmtcs.3638 Disanto, Filippo Rinaldi, Simone Disanto, Filippo Rinaldi, Simone <![CDATA[A permutomino of size n is a polyomino determined by particular pairs $(\pi_1, \pi_2)$ of permutations of length $n$, such that $\pi_1(i) \neq \pi_2(i)$, for $1 \leq i \leq n$. In this paper we consider the class of convex permutominoes which are symmetric with respect to the diagonal $x = y$. We determine the number of these permutominoes according to the dimension and we characterize the class of permutations associated to these objects as particular involutions of length $n$.]]> 0 <![CDATA[In type $A$, the $q,t$-Fuß-Catalan numbers $\mathrm{Cat}_n^{(m)}(q,t)$ can be defined as a bigraded Hilbert series of a module associated to the symmetric group $\mathcal{S}_n$. We generalize this construction to (finite) complex reflection groups and exhibit some nice conjectured algebraic and combinatorial properties of these polynomials in $q$ and $t$. Finally, we present an idea how these polynomials could be related to some graded Hilbert series of modules arising in the context of rational Cherednik algebras. This is work in progress.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3639 https://doi.org/10.46298/dmtcs.3639 Stump, Christian Stump, Christian <![CDATA[In type $A$, the $q,t$-Fuß-Catalan numbers $\mathrm{Cat}_n^{(m)}(q,t)$ can be defined as a bigraded Hilbert series of a module associated to the symmetric group $\mathcal{S}_n$. We generalize this construction to (finite) complex reflection groups and exhibit some nice conjectured algebraic and combinatorial properties of these polynomials in $q$ and $t$. Finally, we present an idea how these polynomials could be related to some graded Hilbert series of modules arising in the context of rational Cherednik algebras. This is work in progress.]]> 0 <![CDATA[In this paper, we study flag structures of matroid base polytopes. We describe faces of matroid base polytopes in terms of matroid data, and give conditions for hyperplane splits of matroid base polytopes. Also, we show how the $\textbf{cd}$-index of a polytope can be expressed when a polytope is cut by a hyperplane, and apply these to the $\textbf{cd}$-index of a matroid base polytope of a rank $2$ matroid.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3640 https://doi.org/10.46298/dmtcs.3640 Kim, Sangwook Kim, Sangwook <![CDATA[In this paper, we study flag structures of matroid base polytopes. We describe faces of matroid base polytopes in terms of matroid data, and give conditions for hyperplane splits of matroid base polytopes. Also, we show how the $\textbf{cd}$-index of a polytope can be expressed when a polytope is cut by a hyperplane, and apply these to the $\textbf{cd}$-index of a matroid base polytope of a rank $2$ matroid.]]> 0 <![CDATA[In this paper, we study the distribution of distances in random Apollonian network structures (RANS), a family of graphs which has a one-to-one correspondence with planar ternary trees. Using multivariate generating functions that express all information on distances, and singularity analysis for evaluating the coefficients of these functions, we prove a Rayleigh limit distribution for distances to an outermost vertex, and show that the average value of the distance between any pair of vertices in a RANS of order $n$ is asymptotically $\sqrt{n}$.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3641 https://doi.org/10.46298/dmtcs.3641 Bodini, Olivier Darrasse, Alexis Soria, Michèle Bodini, Olivier Darrasse, Alexis Soria, Michèle <![CDATA[In this paper, we study the distribution of distances in random Apollonian network structures (RANS), a family of graphs which has a one-to-one correspondence with planar ternary trees. Using multivariate generating functions that express all information on distances, and singularity analysis for evaluating the coefficients of these functions, we prove a Rayleigh limit distribution for distances to an outermost vertex, and show that the average value of the distance between any pair of vertices in a RANS of order $n$ is asymptotically $\sqrt{n}$.]]> 0 <![CDATA[A $k$-triangulation of a convex polygon is a maximal set of diagonals so that no $k+1$ of them mutually cross. $k$-triangulations have received attention in recent literature, with motivation coming from several interpretations of them. We present a new way of looking at $k$-triangulations, where certain star polygons naturally generalize triangles for $k$-triangulations. With this tool we give new, direct proofs of the fundamental properties of $k$-triangulations (number of edges, definition of flip). This interpretation also opens up new avenues of research that we briefly explore in the last section.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3642 https://doi.org/10.46298/dmtcs.3642 Pilaud, Vincent Santos, Francisco Pilaud, Vincent Santos, Francisco <![CDATA[A $k$-triangulation of a convex polygon is a maximal set of diagonals so that no $k+1$ of them mutually cross. $k$-triangulations have received attention in recent literature, with motivation coming from several interpretations of them. We present a new way of looking at $k$-triangulations, where certain star polygons naturally generalize triangles for $k$-triangulations. With this tool we give new, direct proofs of the fundamental properties of $k$-triangulations (number of edges, definition of flip). This interpretation also opens up new avenues of research that we briefly explore in the last section.]]> 0 <![CDATA[We define two classes of multiple basic hypergeometric series $V_{k,t}(a,q)$ and $W_{k,t}(a,q)$ which generalize multiple series studied by Agarwal, Andrews, and Bressoud. We show how to interpret these series as generating functions for special restricted lattice paths and for $n$-color overpartitions with weighted difference conditions. We also point out that some specializations of our series can be written as infinite products, which leads to combinatorial identities linking $n$-color overpartitions with ordinary partitions or overpartitions.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3643 https://doi.org/10.46298/dmtcs.3643 Mallet, Olivier Mallet, Olivier <![CDATA[We define two classes of multiple basic hypergeometric series $V_{k,t}(a,q)$ and $W_{k,t}(a,q)$ which generalize multiple series studied by Agarwal, Andrews, and Bressoud. We show how to interpret these series as generating functions for special restricted lattice paths and for $n$-color overpartitions with weighted difference conditions. We also point out that some specializations of our series can be written as infinite products, which leads to combinatorial identities linking $n$-color overpartitions with ordinary partitions or overpartitions.]]> 0 <![CDATA[We present new conjectures on the distribution of link patterns for fully-packed loop (FPL) configurations that are invariant, or almost invariant, under a quarter turn rotation, extending previous conjectures of Razumov and Stroganov and of de Gier. We prove a special case, showing that the link pattern that is conjectured to be the rarest does have the prescribed probability. As a byproduct, we get a formula for the enumeration of a new class of quasi-symmetry of plane partitions.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3644 https://doi.org/10.46298/dmtcs.3644 Duchon, Philippe Duchon, Philippe <![CDATA[We present new conjectures on the distribution of link patterns for fully-packed loop (FPL) configurations that are invariant, or almost invariant, under a quarter turn rotation, extending previous conjectures of Razumov and Stroganov and of de Gier. We prove a special case, showing that the link pattern that is conjectured to be the rarest does have the prescribed probability. As a byproduct, we get a formula for the enumeration of a new class of quasi-symmetry of plane partitions.]]> 0 <![CDATA[The Kazhdan-Lusztig polynomials for finite Weyl groups arise in representation theory as well as the geometry of Schubert varieties. It was proved very soon after their introduction that they have nonnegative integer coefficients, but no simple all positive interpretation for them is known in general. Deodhar has given a framework, which generally involves recursion, to express the Kazhdan-Lusztig polynomials in a very attractive form. We use a new kind of pattern-avoidance that can be defined for general Coxeter groups to characterize when Deodhar's algorithm yields a non-recursive combinatorial formula for Kazhdan-Lusztig polynomials $P_{x,w}(q)$ of finite Weyl groups. This generalizes results of Billey-Warrington which identified the $321$-hexagon-avoiding permutations, and Fan-Green which identified the fully-tight Coxeter groups. We also show that the leading coefficient known as $\mu (x,w)$ for these Kazhdan―Lusztig polynomials is always either $0$ or $1$. Finally, we generalize the simple combinatorial formula for the Kazhdan―Lusztig polynomials of the $321$-hexagon-avoiding permutations to the case when $w$ is hexagon avoiding and maximally clustered.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3645 https://doi.org/10.46298/dmtcs.3645 Jones, Brant Jones, Brant <![CDATA[The Kazhdan-Lusztig polynomials for finite Weyl groups arise in representation theory as well as the geometry of Schubert varieties. It was proved very soon after their introduction that they have nonnegative integer coefficients, but no simple all positive interpretation for them is known in general. Deodhar has given a framework, which generally involves recursion, to express the Kazhdan-Lusztig polynomials in a very attractive form. We use a new kind of pattern-avoidance that can be defined for general Coxeter groups to characterize when Deodhar's algorithm yields a non-recursive combinatorial formula for Kazhdan-Lusztig polynomials $P_{x,w}(q)$ of finite Weyl groups. This generalizes results of Billey-Warrington which identified the $321$-hexagon-avoiding permutations, and Fan-Green which identified the fully-tight Coxeter groups. We also show that the leading coefficient known as $\mu (x,w)$ for these Kazhdan―Lusztig polynomials is always either $0$ or $1$. Finally, we generalize the simple combinatorial formula for the Kazhdan―Lusztig polynomials of the $321$-hexagon-avoiding permutations to the case when $w$ is hexagon avoiding and maximally clustered.]]> 0 <![CDATA[We study graph weights (i.e., graph invariants) which arise naturally in Mayer's theory and Ree-Hoover's theory of virial expansions in the context of a non-ideal gas. We give special attention to the Second Mayer weight $w_M(c)$ and the Ree-Hoover weight $w_{RH}(c)$ of a $2$-connected graph $c$ which arise from the hard-core continuum gas in one dimension. These weights are computed using signed volumes of convex polytopes naturally associated with the graph $c$. Among our results are the values of Mayer's weight and Ree-Hoover's weight for all $2$-connected graphs $b$ of size at most $8$, and explicit formulas for certain infinite families.]]> Mon, 31 Dec 2007 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3646 https://doi.org/10.46298/dmtcs.3646 Kaouche, Amel Leroux, Pierre Kaouche, Amel Leroux, Pierre <![CDATA[We study graph weights (i.e., graph invariants) which arise naturally in Mayer's theory and Ree-Hoover's theory of virial expansions in the context of a non-ideal gas. We give special attention to the Second Mayer weight $w_M(c)$ and the Ree-Hoover weight $w_{RH}(c)$ of a $2$-connected graph $c$ which arise from the hard-core continuum gas in one dimension. These weights are computed using signed volumes of convex polytopes naturally associated with the graph $c$. Among our results are the values of Mayer's weight and Ree-Hoover's weight for all $2$-connected graphs $b$ of size at most $8$, and explicit formulas for certain infinite families.]]> 0 <![CDATA[Let a and b be two positive integers. A culminating path is a path of Z^2 that starts from (0,0), consists of steps (1,a) and (1,-b), stays above the x-axis and ends at the highest ordinate it ever reaches. These paths were first encountered in bioinformatics, in the analysis of similarity search algorithms. They are also related to certain models of Lorentzian gravity in theoretical physics. We first show that the language on a two letter alphabet that naturally encodes culminating paths is not context-free. Then, we focus on the enumeration of culminating paths. A step by step approach, combined with the kernel method, provides a closed form expression for the generating fucntion of culminating paths ending at a (generic) height k. In the case a=b, we derive from this expression the asymptotic behaviour of the number of culminating paths of length n. When a>b, we obtain the asymptotic behaviour by a simpler argument. When a= b, with no precomputation stage nor non-linear storage required. The choice of the best algorithm is not as clear when a]]> Tue, 05 Jun 2007 06:00:00 +0000 https://doi.org/10.46298/dmtcs.438 https://doi.org/10.46298/dmtcs.438 Bousquet-Mélou, Mireille Ponty, Yann Bousquet-Mélou, Mireille Ponty, Yann <![CDATA[Let a and b be two positive integers. A culminating path is a path of Z^2 that starts from (0,0), consists of steps (1,a) and (1,-b), stays above the x-axis and ends at the highest ordinate it ever reaches. These paths were first encountered in bioinformatics, in the analysis of similarity search algorithms. They are also related to certain models of Lorentzian gravity in theoretical physics. We first show that the language on a two letter alphabet that naturally encodes culminating paths is not context-free. Then, we focus on the enumeration of culminating paths. A step by step approach, combined with the kernel method, provides a closed form expression for the generating fucntion of culminating paths ending at a (generic) height k. In the case a=b, we derive from this expression the asymptotic behaviour of the number of culminating paths of length n. When a>b, we obtain the asymptotic behaviour by a simpler argument. When a= b, with no precomputation stage nor non-linear storage required. The choice of the best algorithm is not as clear when a]]> 0 <![CDATA[In a previous paper, we characterized free monoid morphisms preserving finite Lyndon words. In particular, we proved that such a morphism preserves the order on finite words. Here we study morphisms preserving infinite Lyndon words and morphisms preserving the order on infinite words. We characterize them and show relations with morphisms preserving Lyndon words or the order on finite words. We also briefly study morphisms preserving border-free words and those preserving the radix order.]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.411 https://doi.org/10.46298/dmtcs.411 Richomme, Gwenael Richomme, Gwenael <![CDATA[In a previous paper, we characterized free monoid morphisms preserving finite Lyndon words. In particular, we proved that such a morphism preserves the order on finite words. Here we study morphisms preserving infinite Lyndon words and morphisms preserving the order on infinite words. We characterize them and show relations with morphisms preserving Lyndon words or the order on finite words. We also briefly study morphisms preserving border-free words and those preserving the radix order.]]> 0 <![CDATA[Hilbert words correspond to finite approximations of the Hilbert space filling curve. The Hilbert infinite word H is obtained as the limit of these words. It gives a description of the Hilbert (infinite) curve. We give a uniform tag-system to generate automatically H and, by showing that it is almost cube-free, we prove that it cannot be obtained by simply iterating a morphism.]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.416 https://doi.org/10.46298/dmtcs.416 Séébold, Patrice Séébold, Patrice <![CDATA[Hilbert words correspond to finite approximations of the Hilbert space filling curve. The Hilbert infinite word H is obtained as the limit of these words. It gives a description of the Hilbert (infinite) curve. We give a uniform tag-system to generate automatically H and, by showing that it is almost cube-free, we prove that it cannot be obtained by simply iterating a morphism.]]> 0 <![CDATA[A Parry number is a real number β > 1 such that the Rényi β-expansion of 1 is finite or infinite eventually periodic. If this expansion is finite, β is said to be a simple Parry number. Remind that any Pisot number is a Parry number. In a previous work we have determined the complexity of the fixed point uβ of the canonical substitution associated with β-expansions, when β is a simple Parry number. In this paper we consider the case where β is a non-simple Parry number. We determine the structure of infinite left special branches, which are an important tool for the computation of the complexity of uβ. These results allow in particular to obtain the following characterization: the infinite word uβ is Sturmian if and only if β is a quadratic Pisot unit.]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.415 https://doi.org/10.46298/dmtcs.415 Frougny, Christiane Masáková, Zuzana Pelantová, Edita Frougny, Christiane Masáková, Zuzana Pelantová, Edita <![CDATA[A Parry number is a real number β > 1 such that the Rényi β-expansion of 1 is finite or infinite eventually periodic. If this expansion is finite, β is said to be a simple Parry number. Remind that any Pisot number is a Parry number. In a previous work we have determined the complexity of the fixed point uβ of the canonical substitution associated with β-expansions, when β is a simple Parry number. In this paper we consider the case where β is a non-simple Parry number. We determine the structure of infinite left special branches, which are an important tool for the computation of the complexity of uβ. These results allow in particular to obtain the following characterization: the infinite word uβ is Sturmian if and only if β is a quadratic Pisot unit.]]> 0 <![CDATA[The strong chromatic index of a graph is the minimum number of colours needed to colour the edges in such a way that each colour class is an induced matching. In this paper, we present bounds for strong chromatic index of three different products of graphs in term of the strong chromatic index of each factor. For the cartesian product of paths, cycles or complete graphs, we derive sharper results. In particular, strong chromatic indices of d-dimensional grids and of some toroidal grids are given along with approximate results on the strong chromatic index of generalized hypercubes.]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.414 https://doi.org/10.46298/dmtcs.414 Togni, Olivier Togni, Olivier <![CDATA[The strong chromatic index of a graph is the minimum number of colours needed to colour the edges in such a way that each colour class is an induced matching. In this paper, we present bounds for strong chromatic index of three different products of graphs in term of the strong chromatic index of each factor. For the cartesian product of paths, cycles or complete graphs, we derive sharper results. In particular, strong chromatic indices of d-dimensional grids and of some toroidal grids are given along with approximate results on the strong chromatic index of generalized hypercubes.]]> 0 <![CDATA[The notion of pyramidal polycubes, namely the piling-up of bricks of a non-increasing size, generalizes in R^n the concept of trapezoidal polyominoes. In the present paper, we prove that n-dimensional dominoes can tile a pyramidal polycube if and only if the latter is balanced, that is, if the number of white cubes is equal to the number of black ones for a chessboard-like coloration, generalizing the result of [BC92] when n=2.]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.413 https://doi.org/10.46298/dmtcs.413 Bodini, Olivier Jamet, Damien Bodini, Olivier Jamet, Damien <![CDATA[The notion of pyramidal polycubes, namely the piling-up of bricks of a non-increasing size, generalizes in R^n the concept of trapezoidal polyominoes. In the present paper, we prove that n-dimensional dominoes can tile a pyramidal polycube if and only if the latter is balanced, that is, if the number of white cubes is equal to the number of black ones for a chessboard-like coloration, generalizing the result of [BC92] when n=2.]]> 0 <![CDATA[Smooth words are connected to the Kolakoski sequence. We construct the maximal and the minimal in nite smooth words, with respect to the lexicographical order. The naive algorithm generating them is improved by using a reduction of the De Bruijn graph of their factors. We also study their Lyndon factorizations. Finally, we show that the minimal smooth word over the alphabet f1; 3g belongs to the orbit of the Fibonacci word.]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.412 https://doi.org/10.46298/dmtcs.412 Brlek, Srečko Melançon, Guy Paquin, Geneviève Brlek, Srečko Melançon, Guy Paquin, Geneviève <![CDATA[Smooth words are connected to the Kolakoski sequence. We construct the maximal and the minimal in nite smooth words, with respect to the lexicographical order. The naive algorithm generating them is improved by using a reduction of the De Bruijn graph of their factors. We also study their Lyndon factorizations. Finally, we show that the minimal smooth word over the alphabet f1; 3g belongs to the orbit of the Fibonacci word.]]> 0 <![CDATA[This special issue of Discrete Mathematics & Theoretical Computer Science is dedicated to the tenth "Journées montoises d'informatique théorique" conference (Mons theoretical computer science days) which was held, for the first time, at the Institute of Mathematics of the University of Liège, Belgium, From 8th to 11th September 2004. Previous editions of this conference took place in Mons 1990, 1992, 1994, 1998, in Rouen 1991, in Bordeaux 1993, Marseille 1995, Marne-La-Vallée 2000 and Montpellier 2002.

This tenth edition can be considered as a widely international one. We were lucky to have almost 85 participants from fourteen different countries: Austria, Belgium, Burkina Faso, Canada, Czech republic, Finland, France, Germany, Israel, Italy, Japan, Norway, Poland and Portugal. The main proportion of researchers participating to this event was coming from France and Italy where a long tradition of combinatorics on words is well established. During four days, 42 contributed talks and 7 invited talks were given, the main topics being combinatorics on words, numeration systems, automata and formal languages theory, coding theory, verification, bio-informatics, number theory, grammars, text algorithms, symbolic dynamics and tilings. The invited speakers were: J. Cassaigne (CNRS, Luminy-Marseille), D. Caucal (IRISIA-CNRS, Rennes), C. Frougny (LIAFA, Université Paris 8), T. Helleseth (University of Bergen), S. Langerman (FNRS, Université Libre de Bruxelles), F. Neven (Limburgs Universitair Centrum, Diepenbeek), M.-F. Sagot (Inria Rhône-Alpes, Université Lyon I).

We would like to thanks all the participants, the invited speakers and the anonymous referees who made possible this event and special issue. Each paper has been refereed using high scientific standard by two independent referees. Readers of this special issue may wonder why it took so long to obtain it. We have encountered some problems with the formerly chosen journal and for the benefit of the contributors to this issue, we have chosen Discrete Mathematics & Theoretical Computer Science to publish their work.]]>

Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.396 https://doi.org/10.46298/dmtcs.396 Bruyère, Véronique Rigo, Michel Bruyère, Véronique Rigo, Michel <![CDATA[This special issue of Discrete Mathematics & Theoretical Computer Science is dedicated to the tenth "Journées montoises d'informatique théorique" conference (Mons theoretical computer science days) which was held, for the first time, at the Institute of Mathematics of the University of Liège, Belgium, From 8th to 11th September 2004. Previous editions of this conference took place in Mons 1990, 1992, 1994, 1998, in Rouen 1991, in Bordeaux 1993, Marseille 1995, Marne-La-Vallée 2000 and Montpellier 2002.

This tenth edition can be considered as a widely international one. We were lucky to have almost 85 participants from fourteen different countries: Austria, Belgium, Burkina Faso, Canada, Czech republic, Finland, France, Germany, Israel, Italy, Japan, Norway, Poland and Portugal. The main proportion of researchers participating to this event was coming from France and Italy where a long tradition of combinatorics on words is well established. During four days, 42 contributed talks and 7 invited talks were given, the main topics being combinatorics on words, numeration systems, automata and formal languages theory, coding theory, verification, bio-informatics, number theory, grammars, text algorithms, symbolic dynamics and tilings. The invited speakers were: J. Cassaigne (CNRS, Luminy-Marseille), D. Caucal (IRISIA-CNRS, Rennes), C. Frougny (LIAFA, Université Paris 8), T. Helleseth (University of Bergen), S. Langerman (FNRS, Université Libre de Bruxelles), F. Neven (Limburgs Universitair Centrum, Diepenbeek), M.-F. Sagot (Inria Rhône-Alpes, Université Lyon I).

We would like to thanks all the participants, the invited speakers and the anonymous referees who made possible this event and special issue. Each paper has been refereed using high scientific standard by two independent referees. Readers of this special issue may wonder why it took so long to obtain it. We have encountered some problems with the formerly chosen journal and for the benefit of the contributors to this issue, we have chosen Discrete Mathematics & Theoretical Computer Science to publish their work.]]>

0 <![CDATA[Many results are already known, concerning the palindromic factors and the palindomic prefixes of Standard billiard words, i.e., Sturmian words and billiard words in any dimension, starting at the origin. We give new geometrical proofs of these results, especially for the existence in any dimension of Standard billiard words with arbitrary long palindromic prefixes.]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.410 https://doi.org/10.46298/dmtcs.410 Borel, Jean-Pierre Borel, Jean-Pierre <![CDATA[Many results are already known, concerning the palindromic factors and the palindomic prefixes of Standard billiard words, i.e., Sturmian words and billiard words in any dimension, starting at the origin. We give new geometrical proofs of these results, especially for the existence in any dimension of Standard billiard words with arbitrary long palindromic prefixes.]]> 0 <![CDATA[Graphs and Algorithms]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.409 https://doi.org/10.46298/dmtcs.409 Chang, An Shiu, Wai Chee Chang, An Shiu, Wai Chee <![CDATA[Graphs and Algorithms]]> 0 <![CDATA[Exponential bounds and tail estimates are derived for additive random recursive sequences, which typically arise as functionals of recursive structures, of random trees or in recursive algorithms. In particular they arise as parameters of divide and conquer type algorithms. We derive tail bounds from estimates of the Laplace transforms and of the moment sequences. For the proof we use some classical exponential bounds and some variants of the induction method. The paper generalizes results of Rösler (% \citeyearNPRoesler:91, % \citeyearNPRoesler:92) and % \citeNNeininger:05 on subgaussian tails to more general classes of additive random recursive sequences. It also gives sufficient conditions for tail bounds of the form \exp(-a t^p) which are based on a characterization of \citeNKasahara:78.]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.408 https://doi.org/10.46298/dmtcs.408 Rüschendorf, Ludger Schopp, Eva-Maria Rüschendorf, Ludger Schopp, Eva-Maria <![CDATA[Exponential bounds and tail estimates are derived for additive random recursive sequences, which typically arise as functionals of recursive structures, of random trees or in recursive algorithms. In particular they arise as parameters of divide and conquer type algorithms. We derive tail bounds from estimates of the Laplace transforms and of the moment sequences. For the proof we use some classical exponential bounds and some variants of the induction method. The paper generalizes results of Rösler (% \citeyearNPRoesler:91, % \citeyearNPRoesler:92) and % \citeNNeininger:05 on subgaussian tails to more general classes of additive random recursive sequences. It also gives sufficient conditions for tail bounds of the form \exp(-a t^p) which are based on a characterization of \citeNKasahara:78.]]> 0 <![CDATA[In this paper, we focus on applications having quantitative QoS (Quality of Service) requirements on their end-to-end response time (or jitter). We propose a solution allowing the coexistence of two types of quantitative QoS garantees, deterministic and probabilistic, while providing a high resource utilization. Our solution combines the advantages of the deterministic approach and the probabilistic one. The deterministic approach is based on a worst case analysis. The probabilistic approach uses a mathematical model to obtain the probability that the response time exceeds a given value. We assume that flows are scheduled according to non-preemptive FP/FIFO. The packet with the highest fixed priority is scheduled first. If two packets share the same priority, the packet arrived first is scheduled first. We make no particular assumption concerning the flow priority and the nature of the QoS guarantee requested by the flow. An admission control derived from these results is then proposed, allowing each flow to receive a quantitative QoS guarantee adapted to its QoS requirements. An example illustrates the merits of the coexistence of deterministic and probabilistic QoS guarantees.]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.407 https://doi.org/10.46298/dmtcs.407 Minet, Pascale Martin, Steven Saidane, Leila Azouz Azzaz, Skander Minet, Pascale Martin, Steven Saidane, Leila Azouz Azzaz, Skander <![CDATA[In this paper, we focus on applications having quantitative QoS (Quality of Service) requirements on their end-to-end response time (or jitter). We propose a solution allowing the coexistence of two types of quantitative QoS garantees, deterministic and probabilistic, while providing a high resource utilization. Our solution combines the advantages of the deterministic approach and the probabilistic one. The deterministic approach is based on a worst case analysis. The probabilistic approach uses a mathematical model to obtain the probability that the response time exceeds a given value. We assume that flows are scheduled according to non-preemptive FP/FIFO. The packet with the highest fixed priority is scheduled first. If two packets share the same priority, the packet arrived first is scheduled first. We make no particular assumption concerning the flow priority and the nature of the QoS guarantee requested by the flow. An admission control derived from these results is then proposed, allowing each flow to receive a quantitative QoS guarantee adapted to its QoS requirements. An example illustrates the merits of the coexistence of deterministic and probabilistic QoS guarantees.]]> 0 <![CDATA[The Associative Language Description model (ALD) is a combination of locally testable and constituent structure ideas. It is consistent with current views on brain organization and can rather conveniently describe typical technical languages such as Pascal or HTML. ALD languages are strictly enclosed in context-free languages but in practice the ALD model equals CF grammars in explanatory adequacy. Various properties of ALD have been investigated, but many theoretical questions are still open. For instance, it is unknown, at the present, whether the ALD family includes the regular languages. Here it is proved that several known classes of regular languages are ALD: threshold locally testable languages, group languages, positive commutative languages and commutative languages on 2-letter alphabets. Moreover, we show that there is an ALD language in each level of (restricted) star height hierarchy. These results seem to show that ALD languages are well-distributed over the class of regular languages.]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.406 https://doi.org/10.46298/dmtcs.406 Anselmo, Marcella Cherubini, Alessandra San Pietro, Pierluigi Anselmo, Marcella Cherubini, Alessandra San Pietro, Pierluigi <![CDATA[The Associative Language Description model (ALD) is a combination of locally testable and constituent structure ideas. It is consistent with current views on brain organization and can rather conveniently describe typical technical languages such as Pascal or HTML. ALD languages are strictly enclosed in context-free languages but in practice the ALD model equals CF grammars in explanatory adequacy. Various properties of ALD have been investigated, but many theoretical questions are still open. For instance, it is unknown, at the present, whether the ALD family includes the regular languages. Here it is proved that several known classes of regular languages are ALD: threshold locally testable languages, group languages, positive commutative languages and commutative languages on 2-letter alphabets. Moreover, we show that there is an ALD language in each level of (restricted) star height hierarchy. These results seem to show that ALD languages are well-distributed over the class of regular languages.]]> 0 <![CDATA[In this article, we are dealing with β-numeration, which is a generalization of numeration in a non-integer base. We consider the class of simple Parry numbers such that dβ(1) = 0.k1d-1 kd with d ∈ ℕ, d ≥ 2 and k1 ≥ kd ≥ 1. We prove that these elements define Rauzy fractals that are stable under a central symmetry. We use this result to compute, for several cases of cubic Pisot units, the maximal length among the lengths of the finite β-fractional parts of sums of two β-integers, denoted by L_⊕. In particular, we prove that L_⊕ = 5 in the Tribonacci case.]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.405 https://doi.org/10.46298/dmtcs.405 Bernat, Julien Bernat, Julien <![CDATA[In this article, we are dealing with β-numeration, which is a generalization of numeration in a non-integer base. We consider the class of simple Parry numbers such that dβ(1) = 0.k1d-1 kd with d ∈ ℕ, d ≥ 2 and k1 ≥ kd ≥ 1. We prove that these elements define Rauzy fractals that are stable under a central symmetry. We use this result to compute, for several cases of cubic Pisot units, the maximal length among the lengths of the finite β-fractional parts of sums of two β-integers, denoted by L_⊕. In particular, we prove that L_⊕ = 5 in the Tribonacci case.]]> 0 <![CDATA[In this paper we study bisimulations on dynamical systems through a given partition. Our aim is to give a new vision of the notion of bisimulation by using words. To achieve this goal, we encode the trajectories of the transition system as words. This method was introduced in our paper ''On o-minimal hybrid systems'' in order to give a new proof of the existence of a finite bisimulation for o-minimal hybrid systems (as previously proved in a paper by Lafferriere G., Pappas G.J. and Sastry S.). Here we want to provide a systematic study of this method in order to obtain a procedure for building finite bisimulations based on words.]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.404 https://doi.org/10.46298/dmtcs.404 Brihaye, Thomas Brihaye, Thomas <![CDATA[In this paper we study bisimulations on dynamical systems through a given partition. Our aim is to give a new vision of the notion of bisimulation by using words. To achieve this goal, we encode the trajectories of the transition system as words. This method was introduced in our paper ''On o-minimal hybrid systems'' in order to give a new proof of the existence of a finite bisimulation for o-minimal hybrid systems (as previously proved in a paper by Lafferriere G., Pappas G.J. and Sastry S.). Here we want to provide a systematic study of this method in order to obtain a procedure for building finite bisimulations based on words.]]> 0 <![CDATA[In a recent paper Broutin and Devroye (2005) have studied the height of a class of edge-weighted random trees.This is a class of trees growing in continuous time which includes many wellknown trees as examples. In this paper we derive a limit theorem for the internal path length for this class of trees.For the proof we extend a limit theorem in Neininger and Rüschendorf (2004) to recursive sequences of random variables with continuous time parameter.]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.403 https://doi.org/10.46298/dmtcs.403 Rüschendorf, Ludger Schopp, Eva-Maria Rüschendorf, Ludger Schopp, Eva-Maria <![CDATA[In a recent paper Broutin and Devroye (2005) have studied the height of a class of edge-weighted random trees.This is a class of trees growing in continuous time which includes many wellknown trees as examples. In this paper we derive a limit theorem for the internal path length for this class of trees.For the proof we extend a limit theorem in Neininger and Rüschendorf (2004) to recursive sequences of random variables with continuous time parameter.]]> 0 <![CDATA[We generalize (and hence trivialize and routinize) numerous explicit evaluations of determinants and pfaffians due to Kuperberg, as well as a determinant of Tsuchiya. The level of generality of our statements render their proofs easy and routine, by using Dodgson Condensation and/or Krattenthaler's factor exhaustion method.]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.402 https://doi.org/10.46298/dmtcs.402 Amdeberhan, Tewodros Zeilberger, Doron Amdeberhan, Tewodros Zeilberger, Doron <![CDATA[We generalize (and hence trivialize and routinize) numerous explicit evaluations of determinants and pfaffians due to Kuperberg, as well as a determinant of Tsuchiya. The level of generality of our statements render their proofs easy and routine, by using Dodgson Condensation and/or Krattenthaler's factor exhaustion method.]]> 0 <![CDATA[An undirected graph G=(V,E) is a probe split graph if its vertex set can be partitioned into two sets, N (non-probes) and P (probes) where N is independent and there exists E' ⊆ N× N such that G'=(V,E∪ E') is a split graph. Recently Chang et al. gave an O(V4(V+E)) time recognition algorithm for probe split graphs. In this article we give O(V2+VE) time recognition algorithms and characterisations by forbidden induced subgraphs both for the case when the partition into probes and non-probes is given, and when it is not given.]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.401 https://doi.org/10.46298/dmtcs.401 Le, Van Bang Ridder, H. N., Le, Van Bang Ridder, H. N., <![CDATA[An undirected graph G=(V,E) is a probe split graph if its vertex set can be partitioned into two sets, N (non-probes) and P (probes) where N is independent and there exists E' ⊆ N× N such that G'=(V,E∪ E') is a split graph. Recently Chang et al. gave an O(V4(V+E)) time recognition algorithm for probe split graphs. In this article we give O(V2+VE) time recognition algorithms and characterisations by forbidden induced subgraphs both for the case when the partition into probes and non-probes is given, and when it is not given.]]> 0 <![CDATA[We define a morphism based upon a Latin square that generalizes the Thue-Morse morphism. We prove that fixed points of this morphism are overlap-free sequences, generalizing results of Allouche - Shallit and Frid.]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.400 https://doi.org/10.46298/dmtcs.400 Tompkins, C. Robinson Tompkins, C. Robinson <![CDATA[We define a morphism based upon a Latin square that generalizes the Thue-Morse morphism. We prove that fixed points of this morphism are overlap-free sequences, generalizing results of Allouche - Shallit and Frid.]]> 0 <![CDATA[We analyse the asymptotic behaviour in the mean of a non-commutative rational series, which originates from differential cryptanalysis, using tools from probability theory, and from analytic number theory. We derive a Fourier representation of a first-order summation function obtained by interpreting this rational series as a non-classical rational sequence via the octal numeration system. The method is applicable to a wide class of sequences rational with respect to a numeration system essentially under the condition that they admit a linear representation with nonnegative coefficients.]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.399 https://doi.org/10.46298/dmtcs.399 Dumas, Philippe Lipmaa, Helger Wallén, Johan Dumas, Philippe Lipmaa, Helger Wallén, Johan <![CDATA[We analyse the asymptotic behaviour in the mean of a non-commutative rational series, which originates from differential cryptanalysis, using tools from probability theory, and from analytic number theory. We derive a Fourier representation of a first-order summation function obtained by interpreting this rational series as a non-classical rational sequence via the octal numeration system. The method is applicable to a wide class of sequences rational with respect to a numeration system essentially under the condition that they admit a linear representation with nonnegative coefficients.]]> 0 <![CDATA[Partial cubes are graphs isometrically embeddable into hypercubes. Three infinite families and a few sporadic examples of cubic partial cubes are known. The concept of a tribe is introduced as means to systematize the known examples and establish relations among them. Efficient methods of computation of tribes are developed and several concrete tribes, that include known, as well as new cubic partial cubes, are computed by hand and with the use of a computer.]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.398 https://doi.org/10.46298/dmtcs.398 Klavžar, Sandi Shpectorov, Sergey Klavžar, Sandi Shpectorov, Sergey <![CDATA[Partial cubes are graphs isometrically embeddable into hypercubes. Three infinite families and a few sporadic examples of cubic partial cubes are known. The concept of a tribe is introduced as means to systematize the known examples and establish relations among them. Efficient methods of computation of tribes are developed and several concrete tribes, that include known, as well as new cubic partial cubes, are computed by hand and with the use of a computer.]]> 0 <![CDATA[For certain generalized Thue-Morse words t, we compute the critical exponent, i.e., the supremum of the set of rational numbers that are exponents of powers in t, and determine exactly the occurrences of powers realizing it.]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.397 https://doi.org/10.46298/dmtcs.397 Massé, Alexandre B. Brlek, Srečko Glen, Amy Labbé, Sébastien Massé, Alexandre B. Brlek, Srečko Glen, Amy Labbé, Sébastien <![CDATA[For certain generalized Thue-Morse words t, we compute the critical exponent, i.e., the supremum of the set of rational numbers that are exponents of powers in t, and determine exactly the occurrences of powers realizing it.]]> 0 <![CDATA[A binary sequence of zeros and ones is called a (d; k)-sequence if it does not contain runs of zeros of length either lessthan d or greater than k, where d and k are arbitrary, but fixed, non-negative integers and d < k. Such sequences find requires that (d; k)-sequences do not contain a specific pattern w. Therefore, distribution results concerning pattern occurrence in (d; k)-sequences are of interest. In this paper we study the distribution of the waiting time until the r-th occurrence of a pattern w in a random (d; k)-sequence generated by a Markov source. Numerical examples are also provided.]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.382 https://doi.org/10.46298/dmtcs.382 Stefanov, Valeri T. Szpankowski, Wojciech Stefanov, Valeri T. Szpankowski, Wojciech <![CDATA[A binary sequence of zeros and ones is called a (d; k)-sequence if it does not contain runs of zeros of length either lessthan d or greater than k, where d and k are arbitrary, but fixed, non-negative integers and d < k. Such sequences find requires that (d; k)-sequences do not contain a specific pattern w. Therefore, distribution results concerning pattern occurrence in (d; k)-sequences are of interest. In this paper we study the distribution of the waiting time until the r-th occurrence of a pattern w in a random (d; k)-sequence generated by a Markov source. Numerical examples are also provided.]]> 0 <![CDATA[A word w is called a synchronizing (recurrent, reset, directable) word of a deterministic finite automaton (DFA) if w brings all states of the automaton to some specific state; a DFA that has a synchronizing word is said to be synchronizable. Cerny conjectured in 1964 that every n-state synchronizable DFA possesses a synchronizing word of length at most (n-1)2. We consider automata with aperiodic transition monoid (such automata are called aperiodic). We show that every synchronizable n-state aperiodic DFA has a synchronizing word of length at most n(n-1)/2. Thus, for aperiodic automata as well as for automata accepting only star-free languages, the Cerny conjecture holds true.]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.395 https://doi.org/10.46298/dmtcs.395 Trahtman, A. N. Trahtman, A. N. <![CDATA[A word w is called a synchronizing (recurrent, reset, directable) word of a deterministic finite automaton (DFA) if w brings all states of the automaton to some specific state; a DFA that has a synchronizing word is said to be synchronizable. Cerny conjectured in 1964 that every n-state synchronizable DFA possesses a synchronizing word of length at most (n-1)2. We consider automata with aperiodic transition monoid (such automata are called aperiodic). We show that every synchronizable n-state aperiodic DFA has a synchronizing word of length at most n(n-1)/2. Thus, for aperiodic automata as well as for automata accepting only star-free languages, the Cerny conjecture holds true.]]> 0 <![CDATA[We study properties of β-numeration systems, where β > 1 is the real root of the polynomial x3 - mx2 - x - 1, m ∈ ℕ, m ≥ 1. We consider arithmetic operations on the set of β-integers, i.e., on the set of numbers whose greedy expansion in base β has no fractional part. We show that the number of fractional digits arising under addition of β-integers is at most 5 for m ≥ 3 and 6 for m = 2, whereas under multiplication it is at most 6 for all m ≥ 2. We thus generalize the results known for Tribonacci numeration system, i.e., for m = 1. We summarize the combinatorial properties of infinite words naturally defined by β-integers. We point out the differences between the structure of β-integers in cases m = 1 and m ≥ 2.]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.394 https://doi.org/10.46298/dmtcs.394 Ambrož, Petr Masáková, Zuzana Pelantová, Edita Ambrož, Petr Masáková, Zuzana Pelantová, Edita <![CDATA[We study properties of β-numeration systems, where β > 1 is the real root of the polynomial x3 - mx2 - x - 1, m ∈ ℕ, m ≥ 1. We consider arithmetic operations on the set of β-integers, i.e., on the set of numbers whose greedy expansion in base β has no fractional part. We show that the number of fractional digits arising under addition of β-integers is at most 5 for m ≥ 3 and 6 for m = 2, whereas under multiplication it is at most 6 for all m ≥ 2. We thus generalize the results known for Tribonacci numeration system, i.e., for m = 1. We summarize the combinatorial properties of infinite words naturally defined by β-integers. We point out the differences between the structure of β-integers in cases m = 1 and m ≥ 2.]]> 0 <![CDATA[At the 4th Conference on Combinatorics on Words, Christophe Reutenauer posed the question of whether the dual reflected order yields a Gray code on the Lyndon family. In this paper we give a positive answer. More precisely, we present an O(1)-average-time algorithm for generating length n binary pre-necklaces, necklaces and Lyndon words in Gray code order.]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.393 https://doi.org/10.46298/dmtcs.393 Vajnovszki, Vincent Vajnovszki, Vincent <![CDATA[At the 4th Conference on Combinatorics on Words, Christophe Reutenauer posed the question of whether the dual reflected order yields a Gray code on the Lyndon family. In this paper we give a positive answer. More precisely, we present an O(1)-average-time algorithm for generating length n binary pre-necklaces, necklaces and Lyndon words in Gray code order.]]> 0 <![CDATA[In his thesis Baire defined functions of Baire class 1. A function f is of Baire class 1 if it is the pointwise limit of a sequence of continuous functions. Baire proves the following theorem. A function f is not of class 1 if and only if there exists a closed nonempty set F such that the restriction of f to F has no point of continuity. We prove the automaton version of this theorem. An ω-rational function is not of class 1 if and only if there exists a closed nonempty set F recognized by a Büchi automaton such that the restriction of f to F has no point of continuity. This gives us the opportunity for a discussion on Hausdorff's analysis of Δ°2, ordinals, transfinite induction and some applications of computer science.]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.392 https://doi.org/10.46298/dmtcs.392 Simonnet, Pierre Cagnard, Benoit Simonnet, Pierre Cagnard, Benoit <![CDATA[In his thesis Baire defined functions of Baire class 1. A function f is of Baire class 1 if it is the pointwise limit of a sequence of continuous functions. Baire proves the following theorem. A function f is not of class 1 if and only if there exists a closed nonempty set F such that the restriction of f to F has no point of continuity. We prove the automaton version of this theorem. An ω-rational function is not of class 1 if and only if there exists a closed nonempty set F recognized by a Büchi automaton such that the restriction of f to F has no point of continuity. This gives us the opportunity for a discussion on Hausdorff's analysis of Δ°2, ordinals, transfinite induction and some applications of computer science.]]> 0 <![CDATA[The grundy numbering of a graph is the maximum number of colors used by on-line first-fit coloring, under the worst order of arrival of vertices. The grundy numbering problem is to find this ordering. We prove that there is a constant c>1 so that approximating the grundy numbering problem within c is not possible, unless NP ⊆ RP]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.391 https://doi.org/10.46298/dmtcs.391 Kortsarz, Guy Kortsarz, Guy <![CDATA[The grundy numbering of a graph is the maximum number of colors used by on-line first-fit coloring, under the worst order of arrival of vertices. The grundy numbering problem is to find this ordering. We prove that there is a constant c>1 so that approximating the grundy numbering problem within c is not possible, unless NP ⊆ RP]]> 0 <![CDATA[This work is concerned with the perimeter enumeration of column-convex polyominoes. We consider both the rectangular lattice and the hexagonal lattice. For the rectangular lattice, two formulas for the generating function (gf) already exist and, to all appearances, neither of them admits of a further simplification. We first rederive those two formulas (so as to make the paper self-contained), and then we enrich the rectangular lattice gf with some additional variables. That done, we make a change of variables, which (practically) produces the hexagonal lattice gf. This latter gf was first found by Lin and Wu in 1990. However, our present formula, in addition to having a simpler form, also allows a substantially easier Taylor series expansion. As to the methods, our one is descended from algebraic languages, whereas Lin and Wu used the Temperley methodology.]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.390 https://doi.org/10.46298/dmtcs.390 Feretić, Svjetlan Feretić, Svjetlan <![CDATA[This work is concerned with the perimeter enumeration of column-convex polyominoes. We consider both the rectangular lattice and the hexagonal lattice. For the rectangular lattice, two formulas for the generating function (gf) already exist and, to all appearances, neither of them admits of a further simplification. We first rederive those two formulas (so as to make the paper self-contained), and then we enrich the rectangular lattice gf with some additional variables. That done, we make a change of variables, which (practically) produces the hexagonal lattice gf. This latter gf was first found by Lin and Wu in 1990. However, our present formula, in addition to having a simpler form, also allows a substantially easier Taylor series expansion. As to the methods, our one is descended from algebraic languages, whereas Lin and Wu used the Temperley methodology.]]> 0 <![CDATA[The β-numeration, born with the works of Rényi and Parry, provides a generalization of the notions of integers, decimal numbers and rational numbers by expanding real numbers in base β, where β>1 is not an integer. One of the main differences with the case of numeration in integral base is that the sets which play the role of integers, decimal numbers and rational numbers in base β are not stable under addition or multiplication. In particular, a fractional part may appear when one adds or multiplies two integers in base β. When β is a Pisot number, which corresponds to the most studied case, the lengths of the finite fractional parts that may appear when one adds or multiplies two integers in base β are bounded by constants which only depend on β. We prove that, for any Perron number β, the set of finite or ultimately periodic fractional parts of the sum, or the product, of two integers in base β is finite. Additionally, we prove that it is possible to compute this set for the case of addition when β is a Parry number. As a consequence, we deduce that, when β is a Perron number, there exist bounds, which only depend on β, for the lengths of the finite fractional parts that may appear when one adds or multiplies two integers in base β. Moreover, when β is a Parry number, the bound associated with the case of addition can be explicitly computed.]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.389 https://doi.org/10.46298/dmtcs.389 Bernat, Julien Bernat, Julien <![CDATA[The β-numeration, born with the works of Rényi and Parry, provides a generalization of the notions of integers, decimal numbers and rational numbers by expanding real numbers in base β, where β>1 is not an integer. One of the main differences with the case of numeration in integral base is that the sets which play the role of integers, decimal numbers and rational numbers in base β are not stable under addition or multiplication. In particular, a fractional part may appear when one adds or multiplies two integers in base β. When β is a Pisot number, which corresponds to the most studied case, the lengths of the finite fractional parts that may appear when one adds or multiplies two integers in base β are bounded by constants which only depend on β. We prove that, for any Perron number β, the set of finite or ultimately periodic fractional parts of the sum, or the product, of two integers in base β is finite. Additionally, we prove that it is possible to compute this set for the case of addition when β is a Parry number. As a consequence, we deduce that, when β is a Perron number, there exist bounds, which only depend on β, for the lengths of the finite fractional parts that may appear when one adds or multiplies two integers in base β. Moreover, when β is a Parry number, the bound associated with the case of addition can be explicitly computed.]]> 0 <![CDATA[We consider questions concerning the tileability of orthogonal polygons with colored dominoes. A colored domino is a rotatable 2 × 1 rectangle that is partitioned into two unit squares, which are called faces, each of which is assigned a color. In a colored domino tiling of an orthogonal polygon P, a set of dominoes completely covers P such that no dominoes overlap and so that adjacent faces have the same color. We demonstrated that for simple layout polygons that can be tiled with colored dominoes, two colors are always sufficient. We also show that for tileable non-simple layout polygons, four colors are always sufficient and sometimes necessary. We describe an O(n) time algorithm for computing a colored domino tiling of a simple orthogonal polygon, if such a tiling exists, where n is the number of dominoes used in the tiling. We also show that deciding whether or not a non-simple orthogonal polygon can be tiled with colored dominoes is NP-complete.]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.388 https://doi.org/10.46298/dmtcs.388 Worman, Chris Yang, Boting Worman, Chris Yang, Boting <![CDATA[We consider questions concerning the tileability of orthogonal polygons with colored dominoes. A colored domino is a rotatable 2 × 1 rectangle that is partitioned into two unit squares, which are called faces, each of which is assigned a color. In a colored domino tiling of an orthogonal polygon P, a set of dominoes completely covers P such that no dominoes overlap and so that adjacent faces have the same color. We demonstrated that for simple layout polygons that can be tiled with colored dominoes, two colors are always sufficient. We also show that for tileable non-simple layout polygons, four colors are always sufficient and sometimes necessary. We describe an O(n) time algorithm for computing a colored domino tiling of a simple orthogonal polygon, if such a tiling exists, where n is the number of dominoes used in the tiling. We also show that deciding whether or not a non-simple orthogonal polygon can be tiled with colored dominoes is NP-complete.]]> 0 <![CDATA[A graph class has few cliques if there is a polynomial bound on the number of maximal cliques contained in any member of the class. This restriction is equivalent to the requirement that any graph in the class has a polynomial sized intersection representation that satisfies the Helly property. On any such class of graphs, some problems that are NP-complete on general graphs, such as the maximum clique problem and the maximum weighted clique problem, admit polynomial time algorithms. Other problems, such as the vertex clique cover and edge clique cover problems remain NP-complete on these classes. Several classes of graphs which have few cliques are discussed, and the complexity of some partitioning and covering problems are determined for the class of all graphs which have fewer cliques than a given polynomial bound.]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.387 https://doi.org/10.46298/dmtcs.387 Rosgen, Bill Stewart, Lorna Rosgen, Bill Stewart, Lorna <![CDATA[A graph class has few cliques if there is a polynomial bound on the number of maximal cliques contained in any member of the class. This restriction is equivalent to the requirement that any graph in the class has a polynomial sized intersection representation that satisfies the Helly property. On any such class of graphs, some problems that are NP-complete on general graphs, such as the maximum clique problem and the maximum weighted clique problem, admit polynomial time algorithms. Other problems, such as the vertex clique cover and edge clique cover problems remain NP-complete on these classes. Several classes of graphs which have few cliques are discussed, and the complexity of some partitioning and covering problems are determined for the class of all graphs which have fewer cliques than a given polynomial bound.]]> 0 <![CDATA[In words, generated by independent geometrically distributed random variables, we study the lth descent, which is, roughly speaking, the lth occurrence of a neighbouring pair ab with a>b. The value a is called the initial height, and b the end height. We study these two random variables (and some similar ones) by combinatorial and probabilistic tools. We find in all instances a generating function Ψ(v,u), where the coefficient of vjui refers to the jth descent (ascent), and i to the initial (end) height. From this, various conclusions can be drawn, in particular expected values. In the probabilistic part, a Markov chain model is used, which allows to get explicit expressions for the heights of the second descent. In principle, one could go further, but the complexity of the results forbids it. This is extended to permutations of a large number of elements. Methods from q-analysis are used to simplify the expressions. This is the reason that we confine ourselves to the geometric distribution only. For general discrete distributions, no such tools are available.]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.386 https://doi.org/10.46298/dmtcs.386 Louchard, Guy Prodinger, Helmut Louchard, Guy Prodinger, Helmut <![CDATA[In words, generated by independent geometrically distributed random variables, we study the lth descent, which is, roughly speaking, the lth occurrence of a neighbouring pair ab with a>b. The value a is called the initial height, and b the end height. We study these two random variables (and some similar ones) by combinatorial and probabilistic tools. We find in all instances a generating function Ψ(v,u), where the coefficient of vjui refers to the jth descent (ascent), and i to the initial (end) height. From this, various conclusions can be drawn, in particular expected values. In the probabilistic part, a Markov chain model is used, which allows to get explicit expressions for the heights of the second descent. In principle, one could go further, but the complexity of the results forbids it. This is extended to permutations of a large number of elements. Methods from q-analysis are used to simplify the expressions. This is the reason that we confine ourselves to the geometric distribution only. For general discrete distributions, no such tools are available.]]> 0 <![CDATA[Let G be a graph with n vertices, with independence number α, and with no Kt+1-minor for some t ≥ 5. It is proved that (2α - 1)(2t - 5) ≥ 2n - 5. This improves upon the previous best bound whenever n≥2/5t2.]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.385 https://doi.org/10.46298/dmtcs.385 Wood, David R. Wood, David R. <![CDATA[Let G be a graph with n vertices, with independence number α, and with no Kt+1-minor for some t ≥ 5. It is proved that (2α - 1)(2t - 5) ≥ 2n - 5. This improves upon the previous best bound whenever n≥2/5t2.]]> 0 <![CDATA[Let G=(V,E) be a connected graph with a weight function w: V \to \mathbbZ₊, and let q ≥q 2 be a positive integer. For X⊆ V, let w(X) denote the sum of the weights of the vertices in X. We consider the following problem on G: find a q-partition P=(V₁,V₂, \ldots, V_q) of V such that G[V_i] is connected (1≤q i≤q q) and P maximizes \rm min\w(V_i): 1≤q i≤q q\. This problem is called \textitMax Balanced Connected q-Partition and is denoted by BCP_q. We show that for q≥q 2 the problem BCP_q is NP-hard in the strong sense, even on q-connected graphs, and therefore does not admit a FPTAS, unless \rm P=\rm NP. We also show another inapproximability result for BCP₂ on arbitrary graphs. On q-connected graphs, for q=2 the best result is a \frac43-approximation algorithm obtained by Chleb\'ıková; for q=3 and q=4 we present 2-approximation algorithms. When q is not fixed (it is part of the instance), the corresponding problem is called \textitMax Balanced Connected Partition, and denoted as BCP. We show that BCP does not admit an approximation algorithm with ratio smaller than 6/5, unless \rm P=\rm NP.]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.384 https://doi.org/10.46298/dmtcs.384 Chataigner, Frédéric Salgado, Liliane R. B. Wakabayashi, Yoshiko Chataigner, Frédéric Salgado, Liliane R. B. Wakabayashi, Yoshiko <![CDATA[Let G=(V,E) be a connected graph with a weight function w: V \to \mathbbZ₊, and let q ≥q 2 be a positive integer. For X⊆ V, let w(X) denote the sum of the weights of the vertices in X. We consider the following problem on G: find a q-partition P=(V₁,V₂, \ldots, V_q) of V such that G[V_i] is connected (1≤q i≤q q) and P maximizes \rm min\w(V_i): 1≤q i≤q q\. This problem is called \textitMax Balanced Connected q-Partition and is denoted by BCP_q. We show that for q≥q 2 the problem BCP_q is NP-hard in the strong sense, even on q-connected graphs, and therefore does not admit a FPTAS, unless \rm P=\rm NP. We also show another inapproximability result for BCP₂ on arbitrary graphs. On q-connected graphs, for q=2 the best result is a \frac43-approximation algorithm obtained by Chleb\'ıková; for q=3 and q=4 we present 2-approximation algorithms. When q is not fixed (it is part of the instance), the corresponding problem is called \textitMax Balanced Connected Partition, and denoted as BCP. We show that BCP does not admit an approximation algorithm with ratio smaller than 6/5, unless \rm P=\rm NP.]]> 0 <![CDATA[We consider the problem of finding a balanced ordering of the vertices of a graph. More precisely, we want to minimise the sum, taken over all vertices v, of the difference between the number of neighbours to the left and right of v. This problem, which has applications in graph drawing, was recently introduced by Biedl et al. [Discrete Applied Math. 148:27―48, 2005]. They proved that the problem is solvable in polynomial time for graphs with maximum degree three, but NP-hard for graphs with maximum degree six. One of our main results is to close the gap in these results, by proving NP-hardness for graphs with maximum degree four. Furthermore, we prove that the problem remains NP-hard for planar graphs with maximum degree four and for 5-regular graphs. On the other hand, we introduce a polynomial time algorithm that determines whetherthere is a vertex ordering with total imbalance smaller than a fixed constant, and a polynomial time algorithm that determines whether a given multigraph with even degrees has an 'almost balanced' ordering.]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.383 https://doi.org/10.46298/dmtcs.383 Kára, Jan Kratochvil, Jan Wood, David R. Kára, Jan Kratochvil, Jan Wood, David R. <![CDATA[We consider the problem of finding a balanced ordering of the vertices of a graph. More precisely, we want to minimise the sum, taken over all vertices v, of the difference between the number of neighbours to the left and right of v. This problem, which has applications in graph drawing, was recently introduced by Biedl et al. [Discrete Applied Math. 148:27―48, 2005]. They proved that the problem is solvable in polynomial time for graphs with maximum degree three, but NP-hard for graphs with maximum degree six. One of our main results is to close the gap in these results, by proving NP-hardness for graphs with maximum degree four. Furthermore, we prove that the problem remains NP-hard for planar graphs with maximum degree four and for 5-regular graphs. On the other hand, we introduce a polynomial time algorithm that determines whetherthere is a vertex ordering with total imbalance smaller than a fixed constant, and a polynomial time algorithm that determines whether a given multigraph with even degrees has an 'almost balanced' ordering.]]> 0 <![CDATA[International audience]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.381 https://doi.org/10.46298/dmtcs.381 Kocábová, Petra Masáková, Zuzana Pelantová, Edita Kocábová, Petra Masáková, Zuzana Pelantová, Edita <![CDATA[International audience]]> 0 <![CDATA[Motivated by the study of decipherability conditions for codes weaker than Unique Decipherability (UD), we introduce the notion of coding partition. Such a notion generalizes that of UD code and, for codes that are not UD, allows to recover the ''unique decipherability" at the level of the classes of the partition. By tacking into account the natural order between the partitions, we define the characteristic partition of a code X as the finest coding partition of X. This leads to introduce the canonical decomposition of a code in at most one unambiguous component and other (if any) totally ambiguous components. In the case the code is finite, we give an algorithm for computing its canonical partition. This, in particular, allows to decide whether a given partition of a finite code X is a coding partition. This last problem is then approached in the case the code is a rational set. We prove its decidability under the hypothesis that the partition contains a finite number of classes and each class is a rational set. Moreover we conjecture that the canonical partition satisfies such a hypothesis. Finally we consider also some relationships between coding partitions and varieties of codes.]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.380 https://doi.org/10.46298/dmtcs.380 Burderi, Fabio Restivo, Antonio Burderi, Fabio Restivo, Antonio <![CDATA[Motivated by the study of decipherability conditions for codes weaker than Unique Decipherability (UD), we introduce the notion of coding partition. Such a notion generalizes that of UD code and, for codes that are not UD, allows to recover the ''unique decipherability" at the level of the classes of the partition. By tacking into account the natural order between the partitions, we define the characteristic partition of a code X as the finest coding partition of X. This leads to introduce the canonical decomposition of a code in at most one unambiguous component and other (if any) totally ambiguous components. In the case the code is finite, we give an algorithm for computing its canonical partition. This, in particular, allows to decide whether a given partition of a finite code X is a coding partition. This last problem is then approached in the case the code is a rational set. We prove its decidability under the hypothesis that the partition contains a finite number of classes and each class is a rational set. Moreover we conjecture that the canonical partition satisfies such a hypothesis. Finally we consider also some relationships between coding partitions and varieties of codes.]]> 0 <![CDATA[We describe a general algorithm for generating various families of ribbon tableaux and computing their spin polynomials. This algorithm is derived from a new matricial coding. An advantage of this new notation lies in the fact that it permits one to generate ribbon tableaux with skew shapes.]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.379 https://doi.org/10.46298/dmtcs.379 Descouens, Francois Descouens, Francois <![CDATA[We describe a general algorithm for generating various families of ribbon tableaux and computing their spin polynomials. This algorithm is derived from a new matricial coding. An advantage of this new notation lies in the fact that it permits one to generate ribbon tableaux with skew shapes.]]> 0 <![CDATA[The construction of the universal ring of Witt vectors is related to Lazard's factorizations of free monoids by means of a noncommutative analogue. This is done by associating to a code a specialization of noncommutative symmetric functions.]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/dmtcs.378 https://doi.org/10.46298/dmtcs.378 Luque, Jean-Gabriel Thibon, Jean-Yves Luque, Jean-Gabriel Thibon, Jean-Yves <![CDATA[The construction of the universal ring of Witt vectors is related to Lazard's factorizations of free monoids by means of a noncommutative analogue. This is done by associating to a code a specialization of noncommutative symmetric functions.]]> 0 <![CDATA[We analyze nearest neighbor one-dimensional quantum random walks with arbitrary unitary coin-flip matrices. Using a multivariate generating function analysis we give a simplified proof of a known phenomenon, namely that the walk has linear speed rather than the diffusive behavior observed in classical random walks. We also obtain exact formulae for the leading asymptotic term of the wave function and the location probabilities.]]> Sun, 31 Dec 2006 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3533 https://doi.org/10.46298/dmtcs.3533 Bressler, Andrew Pemantle, Robin Bressler, Andrew Pemantle, Robin <![CDATA[We analyze nearest neighbor one-dimensional quantum random walks with arbitrary unitary coin-flip matrices. Using a multivariate generating function analysis we give a simplified proof of a known phenomenon, namely that the walk has linear speed rather than the diffusive behavior observed in classical random walks. We also obtain exact formulae for the leading asymptotic term of the wave function and the location probabilities.]]> 0 <![CDATA[We propose and analyze an algorithm to approximate distribution functions and densities of perpetuities. Our algorithm refines an earlier approach based on iterating discretized versions of the fixed point equation that defines the perpetuity. We significantly reduce the complexity of the earlier algorithm. Also one particular perpetuity arising in the analysis of the selection algorithm Quickselect is studied in more detail. Our approach works well for distribution functions. For densities we have weaker error bounds although computer experiments indicate that densities can also be well approximated.]]> Sun, 31 Dec 2006 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3544 https://doi.org/10.46298/dmtcs.3544 Knape, Margarete Neininger, Ralph Knape, Margarete Neininger, Ralph <![CDATA[We propose and analyze an algorithm to approximate distribution functions and densities of perpetuities. Our algorithm refines an earlier approach based on iterating discretized versions of the fixed point equation that defines the perpetuity. We significantly reduce the complexity of the earlier algorithm. Also one particular perpetuity arising in the analysis of the selection algorithm Quickselect is studied in more detail. Our approach works well for distribution functions. For densities we have weaker error bounds although computer experiments indicate that densities can also be well approximated.]]> 0 <![CDATA[In this paper, we give the multivariate generating function counting texts according to their length and to the number of occurrences of words from a finite set. The application of the inclusion-exclusion principle to word counting due to Goulden and Jackson (1979, 1983) is used to derive the result. Unlike some other techniques which suppose that the set of words is reduced (i..e., where no two words are factor of one another), the finite set can be chosen arbitrarily. Noonan and Zeilberger (1999) already provided a MAPLE package treating the non-reduced case, without giving an expression of the generating function or a detailed proof. We give a complete proof validating the use of the inclusion-exclusion principle and compare the complexity of the method proposed here with the one using automata for solving the problem.]]> Sun, 31 Dec 2006 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3543 https://doi.org/10.46298/dmtcs.3543 Bassino, Frédérique Clément, Julien Fayolle, J. Nicodème, P. Bassino, Frédérique Clément, Julien Fayolle, J. Nicodème, P. <![CDATA[In this paper, we give the multivariate generating function counting texts according to their length and to the number of occurrences of words from a finite set. The application of the inclusion-exclusion principle to word counting due to Goulden and Jackson (1979, 1983) is used to derive the result. Unlike some other techniques which suppose that the set of words is reduced (i..e., where no two words are factor of one another), the finite set can be chosen arbitrarily. Noonan and Zeilberger (1999) already provided a MAPLE package treating the non-reduced case, without giving an expression of the generating function or a detailed proof. We give a complete proof validating the use of the inclusion-exclusion principle and compare the complexity of the method proposed here with the one using automata for solving the problem.]]> 0 <![CDATA[In 1992, A. Ehrenfeucht and J. Mycielski defined a seemingly pseudorandom binary sequence which has since been termed the EM-sequence. The balance conjecture for the EM-sequence, still open, is the conjecture that the sequence of EM-sequence initial segment averages converges to $1/2$. In this paper, we do not prove the balance conjecture but we do make some progress concerning it, namely, we prove that every limit point of the aforementioned sequence of averages lies in the interval $[1/4,3/4]$, improving the best previous result that every such limit point belongs to the interval $[0.11,0.89]$. Our approach is novel and exploits an analysis of the growth behavior as $n \to \infty$ of the rooted tree formed by the binary strings appearing at least twice as substrings of the length $n$ initial segment of the EM-sequence.]]> Sun, 31 Dec 2006 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3542 https://doi.org/10.46298/dmtcs.3542 Kieffer, John C. Szpankowski, W. Kieffer, John C. Szpankowski, W. <![CDATA[In 1992, A. Ehrenfeucht and J. Mycielski defined a seemingly pseudorandom binary sequence which has since been termed the EM-sequence. The balance conjecture for the EM-sequence, still open, is the conjecture that the sequence of EM-sequence initial segment averages converges to $1/2$. In this paper, we do not prove the balance conjecture but we do make some progress concerning it, namely, we prove that every limit point of the aforementioned sequence of averages lies in the interval $[1/4,3/4]$, improving the best previous result that every such limit point belongs to the interval $[0.11,0.89]$. Our approach is novel and exploits an analysis of the growth behavior as $n \to \infty$ of the rooted tree formed by the binary strings appearing at least twice as substrings of the length $n$ initial segment of the EM-sequence.]]> 0 <![CDATA[In this paper, we study a restricted version of the position restricted pattern matching problem introduced and studied by Mäkinen and Navarro [Position-Restricted Substring Searching, LATIN 2006]. In the problem handled in this paper, we are interested in those occurrences of the pattern that lies in a suffix or in a prefix of the given text. We achieve optimal query time for our problem against a data structure which is an extension of the classic suffix tree data structure. The time and space complexity of the data structure is dominated by that of the suffix tree. Notably, the (best) algorithm by Mäkinen and Navarro, if applied to our problem, gives sub-optimal query time and the corresponding data structure also requires more time and space.]]> Sun, 31 Dec 2006 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3541 https://doi.org/10.46298/dmtcs.3541 Crochemore, Maxime Iliopoulos, Costas S. Rahman, M. Sohel Crochemore, Maxime Iliopoulos, Costas S. Rahman, M. Sohel <![CDATA[In this paper, we study a restricted version of the position restricted pattern matching problem introduced and studied by Mäkinen and Navarro [Position-Restricted Substring Searching, LATIN 2006]. In the problem handled in this paper, we are interested in those occurrences of the pattern that lies in a suffix or in a prefix of the given text. We achieve optimal query time for our problem against a data structure which is an extension of the classic suffix tree data structure. The time and space complexity of the data structure is dominated by that of the suffix tree. Notably, the (best) algorithm by Mäkinen and Navarro, if applied to our problem, gives sub-optimal query time and the corresponding data structure also requires more time and space.]]> 0 <![CDATA[In 1999, Chan proposed an algorithm to solve a given optimization problem: express the solution as the minimum of the solutions of several subproblems and apply the classical randomized algorithm for finding the minimum of $r$ numbers. If the decision versions of the subproblems are easier to solve than the subproblems themselves, then a faster algorithm for the optimization problem may be obtained with randomization. In this paper we present a precise probabilistic analysis of Chan's technique.]]> Sun, 31 Dec 2006 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3540 https://doi.org/10.46298/dmtcs.3540 Cardinal, Jean Langerman, Stefan Louchard, Guy Cardinal, Jean Langerman, Stefan Louchard, Guy <![CDATA[In 1999, Chan proposed an algorithm to solve a given optimization problem: express the solution as the minimum of the solutions of several subproblems and apply the classical randomized algorithm for finding the minimum of $r$ numbers. If the decision versions of the subproblems are easier to solve than the subproblems themselves, then a faster algorithm for the optimization problem may be obtained with randomization. In this paper we present a precise probabilistic analysis of Chan's technique.]]> 0 <![CDATA[For a large number of random Boolean constraint satisfaction problems, such as random $k$-SAT, we study how the number of locally maximal solutions evolves when constraints are added. We give the exponential order of the expected number of these distinguished solutions and prove it depends on the sensitivity of the allowed constraint functions only. As a by-product we provide a general tool for computing an upper bound of the satisfiability threshold for any problem of a large class of random Boolean CSPs.]]> Sun, 31 Dec 2006 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3539 https://doi.org/10.46298/dmtcs.3539 Creignou, Nadia Daudé, Hervé Dubois, Olivier Creignou, Nadia Daudé, Hervé Dubois, Olivier <![CDATA[For a large number of random Boolean constraint satisfaction problems, such as random $k$-SAT, we study how the number of locally maximal solutions evolves when constraints are added. We give the exponential order of the expected number of these distinguished solutions and prove it depends on the sensitivity of the allowed constraint functions only. As a by-product we provide a general tool for computing an upper bound of the satisfiability threshold for any problem of a large class of random Boolean CSPs.]]> 0 <![CDATA[Finding a satisfying assignment for a $k$-CNF formula $(k \geq 3)$, assuming such exists, is a notoriously hard problem. In this work we consider the uniform distribution over satisfiable $k$-CNF formulas with a linear number of clauses (clause-variable ratio greater than some constant). We rigorously analyze the structure of the space of satisfying assignments of a random formula in that distribution, showing that basically all satisfying assignments are clustered in one cluster, and agree on all but a small, though linear, number of variables. This observation enables us to describe a polynomial time algorithm that finds $\textit{whp}$ a satisfying assignment for such formulas, thus asserting that most satisfiable $k$-CNF formulas are easy (whenever the clause-variable ratio is greater than some constant). This should be contrasted with the setting of very sparse $k$-CNF formulas (which are satisfiable $\textit{whp}$), where experimental results show some regime of clause density to be difficult for many SAT heuristics. One explanation for this phenomena, backed up by partially non-rigorous analytical tools from statistical physics, is the complicated clustering of the solution space at that regime, unlike the more "regular" structure that denser formulas possess. Thus in some sense, our result rigorously supports this explanation.]]> Sun, 31 Dec 2006 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3538 https://doi.org/10.46298/dmtcs.3538 Coja-Oghlan, Amin Krivelevich, Michael Vilenchik, Dan Coja-Oghlan, Amin Krivelevich, Michael Vilenchik, Dan <![CDATA[Finding a satisfying assignment for a $k$-CNF formula $(k \geq 3)$, assuming such exists, is a notoriously hard problem. In this work we consider the uniform distribution over satisfiable $k$-CNF formulas with a linear number of clauses (clause-variable ratio greater than some constant). We rigorously analyze the structure of the space of satisfying assignments of a random formula in that distribution, showing that basically all satisfying assignments are clustered in one cluster, and agree on all but a small, though linear, number of variables. This observation enables us to describe a polynomial time algorithm that finds $\textit{whp}$ a satisfying assignment for such formulas, thus asserting that most satisfiable $k$-CNF formulas are easy (whenever the clause-variable ratio is greater than some constant). This should be contrasted with the setting of very sparse $k$-CNF formulas (which are satisfiable $\textit{whp}$), where experimental results show some regime of clause density to be difficult for many SAT heuristics. One explanation for this phenomena, backed up by partially non-rigorous analytical tools from statistical physics, is the complicated clustering of the solution space at that regime, unlike the more "regular" structure that denser formulas possess. Thus in some sense, our result rigorously supports this explanation.]]> 0 <![CDATA[An $f(n)$ $\textit{dominance bound}$ on a heuristic for some problem is a guarantee that the heuristic always returns a solution not worse than at least $f(n)$ solutions. In this paper, we analyze several heuristics for $\textit{Vertex Cover}$, $\textit{Set Cover}$, and $\textit{Knapsack}$ for dominance bounds. In particular, we show that the well-known $\textit{maximal matching}$ heuristic of $\textit{Vertex Cover}$ provides an excellent dominance bound. We introduce new general analysis techniques which apply to a wide range of problems and heuristics for this measure. Certain general results relating approximation ratio and combinatorial dominance guarantees for optimization problems over subsets are established. We prove certain limitations on the combinatorial dominance guarantees of polynomial-time approximation schemes (PTAS), and give inapproximability results for the problems above.]]> Sun, 31 Dec 2006 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3537 https://doi.org/10.46298/dmtcs.3537 Berend, Daniel Skiena, Steven S. Twitto, Yochai Berend, Daniel Skiena, Steven S. Twitto, Yochai <![CDATA[An $f(n)$ $\textit{dominance bound}$ on a heuristic for some problem is a guarantee that the heuristic always returns a solution not worse than at least $f(n)$ solutions. In this paper, we analyze several heuristics for $\textit{Vertex Cover}$, $\textit{Set Cover}$, and $\textit{Knapsack}$ for dominance bounds. In particular, we show that the well-known $\textit{maximal matching}$ heuristic of $\textit{Vertex Cover}$ provides an excellent dominance bound. We introduce new general analysis techniques which apply to a wide range of problems and heuristics for this measure. Certain general results relating approximation ratio and combinatorial dominance guarantees for optimization problems over subsets are established. We prove certain limitations on the combinatorial dominance guarantees of polynomial-time approximation schemes (PTAS), and give inapproximability results for the problems above.]]> 0 <![CDATA[We characterize the asymptotics of heights of the trees of de la Briandais and the ternary search trees (TST) of Bentley and Sedgewick. Our proof is based on a new analysis of the structure of tries that distinguishes the bulk of the tree, called the $\textit{core}$, and the long trees hanging down the core, called the $\textit{spaghettis}$.]]> Sun, 31 Dec 2006 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3536 https://doi.org/10.46298/dmtcs.3536 Broutin, N. Devroye, L. Broutin, N. Devroye, L. <![CDATA[We characterize the asymptotics of heights of the trees of de la Briandais and the ternary search trees (TST) of Bentley and Sedgewick. Our proof is based on a new analysis of the structure of tries that distinguishes the bulk of the tree, called the $\textit{core}$, and the long trees hanging down the core, called the $\textit{spaghettis}$.]]> 0 <![CDATA[We study the average behavior of variants of the UNION-FIND algorithm to maintain partitions of a finite set under the random spanning tree model. By applying the method of moments we can characterize the limiting distribution of the total costs of the algorithms "Quick Find Weighted'' and "Quick Find Biased'' extending the analysis of Knuth and Schönhage, Yao, and Chassaing and Marchand.]]> Sun, 31 Dec 2006 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3535 https://doi.org/10.46298/dmtcs.3535 Kuba, Markus Panholzer, Alois Kuba, Markus Panholzer, Alois <![CDATA[We study the average behavior of variants of the UNION-FIND algorithm to maintain partitions of a finite set under the random spanning tree model. By applying the method of moments we can characterize the limiting distribution of the total costs of the algorithms "Quick Find Weighted'' and "Quick Find Biased'' extending the analysis of Knuth and Schönhage, Yao, and Chassaing and Marchand.]]> 0 <![CDATA[Let $\sigma$ be a random permutation chosen uniformly over the symmetric group $\mathfrak{S}_n$. We study a new "process-valued" statistic of $\sigma$, which appears in the domain of computational biology to construct tests of similarity between ordered lists of genes. More precisely, we consider the following "partial sums": $Y^{(n)}_{p,q} = \mathrm{card} \{1 \leq i \leq p : \sigma_i \leq q \}$ for $0 \leq p,q \leq n$. We show that a suitable normalization of $Y^{(n)}$ converges weakly to a bivariate tied down brownian bridge on $[0,1]^2$, i.e. a continuous centered gaussian process $X^{\infty}_{s,t}$ of covariance: $\mathbb{E}[X^{\infty}_{s,t}X^{\infty}_{s',t'}] = (min(s,s')-ss')(min(t,t')-tt')$.]]> Sun, 31 Dec 2006 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3534 https://doi.org/10.46298/dmtcs.3534 Chapuy, Guillaume Chapuy, Guillaume <![CDATA[Let $\sigma$ be a random permutation chosen uniformly over the symmetric group $\mathfrak{S}_n$. We study a new "process-valued" statistic of $\sigma$, which appears in the domain of computational biology to construct tests of similarity between ordered lists of genes. More precisely, we consider the following "partial sums": $Y^{(n)}_{p,q} = \mathrm{card} \{1 \leq i \leq p : \sigma_i \leq q \}$ for $0 \leq p,q \leq n$. We show that a suitable normalization of $Y^{(n)}$ converges weakly to a bivariate tied down brownian bridge on $[0,1]^2$, i.e. a continuous centered gaussian process $X^{\infty}_{s,t}$ of covariance: $\mathbb{E}[X^{\infty}_{s,t}X^{\infty}_{s',t'}] = (min(s,s')-ss')(min(t,t')-tt')$.]]> 0 <![CDATA[One-sided variations on path length in a trie (a sort of digital trees) are investigated: They include imbalance factors, climbing under different strategies, and key sampling. For the imbalance factor accurate asymptotics for the mean are derived for a randomly chosen key in the trie via poissonization and the Mellin transform, and the inverse of the two operations. It is also shown from an analysis of the moving poles of the Mellin transform of the poissonized moment generating function that the imbalance factor (under appropriate centering and scaling) follows a Gaussian limit law. The method extends to several variations of sampling keys from a trie and we sketch results of climbing under different strategies. The exact probability distribution is computed in one case, to demonstrate that such calculations can be done, at least in principle.]]> Sun, 31 Dec 2006 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3522 https://doi.org/10.46298/dmtcs.3522 Christophi, Costas A. Mahmoud, Hosam M. Christophi, Costas A. Mahmoud, Hosam M. <![CDATA[One-sided variations on path length in a trie (a sort of digital trees) are investigated: They include imbalance factors, climbing under different strategies, and key sampling. For the imbalance factor accurate asymptotics for the mean are derived for a randomly chosen key in the trie via poissonization and the Mellin transform, and the inverse of the two operations. It is also shown from an analysis of the moving poles of the Mellin transform of the poissonized moment generating function that the imbalance factor (under appropriate centering and scaling) follows a Gaussian limit law. The method extends to several variations of sampling keys from a trie and we sketch results of climbing under different strategies. The exact probability distribution is computed in one case, to demonstrate that such calculations can be done, at least in principle.]]> 0 <![CDATA[We derive asymptotics for the moments of the height distribution of watermelons with $p$ branches with wall. This generalises a famous result by de Bruijn, Knuth and Rice on the average height of planted plane trees, and a result by Fulmek on the average height of watermelons with two branches.]]> Sun, 31 Dec 2006 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3532 https://doi.org/10.46298/dmtcs.3532 Feierl, Thomas Feierl, Thomas <![CDATA[We derive asymptotics for the moments of the height distribution of watermelons with $p$ branches with wall. This generalises a famous result by de Bruijn, Knuth and Rice on the average height of planted plane trees, and a result by Fulmek on the average height of watermelons with two branches.]]> 0 <![CDATA[Let $\sum_{\mathbf{n} \in \mathbb{N}^d} F_{\mathbf{n}} \mathbf{x}^{\mathbf{n}}$ be a multivariate generating function that converges in a neighborhood of the origin of $\mathbb{C}^d$. We present a new, multivariate method for computing the asymptotics of the diagonal coefficients $F_{a_1n,\ldots,a_dn}$ and show its superiority over the standard, univariate diagonal method. Several examples are given in detail.]]> Sun, 31 Dec 2006 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3531 https://doi.org/10.46298/dmtcs.3531 Raichev, Alexander Wilson, Mark C. Raichev, Alexander Wilson, Mark C. <![CDATA[Let $\sum_{\mathbf{n} \in \mathbb{N}^d} F_{\mathbf{n}} \mathbf{x}^{\mathbf{n}}$ be a multivariate generating function that converges in a neighborhood of the origin of $\mathbb{C}^d$. We present a new, multivariate method for computing the asymptotics of the diagonal coefficients $F_{a_1n,\ldots,a_dn}$ and show its superiority over the standard, univariate diagonal method. Several examples are given in detail.]]> 0 <![CDATA[We consider the following stochastic bin packing process: the items arrive continuously over time to a server and are packed into bins of unit size according to an online algorithm. The unpacked items form a queue. The items have random sizes with symmetric distribution. Our first contribution identifies some monotonicity properties of the queueing system that allow to derive bounds on the queue size for First Fit and Best Fit algorithms. As a direct application, we show how to compute the stability region under very general conditions on the input process. Our second contribution is a study of the queueing system under heavy load. We show how the monotonicity properties allow one to derive bounds for the speed at which the stationary queue length tends to infinity when the load approaches one. In the case of Best Fit, these bounds are tight. Our analysis shows connections between our dynamic model, average-case results on the classical bin packing problem and planar matching problems.]]> Sun, 31 Dec 2006 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3530 https://doi.org/10.46298/dmtcs.3530 Lelarge, Marc Lelarge, Marc <![CDATA[We consider the following stochastic bin packing process: the items arrive continuously over time to a server and are packed into bins of unit size according to an online algorithm. The unpacked items form a queue. The items have random sizes with symmetric distribution. Our first contribution identifies some monotonicity properties of the queueing system that allow to derive bounds on the queue size for First Fit and Best Fit algorithms. As a direct application, we show how to compute the stability region under very general conditions on the input process. Our second contribution is a study of the queueing system under heavy load. We show how the monotonicity properties allow one to derive bounds for the speed at which the stationary queue length tends to infinity when the load approaches one. In the case of Best Fit, these bounds are tight. Our analysis shows connections between our dynamic model, average-case results on the classical bin packing problem and planar matching problems.]]> 0 <![CDATA[Let $P(z)$ and $Q(y)$ be polynomials of the same degree $k \geq 1$ in the complex variables $z$ and $y$, respectively. In this extended abstract we study the non-linear functional equation $P(z)=Q(y(z))$, where $y(z)$ is restricted to be analytic in a neighborhood of $z=0$. We provide sufficient conditions to ensure that all the roots of $Q(y)$ are contained within the range of $y(z)$ as well as to have $y(z)=z$ as the unique analytic solution of the non-linear equation. Our results are motivated from uniqueness considerations of polynomial canonical representations of the phase or amplitude terms of oscillatory integrals encountered in the asymptotic analysis of the coefficients of mixed powers and multivariable generating functions via saddle-point methods. Uniqueness shall prove important for developing algorithms to determine the Taylor coefficients of the terms appearing in these representations. The uniqueness of Levinson's polynomial canonical representations of analytic functions in several variables follows as a corollary of our one-complex variables results.]]> Sun, 31 Dec 2006 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3529 https://doi.org/10.46298/dmtcs.3529 Lladser, Manuel Lladser, Manuel <![CDATA[Let $P(z)$ and $Q(y)$ be polynomials of the same degree $k \geq 1$ in the complex variables $z$ and $y$, respectively. In this extended abstract we study the non-linear functional equation $P(z)=Q(y(z))$, where $y(z)$ is restricted to be analytic in a neighborhood of $z=0$. We provide sufficient conditions to ensure that all the roots of $Q(y)$ are contained within the range of $y(z)$ as well as to have $y(z)=z$ as the unique analytic solution of the non-linear equation. Our results are motivated from uniqueness considerations of polynomial canonical representations of the phase or amplitude terms of oscillatory integrals encountered in the asymptotic analysis of the coefficients of mixed powers and multivariable generating functions via saddle-point methods. Uniqueness shall prove important for developing algorithms to determine the Taylor coefficients of the terms appearing in these representations. The uniqueness of Levinson's polynomial canonical representations of analytic functions in several variables follows as a corollary of our one-complex variables results.]]> 0 <![CDATA[It is well known that many distributions that arise in the analysis of algorithms have an asymptotically fluctuating behaviour in the sense that we do not have 'full' convergence, but only convergence along suitable subsequences as the size of the input to the algorithm tends to infinity. We are interested in constructions that display such behaviour via an ordinarily convergent background process in the sense that the periodicities arise from this process by deterministic transformations, typically involving discretization as a decisive step. This leads to structural representations of the resulting family of limit distributions along subsequences, which in turn may give access to their properties, such as the tail behaviour (unsuccessful search in digital search trees) or the dependence on parameters of the algorithm (success probability in a selection algorithm).]]> Sun, 31 Dec 2006 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3528 https://doi.org/10.46298/dmtcs.3528 Grübel, Rudolf Grübel, Rudolf <![CDATA[It is well known that many distributions that arise in the analysis of algorithms have an asymptotically fluctuating behaviour in the sense that we do not have 'full' convergence, but only convergence along suitable subsequences as the size of the input to the algorithm tends to infinity. We are interested in constructions that display such behaviour via an ordinarily convergent background process in the sense that the periodicities arise from this process by deterministic transformations, typically involving discretization as a decisive step. This leads to structural representations of the resulting family of limit distributions along subsequences, which in turn may give access to their properties, such as the tail behaviour (unsuccessful search in digital search trees) or the dependence on parameters of the algorithm (success probability in a selection algorithm).]]> 0 <![CDATA[This paper deals with some very simple interacting particle systems, \emphelementary cellular automata, in the fully asynchronous dynamics: at each time step, a cell is randomly picked, and updated. When the initial configuration is simple, we describe the asymptotic behavior of the random walks performed by the borders of the black/white regions. Following a classification introduced by Fatès \emphet al., we show that four kinds of asymptotic behavior arise, two of them being related to Brownian motion.]]> Sun, 31 Dec 2006 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3527 https://doi.org/10.46298/dmtcs.3527 Chassaing, Philippe Gerin, Lucas Chassaing, Philippe Gerin, Lucas <![CDATA[This paper deals with some very simple interacting particle systems, \emphelementary cellular automata, in the fully asynchronous dynamics: at each time step, a cell is randomly picked, and updated. When the initial configuration is simple, we describe the asymptotic behavior of the random walks performed by the borders of the black/white regions. Following a classification introduced by Fatès \emphet al., we show that four kinds of asymptotic behavior arise, two of them being related to Brownian motion.]]> 0 <![CDATA[In our previous work [paper1], we derived an asymptotic expression for the probability that a random decomposable combinatorial structure of size n in the \exp -\log class has a given restricted pattern. In this paper, under similar conditions, we provide the probability that a random decomposable combinatorial structure has a given restricted pattern and the size of its rth smallest component is bigger than k, for r,k given integers. Our studies apply to labeled and unlabeled structures. We also give several concrete examples.]]> Sun, 31 Dec 2006 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3526 https://doi.org/10.46298/dmtcs.3526 Dong, Li Gao, Zhicheng Panario, Daniel Dong, Li Gao, Zhicheng Panario, Daniel <![CDATA[In our previous work [paper1], we derived an asymptotic expression for the probability that a random decomposable combinatorial structure of size n in the \exp -\log class has a given restricted pattern. In this paper, under similar conditions, we provide the probability that a random decomposable combinatorial structure has a given restricted pattern and the size of its rth smallest component is bigger than k, for r,k given integers. Our studies apply to labeled and unlabeled structures. We also give several concrete examples.]]> 0 <![CDATA[We study a random walk with positive drift in the first quadrant of the plane. For a given connected region $\mathcal{C}$ of the first quadrant, we analyze the number of paths contained in $\mathcal{C}$ and the first exit time from $\mathcal{C}$. In our case, region $\mathcal{C}$ is bounded by two crossing lines. It is noted that such a walk is equivalent to a path in a tree from the root to a leaf not exceeding a given height. If this tree is the parsing tree of the Tunstall or Khodak variable-to-fixed code, then the exit time of the underlying random walk corresponds to the phrase length not exceeding a given length. We derive precise asymptotics of the number of paths and the asymptotic distribution of the exit time. Even for such a simple walk, the analysis turns out to be quite sophisticated and it involves Mellin transforms, Tauberian theorems, and infinite number of saddle points.]]> Sun, 31 Dec 2006 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3525 https://doi.org/10.46298/dmtcs.3525 Drmota, Michael Szpankowski, Wojciech Drmota, Michael Szpankowski, Wojciech <![CDATA[We study a random walk with positive drift in the first quadrant of the plane. For a given connected region $\mathcal{C}$ of the first quadrant, we analyze the number of paths contained in $\mathcal{C}$ and the first exit time from $\mathcal{C}$. In our case, region $\mathcal{C}$ is bounded by two crossing lines. It is noted that such a walk is equivalent to a path in a tree from the root to a leaf not exceeding a given height. If this tree is the parsing tree of the Tunstall or Khodak variable-to-fixed code, then the exit time of the underlying random walk corresponds to the phrase length not exceeding a given length. We derive precise asymptotics of the number of paths and the asymptotic distribution of the exit time. Even for such a simple walk, the analysis turns out to be quite sophisticated and it involves Mellin transforms, Tauberian theorems, and infinite number of saddle points.]]> 0 <![CDATA[Upper and lower bounds for the tail probabilities of the Wiener index of random binary search trees are given. For upper bounds the moment generating function of the vector of Wiener index and internal path length is estimated. For the lower bounds a tree class with sufficiently large probability and atypically large Wiener index is constructed. The methods are also applicable to related random search trees.]]> Sun, 31 Dec 2006 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3524 https://doi.org/10.46298/dmtcs.3524 Khan, Tämur Ali Neininger, Ralph Khan, Tämur Ali Neininger, Ralph <![CDATA[Upper and lower bounds for the tail probabilities of the Wiener index of random binary search trees are given. For upper bounds the moment generating function of the vector of Wiener index and internal path length is estimated. For the lower bounds a tree class with sufficiently large probability and atypically large Wiener index is constructed. The methods are also applicable to related random search trees.]]> 0 <![CDATA[We consider samples of n geometric random variables $(Γ _1, Γ _2, \dots Γ _n)$ where $\mathbb{P}\{Γ _j=i\}=pq^{i-1}$, for $1≤j ≤n$, with $p+q=1$. The parameter we study is the position of the first occurrence of the maximum value in a such a sample. We derive a probability generating function for this position with which we compute the first two (factorial) moments. The asymptotic technique known as Rice's method then yields the main terms as well as the Fourier expansions of the fluctuating functions arising in the expected value and the variance.]]> Sun, 31 Dec 2006 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3523 https://doi.org/10.46298/dmtcs.3523 Archibald, Margaret Knopfmacher, Arnold Archibald, Margaret Knopfmacher, Arnold <![CDATA[We consider samples of n geometric random variables $(Γ _1, Γ _2, \dots Γ _n)$ where $\mathbb{P}\{Γ _j=i\}=pq^{i-1}$, for $1≤j ≤n$, with $p+q=1$. The parameter we study is the position of the first occurrence of the maximum value in a such a sample. We derive a probability generating function for this position with which we compute the first two (factorial) moments. The asymptotic technique known as Rice's method then yields the main terms as well as the Fourier expansions of the fluctuating functions arising in the expected value and the variance.]]> 0 <![CDATA[Permutation tableaux are new objects that were introduced by Postnikov in the context of enumeration of the totally positive Grassmannian cells. They are known to be in bijection with permutations and recently, they have been connected to PASEP model used in statistical physics. Properties of permutation tableaux became a focus of a considerable research activity. In this paper we study properties of basic statistics defined on permutation tableaux. We present a simple and unified approach based on probabilistic techniques and use it to compute the expected values of basic statistics defined on permutation tableaux. We also provide a non―bijective and very simple proof that there are n! permutation tableaux of length n.]]> Sun, 31 Dec 2006 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3518 https://doi.org/10.46298/dmtcs.3518 Corteel, Sylvie Hitczenko, Pawel Corteel, Sylvie Hitczenko, Pawel <![CDATA[Permutation tableaux are new objects that were introduced by Postnikov in the context of enumeration of the totally positive Grassmannian cells. They are known to be in bijection with permutations and recently, they have been connected to PASEP model used in statistical physics. Properties of permutation tableaux became a focus of a considerable research activity. In this paper we study properties of basic statistics defined on permutation tableaux. We present a simple and unified approach based on probabilistic techniques and use it to compute the expected values of basic statistics defined on permutation tableaux. We also provide a non―bijective and very simple proof that there are n! permutation tableaux of length n.]]> 0 <![CDATA[In this work we analyze a class of diminishing 2×2 Pólya-Eggenberger urn models with ball replacement matrix M given by $M= \binom{ -a \,0}{c -d}, a,d∈\mathbb{N}$ and $c∈\mathbb{N} _0$. We obtain limit laws for this class of 2×2 urns by giving estimates for the moments of the considered random variables. As a special instance we obtain limit laws for the pills problem, proposed by Knuth and McCarthy, which corresponds to the special case $a=c=d=1$. Furthermore, we also obtain limit laws for the well known sampling without replacement urn, $a=d=1$ and $c=0$, and corresponding generalizations, $a,d∈\mathbb{N}$ and $c=0$.]]> Sun, 31 Dec 2006 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3519 https://doi.org/10.46298/dmtcs.3519 Kuba, Markus Panholzer, Alois Kuba, Markus Panholzer, Alois <![CDATA[In this work we analyze a class of diminishing 2×2 Pólya-Eggenberger urn models with ball replacement matrix M given by $M= \binom{ -a \,0}{c -d}, a,d∈\mathbb{N}$ and $c∈\mathbb{N} _0$. We obtain limit laws for this class of 2×2 urns by giving estimates for the moments of the considered random variables. As a special instance we obtain limit laws for the pills problem, proposed by Knuth and McCarthy, which corresponds to the special case $a=c=d=1$. Furthermore, we also obtain limit laws for the well known sampling without replacement urn, $a=d=1$ and $c=0$, and corresponding generalizations, $a,d∈\mathbb{N}$ and $c=0$.]]> 0 <![CDATA[The minimal length of a plateau (a sequence of horizontal steps, preceded by an up- and followed by a down-step) in a Motzkin path is known to be of interest in the study of secondary structures which in turn appear in mathematical biology. We will treat this and the related parameters maximal plateau length, horizontal segment and maximal horizontal segment as well as some similar parameters in unary-binary trees by a pure generating functions approach―-Motzkin paths are derived from Dyck paths by a substitution process. Furthermore, we provide a pretty general analytic method to obtain means and limiting distributions for these parameters. It turns out that the maximal plateau and the maximal horizontal segment follow a Gumbel distribution.]]> Sun, 31 Dec 2006 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3520 https://doi.org/10.46298/dmtcs.3520 Prodinger, Helmut Wagner, Stephan Prodinger, Helmut Wagner, Stephan <![CDATA[The minimal length of a plateau (a sequence of horizontal steps, preceded by an up- and followed by a down-step) in a Motzkin path is known to be of interest in the study of secondary structures which in turn appear in mathematical biology. We will treat this and the related parameters maximal plateau length, horizontal segment and maximal horizontal segment as well as some similar parameters in unary-binary trees by a pure generating functions approach―-Motzkin paths are derived from Dyck paths by a substitution process. Furthermore, we provide a pretty general analytic method to obtain means and limiting distributions for these parameters. It turns out that the maximal plateau and the maximal horizontal segment follow a Gumbel distribution.]]> 0 <![CDATA[Random Apollonian networks have been recently introduced for representing real graphs. In this paper we study a modified version: random Apollonian network structures (RANS), which preserve the interesting properties of real graphs and can be handled with powerful tools of random generation. We exhibit a bijection between RANS and ternary trees, that transforms the degree of nodes in a RANS into the size of particular subtrees. The distribution of degrees in RANS can thus be analysed within a bivariate Boltzmann model for the generation of random trees, and we show that it has a Catalan form which reduces to a power law with an exponential cutoff: $α ^k k^{-3/2}$, with $α = 8/9$. We also show analogous distributions for the degree in RANS of higher dimension, related to trees of higher arity.]]> Sun, 31 Dec 2006 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3521 https://doi.org/10.46298/dmtcs.3521 Darrasse, Alexis Soria, Michèle Darrasse, Alexis Soria, Michèle <![CDATA[Random Apollonian networks have been recently introduced for representing real graphs. In this paper we study a modified version: random Apollonian network structures (RANS), which preserve the interesting properties of real graphs and can be handled with powerful tools of random generation. We exhibit a bijection between RANS and ternary trees, that transforms the degree of nodes in a RANS into the size of particular subtrees. The distribution of degrees in RANS can thus be analysed within a bivariate Boltzmann model for the generation of random trees, and we show that it has a Catalan form which reduces to a power law with an exponential cutoff: $α ^k k^{-3/2}$, with $α = 8/9$. We also show analogous distributions for the degree in RANS of higher dimension, related to trees of higher arity.]]> 0 <![CDATA[This extended abstract describes and analyses a near-optimal probabilistic algorithm, HYPERLOGLOG, dedicated to estimating the number of \emphdistinct elements (the cardinality) of very large data ensembles. Using an auxiliary memory of m units (typically, "short bytes''), HYPERLOGLOG performs a single pass over the data and produces an estimate of the cardinality such that the relative accuracy (the standard error) is typically about $1.04/\sqrt{m}$. This improves on the best previously known cardinality estimator, LOGLOG, whose accuracy can be matched by consuming only 64% of the original memory. For instance, the new algorithm makes it possible to estimate cardinalities well beyond $10^9$ with a typical accuracy of 2% while using a memory of only 1.5 kilobytes. The algorithm parallelizes optimally and adapts to the sliding window model.]]> Sun, 31 Dec 2006 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3545 https://doi.org/10.46298/dmtcs.3545 Flajolet, Philippe Fusy, Éric Gandouet, Olivier Meunier, Frédéric Flajolet, Philippe Fusy, Éric Gandouet, Olivier Meunier, Frédéric <![CDATA[This extended abstract describes and analyses a near-optimal probabilistic algorithm, HYPERLOGLOG, dedicated to estimating the number of \emphdistinct elements (the cardinality) of very large data ensembles. Using an auxiliary memory of m units (typically, "short bytes''), HYPERLOGLOG performs a single pass over the data and produces an estimate of the cardinality such that the relative accuracy (the standard error) is typically about $1.04/\sqrt{m}$. This improves on the best previously known cardinality estimator, LOGLOG, whose accuracy can be matched by consuming only 64% of the original memory. For instance, the new algorithm makes it possible to estimate cardinalities well beyond $10^9$ with a typical accuracy of 2% while using a memory of only 1.5 kilobytes. The algorithm parallelizes optimally and adapts to the sliding window model.]]> 0 <![CDATA[Message passing algorithms are popular in many combinatorial optimization problems. For example, experimental results show that \emphsurvey propagation (a certain message passing algorithm) is effective in finding proper k-colorings of random graphs in the near-threshold regime. In 1962 Gallager introduced the concept of Low Density Parity Check (LDPC) codes, and suggested a simple decoding algorithm based on message passing. In 1994 Alon and Kahale exhibited a coloring algorithm and proved its usefulness for finding a k-coloring of graphs drawn from a certain planted-solution distribution over k-colorable graphs. In this work we show an interpretation of Alon and Kahale's coloring algorithm in light of Gallager's decoding algorithm, thus showing a connection between the two problems - coloring and decoding. This also provides a rigorous evidence for the usefulness of the message passing paradigm for the graph coloring problem.]]> Sun, 31 Dec 2006 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3546 https://doi.org/10.46298/dmtcs.3546 Ben-Shimon, Sonny Vilenchik, Dan Ben-Shimon, Sonny Vilenchik, Dan <![CDATA[Message passing algorithms are popular in many combinatorial optimization problems. For example, experimental results show that \emphsurvey propagation (a certain message passing algorithm) is effective in finding proper k-colorings of random graphs in the near-threshold regime. In 1962 Gallager introduced the concept of Low Density Parity Check (LDPC) codes, and suggested a simple decoding algorithm based on message passing. In 1994 Alon and Kahale exhibited a coloring algorithm and proved its usefulness for finding a k-coloring of graphs drawn from a certain planted-solution distribution over k-colorable graphs. In this work we show an interpretation of Alon and Kahale's coloring algorithm in light of Gallager's decoding algorithm, thus showing a connection between the two problems - coloring and decoding. This also provides a rigorous evidence for the usefulness of the message passing paradigm for the graph coloring problem.]]> 0 <![CDATA[We answer an old question: what are possible growth rates of the expected number of vector-maximal points in a uniform sample from a polytope.]]> Sun, 31 Dec 2006 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3547 https://doi.org/10.46298/dmtcs.3547 Baryshnikov, Yu. Baryshnikov, Yu. <![CDATA[We answer an old question: what are possible growth rates of the expected number of vector-maximal points in a uniform sample from a polytope.]]> 0 <![CDATA[We study the space requirements of a sorting algorithm where only items that at the end will be adjacent are kept together. This is equivalent to the following combinatorial problem: Consider a string of fixed length n that starts as a string of 0's, and then evolves by changing each 0 to 1, with the n changes done in random order. What is the maximal number of runs of 1's? We give asymptotic results for the distribution and mean. It turns out that, as in many problems involving a maximum, the maximum is asymptotically normal, with fluctuations of order $n^{1/2}$, and to the first order well approximated by the number of runs at the instance when the expectation is maximized, in this case when half the elements have changed to 1; there is also a second order term of order $n^{1/3}$. We also treat some variations, including priority queues and sock-sorting.]]> Sun, 31 Dec 2006 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3548 https://doi.org/10.46298/dmtcs.3548 Janson, Svante Janson, Svante <![CDATA[We study the space requirements of a sorting algorithm where only items that at the end will be adjacent are kept together. This is equivalent to the following combinatorial problem: Consider a string of fixed length n that starts as a string of 0's, and then evolves by changing each 0 to 1, with the n changes done in random order. What is the maximal number of runs of 1's? We give asymptotic results for the distribution and mean. It turns out that, as in many problems involving a maximum, the maximum is asymptotically normal, with fluctuations of order $n^{1/2}$, and to the first order well approximated by the number of runs at the instance when the expectation is maximized, in this case when half the elements have changed to 1; there is also a second order term of order $n^{1/3}$. We also treat some variations, including priority queues and sock-sorting.]]> 0 <![CDATA[The Gaussian algorithm for lattice reduction in dimension 2 is precisely analysed under a class of realistic probabilistic models, which are of interest when applying the Gauss algorithm "inside'' the LLL algorithm. The proofs deal with the underlying dynamical systems and transfer operators. All the main parameters are studied: execution parameters which describe the behaviour of the algorithm itself as well as output parameters, which describe the geometry of reduced bases.]]> Sun, 31 Dec 2006 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3549 https://doi.org/10.46298/dmtcs.3549 Vallée, Brigitte Vera, Antonio Vallée, Brigitte Vera, Antonio <![CDATA[The Gaussian algorithm for lattice reduction in dimension 2 is precisely analysed under a class of realistic probabilistic models, which are of interest when applying the Gauss algorithm "inside'' the LLL algorithm. The proofs deal with the underlying dynamical systems and transfer operators. All the main parameters are studied: execution parameters which describe the behaviour of the algorithm itself as well as output parameters, which describe the geometry of reduced bases.]]> 0 <![CDATA[We study the Hamming distance from polynomials to classes of polynomials that share certain properties of irreducible polynomials. The results give insight into whether or not irreducible polynomials can be effectively modeled by these more general classes of polynomials. For example, we prove that the number of degree $n$ polynomials of Hamming distance one from a randomly chosen set of $\lfloor 2^n/n \rfloor$ odd density polynomials, each of degree $n$ and each with non-zero constant term, is asymptotically $(1-e^{-4}) 2^{n-2}$, and this appears to be inconsistent with the numbers for irreducible polynomials. We also conjecture that there is a constant $c$ such that every polynomial has Hamming distance at most $c$ from an irreducible polynomial. Using exhaustive lists of irreducible polynomials over $\mathbb{F}_2$ for degrees $1 ≤ n ≤ 32$, we count the number of polynomials with a given Hamming distance to some irreducible polynomial of the same degree. Our work is based on this "empirical" study.]]> Sun, 31 Dec 2006 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3550 https://doi.org/10.46298/dmtcs.3550 Lee, Gilbert Ruskey, Frank Williams, Aaron Lee, Gilbert Ruskey, Frank Williams, Aaron <![CDATA[We study the Hamming distance from polynomials to classes of polynomials that share certain properties of irreducible polynomials. The results give insight into whether or not irreducible polynomials can be effectively modeled by these more general classes of polynomials. For example, we prove that the number of degree $n$ polynomials of Hamming distance one from a randomly chosen set of $\lfloor 2^n/n \rfloor$ odd density polynomials, each of degree $n$ and each with non-zero constant term, is asymptotically $(1-e^{-4}) 2^{n-2}$, and this appears to be inconsistent with the numbers for irreducible polynomials. We also conjecture that there is a constant $c$ such that every polynomial has Hamming distance at most $c$ from an irreducible polynomial. Using exhaustive lists of irreducible polynomials over $\mathbb{F}_2$ for degrees $1 ≤ n ≤ 32$, we count the number of polynomials with a given Hamming distance to some irreducible polynomial of the same degree. Our work is based on this "empirical" study.]]> 0 <![CDATA[A two-parameter family of random permutations of $[n]$ is introduced, with distribution conditionally uniform given the counts of upper and lower records. The family interpolates between two versions of Ewens' distribution. A distinguished role of the family is determined by the fact that every sequence of coherent permutations $(π _n,n=1,2,\ldots)$ with the indicated kind of sufficiency is obtainable by randomisation of the parameters. Generating algorithms and asymptotic properties of the permutations follow from the representation via initial ranks.]]> Sun, 31 Dec 2006 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3551 https://doi.org/10.46298/dmtcs.3551 Gnedin, Alexander Gnedin, Alexander <![CDATA[A two-parameter family of random permutations of $[n]$ is introduced, with distribution conditionally uniform given the counts of upper and lower records. The family interpolates between two versions of Ewens' distribution. A distinguished role of the family is determined by the fact that every sequence of coherent permutations $(π _n,n=1,2,\ldots)$ with the indicated kind of sufficiency is obtainable by randomisation of the parameters. Generating algorithms and asymptotic properties of the permutations follow from the representation via initial ranks.]]> 0 <![CDATA[This work is devoted to the understanding of properties of random graphs from graph classes with structural constraints. We propose a method that is based on the analysis of the behaviour of Boltzmann sampler algorithms, and may be used to obtain precise estimates for the maximum degree and maximum size of a biconnected block of a "typical'' member of the class in question. We illustrate how our method works on several graph classes, namely dissections and triangulations of convex polygons, embedded trees, and block and cactus graphs.]]> Sun, 31 Dec 2006 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3552 https://doi.org/10.46298/dmtcs.3552 Panagiotou, Konstantinos Weißl, Andreas Panagiotou, Konstantinos Weißl, Andreas <![CDATA[This work is devoted to the understanding of properties of random graphs from graph classes with structural constraints. We propose a method that is based on the analysis of the behaviour of Boltzmann sampler algorithms, and may be used to obtain precise estimates for the maximum degree and maximum size of a biconnected block of a "typical'' member of the class in question. We illustrate how our method works on several graph classes, namely dissections and triangulations of convex polygons, embedded trees, and block and cactus graphs.]]> 0 <![CDATA[We consider words with letters from a $q-ary$ alphabet $\mathcal{A}$. The kth subword complexity of a word $w ∈\mathcal{A}^*$ is the number of distinct subwords of length $k$ that appear as contiguous subwords of $w$. We analyze subword complexity from both combinatorial and probabilistic viewpoints. Our first main result is a precise analysis of the expected $kth$ subword complexity of a randomly-chosen word $w ∈\mathcal{A}^n$. Our other main result describes, for $w ∈\mathcal{A}^*$, the degree to which one understands the set of all subwords of $w$, provided that one knows only the set of all subwords of some particular length $k$. Our methods rely upon a precise characterization of overlaps between words of length $k$. We use three kinds of correlation polynomials of words of length $k$: unweighted correlation polynomials; correlation polynomials associated to a Bernoulli source; and generalized multivariate correlation polynomials. We survey previously-known results about such polynomials, and we also present some new results concerning correlation polynomials.]]> Sun, 31 Dec 2006 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3553 https://doi.org/10.46298/dmtcs.3553 Gheorghiciuc, Irina Ward, Mark Daniel Gheorghiciuc, Irina Ward, Mark Daniel <![CDATA[We consider words with letters from a $q-ary$ alphabet $\mathcal{A}$. The kth subword complexity of a word $w ∈\mathcal{A}^*$ is the number of distinct subwords of length $k$ that appear as contiguous subwords of $w$. We analyze subword complexity from both combinatorial and probabilistic viewpoints. Our first main result is a precise analysis of the expected $kth$ subword complexity of a randomly-chosen word $w ∈\mathcal{A}^n$. Our other main result describes, for $w ∈\mathcal{A}^*$, the degree to which one understands the set of all subwords of $w$, provided that one knows only the set of all subwords of some particular length $k$. Our methods rely upon a precise characterization of overlaps between words of length $k$. We use three kinds of correlation polynomials of words of length $k$: unweighted correlation polynomials; correlation polynomials associated to a Bernoulli source; and generalized multivariate correlation polynomials. We survey previously-known results about such polynomials, and we also present some new results concerning correlation polynomials.]]> 0 <![CDATA[We study the problem of efficiently sampling k-colorings of bipartite graphs. We show that a class of markov chains cannot be used as efficient samplers. Precisely, we show that, for any k, 6 ≤ k ≤ n^\1/3-ε \, ε > 0 fixed, \emphalmost every bipartite graph on n+n vertices is such that the mixing time of any markov chain asymptotically uniform on its k-colorings is exponential in n/k^2 (if it is allowed to only change the colors of O(n/k) vertices in a single transition step). This kind of exponential time mixing is called \emphtorpid mixing. As a corollary, we show that there are (for every n) bipartite graphs on 2n vertices with Δ (G) = Ω (\ln n) such that for every k, 6 ≤ k ≤ Δ /(6 \ln Δ ), each member of a large class of chains mixes torpidly. While, for fixed k, such negative results are implied by the work of CDF, our results are more general in that they allow k to grow with n. We also show that these negative results hold true for H-colorings of bipartite graphs provided H contains a spanning complete bipartite subgraph. We also present explicit examples of colorings (k-colorings or H-colorings) which admit 1-cautious chains that are ergodic and are shown to have exponential mixing time. While, for fixed k or fixed H, such negative results are implied by the work of CDF, our results are more general in that they allow k or H to vary with n.]]> Sun, 01 Jan 2006 07:00:00 +0000 https://doi.org/10.46298/dmtcs.368 https://doi.org/10.46298/dmtcs.368 Balasubramanian, R. Subramanian, C.R. Balasubramanian, R. Subramanian, C.R. <![CDATA[We study the problem of efficiently sampling k-colorings of bipartite graphs. We show that a class of markov chains cannot be used as efficient samplers. Precisely, we show that, for any k, 6 ≤ k ≤ n^\1/3-ε \, ε > 0 fixed, \emphalmost every bipartite graph on n+n vertices is such that the mixing time of any markov chain asymptotically uniform on its k-colorings is exponential in n/k^2 (if it is allowed to only change the colors of O(n/k) vertices in a single transition step). This kind of exponential time mixing is called \emphtorpid mixing. As a corollary, we show that there are (for every n) bipartite graphs on 2n vertices with Δ (G) = Ω (\ln n) such that for every k, 6 ≤ k ≤ Δ /(6 \ln Δ ), each member of a large class of chains mixes torpidly. While, for fixed k, such negative results are implied by the work of CDF, our results are more general in that they allow k to grow with n. We also show that these negative results hold true for H-colorings of bipartite graphs provided H contains a spanning complete bipartite subgraph. We also present explicit examples of colorings (k-colorings or H-colorings) which admit 1-cautious chains that are ergodic and are shown to have exponential mixing time. While, for fixed k or fixed H, such negative results are implied by the work of CDF, our results are more general in that they allow k or H to vary with n.]]> 0 <![CDATA[Let n, s be positive integers such that 2 ≤ s < n and s = n/2 . An undirected double-loop network G(n; 1, s) is an undirected graph (V,E), where V =Zn={0, 1, 2, . . . , n−1} and E={(i, i+1 (mod n)), (i, i+s (mod n)) | i ∈ Z}. It is a circulant graph with n nodes and degree 4. In this paper, the sufficient and necessary conditions for a class of undirected double-loop networks to be optimal are presented. By these conditions, 6 new optimal and 5 new suboptimal infinite families of undirected double-loop networks are given.]]> Sun, 01 Jan 2006 07:00:00 +0000 https://doi.org/10.46298/dmtcs.377 https://doi.org/10.46298/dmtcs.377 Chen, Bao-Xing Meng, Ji-Xiang Xiao, Wen-Jun Chen, Bao-Xing Meng, Ji-Xiang Xiao, Wen-Jun <![CDATA[Let n, s be positive integers such that 2 ≤ s < n and s = n/2 . An undirected double-loop network G(n; 1, s) is an undirected graph (V,E), where V =Zn={0, 1, 2, . . . , n−1} and E={(i, i+1 (mod n)), (i, i+s (mod n)) | i ∈ Z}. It is a circulant graph with n nodes and degree 4. In this paper, the sufficient and necessary conditions for a class of undirected double-loop networks to be optimal are presented. By these conditions, 6 new optimal and 5 new suboptimal infinite families of undirected double-loop networks are given.]]> 0 <![CDATA[Let ℕ be the set of all positive integers and let A be any ordered subset of ℕ. Recently, Heubach and Mansour enumerated the number of compositions of n with m parts in A that contain the subword τ exactly r times, where τ∈{111,112,221,123}. Our aims are (1) to generalize the above results, i.e., to enumerate the number of compositions of n with m parts in A that contain an ℓ-letter subword, and (2) to analyze the number of compositions of n with m parts that avoid an ℓ-letter pattern, for given ℓ. We use tools such as asymptotic analysis of generating functions leading to Gaussian asymptotic.]]> Sun, 01 Jan 2006 07:00:00 +0000 https://doi.org/10.46298/dmtcs.376 https://doi.org/10.46298/dmtcs.376 Mansour, Toufik Sirhan, Basel O. Mansour, Toufik Sirhan, Basel O. <![CDATA[Let ℕ be the set of all positive integers and let A be any ordered subset of ℕ. Recently, Heubach and Mansour enumerated the number of compositions of n with m parts in A that contain the subword τ exactly r times, where τ∈{111,112,221,123}. Our aims are (1) to generalize the above results, i.e., to enumerate the number of compositions of n with m parts in A that contain an ℓ-letter subword, and (2) to analyze the number of compositions of n with m parts that avoid an ℓ-letter pattern, for given ℓ. We use tools such as asymptotic analysis of generating functions leading to Gaussian asymptotic.]]> 0 <![CDATA[We study d-records in sequences generated by independent geometric random variables and derive explicit and asymptotic formulæ for expectation and variance. Informally speaking, a d-record occurs, when one computes the d-largest values, and the variable maintaining it changes its value while the sequence is scanned from left to right. This is done for the "strict model," but a "weak model" is also briefly investigated. We also discuss the limit q → 1 (q the parameter of the geometric distribution), which leads to the model of random permutations.]]> Sun, 01 Jan 2006 07:00:00 +0000 https://doi.org/10.46298/dmtcs.375 https://doi.org/10.46298/dmtcs.375 Prodinger, Helmut Prodinger, Helmut <![CDATA[We study d-records in sequences generated by independent geometric random variables and derive explicit and asymptotic formulæ for expectation and variance. Informally speaking, a d-record occurs, when one computes the d-largest values, and the variable maintaining it changes its value while the sequence is scanned from left to right. This is done for the "strict model," but a "weak model" is also briefly investigated. We also discuss the limit q → 1 (q the parameter of the geometric distribution), which leads to the model of random permutations.]]> 0 <![CDATA[We introduce a new class of morphisms for event structures. The category obtained is cartesian closed, and a natural notion of quotient event structure is defined within it. We study in particular the topological space of maximal configurations of quotient event structures. We introduce the compression of event structures as an example of quotient: the compression of an event structure E is a minimal event structure with the same space of maximal configurations as E.]]> Sun, 01 Jan 2006 07:00:00 +0000 https://doi.org/10.46298/dmtcs.374 https://doi.org/10.46298/dmtcs.374 Abbes, Samy Abbes, Samy <![CDATA[We introduce a new class of morphisms for event structures. The category obtained is cartesian closed, and a natural notion of quotient event structure is defined within it. We study in particular the topological space of maximal configurations of quotient event structures. We introduce the compression of event structures as an example of quotient: the compression of an event structure E is a minimal event structure with the same space of maximal configurations as E.]]> 0 <![CDATA[As a generalization of connected domination in a graph G we consider domination by sets having at most k components. The order γ _c^k (G) of such a smallest set we relate to γ _c(G), the order of a smallest connected dominating set. For a tree T we give bounds on γ _c^k (T) in terms of minimum valency and diameter. For trees the inequality γ _c^k (T)≤ n-k-1 is known to hold, we determine the class of trees, for which equality holds.]]> Sun, 01 Jan 2006 07:00:00 +0000 https://doi.org/10.46298/dmtcs.373 https://doi.org/10.46298/dmtcs.373 Kouider, Mekkia Vestergaard, Preben Dahl Kouider, Mekkia Vestergaard, Preben Dahl <![CDATA[As a generalization of connected domination in a graph G we consider domination by sets having at most k components. The order γ _c^k (G) of such a smallest set we relate to γ _c(G), the order of a smallest connected dominating set. For a tree T we give bounds on γ _c^k (T) in terms of minimum valency and diameter. For trees the inequality γ _c^k (T)≤ n-k-1 is known to hold, we determine the class of trees, for which equality holds.]]> 0 <![CDATA[The Maximum Weight Stable Set (MWS) Problem is one of the fundamental problems on graphs. It is well-known to be NP-complete for triangle-free graphs, and Mosca has shown that it is solvable in polynomial time when restricted to P6- and triangle-free graphs. We give a complete structure analysis of (nonbipartite) P6- and triangle-free graphs which are prime in the sense of modular decomposition. It turns out that the structure of these graphs is extremely simple implying bounded clique-width and thus, efficient algorithms exist for all problems expressible in terms of Monadic Second Order Logic with quantification only over vertex predicates. The problems Vertex Cover, MWS, Maximum Clique, Minimum Dominating Set, Steiner Tree, and Maximum Induced Matching are among them. Our results improve the previous one on the MWS problem by Mosca with respect to structure and time bound but also extends a previous result by Fouquet, Giakoumakis and Vanherpe which have shown that bipartite P6-free graphs have bounded clique-width. Moreover, it covers a result by Randerath, Schiermeyer and Tewes on polynomial time 3-colorability of P6- and triangle-free graphs.]]> Sun, 01 Jan 2006 07:00:00 +0000 https://doi.org/10.46298/dmtcs.372 https://doi.org/10.46298/dmtcs.372 Brandstädt, Andreas Klembt, Tilo Mahfud, Suhail Brandstädt, Andreas Klembt, Tilo Mahfud, Suhail <![CDATA[The Maximum Weight Stable Set (MWS) Problem is one of the fundamental problems on graphs. It is well-known to be NP-complete for triangle-free graphs, and Mosca has shown that it is solvable in polynomial time when restricted to P6- and triangle-free graphs. We give a complete structure analysis of (nonbipartite) P6- and triangle-free graphs which are prime in the sense of modular decomposition. It turns out that the structure of these graphs is extremely simple implying bounded clique-width and thus, efficient algorithms exist for all problems expressible in terms of Monadic Second Order Logic with quantification only over vertex predicates. The problems Vertex Cover, MWS, Maximum Clique, Minimum Dominating Set, Steiner Tree, and Maximum Induced Matching are among them. Our results improve the previous one on the MWS problem by Mosca with respect to structure and time bound but also extends a previous result by Fouquet, Giakoumakis and Vanherpe which have shown that bipartite P6-free graphs have bounded clique-width. Moreover, it covers a result by Randerath, Schiermeyer and Tewes on polynomial time 3-colorability of P6- and triangle-free graphs.]]> 0 <![CDATA[A multicoloring of a weighted graph G is an assignment of sets of colors to the vertices of G so that two adjacent vertices receive two disjoint sets of colors. A multicoloring problem on G is to find a multicoloring of G. In particular, we are interested in a minimum multicoloring that uses the least total number of colors. The main focus of this work is to obtain upper bounds on the weighted chromatic number of some classes of graphs in terms of the weighted clique number. We first propose an 11/6-approximation algorithm for multicoloring any weighted planar graph. We then study the multicoloring problem on powers of square and triangular meshes. Among other results, we show that the infinite triangular mesh is an induced subgraph of the fourth power of the infinite square mesh and we present 2-approximation algorithms for multicoloring a power square mesh and the second power of a triangular mesh, 3-approximation algorithms for multicoloring powers of semi-toroidal meshes and of triangular meshes and 4-approximation algorithm for multicoloring the power of a toroidal mesh. We also give similar algorithms for the Cartesian product of powers of paths and of cycles.]]> Sun, 01 Jan 2006 07:00:00 +0000 https://doi.org/10.46298/dmtcs.371 https://doi.org/10.46298/dmtcs.371 Kchikech, Mustapha Togni, Olivier Kchikech, Mustapha Togni, Olivier <![CDATA[A multicoloring of a weighted graph G is an assignment of sets of colors to the vertices of G so that two adjacent vertices receive two disjoint sets of colors. A multicoloring problem on G is to find a multicoloring of G. In particular, we are interested in a minimum multicoloring that uses the least total number of colors. The main focus of this work is to obtain upper bounds on the weighted chromatic number of some classes of graphs in terms of the weighted clique number. We first propose an 11/6-approximation algorithm for multicoloring any weighted planar graph. We then study the multicoloring problem on powers of square and triangular meshes. Among other results, we show that the infinite triangular mesh is an induced subgraph of the fourth power of the infinite square mesh and we present 2-approximation algorithms for multicoloring a power square mesh and the second power of a triangular mesh, 3-approximation algorithms for multicoloring powers of semi-toroidal meshes and of triangular meshes and 4-approximation algorithm for multicoloring the power of a toroidal mesh. We also give similar algorithms for the Cartesian product of powers of paths and of cycles.]]> 0 <![CDATA[In this paper, we consider the recognition problem on three classes of perfectly orderable graphs, namely, the HH-free, the HHD-free, and the Welsh-Powell opposition graphs (or WPO-graphs). In particular, we prove properties of the chordal completion of a graph and show that a modified version of the classic linear-time algorithm for testing for a perfect elimination ordering can be efficiently used to determine in O(n min \m α (n,n), m + n^2 log n\) time whether a given graph G on n vertices and m edges contains a house or a hole; this implies an O(n min \m α (n,n), m + n^2 log n\)-time and O(n+m)-space algorithm for recognizing HH-free graphs, and in turn leads to an HHD-free graph recognition algorithm exhibiting the same time and space complexity. We also show that determining whether the complement øverlineG of the graph G is HH-free can be efficiently resolved in O(n m) time using O(n^2) space, which leads to an O(n m)-time and O(n^2)-space algorithm for recognizing WPO-graphs. The previously best algorithms for recognizing HH-free, HHD-free, and WPO-graphs required O(n^3) time and O(n^2) space.]]> Sun, 01 Jan 2006 07:00:00 +0000 https://doi.org/10.46298/dmtcs.370 https://doi.org/10.46298/dmtcs.370 Nikolopoulos, Stavros D. Palios, Leonidas Nikolopoulos, Stavros D. Palios, Leonidas <![CDATA[In this paper, we consider the recognition problem on three classes of perfectly orderable graphs, namely, the HH-free, the HHD-free, and the Welsh-Powell opposition graphs (or WPO-graphs). In particular, we prove properties of the chordal completion of a graph and show that a modified version of the classic linear-time algorithm for testing for a perfect elimination ordering can be efficiently used to determine in O(n min \m α (n,n), m + n^2 log n\) time whether a given graph G on n vertices and m edges contains a house or a hole; this implies an O(n min \m α (n,n), m + n^2 log n\)-time and O(n+m)-space algorithm for recognizing HH-free graphs, and in turn leads to an HHD-free graph recognition algorithm exhibiting the same time and space complexity. We also show that determining whether the complement øverlineG of the graph G is HH-free can be efficiently resolved in O(n m) time using O(n^2) space, which leads to an O(n m)-time and O(n^2)-space algorithm for recognizing WPO-graphs. The previously best algorithms for recognizing HH-free, HHD-free, and WPO-graphs required O(n^3) time and O(n^2) space.]]> 0 <![CDATA[We survey multivariate limit theorems in the framework of the contraction method for recursive sequences as arising in the analysis of algorithms, random trees or branching processes. We compare and improve various general conditions under which limit laws can be obtained, state related open problems and give applications to the analysis of algorithms and branching recurrences.]]> Sun, 01 Jan 2006 07:00:00 +0000 https://doi.org/10.46298/dmtcs.369 https://doi.org/10.46298/dmtcs.369 Neininger, Ralph Rüschendorf, Ludger Neininger, Ralph Rüschendorf, Ludger <![CDATA[We survey multivariate limit theorems in the framework of the contraction method for recursive sequences as arising in the analysis of algorithms, random trees or branching processes. We compare and improve various general conditions under which limit laws can be obtained, state related open problems and give applications to the analysis of algorithms and branching recurrences.]]> 0 <![CDATA[In this paper we study a semi on-line version of the classical multiprocessor scheduling problem on two identical processors. We assume that the sum of the tasks and an upper bound gamma on the size of each task are known. Each task has to be assigned upon arrival and the assignment cannot be changed later. The objective is the minimization of the maximum completion time on the processors. In this paper we propose new algorithms and improve known lower and upper bounds on the competitive ratio. Algorithms and bounds depend on the value of gamma. An optimal algorithm is obtained for gamma in the interval [ 1/n,2(n+1)/n(2n+1) ] and gamma = (2n-1)/2n(n-1), where n is any integer value larger or equal 2.]]> Sun, 01 Jan 2006 07:00:00 +0000 https://doi.org/10.46298/dmtcs.367 https://doi.org/10.46298/dmtcs.367 Angelelli, Enrico Speranza, Maria Grazia Tuza, Zsolt Angelelli, Enrico Speranza, Maria Grazia Tuza, Zsolt <![CDATA[In this paper we study a semi on-line version of the classical multiprocessor scheduling problem on two identical processors. We assume that the sum of the tasks and an upper bound gamma on the size of each task are known. Each task has to be assigned upon arrival and the assignment cannot be changed later. The objective is the minimization of the maximum completion time on the processors. In this paper we propose new algorithms and improve known lower and upper bounds on the competitive ratio. Algorithms and bounds depend on the value of gamma. An optimal algorithm is obtained for gamma in the interval [ 1/n,2(n+1)/n(2n+1) ] and gamma = (2n-1)/2n(n-1), where n is any integer value larger or equal 2.]]> 0 <![CDATA[Let A be a finite subset of ℤ2. We say A tiles ℤ2 with the translation set C, if any integer z∈ℤ2 can be represented as z1+z2, z1∈ A, z2∈ C in an unique way. In this case we call A a ℤ2-tile and write A ⊕ C = ℤ2. A tile A is said to be a normal ℤ2-tile if there exists a periodic set C such that A ⊕ C = ℤ2. We characterize all normal ℤ2-tiles with prime cardinality.]]> Sun, 01 Jan 2006 07:00:00 +0000 https://doi.org/10.46298/dmtcs.366 https://doi.org/10.46298/dmtcs.366 Rao, Hui Xue, Yu-Mei Rao, Hui Xue, Yu-Mei <![CDATA[Let A be a finite subset of ℤ2. We say A tiles ℤ2 with the translation set C, if any integer z∈ℤ2 can be represented as z1+z2, z1∈ A, z2∈ C in an unique way. In this case we call A a ℤ2-tile and write A ⊕ C = ℤ2. A tile A is said to be a normal ℤ2-tile if there exists a periodic set C such that A ⊕ C = ℤ2. We characterize all normal ℤ2-tiles with prime cardinality.]]> 0 <![CDATA[We prove that the topological entropy of subshifts having decidable language is uncomputable in the following sense: For no error bound less than 1/4 does there exists a program that, given a decision procedure for the language of a subshift as input, will approximate the entropy of the subshift within the error bound. In addition, we prove that not only is the topological entropy of sofic shifts computable to arbitary precision (a well-known fact), but all standard comparisons of the topological entropy with rational numbers are decidable.]]> Sun, 01 Jan 2006 07:00:00 +0000 https://doi.org/10.46298/dmtcs.365 https://doi.org/10.46298/dmtcs.365 Simonsen, Jakob Grue Simonsen, Jakob Grue <![CDATA[We prove that the topological entropy of subshifts having decidable language is uncomputable in the following sense: For no error bound less than 1/4 does there exists a program that, given a decision procedure for the language of a subshift as input, will approximate the entropy of the subshift within the error bound. In addition, we prove that not only is the topological entropy of sofic shifts computable to arbitary precision (a well-known fact), but all standard comparisons of the topological entropy with rational numbers are decidable.]]> 0 <![CDATA[The K_n-complement of a graph G, denoted by K_n-G, is defined as the graph obtained from the complete graph K_n by removing a set of edges that span G; if G has n vertices, then K_n-G coincides with the complement øverlineG of the graph G. In this paper we extend the previous notion and derive determinant based formulas for the number of spanning trees of graphs of the form K_n^m #x00b1 G, where K_n^m is the complete multigraph on n vertices with exactly m edges joining every pair of vertices and G is a multigraph spanned by a set of edges of K_n^m; the graph K_n^m + G (resp. K_n^m - G) is obtained from K_n^m by adding (resp. removing) the edges of G. Moreover, we derive determinant based formulas for graphs that result from K_n^m by adding and removing edges of multigraphs spanned by sets of edges of the graph K_n^m. We also prove closed formulas for the number of spanning tree of graphs of the form K_n^m #x00b1 G, where G is (i) a complete multipartite graph, and (ii) a multi-star graph. Our results generalize previous results and extend the family of graphs admitting formulas for the number of their spanning trees.]]> Sun, 01 Jan 2006 07:00:00 +0000 https://doi.org/10.46298/dmtcs.364 https://doi.org/10.46298/dmtcs.364 Nikolopoulos, Stavros D. Papadopoulos, Charis Nikolopoulos, Stavros D. Papadopoulos, Charis <![CDATA[The K_n-complement of a graph G, denoted by K_n-G, is defined as the graph obtained from the complete graph K_n by removing a set of edges that span G; if G has n vertices, then K_n-G coincides with the complement øverlineG of the graph G. In this paper we extend the previous notion and derive determinant based formulas for the number of spanning trees of graphs of the form K_n^m #x00b1 G, where K_n^m is the complete multigraph on n vertices with exactly m edges joining every pair of vertices and G is a multigraph spanned by a set of edges of K_n^m; the graph K_n^m + G (resp. K_n^m - G) is obtained from K_n^m by adding (resp. removing) the edges of G. Moreover, we derive determinant based formulas for graphs that result from K_n^m by adding and removing edges of multigraphs spanned by sets of edges of the graph K_n^m. We also prove closed formulas for the number of spanning tree of graphs of the form K_n^m #x00b1 G, where G is (i) a complete multipartite graph, and (ii) a multi-star graph. Our results generalize previous results and extend the family of graphs admitting formulas for the number of their spanning trees.]]> 0 <![CDATA[For words of length n, generated by independent geometric random variables, we study the average initial and end heights of the first descent in the word. In addition we compute the average initial and end height of the first descent for a random permutation of n letters.]]> Sun, 01 Jan 2006 07:00:00 +0000 https://doi.org/10.46298/dmtcs.363 https://doi.org/10.46298/dmtcs.363 Knopfmacher, Arnold Prodinger, Helmut Knopfmacher, Arnold Prodinger, Helmut <![CDATA[For words of length n, generated by independent geometric random variables, we study the average initial and end heights of the first descent in the word. In addition we compute the average initial and end height of the first descent for a random permutation of n letters.]]> 0 <![CDATA[An interval of a permutation is a consecutive substring consisting of consecutive symbols. For example, 4536 is an interval in the permutation 71453682. These arise in genetic applications. For the applications, it makes sense to generalize so as to allow gaps of bounded size δ -1, both in the locations and the symbols. For example, 4527 has gaps bounded by 1 (since 3 and 6 are missing) and is therefore a δ -interval of 389415627 for δ =2. After analyzing the distribution of the number of intervals of a uniform random permutation, we study the number of 2-intervals. This is exponentially large, but tightly clustered around its mean. Perhaps surprisingly, the quenched and annealed means are the same. Our analysis is via a multivariate generating function enumerating pairs of potential 2-intervals by size and intersection size.\par]]> Sun, 01 Jan 2006 07:00:00 +0000 https://doi.org/10.46298/dmtcs.362 https://doi.org/10.46298/dmtcs.362 Corteel, Sylvie Louchard, Guy Pemantle, Robin Corteel, Sylvie Louchard, Guy Pemantle, Robin <![CDATA[An interval of a permutation is a consecutive substring consisting of consecutive symbols. For example, 4536 is an interval in the permutation 71453682. These arise in genetic applications. For the applications, it makes sense to generalize so as to allow gaps of bounded size δ -1, both in the locations and the symbols. For example, 4527 has gaps bounded by 1 (since 3 and 6 are missing) and is therefore a δ -interval of 389415627 for δ =2. After analyzing the distribution of the number of intervals of a uniform random permutation, we study the number of 2-intervals. This is exponentially large, but tightly clustered around its mean. Perhaps surprisingly, the quenched and annealed means are the same. Our analysis is via a multivariate generating function enumerating pairs of potential 2-intervals by size and intersection size.\par]]> 0 <![CDATA[The L(h, k)-labeling is an assignment of non negative integer labels to the nodes of a graph such that 'close' nodes have labels which differ by at least k, and 'very close' nodes have labels which differ by at least h. The span of an L(h,k)-labeling is the difference between the largest and the smallest assigned label. We study L(h, k)-labelings of cellular, squared and hexagonal grids, seeking those with minimum span for each value of k and h ≥ k. The L(h,k)-labeling problem has been intensively studied in some special cases, i.e. when k=0 (vertex coloring), h=k (vertex coloring the square of the graph) and h=2k (radio- or λ -coloring) but no results are known in the general case for regular grids. In this paper, we completely solve the L(h,k)-labeling problem on regular grids, finding exact values of the span for each value of h and k; only in a small interval we provide different upper and lower bounds.]]> Sun, 01 Jan 2006 07:00:00 +0000 https://doi.org/10.46298/dmtcs.361 https://doi.org/10.46298/dmtcs.361 Calamoneri, Tiziana Calamoneri, Tiziana <![CDATA[The L(h, k)-labeling is an assignment of non negative integer labels to the nodes of a graph such that 'close' nodes have labels which differ by at least k, and 'very close' nodes have labels which differ by at least h. The span of an L(h,k)-labeling is the difference between the largest and the smallest assigned label. We study L(h, k)-labelings of cellular, squared and hexagonal grids, seeking those with minimum span for each value of k and h ≥ k. The L(h,k)-labeling problem has been intensively studied in some special cases, i.e. when k=0 (vertex coloring), h=k (vertex coloring the square of the graph) and h=2k (radio- or λ -coloring) but no results are known in the general case for regular grids. In this paper, we completely solve the L(h,k)-labeling problem on regular grids, finding exact values of the span for each value of h and k; only in a small interval we provide different upper and lower bounds.]]> 0 <![CDATA[We consider the problem of generating a random q-colouring of a graph G=(V,E). We consider the simple Glauber Dynamics chain. We show that if for all v ∈ V the average degree of the subgraph H_v induced by the neighbours of v ∈ V is #x226a Δ where Δ is the maximum degree and Δ >c_1\ln n then for sufficiently large c_1, this chain mixes rapidly provided q/Δ >α , where α #x2248 1.763 is the root of α = e^\1/α \. For this class of graphs, which includes planar graphs, triangle free graphs and random graphs G_\n,p\ with p #x226a 1, this beats the 11Δ /6 bound of Vigoda for general graphs.]]> Sun, 01 Jan 2006 07:00:00 +0000 https://doi.org/10.46298/dmtcs.360 https://doi.org/10.46298/dmtcs.360 Frieze, Alan Vera, Juan Frieze, Alan Vera, Juan <![CDATA[We consider the problem of generating a random q-colouring of a graph G=(V,E). We consider the simple Glauber Dynamics chain. We show that if for all v ∈ V the average degree of the subgraph H_v induced by the neighbours of v ∈ V is #x226a Δ where Δ is the maximum degree and Δ >c_1\ln n then for sufficiently large c_1, this chain mixes rapidly provided q/Δ >α , where α #x2248 1.763 is the root of α = e^\1/α \. For this class of graphs, which includes planar graphs, triangle free graphs and random graphs G_\n,p\ with p #x226a 1, this beats the 11Δ /6 bound of Vigoda for general graphs.]]> 0 <![CDATA[We study the online specialization problem, where items arrive in an online fashion for processing by one of n different methods. Each method has two costs: a processing cost (paid once for each item processed), and a set-up cost (paid only once, on the method's first use). There are n possible types of items; an item's type determines the set of methods available to process it. Each method has a different degree of specialization. Highly specialized methods can process few item types while generic methods may process all item types. This is a generalization of ski-rental and closely related to the capital investment problem of Y. Azar, Y. Bartal, E. Feuerstein, A. Fiat, S. Leonardi, and A. Rosen. On capital investment. In Algorithmica, 25(1):22-36, 1999.. We primarily study the case where method i+1 is always more specialized than method i and the set-up cost for a more specialized method is always higher than that of a less specialized method. We describe an algorithm with competitive ratio O(log(n)), and also show an Ω (log(n)) lower bound on the competitive ratio for this problem; this shows our ratio is tight up to constant factors.]]> Sun, 01 Jan 2006 07:00:00 +0000 https://doi.org/10.46298/dmtcs.359 https://doi.org/10.46298/dmtcs.359 Hong, Edwin S. Hong, Edwin S. <![CDATA[We study the online specialization problem, where items arrive in an online fashion for processing by one of n different methods. Each method has two costs: a processing cost (paid once for each item processed), and a set-up cost (paid only once, on the method's first use). There are n possible types of items; an item's type determines the set of methods available to process it. Each method has a different degree of specialization. Highly specialized methods can process few item types while generic methods may process all item types. This is a generalization of ski-rental and closely related to the capital investment problem of Y. Azar, Y. Bartal, E. Feuerstein, A. Fiat, S. Leonardi, and A. Rosen. On capital investment. In Algorithmica, 25(1):22-36, 1999.. We primarily study the case where method i+1 is always more specialized than method i and the set-up cost for a more specialized method is always higher than that of a less specialized method. We describe an algorithm with competitive ratio O(log(n)), and also show an Ω (log(n)) lower bound on the competitive ratio for this problem; this shows our ratio is tight up to constant factors.]]> 0 <![CDATA[For a skip list variant, introduced by Cho and Sahni, we analyse what is the analogue of horizontal plus vertical search cost in the original skip list model. While the average in Pugh's original version behaves like $Q \log_Q n$, with $Q = \frac{1}{q}$ a parameter, it is here given by $(Q+1) \log_Q n$.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3476 https://doi.org/10.46298/dmtcs.3476 Louchard, Guy Prodinger, Helmut Louchard, Guy Prodinger, Helmut <![CDATA[For a skip list variant, introduced by Cho and Sahni, we analyse what is the analogue of horizontal plus vertical search cost in the original skip list model. While the average in Pugh's original version behaves like $Q \log_Q n$, with $Q = \frac{1}{q}$ a parameter, it is here given by $(Q+1) \log_Q n$.]]> 0 <![CDATA[We study a multi-type branching process in i.i.d. random environment. Assuming that the associated random walk satisfies the Doney-Spitzer condition, we find the asymptotics of the survival probability at time $n$ as $n \to \infty$.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3477 https://doi.org/10.46298/dmtcs.3477 Dyakonova, Elena Dyakonova, Elena <![CDATA[We study a multi-type branching process in i.i.d. random environment. Assuming that the associated random walk satisfies the Doney-Spitzer condition, we find the asymptotics of the survival probability at time $n$ as $n \to \infty$.]]> 0 <![CDATA[In this paper we consider the class of $\textit{permutominoes}$, i.e. a special class of polyominoes which are determined by a pair of permutations having the same size. We give a characterization of the permutations associated with convex permutominoes, and then we enumerate various classes of convex permutominoes, including parallelogram, directed-convex, and stack ones.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3478 https://doi.org/10.46298/dmtcs.3478 Fanti, I. Frosini, A. Grazzini, E. Pinzani, R. Rinaldi, S. Fanti, I. Frosini, A. Grazzini, E. Pinzani, R. Rinaldi, S. <![CDATA[In this paper we consider the class of $\textit{permutominoes}$, i.e. a special class of polyominoes which are determined by a pair of permutations having the same size. We give a characterization of the permutations associated with convex permutominoes, and then we enumerate various classes of convex permutominoes, including parallelogram, directed-convex, and stack ones.]]> 0 <![CDATA[For a class of random partitions of an infinite set a de Finetti-type representation is derived, and in one special case a central limit theorem for the number of blocks is shown.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3479 https://doi.org/10.46298/dmtcs.3479 Gnedin, Alexander Gnedin, Alexander <![CDATA[For a class of random partitions of an infinite set a de Finetti-type representation is derived, and in one special case a central limit theorem for the number of blocks is shown.]]> 0 <![CDATA[Let $S$ be a set of $d$-dimensional row vectors with entries in a $q$-ary alphabet. A matrix $M$ with entries in the same $q$-ary alphabet is $S$-constrained if every set of $d$ columns of $M$ contains as a submatrix a copy of the vectors in $S$, up to permutation. For a given set $S$ of $d$-dimensional vectors, we compute the asymptotic probability for a random matrix $M$ to be $S$-constrained, as the numbers of rows and columns both tend to infinity. If $n$ is the number of columns and $m=m_n$ the number of rows, then the threshold is at $m_n= \alpha_d \log (n)$, where $\alpha_d$ only depends on the dimension $d$ of vectors and not on the particular set $S$. Applications to superimposed codes, shattering classes of functions, and Sidon families of sets are proposed. For $d=2$, an explicit construction of a $S$-constrained matrix is given.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3480 https://doi.org/10.46298/dmtcs.3480 Gravier, Sylvain Ycart, Bernard Gravier, Sylvain Ycart, Bernard <![CDATA[Let $S$ be a set of $d$-dimensional row vectors with entries in a $q$-ary alphabet. A matrix $M$ with entries in the same $q$-ary alphabet is $S$-constrained if every set of $d$ columns of $M$ contains as a submatrix a copy of the vectors in $S$, up to permutation. For a given set $S$ of $d$-dimensional vectors, we compute the asymptotic probability for a random matrix $M$ to be $S$-constrained, as the numbers of rows and columns both tend to infinity. If $n$ is the number of columns and $m=m_n$ the number of rows, then the threshold is at $m_n= \alpha_d \log (n)$, where $\alpha_d$ only depends on the dimension $d$ of vectors and not on the particular set $S$. Applications to superimposed codes, shattering classes of functions, and Sidon families of sets are proposed. For $d=2$, an explicit construction of a $S$-constrained matrix is given.]]> 0 <![CDATA[This paper tackles the enumeration and asymptotics of the area below directed lattice paths (walks on $\mathbb{N}$ with a finite set of jumps). It is a nice surprise (obtained via the "kernel method'') that the generating functions of the moments of the area are algebraic functions, expressible as symmetric functions in terms of the roots of the kernel. For a large class of walks, we give full asymptotics for the average area of excursions ("discrete'' reflected Brownian bridge) and meanders ("discrete'' reflected Brownian motion). We show that drift is not playing any role in the first case. We also generalise previous works related to the number of points below a path and to the area between a path and a line of rational slope.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3481 https://doi.org/10.46298/dmtcs.3481 Banderier, Cyril Gittenberger, Bernhard Banderier, Cyril Gittenberger, Bernhard <![CDATA[This paper tackles the enumeration and asymptotics of the area below directed lattice paths (walks on $\mathbb{N}$ with a finite set of jumps). It is a nice surprise (obtained via the "kernel method'') that the generating functions of the moments of the area are algebraic functions, expressible as symmetric functions in terms of the roots of the kernel. For a large class of walks, we give full asymptotics for the average area of excursions ("discrete'' reflected Brownian bridge) and meanders ("discrete'' reflected Brownian motion). We show that drift is not playing any role in the first case. We also generalise previous works related to the number of points below a path and to the area between a path and a line of rational slope.]]> 0 <![CDATA[Grown simple families of increasing trees are a subclass of increasing trees, which can be constructed by an insertion process. Three such tree families contained in the grown simple families of increasing trees are of particular interest: $\textit{recursive trees}$, $\textit{plane-oriented recursive trees}$ and $\textit{binary increasing trees}$. Here we present a general approach for the analysis of a number of label-based parameters in a random grown simple increasing tree of size $n$ as, e.g., $\textit{the degree of the node labeled j}$, $\textit{the subtree-size of the node labeled j}$, etc. Further we apply the approach to the random variable $X_{n,j,a}$, which counts the number of size-$a$ branches attached to the node labeled $j$ (= subtrees of size $a$ rooted at the children of the node labeled $j$) in a random grown simple increasing tree of size $n$. We can give closed formulæ for the probability distribution and the factorial moments. Furthermore limiting distribution results for $X_{n,j,a}$ are given dependent on the growth behavior of $j=j(n)$ compared to $n$.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3482 https://doi.org/10.46298/dmtcs.3482 Kuba, Markus Panholzer, Alois Kuba, Markus Panholzer, Alois <![CDATA[Grown simple families of increasing trees are a subclass of increasing trees, which can be constructed by an insertion process. Three such tree families contained in the grown simple families of increasing trees are of particular interest: $\textit{recursive trees}$, $\textit{plane-oriented recursive trees}$ and $\textit{binary increasing trees}$. Here we present a general approach for the analysis of a number of label-based parameters in a random grown simple increasing tree of size $n$ as, e.g., $\textit{the degree of the node labeled j}$, $\textit{the subtree-size of the node labeled j}$, etc. Further we apply the approach to the random variable $X_{n,j,a}$, which counts the number of size-$a$ branches attached to the node labeled $j$ (= subtrees of size $a$ rooted at the children of the node labeled $j$) in a random grown simple increasing tree of size $n$. We can give closed formulæ for the probability distribution and the factorial moments. Furthermore limiting distribution results for $X_{n,j,a}$ are given dependent on the growth behavior of $j=j(n)$ compared to $n$.]]> 0 <![CDATA[An example is given which shows that, in general, conditioned Galton-Watson trees cannot be obtained by adding vertices one by one, while this can be done in some important but special cases, as shown by Luczak and Winkler.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3483 https://doi.org/10.46298/dmtcs.3483 Janson, Svante Janson, Svante <![CDATA[An example is given which shows that, in general, conditioned Galton-Watson trees cannot be obtained by adding vertices one by one, while this can be done in some important but special cases, as shown by Luczak and Winkler.]]> 0 <![CDATA[We analyse the distribution of the root pattern of randomly grown multidimensional point quadtrees. In particular, exact, recursive and asymptotic formulas are given for the expected arity of the root.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3484 https://doi.org/10.46298/dmtcs.3484 Labelle, Gilbert Laforest, Louise Provençal, Xavier Labelle, Gilbert Laforest, Louise Provençal, Xavier <![CDATA[We analyse the distribution of the root pattern of randomly grown multidimensional point quadtrees. In particular, exact, recursive and asymptotic formulas are given for the expected arity of the root.]]> 0 <![CDATA[A classification strategy based on $\delta$-patterns is developed via a combinatorial optimization problem related with the maximal clique generation problem on a graph. The proposed solution uses the cross entropy method and has the advantage to be particularly suitable for large datasets. This study is tailored for the particularities of the genomic data.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3485 https://doi.org/10.46298/dmtcs.3485 Alexe, Gabriela Bhanot, Gyan Climescu-Haulica, Adriana Alexe, Gabriela Bhanot, Gyan Climescu-Haulica, Adriana <![CDATA[A classification strategy based on $\delta$-patterns is developed via a combinatorial optimization problem related with the maximal clique generation problem on a graph. The proposed solution uses the cross entropy method and has the advantage to be particularly suitable for large datasets. This study is tailored for the particularities of the genomic data.]]> 0 <![CDATA[The aim of this paper is to extend the analysis of Cuckoo Hashing of Devroye and Morin in 2003. In particular we make several asymptotic results much more precise. We show, that the probability that the construction of a hash table succeeds, is asymptotically $1-c(\varepsilon)/m+O(1/m^2)$ for some explicit $c(\varepsilon)$, where $m$ denotes the size of each of the two tables, $n=m(1- \varepsilon)$ is the number of keys and $\varepsilon \in (0,1)$. The analysis rests on a generating function approach to the so called Cuckoo Graph, a random bipartite graph. We apply a double saddle point method to obtain asymptotic results covering tree sizes, the number of cycles and the probability that no complex component occurs.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3486 https://doi.org/10.46298/dmtcs.3486 Kutzelnigg, Reinhard Kutzelnigg, Reinhard <![CDATA[The aim of this paper is to extend the analysis of Cuckoo Hashing of Devroye and Morin in 2003. In particular we make several asymptotic results much more precise. We show, that the probability that the construction of a hash table succeeds, is asymptotically $1-c(\varepsilon)/m+O(1/m^2)$ for some explicit $c(\varepsilon)$, where $m$ denotes the size of each of the two tables, $n=m(1- \varepsilon)$ is the number of keys and $\varepsilon \in (0,1)$. The analysis rests on a generating function approach to the so called Cuckoo Graph, a random bipartite graph. We apply a double saddle point method to obtain asymptotic results covering tree sizes, the number of cycles and the probability that no complex component occurs.]]> 0 <![CDATA[We present a combinatorial approach of the variance for the number of maxima in hypercubes. This leads to an explicit expression, in terms of Multiple Zeta Values, of the dominant term in the asymptotic expansion of this variance.Moreover, we get an algorithm to compute this expansion, and show that all coefficients occuring belong to the $\mathbb{Q}$-algebra generated by Multiple Zeta Values, and by Euler's constant $\gamma$.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3487 https://doi.org/10.46298/dmtcs.3487 Costermans, Christian Minh, Hoang Ngoc Costermans, Christian Minh, Hoang Ngoc <![CDATA[We present a combinatorial approach of the variance for the number of maxima in hypercubes. This leads to an explicit expression, in terms of Multiple Zeta Values, of the dominant term in the asymptotic expansion of this variance.Moreover, we get an algorithm to compute this expansion, and show that all coefficients occuring belong to the $\mathbb{Q}$-algebra generated by Multiple Zeta Values, and by Euler's constant $\gamma$.]]> 0 <![CDATA[We present a software package that guesses formulas for sequences of, for example, rational numbers or rational functions, given the first few terms. Thereby we extend and complement Christian Krattenthaler’s program $\mathtt{Rate}$ and the relevant parts of Bruno Salvy and Paul Zimmermann’s $\mathtt{GFUN}$.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3488 https://doi.org/10.46298/dmtcs.3488 Rubey, Martin Rubey, Martin <![CDATA[We present a software package that guesses formulas for sequences of, for example, rational numbers or rational functions, given the first few terms. Thereby we extend and complement Christian Krattenthaler’s program $\mathtt{Rate}$ and the relevant parts of Bruno Salvy and Paul Zimmermann’s $\mathtt{GFUN}$.]]> 0 <![CDATA[Consider random graph with $N+ 1$ vertices as follows. The degrees of vertices $1,2,\ldots, N$ are the independent identically distributed random variables $\xi_1, \xi_2, \ldots , \xi_N$ with distribution $\mathbf{P}\{\xi_1 \geq k\}=k^{− \tau},$ $k= 1,2,\ldots,$ $\tau \in (1,2)$,(1) and the vertex $N+1$ has degree $0$, if the sum $\zeta_N=\xi_1+ \ldots +\xi_N$ is even, else degree is $1$. From (1) we get that $p_k=\mathbf{P}\{\xi_1=k\}=k^{−\tau}−(k+ 1)^{−\tau}$, $k= 1,2,\ldots$ Let $G(k_1, \ldots , k_N)$ be a set of graphs with $\xi_1=k_1,\ldots, \xi_N=k_N$. If $g$ is a realization of random graph then $\mathbf{P}\{g \in G(k_1, \ldots , k_N)\}=p_{k_1} \cdot \ldots \cdot p_{k_N}$. The probability distribution on the set of graph is defined such that for a vector $(k_1, \ldots, k_N)$ all graphs, lying in $G(k_1, \ldots , k_N)$, are equiprobable. Studies of the past few years show that such graphs are good random graph models for Internet and other networks topology description (see, for example, H. Reittu and I. Norros (2004)).To build the graph, we have $N$ numbered vertices and incident to vertex $i \xi_i$ stubs, $i= 1, \ldots , N$.All stubs need to be connected to another stub to construct the graph. The stubs are numbered in an arbitrary order from $1$ to $\zeta_N$. Let $\eta_{(N)}$ be the maximum degree of the vertices.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3489 https://doi.org/10.46298/dmtcs.3489 Pavlov, Yuri Pavlov, Yuri <![CDATA[Consider random graph with $N+ 1$ vertices as follows. The degrees of vertices $1,2,\ldots, N$ are the independent identically distributed random variables $\xi_1, \xi_2, \ldots , \xi_N$ with distribution $\mathbf{P}\{\xi_1 \geq k\}=k^{− \tau},$ $k= 1,2,\ldots,$ $\tau \in (1,2)$,(1) and the vertex $N+1$ has degree $0$, if the sum $\zeta_N=\xi_1+ \ldots +\xi_N$ is even, else degree is $1$. From (1) we get that $p_k=\mathbf{P}\{\xi_1=k\}=k^{−\tau}−(k+ 1)^{−\tau}$, $k= 1,2,\ldots$ Let $G(k_1, \ldots , k_N)$ be a set of graphs with $\xi_1=k_1,\ldots, \xi_N=k_N$. If $g$ is a realization of random graph then $\mathbf{P}\{g \in G(k_1, \ldots , k_N)\}=p_{k_1} \cdot \ldots \cdot p_{k_N}$. The probability distribution on the set of graph is defined such that for a vector $(k_1, \ldots, k_N)$ all graphs, lying in $G(k_1, \ldots , k_N)$, are equiprobable. Studies of the past few years show that such graphs are good random graph models for Internet and other networks topology description (see, for example, H. Reittu and I. Norros (2004)).To build the graph, we have $N$ numbered vertices and incident to vertex $i \xi_i$ stubs, $i= 1, \ldots , N$.All stubs need to be connected to another stub to construct the graph. The stubs are numbered in an arbitrary order from $1$ to $\zeta_N$. Let $\eta_{(N)}$ be the maximum degree of the vertices.]]> 0 <![CDATA[We investigate the probability that a sample $\Gamma=(\Gamma_1,\Gamma_2,\ldots,\Gamma_n)$ of independent, identically distributed random variables with a geometric distribution has no elements occurring exactly $j$ times, where $j$ belongs to a specified finite $\textit{'forbidden set'}$ $A$ of multiplicities. Specific choices of the set $A$ enable one to determine the asymptotic probabilities that such a sample has no variable occuring with multiplicity $b$, or which has all multiplicities greater than $b$, for any fixed integer $b \geq 1$.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3490 https://doi.org/10.46298/dmtcs.3490 Archibald, Margaret Knopfmacher, Arnold Archibald, Margaret Knopfmacher, Arnold <![CDATA[We investigate the probability that a sample $\Gamma=(\Gamma_1,\Gamma_2,\ldots,\Gamma_n)$ of independent, identically distributed random variables with a geometric distribution has no elements occurring exactly $j$ times, where $j$ belongs to a specified finite $\textit{'forbidden set'}$ $A$ of multiplicities. Specific choices of the set $A$ enable one to determine the asymptotic probabilities that such a sample has no variable occuring with multiplicity $b$, or which has all multiplicities greater than $b$, for any fixed integer $b \geq 1$.]]> 0 <![CDATA[We establish a fundamental isomorphism between discrete-time balanced urn processes and certain ordinary differential systems, which are nonlinear, autonomous, and of a simple monomial form. As a consequence, all balanced urn processes with balls of two colours are proved to be analytically solvable in finite terms. The corresponding generating functions are expressed in terms of certain Abelian integrals over curves of the Fermat type (which are also hypergeometric functions), together with their inverses. A consequence is the unification of the analyses of many classical models, including those related to the coupon collector's problem, particle transfer (the Ehrenfest model), Friedman's "adverse campaign'' and Pólya's contagion model, as well as the OK Corral model (a basic case of Lanchester's theory of conflicts). In each case, it is possible to quantify very precisely the probable composition of the urn at any discrete instant. We study here in detail "semi-sacrificial'' urns, for which the following are obtained: a Gaussian limiting distribution with speed of convergence estimates as well as a characterization of the large and extreme large deviation regimes. We also work out explicitly the case of $2$-dimensional triangular models, where local limit laws of the stable type are obtained. A few models of dimension three or greater, e.g., "autistic'' (generalized Pólya), cyclic chambers (generalized Ehrenfest), generalized coupon-collector, and triangular urns, are also shown to be exactly solvable.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3506 https://doi.org/10.46298/dmtcs.3506 Flajolet, Philippe Dumas, Philippe Puyhaubert, Vincent Flajolet, Philippe Dumas, Philippe Puyhaubert, Vincent <![CDATA[We establish a fundamental isomorphism between discrete-time balanced urn processes and certain ordinary differential systems, which are nonlinear, autonomous, and of a simple monomial form. As a consequence, all balanced urn processes with balls of two colours are proved to be analytically solvable in finite terms. The corresponding generating functions are expressed in terms of certain Abelian integrals over curves of the Fermat type (which are also hypergeometric functions), together with their inverses. A consequence is the unification of the analyses of many classical models, including those related to the coupon collector's problem, particle transfer (the Ehrenfest model), Friedman's "adverse campaign'' and Pólya's contagion model, as well as the OK Corral model (a basic case of Lanchester's theory of conflicts). In each case, it is possible to quantify very precisely the probable composition of the urn at any discrete instant. We study here in detail "semi-sacrificial'' urns, for which the following are obtained: a Gaussian limiting distribution with speed of convergence estimates as well as a characterization of the large and extreme large deviation regimes. We also work out explicitly the case of $2$-dimensional triangular models, where local limit laws of the stable type are obtained. A few models of dimension three or greater, e.g., "autistic'' (generalized Pólya), cyclic chambers (generalized Ehrenfest), generalized coupon-collector, and triangular urns, are also shown to be exactly solvable.]]> 0 <![CDATA[We determine the spectral dimensions of a variety of ensembles of infinite trees. Common to the ensembles considered is that sample trees have a distinguished infinite spine at whose vertices branches can be attached according to some probability distribution. In particular, we consider a family of ensembles of $\textit{combs}$, whose branches are linear chains, with spectral dimensions varying continuously between $1$ and $3/2$. We also introduce a class of ensembles of infinite trees, called $\textit{generic random trees}$, which are obtained as limits of ensembles of finite trees conditioned to have fixed size $N$, as $N \to \infty$. Among these ensembles is the so-called uniform random tree. We show that generic random trees have spectral dimension $d_s=4/3$.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3507 https://doi.org/10.46298/dmtcs.3507 Durhuus, Bergfinnur Jonsson, Thordur Wheater, John Durhuus, Bergfinnur Jonsson, Thordur Wheater, John <![CDATA[We determine the spectral dimensions of a variety of ensembles of infinite trees. Common to the ensembles considered is that sample trees have a distinguished infinite spine at whose vertices branches can be attached according to some probability distribution. In particular, we consider a family of ensembles of $\textit{combs}$, whose branches are linear chains, with spectral dimensions varying continuously between $1$ and $3/2$. We also introduce a class of ensembles of infinite trees, called $\textit{generic random trees}$, which are obtained as limits of ensembles of finite trees conditioned to have fixed size $N$, as $N \to \infty$. Among these ensembles is the so-called uniform random tree. We show that generic random trees have spectral dimension $d_s=4/3$.]]> 0 <![CDATA[An ordered partition of $[n]:=\{1,2,\ldots, n\}$ is a sequence of disjoint and nonempty subsets, called blocks, whose union is $[n]$. The aim of this paper is to compute some generating functions of ordered partitions by the transfer-matrix method. In particular, we prove several conjectures of Steingrímsson, which assert that the generating function of some statistics of ordered partitions give rise to a natural $q$-analogue of $k!S(n,k)$, where $S(n,k)$ is the Stirling number of the second kind.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3508 https://doi.org/10.46298/dmtcs.3508 Ishikawa, Masao Kasraoui, Anisse Zeng, Jiang Ishikawa, Masao Kasraoui, Anisse Zeng, Jiang <![CDATA[An ordered partition of $[n]:=\{1,2,\ldots, n\}$ is a sequence of disjoint and nonempty subsets, called blocks, whose union is $[n]$. The aim of this paper is to compute some generating functions of ordered partitions by the transfer-matrix method. In particular, we prove several conjectures of Steingrímsson, which assert that the generating function of some statistics of ordered partitions give rise to a natural $q$-analogue of $k!S(n,k)$, where $S(n,k)$ is the Stirling number of the second kind.]]> 0 <![CDATA[A composition of a positive integer $n$ is a finite sequence of positive integers $a_1, a_2, \ldots, a_k$ such that $a_1+a_2+ \cdots +a_k=n$. Let $d$ be a fixed nonnegative integer. We say that we have an ascent of size $d$ or more at position $i$, if $a_{i+1}\geq a_i+d$. We study the average position, initial height and end height of the first ascent of size $d$ or more in compositions of $n$ as $n \to \infty$.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3509 https://doi.org/10.46298/dmtcs.3509 Brennan, Charlotte Knopfmacher, Arnold Brennan, Charlotte Knopfmacher, Arnold <![CDATA[A composition of a positive integer $n$ is a finite sequence of positive integers $a_1, a_2, \ldots, a_k$ such that $a_1+a_2+ \cdots +a_k=n$. Let $d$ be a fixed nonnegative integer. We say that we have an ascent of size $d$ or more at position $i$, if $a_{i+1}\geq a_i+d$. We study the average position, initial height and end height of the first ascent of size $d$ or more in compositions of $n$ as $n \to \infty$.]]> 0 <![CDATA[We solve a problem by V. I. Arnold dealing with "how random" modular arithmetic progressions can be. After making precise how Arnold proposes to measure the randomness of a modular sequence, we show that this measure of randomness takes a simplified form in the case of arithmetic progressions. This simplified expression is then estimated using the methodology of dynamical analysis, which operates with tools coming from dynamical systems theory. In conclusion, this study shows that modular arithmetic progressions are far from behaving like purely random sequences, according to Arnold's definition.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3510 https://doi.org/10.46298/dmtcs.3510 Cesaratto, Eda Plagne, Alain Vallée, Brigitte Cesaratto, Eda Plagne, Alain Vallée, Brigitte <![CDATA[We solve a problem by V. I. Arnold dealing with "how random" modular arithmetic progressions can be. After making precise how Arnold proposes to measure the randomness of a modular sequence, we show that this measure of randomness takes a simplified form in the case of arithmetic progressions. This simplified expression is then estimated using the methodology of dynamical analysis, which operates with tools coming from dynamical systems theory. In conclusion, this study shows that modular arithmetic progressions are far from behaving like purely random sequences, according to Arnold's definition.]]> 0 <![CDATA[A result of Foata and Schützenberger states that two statistics on permutations, the number of inversions and the inverse major index, have the same distribution on a descent class. We give a multivariate generalization of this property: the sorted vectors of the Lehmer code, of the inverse majcode, and of a new code (the inverse saillance code), have the same distribution on a descent class, and their common multivariate generating function is a flagged ribbon Schur function.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3511 https://doi.org/10.46298/dmtcs.3511 Hivert, Florent Novelli, Jean-Christophe Thibon, Jean-Yves Hivert, Florent Novelli, Jean-Christophe Thibon, Jean-Yves <![CDATA[A result of Foata and Schützenberger states that two statistics on permutations, the number of inversions and the inverse major index, have the same distribution on a descent class. We give a multivariate generalization of this property: the sorted vectors of the Lehmer code, of the inverse majcode, and of a new code (the inverse saillance code), have the same distribution on a descent class, and their common multivariate generating function is a flagged ribbon Schur function.]]> 0 <![CDATA[Building on theoretical insights and rich experimental data of our preprints, we present here new theoretical and experimental results in three interrelated approaches to the Collatz problem and its generalizations: \emphalgorithmic decidability, random behavior, and Diophantine representation of related discrete dynamical systems, and their \emphcyclic and divergent properties.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3512 https://doi.org/10.46298/dmtcs.3512 Belaga, Edward G. Mignotte, Maurice Belaga, Edward G. Mignotte, Maurice <![CDATA[Building on theoretical insights and rich experimental data of our preprints, we present here new theoretical and experimental results in three interrelated approaches to the Collatz problem and its generalizations: \emphalgorithmic decidability, random behavior, and Diophantine representation of related discrete dynamical systems, and their \emphcyclic and divergent properties.]]> 0 <![CDATA[Let $X_1,\ldots,X_{n\choose 2}$ be independent identically distributed weights for the edges of $K_n$. If $X_i \neq X_j$ for$ i \neq j$, then there exists a unique minimum weight spanning tree $T$ of $K_n$ with these edge weights. We show that the expected diameter of $T$ is $Θ (n^{1/3})$. This settles a question of [Frieze97].]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3513 https://doi.org/10.46298/dmtcs.3513 Addario-Berry, Louigi Broutin, Nicolas Reed, Bruce Addario-Berry, Louigi Broutin, Nicolas Reed, Bruce <![CDATA[Let $X_1,\ldots,X_{n\choose 2}$ be independent identically distributed weights for the edges of $K_n$. If $X_i \neq X_j$ for$ i \neq j$, then there exists a unique minimum weight spanning tree $T$ of $K_n$ with these edge weights. We show that the expected diameter of $T$ is $Θ (n^{1/3})$. This settles a question of [Frieze97].]]> 0 <![CDATA[We show that a family of generalized meta-Fibonacci sequences arise when counting the number of leaves at the largest level in certain infinite sequences of k-ary trees and restricted compositions of an integer. For this family of generalized meta-Fibonacci sequences and two families of related sequences we derive ordinary generating functions and recurrence relations.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3514 https://doi.org/10.46298/dmtcs.3514 Deugau, Chris Ruskey, Frank Deugau, Chris Ruskey, Frank <![CDATA[We show that a family of generalized meta-Fibonacci sequences arise when counting the number of leaves at the largest level in certain infinite sequences of k-ary trees and restricted compositions of an integer. For this family of generalized meta-Fibonacci sequences and two families of related sequences we derive ordinary generating functions and recurrence relations.]]> 0 <![CDATA[In this article, we study a variant of the coupon collector's problem introducing a notion of a \emphbonus. Suppose that there are c different types of coupons made up of bonus coupons and ordinary coupons, and that a collector gets every coupon with probability 1/c each day. Moreover suppose that every time he gets a bonus coupon he immediately obtains one more coupon. Under this setting, we consider the number of days he needs to collect in order to have at least one of each type. We then give not only the expectation but also the exact distribution represented by a gamma distribution. Moreover we investigate their limits as the Gumbel (double exponential) distribution and the Gauss (normal) distribution.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3515 https://doi.org/10.46298/dmtcs.3515 Nakata, Toshio Kubo, Izumi Nakata, Toshio Kubo, Izumi <![CDATA[In this article, we study a variant of the coupon collector's problem introducing a notion of a \emphbonus. Suppose that there are c different types of coupons made up of bonus coupons and ordinary coupons, and that a collector gets every coupon with probability 1/c each day. Moreover suppose that every time he gets a bonus coupon he immediately obtains one more coupon. Under this setting, we consider the number of days he needs to collect in order to have at least one of each type. We then give not only the expectation but also the exact distribution represented by a gamma distribution. Moreover we investigate their limits as the Gumbel (double exponential) distribution and the Gauss (normal) distribution.]]> 0 <![CDATA[A leader election algorithm is an elimination process that divides recursively into tow subgroups an initial group of n items, eliminates one subgroup and continues the procedure until a subgroup is of size 1. In this paper the biased case is analyzed. We are interested in the cost of the algorithm e. the number of operations needed until the algorithm stops. Using a probabilistic approach, the asymptotic behavior of the algorithm is shown to be related to the behavior of a hitting time of two random sequences on [0,1].]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3516 https://doi.org/10.46298/dmtcs.3516 Mohamed, Hanene Mohamed, Hanene <![CDATA[A leader election algorithm is an elimination process that divides recursively into tow subgroups an initial group of n items, eliminates one subgroup and continues the procedure until a subgroup is of size 1. In this paper the biased case is analyzed. We are interested in the cost of the algorithm e. the number of operations needed until the algorithm stops. Using a probabilistic approach, the asymptotic behavior of the algorithm is shown to be related to the behavior of a hitting time of two random sequences on [0,1].]]> 0 <![CDATA[The purpose of this survey is to present recent results concerning concentration properties of extremal parameters of random discrete structures. A main emphasis is placed on the height and maximum degree of several kinds of random trees. We also provide exponential tail estimates for the height distribution of scale-free trees.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3517 https://doi.org/10.46298/dmtcs.3517 Drmota, Michael Drmota, Michael <![CDATA[The purpose of this survey is to present recent results concerning concentration properties of extremal parameters of random discrete structures. A main emphasis is placed on the height and maximum degree of several kinds of random trees. We also provide exponential tail estimates for the height distribution of scale-free trees.]]> 0 <![CDATA[We consider growing random recursive trees in random environment, in which at each step a new vertex is attached according to a probability distribution that assigns the tree vertices masses proportional to their random weights.The main aim of the paper is to study the asymptotic behavior of the mean numbers of outgoing vertices as the number of steps tends to infinity, under the assumption that the random weights have a product form with independent identically distributed factors.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3491 https://doi.org/10.46298/dmtcs.3491 Borovkov, Konstantin Vatutin, Vladimir Borovkov, Konstantin Vatutin, Vladimir <![CDATA[We consider growing random recursive trees in random environment, in which at each step a new vertex is attached according to a probability distribution that assigns the tree vertices masses proportional to their random weights.The main aim of the paper is to study the asymptotic behavior of the mean numbers of outgoing vertices as the number of steps tends to infinity, under the assumption that the random weights have a product form with independent identically distributed factors.]]> 0 <![CDATA[Giroire has recently proposed an algorithm which returns the $\textit{approximate}$ number of distinct elements in a large sequence of words, under strong constraints coming from the analysis of large data bases. His estimation is based on statistical properties of uniform random variables in $[0,1]$. In this note we propose an optimal estimation, using Kullback information and estimation theory.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3492 https://doi.org/10.46298/dmtcs.3492 Chassaing, Philippe Gerin, Lucas Chassaing, Philippe Gerin, Lucas <![CDATA[Giroire has recently proposed an algorithm which returns the $\textit{approximate}$ number of distinct elements in a large sequence of words, under strong constraints coming from the analysis of large data bases. His estimation is based on statistical properties of uniform random variables in $[0,1]$. In this note we propose an optimal estimation, using Kullback information and estimation theory.]]> 0 <![CDATA[In this paper we study a variant of the Sand Piles Model, where the evolution rule consists of the falling down of one grain to a random column and an avalanche to reach a stable configuration. We prove that the infinite set of all stable configurations have a lattice structure which is a sublattice of Young lattice. At the end, based on a discussion about avalanches, we construct a generating tree of this model and show its strongtly recursive structure.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3493 https://doi.org/10.46298/dmtcs.3493 Phan, Thi Ha Duong Tran, Thi Thu Huong Phan, Thi Ha Duong Tran, Thi Thu Huong <![CDATA[In this paper we study a variant of the Sand Piles Model, where the evolution rule consists of the falling down of one grain to a random column and an avalanche to reach a stable configuration. We prove that the infinite set of all stable configurations have a lattice structure which is a sublattice of Young lattice. At the end, based on a discussion about avalanches, we construct a generating tree of this model and show its strongtly recursive structure.]]> 0 <![CDATA[We show that the number of spanning trees in the finite Sierpiński graph of level $n$ is given by $\sqrt[4]{\frac{3}{20}} (\frac{5}{3})^{-n/2} (\sqrt[4]{540})^{3^n}$. The proof proceeds in two steps: First, we show that the number of spanning trees and two further quantities satisfy a $3$-dimensional polynomial recursion using the self-similar structure. Secondly, it turns out, that the dynamical behavior of the recursion is given by a $2$-dimensional polynomial map, whose iterates can be computed explicitly.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3494 https://doi.org/10.46298/dmtcs.3494 Teufl, Elmar Wagner, Stephan Teufl, Elmar Wagner, Stephan <![CDATA[We show that the number of spanning trees in the finite Sierpiński graph of level $n$ is given by $\sqrt[4]{\frac{3}{20}} (\frac{5}{3})^{-n/2} (\sqrt[4]{540})^{3^n}$. The proof proceeds in two steps: First, we show that the number of spanning trees and two further quantities satisfy a $3$-dimensional polynomial recursion using the self-similar structure. Secondly, it turns out, that the dynamical behavior of the recursion is given by a $2$-dimensional polynomial map, whose iterates can be computed explicitly.]]> 0 <![CDATA[We consider extended binary trees and study the common right and left depth of leaf $j$, where the leaves are labelled from left to right by $0, 1, \ldots, n$, and the common right and left external pathlength of binary trees of size $n$. Under the random tree model, i.e., the Catalan model, we characterize the common limiting distribution of the suitably scaled left depth and the difference between the right and the left depth of leaf $j$ in a random size-$n$ binary tree when $j \sim \rho n$ with $0< \rho < 1$, as well as the common limiting distribution of the suitably scaled left external pathlength and the difference between the right and the left external pathlength of a random size-$n$ binary tree.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3495 https://doi.org/10.46298/dmtcs.3495 Panholzer, Alois Panholzer, Alois <![CDATA[We consider extended binary trees and study the common right and left depth of leaf $j$, where the leaves are labelled from left to right by $0, 1, \ldots, n$, and the common right and left external pathlength of binary trees of size $n$. Under the random tree model, i.e., the Catalan model, we characterize the common limiting distribution of the suitably scaled left depth and the difference between the right and the left depth of leaf $j$ in a random size-$n$ binary tree when $j \sim \rho n$ with $0< \rho < 1$, as well as the common limiting distribution of the suitably scaled left external pathlength and the difference between the right and the left external pathlength of a random size-$n$ binary tree.]]> 0 <![CDATA[Random sequences from alphabet $\{1, \ldots,r\}$ are examined where repeated letters are allowed. Binary search trees are formed from these, and the average left-going depth of the first $1$ is found. Next, the right-going depth of the first $r$ is examined, and finally a merge (or 'shuffle') operator is used to obtain the average depth of an arbitrary node, which can be expressed in terms of the left-going and right-going depths. The variance of each of these parameters is also found.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3496 https://doi.org/10.46298/dmtcs.3496 Archibald, Margaret Clément, Julien Archibald, Margaret Clément, Julien <![CDATA[Random sequences from alphabet $\{1, \ldots,r\}$ are examined where repeated letters are allowed. Binary search trees are formed from these, and the average left-going depth of the first $1$ is found. Next, the right-going depth of the first $r$ is examined, and finally a merge (or 'shuffle') operator is used to obtain the average depth of an arbitrary node, which can be expressed in terms of the left-going and right-going depths. The variance of each of these parameters is also found.]]> 0 <![CDATA[A tree is called $k$-decomposable if it has a spanning forest whose components are all of size $k$. Analogously, a tree is called $T$-decomposable for a fixed tree $T$ if it has a spanning forest whose components are all isomorphic to $T$. In this paper, we use a generating functions approach to derive exact and asymptotic results on the number of $k$-decomposable and $T$-decomposable trees from a so-called simply generated family of trees - we find that there is a surprisingly simple functional equation for the counting series of $k$-decomposable trees. In particular, we will study the limit case when $k$ goes to $\infty$. It turns out that the ratio of $k$-decomposable trees increases when $k$ becomes large.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3497 https://doi.org/10.46298/dmtcs.3497 Wagner, Stephan G. Wagner, Stephan G. <![CDATA[A tree is called $k$-decomposable if it has a spanning forest whose components are all of size $k$. Analogously, a tree is called $T$-decomposable for a fixed tree $T$ if it has a spanning forest whose components are all isomorphic to $T$. In this paper, we use a generating functions approach to derive exact and asymptotic results on the number of $k$-decomposable and $T$-decomposable trees from a so-called simply generated family of trees - we find that there is a surprisingly simple functional equation for the counting series of $k$-decomposable trees. In particular, we will study the limit case when $k$ goes to $\infty$. It turns out that the ratio of $k$-decomposable trees increases when $k$ becomes large.]]> 0 <![CDATA[We investigate class of well-poised basic hypergeometric series $\tilde{J}_{k,i}(a;x;q)$, interpreting these series as generating functions for overpartitions defined by multiplicity conditions. We also show how to interpret the $\tilde{J}_{k,i}(a;1;q)$ as generating functions for overpartitions whose successive ranks are bounded, for overpartitions that are invariant under a certain class of conjugations, and for special restricted lattice paths. We highlight the cases $(a,q) \to (1/q,q)$, $(1/q,q^2)$, and $(0,q)$, where some of the functions $\tilde{J}_{k,i}(a;x;q)$ become infinite products. The latter case corresponds to Bressoud's family of Rogers-Ramanujan identities for even moduli.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3498 https://doi.org/10.46298/dmtcs.3498 Corteel, Sylvie Lovejoy, Jeremy Mallet, Olivier Corteel, Sylvie Lovejoy, Jeremy Mallet, Olivier <![CDATA[We investigate class of well-poised basic hypergeometric series $\tilde{J}_{k,i}(a;x;q)$, interpreting these series as generating functions for overpartitions defined by multiplicity conditions. We also show how to interpret the $\tilde{J}_{k,i}(a;1;q)$ as generating functions for overpartitions whose successive ranks are bounded, for overpartitions that are invariant under a certain class of conjugations, and for special restricted lattice paths. We highlight the cases $(a,q) \to (1/q,q)$, $(1/q,q^2)$, and $(0,q)$, where some of the functions $\tilde{J}_{k,i}(a;x;q)$ become infinite products. The latter case corresponds to Bressoud's family of Rogers-Ramanujan identities for even moduli.]]> 0 <![CDATA[We present a bijection between the set $\mathcal{A}_n$ of deterministic and accessible automata with $n$ states on a $k$-letters alphabet and some diagrams, which can themselves be represented as partitions of the set $[\![ 1..(kn+1) ]\!]$ into $n$ non-empty parts. This combinatorial construction shows that the asymptotic order of the cardinality of $\mathcal{A}_n$ is related to the Stirling number $\{^{kn}_n\}$. Our bijective approach also yields an efficient random sampler of automata with $n$ states, of complexity $O(n^{3/2})$, using the framework of Boltzmann samplers.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3499 https://doi.org/10.46298/dmtcs.3499 Bassino, Frédérique Nicaud, Cyril Bassino, Frédérique Nicaud, Cyril <![CDATA[We present a bijection between the set $\mathcal{A}_n$ of deterministic and accessible automata with $n$ states on a $k$-letters alphabet and some diagrams, which can themselves be represented as partitions of the set $[\![ 1..(kn+1) ]\!]$ into $n$ non-empty parts. This combinatorial construction shows that the asymptotic order of the cardinality of $\mathcal{A}_n$ is related to the Stirling number $\{^{kn}_n\}$. Our bijective approach also yields an efficient random sampler of automata with $n$ states, of complexity $O(n^{3/2})$, using the framework of Boltzmann samplers.]]> 0 <![CDATA[The aim of this paper is counting the probability that a random modal formula is a tautology. We examine $\{ \to,\Box \}$ fragment of two modal logics $\mathbf{S5}$ and $\mathbf{S4}$ over the language with one propositional variable. Any modal formula written in such a language may be interpreted as a unary binary tree. As it is known, there are finitely many different formulas written in one variable in the logic $\mathbf{S5}$ and this is the key to count the proportion of tautologies of $\mathbf{S5}$ among all formulas. Although the logic $\mathbf{S4}$ does not have this property, there exist its normal extensions having finitely many non-equivalent formulas.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3500 https://doi.org/10.46298/dmtcs.3500 Kostrzycka, Zofia Kostrzycka, Zofia <![CDATA[The aim of this paper is counting the probability that a random modal formula is a tautology. We examine $\{ \to,\Box \}$ fragment of two modal logics $\mathbf{S5}$ and $\mathbf{S4}$ over the language with one propositional variable. Any modal formula written in such a language may be interpreted as a unary binary tree. As it is known, there are finitely many different formulas written in one variable in the logic $\mathbf{S5}$ and this is the key to count the proportion of tautologies of $\mathbf{S5}$ among all formulas. Although the logic $\mathbf{S4}$ does not have this property, there exist its normal extensions having finitely many non-equivalent formulas.]]> 0 <![CDATA[Given an integer $m \geq 1$, let $\| \cdot \|$ be a norm in $\mathbb{R}^{m+1}$ and let $\mathbb{S}_+^m$ denote the set of points $\mathbf{d}=(d_0,\ldots,d_m)$ in $\mathbb{R}^{m+1}$ with nonnegative coordinates and such that $\| \mathbf{d} \|=1$. Consider for each $1 \leq j \leq m$ a function $f_j(z)$ that is analytic in an open neighborhood of the point $z=0$ in the complex plane and with possibly negative Taylor coefficients. Given $\mathbf{n}=(n_0,\ldots,n_m)$ in $\mathbb{Z}^{m+1}$ with nonnegative coordinates, we develop a method to systematically associate a parameter-varying integral to study the asymptotic behavior of the coefficient of $z^{n_0}$ of the Taylor series of $\prod_{j=1}^m \{f_j(z)\}^{n_j}$, as $\| \mathbf{n} \| \to \infty$. The associated parameter-varying integral has a phase term with well specified properties that make the asymptotic analysis of the integral amenable to saddle-point methods: for many $\mathbf{d} \in \mathbb{S}_+^m$, these methods ensure uniform asymptotic expansions for $[z^{n_0}] \prod_{j=1}^m \{f_j(z)\}^{n_j}$ provided that $\mathbf{n}/ \| \mathbf{n} \|$ stays sufficiently close to $\mathbf{d}$ as $\| \mathbf{n} \| \to \infty$. Our method finds applications in studying the asymptotic behavior of the coefficients of a certain multivariable generating functions as well as in problems related to the Lagrange inversion formula for instance in the context random planar maps.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3501 https://doi.org/10.46298/dmtcs.3501 Lladser, Manuel Lladser, Manuel <![CDATA[Given an integer $m \geq 1$, let $\| \cdot \|$ be a norm in $\mathbb{R}^{m+1}$ and let $\mathbb{S}_+^m$ denote the set of points $\mathbf{d}=(d_0,\ldots,d_m)$ in $\mathbb{R}^{m+1}$ with nonnegative coordinates and such that $\| \mathbf{d} \|=1$. Consider for each $1 \leq j \leq m$ a function $f_j(z)$ that is analytic in an open neighborhood of the point $z=0$ in the complex plane and with possibly negative Taylor coefficients. Given $\mathbf{n}=(n_0,\ldots,n_m)$ in $\mathbb{Z}^{m+1}$ with nonnegative coordinates, we develop a method to systematically associate a parameter-varying integral to study the asymptotic behavior of the coefficient of $z^{n_0}$ of the Taylor series of $\prod_{j=1}^m \{f_j(z)\}^{n_j}$, as $\| \mathbf{n} \| \to \infty$. The associated parameter-varying integral has a phase term with well specified properties that make the asymptotic analysis of the integral amenable to saddle-point methods: for many $\mathbf{d} \in \mathbb{S}_+^m$, these methods ensure uniform asymptotic expansions for $[z^{n_0}] \prod_{j=1}^m \{f_j(z)\}^{n_j}$ provided that $\mathbf{n}/ \| \mathbf{n} \|$ stays sufficiently close to $\mathbf{d}$ as $\| \mathbf{n} \| \to \infty$. Our method finds applications in studying the asymptotic behavior of the coefficients of a certain multivariable generating functions as well as in problems related to the Lagrange inversion formula for instance in the context random planar maps.]]> 0 <![CDATA[In computational biology, a large amount of problems, such as pattern discovery, deals with the comparison of several sequences (of nucleotides, proteins or genes for instance). Very often, algorithms that address this problem use score functions that reflect a notion of similarity between the sequences. The most efficient methods take benefit from theoretical knowledge of the classical behavior of these score functions such as their mean, their variance, and sometime their asymptotic distribution in a given probabilistic model. In this paper, we study a recent family of score functions introduced in Mancheron 2003, which allows to compare two words having the same length. Here, the similarity takes into account all matches and mismatches between two sequences and not only the longest common subsequence as in the case of classical algorithms such as BLAST or FASTA. Based on generating functions, we provide closed formulas for the mean and the variance of these functions in an independent probabilistic model. Finally, we prove that every function in this family asymptotically behaves as a Gaussian random variable.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3502 https://doi.org/10.46298/dmtcs.3502 Bourdon, Jérémie Mancheron, Alban Bourdon, Jérémie Mancheron, Alban <![CDATA[In computational biology, a large amount of problems, such as pattern discovery, deals with the comparison of several sequences (of nucleotides, proteins or genes for instance). Very often, algorithms that address this problem use score functions that reflect a notion of similarity between the sequences. The most efficient methods take benefit from theoretical knowledge of the classical behavior of these score functions such as their mean, their variance, and sometime their asymptotic distribution in a given probabilistic model. In this paper, we study a recent family of score functions introduced in Mancheron 2003, which allows to compare two words having the same length. Here, the similarity takes into account all matches and mismatches between two sequences and not only the longest common subsequence as in the case of classical algorithms such as BLAST or FASTA. Based on generating functions, we provide closed formulas for the mean and the variance of these functions in an independent probabilistic model. Finally, we prove that every function in this family asymptotically behaves as a Gaussian random variable.]]> 0 <![CDATA[In this paper, we discuss the problem of estimating the number of "elephants'' in a stream of IP packets. First, the problem is formulated in the context of multisets. Next, we explore some of the theoretical space complexity of this problem, and it is shown that it cannot be solved with less than $\Omega (n)$ units of memory in general, $n$ being the number of different elements in the multiset. Finally, we describe an algorithm, based on Durand-Flajolet's LOGLOG algorithm coupled with a thinning of the packet stream, which returns an estimator of the number of elephants using a small amount of memory. This algorithm allows a good estimation for particular families of random multiset. The mean and variance of this estimator are computed. The algorithm is then tested on synthetic data.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3503 https://doi.org/10.46298/dmtcs.3503 Gandouet, Olivier Jean-Marie, Alain Gandouet, Olivier Jean-Marie, Alain <![CDATA[In this paper, we discuss the problem of estimating the number of "elephants'' in a stream of IP packets. First, the problem is formulated in the context of multisets. Next, we explore some of the theoretical space complexity of this problem, and it is shown that it cannot be solved with less than $\Omega (n)$ units of memory in general, $n$ being the number of different elements in the multiset. Finally, we describe an algorithm, based on Durand-Flajolet's LOGLOG algorithm coupled with a thinning of the packet stream, which returns an estimator of the number of elephants using a small amount of memory. This algorithm allows a good estimation for particular families of random multiset. The mean and variance of this estimator are computed. The algorithm is then tested on synthetic data.]]> 0 <![CDATA[We explore a similarity between the $n$ by $n$ random assignment problem and the random shortest path problem on the complete graph on $n+1$ vertices. This similarity is a consequence of the proof of the Parisi formula for the assignment problem given by C. Nair, B. Prabhakar and M. Sharma in 2003. We give direct proofs of the analogs for the shortest path problem of some results established by D. Aldous in connection with his $\zeta (2)$ limit theorem for the assignment problem.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3504 https://doi.org/10.46298/dmtcs.3504 Wästlund, Johan Wästlund, Johan <![CDATA[We explore a similarity between the $n$ by $n$ random assignment problem and the random shortest path problem on the complete graph on $n+1$ vertices. This similarity is a consequence of the proof of the Parisi formula for the assignment problem given by C. Nair, B. Prabhakar and M. Sharma in 2003. We give direct proofs of the analogs for the shortest path problem of some results established by D. Aldous in connection with his $\zeta (2)$ limit theorem for the assignment problem.]]> 0 <![CDATA[We show a new invariance principle for the radius and other functionals of a class of conditioned `Boltzmann-Gibbs'-distributed random planar maps. It improves over the more restrictive case of bipartite maps that was discussed in Marckert and Miermont (2006). As in the latter paper, we make use of a bijection between planar maps and a class of labelled multitype trees, due to Bouttier et al. (2004). We also rely on an invariance principle for multitype spatial Galton-Watson trees, which is proved in a companion paper.]]> Sat, 31 Dec 2005 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3505 https://doi.org/10.46298/dmtcs.3505 Miermont, Grégory Miermont, Grégory <![CDATA[We show a new invariance principle for the radius and other functionals of a class of conditioned `Boltzmann-Gibbs'-distributed random planar maps. It improves over the more restrictive case of bipartite maps that was discussed in Marckert and Miermont (2006). As in the latter paper, we make use of a bijection between planar maps and a class of labelled multitype trees, due to Bouttier et al. (2004). We also rely on an invariance principle for multitype spatial Galton-Watson trees, which is proved in a companion paper.]]> 0 <![CDATA[Suppose that p,q,r,s are non-negative integers with m=p+q+r+s. The class X(p,q,r,s) of permutations that contain no pattern of the form α β γ where |α |=r, |γ |=s and β is any arrangement of \1,2,\ldots,p\∪ \m-q+1, m-q+2, \ldots,m\ is considered. A recurrence relation to enumerate the permutations of X(p,q,r,s) is established. The method of proof also shows that X(p,q,r,s)=X(p,q,1,0)X(1,0,r,s) in the sense of permutational composition.\par 2000 MATHEMATICS SUBJECT CLASSIFICATION: 05A05]]> Sat, 01 Jan 2005 07:00:00 +0000 https://doi.org/10.46298/dmtcs.356 https://doi.org/10.46298/dmtcs.356 Atkinson, M. D. Atkinson, M. D. <![CDATA[Suppose that p,q,r,s are non-negative integers with m=p+q+r+s. The class X(p,q,r,s) of permutations that contain no pattern of the form α β γ where |α |=r, |γ |=s and β is any arrangement of \1,2,\ldots,p\∪ \m-q+1, m-q+2, \ldots,m\ is considered. A recurrence relation to enumerate the permutations of X(p,q,r,s) is established. The method of proof also shows that X(p,q,r,s)=X(p,q,1,0)X(1,0,r,s) in the sense of permutational composition.\par 2000 MATHEMATICS SUBJECT CLASSIFICATION: 05A05]]> 0 <![CDATA[We present here algebraic formulas associating a k-automaton to a k-epsilon-automaton. The existence depends on the definition of the star of matrices and of elements in the semiring k. For this reason, we present the theorem which allows the transformation of k-epsilon-automata into k-automata. The two automata have the same behaviour.]]> Sat, 01 Jan 2005 07:00:00 +0000 https://doi.org/10.46298/dmtcs.358 https://doi.org/10.46298/dmtcs.358 Duchamp, Gérard, Kacem, Hatem Hadj Laugerotte, Eric Duchamp, Gérard, Kacem, Hatem Hadj Laugerotte, Eric <![CDATA[We present here algebraic formulas associating a k-automaton to a k-epsilon-automaton. The existence depends on the definition of the star of matrices and of elements in the semiring k. For this reason, we present the theorem which allows the transformation of k-epsilon-automata into k-automata. The two automata have the same behaviour.]]> 0 <![CDATA[A sequence S is potentially K_p,1,1 graphical if it has a realization containing a K_p,1,1 as a subgraph, where K_p,1,1 is a complete 3-partite graph with partition sizes p,1,1. Let σ (K_p,1,1, n) denote the smallest degree sum such that every n-term graphical sequence S with σ (S)≥ σ (K_p,1,1, n) is potentially K_p,1,1 graphical. In this paper, we prove that σ (K_p,1,1, n)≥ 2[((p+1)(n-1)+2)/2] for n ≥ p+2. We conjecture that equality holds for n ≥ 2p+4. We prove that this conjecture is true for p = 3. AMS Subject Classifications: 05C07, 05C35]]> Sat, 01 Jan 2005 07:00:00 +0000 https://doi.org/10.46298/dmtcs.357 https://doi.org/10.46298/dmtcs.357 Lai, Chunhui Lai, Chunhui <![CDATA[A sequence S is potentially K_p,1,1 graphical if it has a realization containing a K_p,1,1 as a subgraph, where K_p,1,1 is a complete 3-partite graph with partition sizes p,1,1. Let σ (K_p,1,1, n) denote the smallest degree sum such that every n-term graphical sequence S with σ (S)≥ σ (K_p,1,1, n) is potentially K_p,1,1 graphical. In this paper, we prove that σ (K_p,1,1, n)≥ 2[((p+1)(n-1)+2)/2] for n ≥ p+2. We conjecture that equality holds for n ≥ 2p+4. We prove that this conjecture is true for p = 3. AMS Subject Classifications: 05C07, 05C35]]> 0 <![CDATA[Let G be a graph with chromatic number χ (G). A vertex colouring of G is \emphacyclic if each bichromatic subgraph is a forest. A \emphstar colouring of G is an acyclic colouring in which each bichromatic subgraph is a star forest. Let χ _a(G) and χ _s(G) denote the acyclic and star chromatic numbers of G. This paper investigates acyclic and star colourings of subdivisions. Let G' be the graph obtained from G by subdividing each edge once. We prove that acyclic (respectively, star) colourings of G' correspond to vertex partitions of G in which each subgraph has small arboricity (chromatic index). It follows that χ _a(G'), χ _s(G') and χ (G) are tied, in the sense that each is bounded by a function of the other. Moreover the binding functions that we establish are all tight. The \emphoriented chromatic number χ ^→(G) of an (undirected) graph G is the maximum, taken over all orientations D of G, of the minimum number of colours in a vertex colouring of D such that between any two colour classes, all edges have the same direction. We prove that χ ^→(G')=χ (G) whenever χ (G)≥ 9.]]> Sat, 01 Jan 2005 07:00:00 +0000 https://doi.org/10.46298/dmtcs.344 https://doi.org/10.46298/dmtcs.344 Wood, David R. Wood, David R. <![CDATA[Let G be a graph with chromatic number χ (G). A vertex colouring of G is \emphacyclic if each bichromatic subgraph is a forest. A \emphstar colouring of G is an acyclic colouring in which each bichromatic subgraph is a star forest. Let χ _a(G) and χ _s(G) denote the acyclic and star chromatic numbers of G. This paper investigates acyclic and star colourings of subdivisions. Let G' be the graph obtained from G by subdividing each edge once. We prove that acyclic (respectively, star) colourings of G' correspond to vertex partitions of G in which each subgraph has small arboricity (chromatic index). It follows that χ _a(G'), χ _s(G') and χ (G) are tied, in the sense that each is bounded by a function of the other. Moreover the binding functions that we establish are all tight. The \emphoriented chromatic number χ ^→(G) of an (undirected) graph G is the maximum, taken over all orientations D of G, of the minimum number of colours in a vertex colouring of D such that between any two colour classes, all edges have the same direction. We prove that χ ^→(G')=χ (G) whenever χ (G)≥ 9.]]> 0 <![CDATA[The purpose of this paper is to solve a special class of combinational games consisting of two-pile counter pickup games for which the maximum number of counters that can be removed on each successive move changes during the play of the games. Two players alternate moving. Each player in his turn first chooses one of the piles, and his choice of piles can change from move to move. He then removes counters from this chosen pile. A function f:Z^+ → Z^+ is given which determines the maximum size of the next move in terms of the current move size. The game ends as soon as one of the two piles is empty, and the winner is the last player to move in the game. The games for which f(k)=k, f(k)=2k, and f(k)=3k use the same formula for computing the smallest winning move size. Here we find all the functions f for which this formula works, and we also give the winning strategy for each function. See Holshouser, A, James Rudzinski and Harold Reiter: Dynamic One-Pile Nim for a discussion of the single pile game.]]> Sat, 01 Jan 2005 07:00:00 +0000 https://doi.org/10.46298/dmtcs.355 https://doi.org/10.46298/dmtcs.355 Holshouser, Arthur Reiter, Harold Holshouser, Arthur Reiter, Harold <![CDATA[The purpose of this paper is to solve a special class of combinational games consisting of two-pile counter pickup games for which the maximum number of counters that can be removed on each successive move changes during the play of the games. Two players alternate moving. Each player in his turn first chooses one of the piles, and his choice of piles can change from move to move. He then removes counters from this chosen pile. A function f:Z^+ → Z^+ is given which determines the maximum size of the next move in terms of the current move size. The game ends as soon as one of the two piles is empty, and the winner is the last player to move in the game. The games for which f(k)=k, f(k)=2k, and f(k)=3k use the same formula for computing the smallest winning move size. Here we find all the functions f for which this formula works, and we also give the winning strategy for each function. See Holshouser, A, James Rudzinski and Harold Reiter: Dynamic One-Pile Nim for a discussion of the single pile game.]]> 0 <![CDATA[Binary unlabeled ordered trees (further called binary trees) were studied at least since Euler, who enumerated them. The number of such trees with n nodes is now known as the Catalan number. Over the years various interesting questions about the statistics of such trees were investigated (e.g., height and path length distributions for a randomly selected tree). Binary trees find an abundance of applications in computer science. However, recently Seroussi posed a new and interesting problem motivated by information theory considerations: how many binary trees of a \emphgiven path length (sum of depths) are there? This question arose in the study of \emphuniversal types of sequences. Two sequences of length p have the same universal type if they generate the same set of phrases in the incremental parsing of the Lempel-Ziv'78 scheme since one proves that such sequences converge to the same empirical distribution. It turns out that the number of distinct types of sequences of length p corresponds to the number of binary (unlabeled and ordered) trees, T_p, of given path length p (and also the number of distinct Lempel-Ziv'78 parsings of length p sequences). We first show that the number of binary trees with given path length p is asymptotically equal to T_p ~ 2^2p/(log_2 p)(1+O(log ^-2/3 p)). Then we establish various limiting distributions for the number of nodes (number of phrases in the Lempel-Ziv'78 scheme) when a tree is selected randomly among all trees of given path length p. Throughout, we use methods of analytic algorithmics such as generating functions and complex asymptotics, as well as methods of applied mathematics such as the WKB method and matched asymptotics.]]> Sat, 01 Jan 2005 07:00:00 +0000 https://doi.org/10.46298/dmtcs.354 https://doi.org/10.46298/dmtcs.354 Knessl, Charles Szpankowski, Wojciech Knessl, Charles Szpankowski, Wojciech <![CDATA[Binary unlabeled ordered trees (further called binary trees) were studied at least since Euler, who enumerated them. The number of such trees with n nodes is now known as the Catalan number. Over the years various interesting questions about the statistics of such trees were investigated (e.g., height and path length distributions for a randomly selected tree). Binary trees find an abundance of applications in computer science. However, recently Seroussi posed a new and interesting problem motivated by information theory considerations: how many binary trees of a \emphgiven path length (sum of depths) are there? This question arose in the study of \emphuniversal types of sequences. Two sequences of length p have the same universal type if they generate the same set of phrases in the incremental parsing of the Lempel-Ziv'78 scheme since one proves that such sequences converge to the same empirical distribution. It turns out that the number of distinct types of sequences of length p corresponds to the number of binary (unlabeled and ordered) trees, T_p, of given path length p (and also the number of distinct Lempel-Ziv'78 parsings of length p sequences). We first show that the number of binary trees with given path length p is asymptotically equal to T_p ~ 2^2p/(log_2 p)(1+O(log ^-2/3 p)). Then we establish various limiting distributions for the number of nodes (number of phrases in the Lempel-Ziv'78 scheme) when a tree is selected randomly among all trees of given path length p. Throughout, we use methods of analytic algorithmics such as generating functions and complex asymptotics, as well as methods of applied mathematics such as the WKB method and matched asymptotics.]]> 0 <![CDATA[Let T=T(A,D) be a self-affine tile in \reals^n defined by an integral expanding matrix A and a digit set D. In connection with canonical number systems, we study connectedness of T when D corresponds to the set of consecutive integers \0,1,..., |det(A)|-1\. It is shown that in \reals^3 and \reals^4, for any integral expanding matrix A, T(A,D) is connected. We also study the connectedness of Pisot dual tilings which play an important role in the study of β -expansion, substitution and symbolic dynamical system. It is shown that each tile generated by a Pisot unit of degree 3 is arcwise connected. This is naturally expected since the digit set consists of consecutive integers as above. However surprisingly, we found families of disconnected Pisot dual tiles of degree 4. Also we give a simple necessary and sufficient condition for the connectedness of the Pisot dual tiles of degree 4. As a byproduct, a complete classification of the β -expansion of 1 for quartic Pisot units is given.]]> Sat, 01 Jan 2005 07:00:00 +0000 https://doi.org/10.46298/dmtcs.353 https://doi.org/10.46298/dmtcs.353 Akiyama, Shigeki Gjini, Nertila Akiyama, Shigeki Gjini, Nertila <![CDATA[Let T=T(A,D) be a self-affine tile in \reals^n defined by an integral expanding matrix A and a digit set D. In connection with canonical number systems, we study connectedness of T when D corresponds to the set of consecutive integers \0,1,..., |det(A)|-1\. It is shown that in \reals^3 and \reals^4, for any integral expanding matrix A, T(A,D) is connected. We also study the connectedness of Pisot dual tilings which play an important role in the study of β -expansion, substitution and symbolic dynamical system. It is shown that each tile generated by a Pisot unit of degree 3 is arcwise connected. This is naturally expected since the digit set consists of consecutive integers as above. However surprisingly, we found families of disconnected Pisot dual tiles of degree 4. Also we give a simple necessary and sufficient condition for the connectedness of the Pisot dual tiles of degree 4. As a byproduct, a complete classification of the β -expansion of 1 for quartic Pisot units is given.]]> 0 <![CDATA[A \emphk-queue layout of a graph G consists of a linear order σ of V(G), and a partition of E(G) into k sets, each of which contains no two edges that are nested in σ . This paper studies queue layouts of graph products and powers]]> Sat, 01 Jan 2005 07:00:00 +0000 https://doi.org/10.46298/dmtcs.352 https://doi.org/10.46298/dmtcs.352 Wood, David R. Wood, David R. <![CDATA[A \emphk-queue layout of a graph G consists of a linear order σ of V(G), and a partition of E(G) into k sets, each of which contains no two edges that are nested in σ . This paper studies queue layouts of graph products and powers]]> 0 <![CDATA[The classes of Raspail (also known as Bipolarizable) and P_4-simplicial graphs were introduced by Hoàng and Reed who showed that both classes are perfectly orderable and admit polynomial-time recognition algorithms HR1. In this paper, we consider the recognition problem on these classes of graphs and present algorithms that solve it in O(n m) time. In particular, we prove properties and show that we can produce bipolarizable and P_4-simplicial orderings on the vertices of the input graph G, if such orderings exist, working only on P_3s that participate in a P_4 of G. The proposed recognition algorithms are simple, use simple data structures and both require O(n + m) space. Additionally, we show how our recognition algorithms can be augmented to provide certificates, whenever they decide that G is not bipolarizable or P_4-simplicial; the augmentation takes O(n + m) time and space. Finally, we include a diagram on class inclusions and the currently best recognition time complexities for a number of perfectly orderable classes of graphs.]]> Sat, 01 Jan 2005 07:00:00 +0000 https://doi.org/10.46298/dmtcs.351 https://doi.org/10.46298/dmtcs.351 Nikolopoulos, Stavros D. Palios, Leonidas Nikolopoulos, Stavros D. Palios, Leonidas <![CDATA[The classes of Raspail (also known as Bipolarizable) and P_4-simplicial graphs were introduced by Hoàng and Reed who showed that both classes are perfectly orderable and admit polynomial-time recognition algorithms HR1. In this paper, we consider the recognition problem on these classes of graphs and present algorithms that solve it in O(n m) time. In particular, we prove properties and show that we can produce bipolarizable and P_4-simplicial orderings on the vertices of the input graph G, if such orderings exist, working only on P_3s that participate in a P_4 of G. The proposed recognition algorithms are simple, use simple data structures and both require O(n + m) space. Additionally, we show how our recognition algorithms can be augmented to provide certificates, whenever they decide that G is not bipolarizable or P_4-simplicial; the augmentation takes O(n + m) time and space. Finally, we include a diagram on class inclusions and the currently best recognition time complexities for a number of perfectly orderable classes of graphs.]]> 0 <![CDATA[We study BVASS (Branching VASS) which extend VASS (Vector Addition Systems with States) by allowing addition transitions that merge two configurations. Runs in BVASS are tree-like structures instead of linear ones as for VASS. We show that the construction of Karp-Miller trees for VASS can be extended to BVASS. This entails that the coverability set for BVASS is computable. This allows us to obtain decidability results for certain classes of equational tree automata with an associative-commutative symbol. Recent independent work by de Groote et al. implies that decidability of reachability in BVASS is equivalent to decidability of provability in MELL (multiplicative exponential linear logic), which is still an open problem. Hence our results are also a step towards answering this question in the affirmative.]]> Sat, 01 Jan 2005 07:00:00 +0000 https://doi.org/10.46298/dmtcs.350 https://doi.org/10.46298/dmtcs.350 Verma, Kumar Neeraj Goubault-Larrecq, Jean Verma, Kumar Neeraj Goubault-Larrecq, Jean <![CDATA[We study BVASS (Branching VASS) which extend VASS (Vector Addition Systems with States) by allowing addition transitions that merge two configurations. Runs in BVASS are tree-like structures instead of linear ones as for VASS. We show that the construction of Karp-Miller trees for VASS can be extended to BVASS. This entails that the coverability set for BVASS is computable. This allows us to obtain decidability results for certain classes of equational tree automata with an associative-commutative symbol. Recent independent work by de Groote et al. implies that decidability of reachability in BVASS is equivalent to decidability of provability in MELL (multiplicative exponential linear logic), which is still an open problem. Hence our results are also a step towards answering this question in the affirmative.]]> 0 <![CDATA[We prove that the incidence chromatic number of every 3-degenerated graph G is at most Δ (G)+4. It is known that the incidence chromatic number of every graph G with maximum average degree mad(G)<3 is at most Δ (G)+3. We show that when Δ (G) ≥ 5, this bound may be decreased to Δ (G)+2. Moreover, we show that for every graph G with mad(G)<22/9 (resp. with mad(G)<16/7 and Δ (G)≥ 4), this bound may be decreased to Δ (G)+2 (resp. to Δ (G)+1).]]> Sat, 01 Jan 2005 07:00:00 +0000 https://doi.org/10.46298/dmtcs.349 https://doi.org/10.46298/dmtcs.349 Dolama, Mohammad Hosseini Sopena, Eric Dolama, Mohammad Hosseini Sopena, Eric <![CDATA[We prove that the incidence chromatic number of every 3-degenerated graph G is at most Δ (G)+4. It is known that the incidence chromatic number of every graph G with maximum average degree mad(G)<3 is at most Δ (G)+3. We show that when Δ (G) ≥ 5, this bound may be decreased to Δ (G)+2. Moreover, we show that for every graph G with mad(G)<22/9 (resp. with mad(G)<16/7 and Δ (G)≥ 4), this bound may be decreased to Δ (G)+2 (resp. to Δ (G)+1).]]> 0 <![CDATA[We classify, up to an isomorphism, the class of all dihedral f-tilings of S^2, whose prototiles are a spherical triangle and a spherical rhombus. The equiangular case was considered and classified in Ana M. Breda and Altino F. Santos, Dihedral f-tilings of the sphere by spherical triangles and equiangular well-centered quadrangles. Here we complete the classification considering the case of non-equiangular rhombi.]]> Sat, 01 Jan 2005 07:00:00 +0000 https://doi.org/10.46298/dmtcs.348 https://doi.org/10.46298/dmtcs.348 Breda, Ana, Santos, Altino F. Breda, Ana, Santos, Altino F. <![CDATA[We classify, up to an isomorphism, the class of all dihedral f-tilings of S^2, whose prototiles are a spherical triangle and a spherical rhombus. The equiangular case was considered and classified in Ana M. Breda and Altino F. Santos, Dihedral f-tilings of the sphere by spherical triangles and equiangular well-centered quadrangles. Here we complete the classification considering the case of non-equiangular rhombi.]]> 0 <![CDATA[A generating method of self-affine tilings for Pisot, unimodular, irreducible substitutions, as well as the fact that the associated substitution dynamical systems are isomorphic to rotations on the torus are established in P. Arnoux and S. Ito. The aim of this paper is to extend these facts in the case where the characteristic polynomial of a substitution is non-irreducible for a special class of substitutions on five letters. Finally we show that the substitution dynamical systems for this class are isomorphic to induced transformations of rotations on the torus.]]> Sat, 01 Jan 2005 07:00:00 +0000 https://doi.org/10.46298/dmtcs.347 https://doi.org/10.46298/dmtcs.347 Ito, Shunji Ei, Hiromi Ito, Shunji Ei, Hiromi <![CDATA[A generating method of self-affine tilings for Pisot, unimodular, irreducible substitutions, as well as the fact that the associated substitution dynamical systems are isomorphic to rotations on the torus are established in P. Arnoux and S. Ito. The aim of this paper is to extend these facts in the case where the characteristic polynomial of a substitution is non-irreducible for a special class of substitutions on five letters. Finally we show that the substitution dynamical systems for this class are isomorphic to induced transformations of rotations on the torus.]]> 0 <![CDATA[A \emphk-stack layout (respectively, \emphk-queuelayout) of a graph consists of a total order of the vertices, and a partition of the edges into k sets of non-crossing (non-nested) edges with respect to the vertex ordering. A \emphk-track layout of a graph consists of a vertex k-colouring, and a total order of each vertex colour class, such that between each pair of colour classes no two edges cross. The \emphstack-number (respectively, \emphqueue-number, \emphtrack-number) of a graph G, denoted by sn(G) (qn(G), tn(G)), is the minimum k such that G has a k-stack (k-queue, k-track) layout.\par This paper studies stack, queue, and track layouts of graph subdivisions. It is known that every graph has a 3-stack subdivision. The best known upper bound on the number of division vertices per edge in a 3-stack subdivision of an n-vertex graph G is improved from O(log n) to O(log min\sn(G),qn(G)\). This result reduces the question of whether queue-number is bounded by stack-number to whether 3-stack graphs have bounded queue number.\par It is proved that every graph has a 2-queue subdivision, a 4-track subdivision, and a mixed 1-stack 1-queue subdivision. All these values are optimal for every non-planar graph. In addition, we characterise those graphs with k-stack, k-queue, and k-track subdivisions, for all values of k. The number of division vertices per edge in the case of 2-queue and 4-track subdivisions, namely O(log qn(G)), is optimal to within a constant factor, for every graph G. \par Applications to 3D polyline grid drawings are presented. For example, it is proved that every graph G has a 3D polyline grid drawing with the vertices on a rectangular prism, and with O(log qn(G)) bends per edge. Finally, we establish a tight relationship between queue layouts and so-called 2-track thickness of bipartite graphs. \par]]> Sat, 01 Jan 2005 07:00:00 +0000 https://doi.org/10.46298/dmtcs.346 https://doi.org/10.46298/dmtcs.346 Dujmović, Vida Wood, David R. Dujmović, Vida Wood, David R. <![CDATA[A \emphk-stack layout (respectively, \emphk-queuelayout) of a graph consists of a total order of the vertices, and a partition of the edges into k sets of non-crossing (non-nested) edges with respect to the vertex ordering. A \emphk-track layout of a graph consists of a vertex k-colouring, and a total order of each vertex colour class, such that between each pair of colour classes no two edges cross. The \emphstack-number (respectively, \emphqueue-number, \emphtrack-number) of a graph G, denoted by sn(G) (qn(G), tn(G)), is the minimum k such that G has a k-stack (k-queue, k-track) layout.\par This paper studies stack, queue, and track layouts of graph subdivisions. It is known that every graph has a 3-stack subdivision. The best known upper bound on the number of division vertices per edge in a 3-stack subdivision of an n-vertex graph G is improved from O(log n) to O(log min\sn(G),qn(G)\). This result reduces the question of whether queue-number is bounded by stack-number to whether 3-stack graphs have bounded queue number.\par It is proved that every graph has a 2-queue subdivision, a 4-track subdivision, and a mixed 1-stack 1-queue subdivision. All these values are optimal for every non-planar graph. In addition, we characterise those graphs with k-stack, k-queue, and k-track subdivisions, for all values of k. The number of division vertices per edge in the case of 2-queue and 4-track subdivisions, namely O(log qn(G)), is optimal to within a constant factor, for every graph G. \par Applications to 3D polyline grid drawings are presented. For example, it is proved that every graph G has a 3D polyline grid drawing with the vertices on a rectangular prism, and with O(log qn(G)) bends per edge. Finally, we establish a tight relationship between queue layouts and so-called 2-track thickness of bipartite graphs. \par]]> 0 <![CDATA[A graph is unfrozen with respect to k independent set if it has an independent set of size k after the addition of any edge. The problem of recognizing such graphs is known to be NP-complete. A graph is maximal if the addition of one edge means it is no longer unfrozen. We designate the problem of recognizing maximal unfrozen graphs as MAX(U(k-SET)) and show that this problem is CO-NP-complete. This partially fills a gap in known complexity cases of maximal NP-complete problems, and raises some interesting open conjectures discussed in the conclusion.]]> Sat, 01 Jan 2005 07:00:00 +0000 https://doi.org/10.46298/dmtcs.345 https://doi.org/10.46298/dmtcs.345 Abbas, Nesrine Culberson, Joseph Stewart, Lorna Abbas, Nesrine Culberson, Joseph Stewart, Lorna <![CDATA[A graph is unfrozen with respect to k independent set if it has an independent set of size k after the addition of any edge. The problem of recognizing such graphs is known to be NP-complete. A graph is maximal if the addition of one edge means it is no longer unfrozen. We designate the problem of recognizing maximal unfrozen graphs as MAX(U(k-SET)) and show that this problem is CO-NP-complete. This partially fills a gap in known complexity cases of maximal NP-complete problems, and raises some interesting open conjectures discussed in the conclusion.]]> 0 <![CDATA[The chordality of a graph with at least one cycle is the length of the longest induced cycle in it. The odd (even) chordality is defined to be the length of the longest induced odd (even) cycle in it. Chordal graphs have chordality at most 3. We show that co-circular-arc graphs and co-circle graphs have even chordality at most 4. We also identify few other classes of graphs having bounded (by a constant) chordality values.]]> Sat, 01 Jan 2005 07:00:00 +0000 https://doi.org/10.46298/dmtcs.343 https://doi.org/10.46298/dmtcs.343 Chandran, Sunil, Lozin, Vadim V. Subramanian, C.R. Chandran, Sunil, Lozin, Vadim V. Subramanian, C.R. <![CDATA[The chordality of a graph with at least one cycle is the length of the longest induced cycle in it. The odd (even) chordality is defined to be the length of the longest induced odd (even) cycle in it. Chordal graphs have chordality at most 3. We show that co-circular-arc graphs and co-circle graphs have even chordality at most 4. We also identify few other classes of graphs having bounded (by a constant) chordality values.]]> 0 <![CDATA[In this article we show the Markov-WZ Method in action as it finds rapidly converging series representations for a given hypergeometric series. We demonstrate the method by finding new representations for log(2),ζ (2) and ζ (3).]]> Sat, 01 Jan 2005 07:00:00 +0000 https://doi.org/10.46298/dmtcs.342 https://doi.org/10.46298/dmtcs.342 Mohammed, Mohamud Mohammed, Mohamud <![CDATA[In this article we show the Markov-WZ Method in action as it finds rapidly converging series representations for a given hypergeometric series. We demonstrate the method by finding new representations for log(2),ζ (2) and ζ (3).]]> 0 <![CDATA[We examine how we can define several probability distributions on the set of Boolean functions on a fixed number of variables, starting from a representation of Boolean expressions by trees. Analytic tools give us a systematic way to prove the existence of probability distributions, the main challenge being the actual computation of the distributions. We finally consider the relations between the probability of a Boolean function and its complexity.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3475 https://doi.org/10.46298/dmtcs.3475 Gardy, Danièle Gardy, Danièle <![CDATA[We examine how we can define several probability distributions on the set of Boolean functions on a fixed number of variables, starting from a representation of Boolean expressions by trees. Analytic tools give us a systematic way to prove the existence of probability distributions, the main challenge being the actual computation of the distributions. We finally consider the relations between the probability of a Boolean function and its complexity.]]> 0 <![CDATA[Let $\mathcal{P}$ be a collection of nontrivial simple paths in a tree $T$. The edge intersection graph of $\mathcal{P}$, denoted by EPT($\mathcal{P}$), has vertex set that corresponds to the members of $\mathcal{P}$, and two vertices are joined by an edge if the corresponding members of $\mathcal{P}$ share a common edge in $T$. An undirected graph $G$ is called an edge intersection graph of paths in a tree, if $G = EPT(\mathcal{P})$ for some $\mathcal{P}$ and $T$. The EPT graphs are useful in network applications. Scheduling undirected calls in a tree or assigning wavelengths to virtual connections in an optical tree network are equivalent to coloring its EPT graph. It is known that recognition and coloring of EPT graphs are NP-complete problems. However, the EPT graphs restricted to host trees of vertex degree 3 are precisely the chordal EPT graphs, and therefore can be colored in polynomial time complexity. We prove a new analogous result that weakly chordal EPT graphs are precisely the EPT graphs with host tree restricted to degree 4. This also implies that the coloring of the edge intersection graph of paths in a degree 4 tree is polynomial. We raise a number of intriguing conjectures regarding related families of graphs.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3439 https://doi.org/10.46298/dmtcs.3439 Golumbic, Martin Charles Lipshteyn, Marina Stern, Michal Golumbic, Martin Charles Lipshteyn, Marina Stern, Michal <![CDATA[Let $\mathcal{P}$ be a collection of nontrivial simple paths in a tree $T$. The edge intersection graph of $\mathcal{P}$, denoted by EPT($\mathcal{P}$), has vertex set that corresponds to the members of $\mathcal{P}$, and two vertices are joined by an edge if the corresponding members of $\mathcal{P}$ share a common edge in $T$. An undirected graph $G$ is called an edge intersection graph of paths in a tree, if $G = EPT(\mathcal{P})$ for some $\mathcal{P}$ and $T$. The EPT graphs are useful in network applications. Scheduling undirected calls in a tree or assigning wavelengths to virtual connections in an optical tree network are equivalent to coloring its EPT graph. It is known that recognition and coloring of EPT graphs are NP-complete problems. However, the EPT graphs restricted to host trees of vertex degree 3 are precisely the chordal EPT graphs, and therefore can be colored in polynomial time complexity. We prove a new analogous result that weakly chordal EPT graphs are precisely the EPT graphs with host tree restricted to degree 4. This also implies that the coloring of the edge intersection graph of paths in a degree 4 tree is polynomial. We raise a number of intriguing conjectures regarding related families of graphs.]]> 0 <![CDATA[We enumerate walks in the plane $\mathbb{R}^2$, with steps East and North, that stop as soon as they reach a given line; these walks are counted according to the distance of the line to the origin, and we study the asymptotic behavior when the line has a fixed slope and moves away from the origin. When the line has a rational slope, we study a more general class of walks, and give exact as well as asymptotic enumerative results; for this, we define a nice bijection from our walks to words of a rational language. For a general slope, asymptotic results are obtained; in this case, the method employed leads us to find asymptotic results for a wider class of walks in $\mathbb{R}^m$.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3449 https://doi.org/10.46298/dmtcs.3449 Nadeau, Philippe Nadeau, Philippe <![CDATA[We enumerate walks in the plane $\mathbb{R}^2$, with steps East and North, that stop as soon as they reach a given line; these walks are counted according to the distance of the line to the origin, and we study the asymptotic behavior when the line has a fixed slope and moves away from the origin. When the line has a rational slope, we study a more general class of walks, and give exact as well as asymptotic enumerative results; for this, we define a nice bijection from our walks to words of a rational language. For a general slope, asymptotic results are obtained; in this case, the method employed leads us to find asymptotic results for a wider class of walks in $\mathbb{R}^m$.]]> 0 <![CDATA[We establish sufficient conditions for a toric lattice $T_{m,n}$ to be Hamiltonian. Also, we give some asymptotics for the number of Hamiltonian cycles in $T_{m,n}$.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3448 https://doi.org/10.46298/dmtcs.3448 Leontiev, Vladimir K. Leontiev, Vladimir K. <![CDATA[We establish sufficient conditions for a toric lattice $T_{m,n}$ to be Hamiltonian. Also, we give some asymptotics for the number of Hamiltonian cycles in $T_{m,n}$.]]> 0 <![CDATA[The local chromatic number of a graph, introduced by Erdős et al., is the minimum number of colors that must appear in the closed neighborhood of some vertex in any proper coloring of the graph. This talk would like to survey some of our recent results on this parameter. We give a lower bound for the local chromatic number in terms of the lower bound of the chromatic number provided by the topological method introduced by Lovász. We show that this bound is tight in many cases. In particular, we determine the local chromatic number of certain odd chromatic Schrijver graphs and generalized Mycielski graphs. We further elaborate on the case of $4$-chromatic graphs and, in particular, on surface quadrangulations.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3447 https://doi.org/10.46298/dmtcs.3447 Simonyi, Gábor Tardos, Gábor Simonyi, Gábor Tardos, Gábor <![CDATA[The local chromatic number of a graph, introduced by Erdős et al., is the minimum number of colors that must appear in the closed neighborhood of some vertex in any proper coloring of the graph. This talk would like to survey some of our recent results on this parameter. We give a lower bound for the local chromatic number in terms of the lower bound of the chromatic number provided by the topological method introduced by Lovász. We show that this bound is tight in many cases. In particular, we determine the local chromatic number of certain odd chromatic Schrijver graphs and generalized Mycielski graphs. We further elaborate on the case of $4$-chromatic graphs and, in particular, on surface quadrangulations.]]> 0 <![CDATA[We improve the lower bounds on Klarner's constant, which describes the exponential growth rate of the number of polyominoes (connected subsets of grid squares) with a given number of squares. We achieve this by analyzing polyominoes on a different surface, a so-called $\textit{twisted cylinder}$ by the transfer matrix method. A bijective representation of the "states'' of partial solutions is crucial for allowing a compact representation of the successive iteration vectors for the transfer matrix method.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3446 https://doi.org/10.46298/dmtcs.3446 Barequet, Gill Moffie, Micha Ribó, Ares Rote, Günter Barequet, Gill Moffie, Micha Ribó, Ares Rote, Günter <![CDATA[We improve the lower bounds on Klarner's constant, which describes the exponential growth rate of the number of polyominoes (connected subsets of grid squares) with a given number of squares. We achieve this by analyzing polyominoes on a different surface, a so-called $\textit{twisted cylinder}$ by the transfer matrix method. A bijective representation of the "states'' of partial solutions is crucial for allowing a compact representation of the successive iteration vectors for the transfer matrix method.]]> 0 <![CDATA[We show that any graph of maximum degree at most $3$ has a two-coloring, such that one color-class is an independent set while the other color induces monochromatic components of order at most $189$. On the other hand for any constant $C$ we exhibit a $4$-regular graph, such that the deletion of any independent set leaves at least one component of order greater than $C$. Similar results are obtained for coloring graphs of given maximum degree with $k+ \ell$ colors such that $k$ parts form an independent set and $\ell$ parts span components of order bounded by a constant. A lot of interesting questions remain open.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3445 https://doi.org/10.46298/dmtcs.3445 Berke, Robert Szabó, Tibor Berke, Robert Szabó, Tibor <![CDATA[We show that any graph of maximum degree at most $3$ has a two-coloring, such that one color-class is an independent set while the other color induces monochromatic components of order at most $189$. On the other hand for any constant $C$ we exhibit a $4$-regular graph, such that the deletion of any independent set leaves at least one component of order greater than $C$. Similar results are obtained for coloring graphs of given maximum degree with $k+ \ell$ colors such that $k$ parts form an independent set and $\ell$ parts span components of order bounded by a constant. A lot of interesting questions remain open.]]> 0 <![CDATA[Hamiltionian chain is a generalisation of hamiltonian cycles for hypergraphs. Among the several possible ways of generalisations this is probably the most strong one, it requires the strongest structure. Since there are many interesting questions about hamiltonian cycles in graphs, we can try to answer these questions for hypergraphs, too. In the present article we give a survey on results about such questions.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3444 https://doi.org/10.46298/dmtcs.3444 Katona, Gyula Y. Katona, Gyula Y. <![CDATA[Hamiltionian chain is a generalisation of hamiltonian cycles for hypergraphs. Among the several possible ways of generalisations this is probably the most strong one, it requires the strongest structure. Since there are many interesting questions about hamiltonian cycles in graphs, we can try to answer these questions for hypergraphs, too. In the present article we give a survey on results about such questions.]]> 0 <![CDATA[A family $\mathcal{P} = \{\pi_1, \ldots , \pi_q\}$ of permutations of $[n]=\{1,\ldots,n\}$ is $\textit{completely}$ $k$-$\textit{scrambling}$ [Spencer, 1972; Füredi, 1996] if for any distinct $k$ points $x_1,\ldots,x_k \in [n]$, permutations $\pi_i$'s in $\mathcal{P}$ produce all $k!$ possible orders on $\pi_i (x_1),\ldots, \pi_i(x_k)$. Let $N^{\ast}(n,k)$ be the minimum size of such a family. This paper focuses on the case $k=3$. By a simple explicit construction, we show the following upper bound, which we express together with the lower bound due to Füredi for comparison. $\frac{2}{ \log _2e} \log_2 n \leq N^{\ast}(n,3) \leq 2\log_2n + (1+o(1)) \log_2 \log _2n$. We also prove the existence of $\lim_{n \to \infty} N^{\ast}(n,3) / \log_2 n = c_3$. Determining the value $c_3$ and proving the existence of $\lim_{n \to \infty} N^{\ast}(n,k) / \log_2 n = c_k$ for $k \geq 4$ remain open.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3443 https://doi.org/10.46298/dmtcs.3443 Tarui, Jun Tarui, Jun <![CDATA[A family $\mathcal{P} = \{\pi_1, \ldots , \pi_q\}$ of permutations of $[n]=\{1,\ldots,n\}$ is $\textit{completely}$ $k$-$\textit{scrambling}$ [Spencer, 1972; Füredi, 1996] if for any distinct $k$ points $x_1,\ldots,x_k \in [n]$, permutations $\pi_i$'s in $\mathcal{P}$ produce all $k!$ possible orders on $\pi_i (x_1),\ldots, \pi_i(x_k)$. Let $N^{\ast}(n,k)$ be the minimum size of such a family. This paper focuses on the case $k=3$. By a simple explicit construction, we show the following upper bound, which we express together with the lower bound due to Füredi for comparison. $\frac{2}{ \log _2e} \log_2 n \leq N^{\ast}(n,3) \leq 2\log_2n + (1+o(1)) \log_2 \log _2n$. We also prove the existence of $\lim_{n \to \infty} N^{\ast}(n,3) / \log_2 n = c_3$. Determining the value $c_3$ and proving the existence of $\lim_{n \to \infty} N^{\ast}(n,k) / \log_2 n = c_k$ for $k \geq 4$ remain open.]]> 0 <![CDATA[Given a family $\{u_i,v_i\}_{i=1}^k$ of pairwise distinct vertices of the $n$-dimensional hypercube $Q_n$ such that the distance of $u_i$ and $v_i$ is odd and $k \leq n-1$, there exists a family $\{P_i\}_{i=1}^k$ of paths such that $u_i$ and $v_i$ are the endvertices of $P_i$ and $\{V(P_i)\}_{i=1}^k$ partitions $V(Q_n)$. This holds for any $n \geq 2$ with one exception in the case when $n=k+1=4$. On the other hand, for any $n \geq 3$ there exist $n$ pairs of vertices satisfying the above condition for which such a family of spanning paths does not exist. We suggest further generalization of this result and explore a relationship to the problem of hamiltonicity of hypercubes with faulty vertices.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3442 https://doi.org/10.46298/dmtcs.3442 Dvořák, Tomáš Gregor, Petr Koubek, Václav Dvořák, Tomáš Gregor, Petr Koubek, Václav <![CDATA[Given a family $\{u_i,v_i\}_{i=1}^k$ of pairwise distinct vertices of the $n$-dimensional hypercube $Q_n$ such that the distance of $u_i$ and $v_i$ is odd and $k \leq n-1$, there exists a family $\{P_i\}_{i=1}^k$ of paths such that $u_i$ and $v_i$ are the endvertices of $P_i$ and $\{V(P_i)\}_{i=1}^k$ partitions $V(Q_n)$. This holds for any $n \geq 2$ with one exception in the case when $n=k+1=4$. On the other hand, for any $n \geq 3$ there exist $n$ pairs of vertices satisfying the above condition for which such a family of spanning paths does not exist. We suggest further generalization of this result and explore a relationship to the problem of hamiltonicity of hypercubes with faulty vertices.]]> 0 <![CDATA[Raspaud and Sopena showed that the oriented chromatic number of a graph with acyclic chromatic number $k$ is at most $k2^{k-1}$. We prove that this bound is tight for $k \geq 3$. We also show that some improper and/or acyclic colorings are $\mathrm{NP}$-complete on a class $\mathcal{C}$ of planar graphs. We try to get the most restrictive conditions on the class $\mathcal{C}$, such as having large girth and small maximum degree. In particular, we obtain the $\mathrm{NP}$-completeness of $3$-$\mathrm{ACYCLIC \space COLORABILITY}$ on bipartite planar graphs with maximum degree $4$, and of $4$-$\mathrm{ACYCLIC \space COLORABILITY}$ on bipartite planar graphs with maximum degree $8$.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3441 https://doi.org/10.46298/dmtcs.3441 Ochem, Pascal Ochem, Pascal <![CDATA[Raspaud and Sopena showed that the oriented chromatic number of a graph with acyclic chromatic number $k$ is at most $k2^{k-1}$. We prove that this bound is tight for $k \geq 3$. We also show that some improper and/or acyclic colorings are $\mathrm{NP}$-complete on a class $\mathcal{C}$ of planar graphs. We try to get the most restrictive conditions on the class $\mathcal{C}$, such as having large girth and small maximum degree. In particular, we obtain the $\mathrm{NP}$-completeness of $3$-$\mathrm{ACYCLIC \space COLORABILITY}$ on bipartite planar graphs with maximum degree $4$, and of $4$-$\mathrm{ACYCLIC \space COLORABILITY}$ on bipartite planar graphs with maximum degree $8$.]]> 0 <![CDATA[A random geometric graph $G_n$ is obtained as follows. We take $X_1, X_2, \ldots, X_n ∈\mathbb{R}^d$ at random (i.i.d. according to some probability distribution ν on $\mathbb{R}^d$). For $i ≠j$ we join $X_i$ and $X_j$ by an edge if $║X_i - X_j ║< r(n)$. We study the properties of the chromatic number $χ _n$ and clique number $ω _n$ of this graph as n becomes large, where we assume that $r(n) →0$. We allow any choice $ν$ that has a bounded density function and $║. ║$ may be any norm on $ℝ^d$. Depending on the choice of $r(n)$, qualitatively different types of behaviour can be observed. We distinguish three main cases, in terms of the key quantity $n r^d$ (which is a measure of the average degree). If $r(n)$ is such that $\frac{nr^d}{ln n} →0$ as $n →∞$ then $\frac{χ _n}{ ω _n} →1$ almost surely. If n $\frac{r^d }{\ln n} →∞$ then $\frac{χ _n }{ ω _n} →1 / δ$ almost surely, where $δ$ is the (translational) packing density of the unit ball $B := \{ x ∈ℝ^d: ║x║< 1 \}$ (i.e. $δ$ is the proportion of $d$-space that can be filled with disjoint translates of $B$). If $\frac{n r^d }{\ln n} →t ∈(0,∞)$ then $\frac{χ _n }{ ω _n}$ tends almost surely to a constant that can be bounded in terms of $δ$ and $t$. These results extend earlier work of McDiarmid and Penrose. The proofs in fact yield separate expressions for $χ _n$ and $ω _n$. We are also able to prove a conjecture by Penrose. This states that when $\frac{n r^d }{ln n} →0$ then the clique number becomes focussed on two adjacent integers, meaning that there exists a sequence $k(n)$ such that $\mathbb{P}( ω _n ∈\{k(n), k(n)+1\}) →1$ as $n →∞$. The analogous result holds for the chromatic number (and for the maximum degree, as was already shown by Penrose in the uniform case).]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3440 https://doi.org/10.46298/dmtcs.3440 McDiarmid, Colin J. H. Müller, Tobias McDiarmid, Colin J. H. Müller, Tobias <![CDATA[A random geometric graph $G_n$ is obtained as follows. We take $X_1, X_2, \ldots, X_n ∈\mathbb{R}^d$ at random (i.i.d. according to some probability distribution ν on $\mathbb{R}^d$). For $i ≠j$ we join $X_i$ and $X_j$ by an edge if $║X_i - X_j ║< r(n)$. We study the properties of the chromatic number $χ _n$ and clique number $ω _n$ of this graph as n becomes large, where we assume that $r(n) →0$. We allow any choice $ν$ that has a bounded density function and $║. ║$ may be any norm on $ℝ^d$. Depending on the choice of $r(n)$, qualitatively different types of behaviour can be observed. We distinguish three main cases, in terms of the key quantity $n r^d$ (which is a measure of the average degree). If $r(n)$ is such that $\frac{nr^d}{ln n} →0$ as $n →∞$ then $\frac{χ _n}{ ω _n} →1$ almost surely. If n $\frac{r^d }{\ln n} →∞$ then $\frac{χ _n }{ ω _n} →1 / δ$ almost surely, where $δ$ is the (translational) packing density of the unit ball $B := \{ x ∈ℝ^d: ║x║< 1 \}$ (i.e. $δ$ is the proportion of $d$-space that can be filled with disjoint translates of $B$). If $\frac{n r^d }{\ln n} →t ∈(0,∞)$ then $\frac{χ _n }{ ω _n}$ tends almost surely to a constant that can be bounded in terms of $δ$ and $t$. These results extend earlier work of McDiarmid and Penrose. The proofs in fact yield separate expressions for $χ _n$ and $ω _n$. We are also able to prove a conjecture by Penrose. This states that when $\frac{n r^d }{ln n} →0$ then the clique number becomes focussed on two adjacent integers, meaning that there exists a sequence $k(n)$ such that $\mathbb{P}( ω _n ∈\{k(n), k(n)+1\}) →1$ as $n →∞$. The analogous result holds for the chromatic number (and for the maximum degree, as was already shown by Penrose in the uniform case).]]> 0 <![CDATA[Using a fixed set of colors $C$, Ann and Ben color the edges of a graph $G$ so that no monochromatic cycle may appear. Ann wins if all edges of $G$ have been colored, while Ben wins if completing a coloring is not possible. The minimum size of $C$ for which Ann has a winning strategy is called the $\textit{game arboricity}$ of $G$, denoted by $A_g(G)$. We prove that $A_g(G) \leq 3k$ for any graph $G$ of arboricity $k$, and that there are graphs such that $A_g(G) \geq 2k-2$. The upper bound is achieved by a suitable version of the activation strategy, used earlier for the vertex coloring game. We also provide other strategie based on induction.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3428 https://doi.org/10.46298/dmtcs.3428 Bartnicki, Tomasz Grytczuk, Jaroslaw Kierstead, Hal Bartnicki, Tomasz Grytczuk, Jaroslaw Kierstead, Hal <![CDATA[Using a fixed set of colors $C$, Ann and Ben color the edges of a graph $G$ so that no monochromatic cycle may appear. Ann wins if all edges of $G$ have been colored, while Ben wins if completing a coloring is not possible. The minimum size of $C$ for which Ann has a winning strategy is called the $\textit{game arboricity}$ of $G$, denoted by $A_g(G)$. We prove that $A_g(G) \leq 3k$ for any graph $G$ of arboricity $k$, and that there are graphs such that $A_g(G) \geq 2k-2$. The upper bound is achieved by a suitable version of the activation strategy, used earlier for the vertex coloring game. We also provide other strategie based on induction.]]> 0 <![CDATA[In this paper we improve the best known bound for the $L(p,1)$-labelling of graphs with given maximal degree.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3438 https://doi.org/10.46298/dmtcs.3438 Gonçalves, Daniel Gonçalves, Daniel <![CDATA[In this paper we improve the best known bound for the $L(p,1)$-labelling of graphs with given maximal degree.]]> 0 <![CDATA[We consider a new type of extremal hypergraph problem: given an $r$-graph $\mathcal{F}$ and an integer $k≥2$ determine the maximum number of edges in an $\mathcal{F}$-free, $k$-colourable $r$-graph on $n$ vertices. Our motivation for studying such problems is that it allows us to give a new upper bound for an old problem due to Turán. We show that a 3-graph in which any four vertices span at most two edges has density less than $\frac{33}{ 100}$, improving previous bounds of $\frac{1}{ 3}$ due to de Caen [1], and $\frac{1}{ 3}-4.5305×10^-6$ due to Mubayi [9].]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3437 https://doi.org/10.46298/dmtcs.3437 Talbot, John Talbot, John <![CDATA[We consider a new type of extremal hypergraph problem: given an $r$-graph $\mathcal{F}$ and an integer $k≥2$ determine the maximum number of edges in an $\mathcal{F}$-free, $k$-colourable $r$-graph on $n$ vertices. Our motivation for studying such problems is that it allows us to give a new upper bound for an old problem due to Turán. We show that a 3-graph in which any four vertices span at most two edges has density less than $\frac{33}{ 100}$, improving previous bounds of $\frac{1}{ 3}$ due to de Caen [1], and $\frac{1}{ 3}-4.5305×10^-6$ due to Mubayi [9].]]> 0 <![CDATA[We analyze the one-dimensional version of Jim Propp's $P$-machine, a simple deterministic process that simulates a random walk on $\mathbb{Z}$. The "output'' of the machine is astonishingly close to the expected behavior of a random walk, even on long intervals of space and time.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3436 https://doi.org/10.46298/dmtcs.3436 Cooper, Joshua Doerr, Benjamin Spencer, Joel Tardos, Gábor Cooper, Joshua Doerr, Benjamin Spencer, Joel Tardos, Gábor <![CDATA[We analyze the one-dimensional version of Jim Propp's $P$-machine, a simple deterministic process that simulates a random walk on $\mathbb{Z}$. The "output'' of the machine is astonishingly close to the expected behavior of a random walk, even on long intervals of space and time.]]> 0 <![CDATA[Before this work, at least 762 inequivalent Hadamard matrices of order 36 were known. We found 7238 Hadamard matrices of order 36 and 522 inequivalent [72,36,12] double-even self-dual codes which are obtained from all 2-(35,17,8) designs with an automorphism of order 3 and 2 fixed points and blocks.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3435 https://doi.org/10.46298/dmtcs.3435 Bouyukliev, Iliya Fack, Veerle Winne, Joost Bouyukliev, Iliya Fack, Veerle Winne, Joost <![CDATA[Before this work, at least 762 inequivalent Hadamard matrices of order 36 were known. We found 7238 Hadamard matrices of order 36 and 522 inequivalent [72,36,12] double-even self-dual codes which are obtained from all 2-(35,17,8) designs with an automorphism of order 3 and 2 fixed points and blocks.]]> 0 <![CDATA[A proper vertex coloring of a non oriented graph $G=(V,E)$ is linear if the graph induced by the vertices of two color classes is a forest of paths. A graph $G$ is $L$-list colorable if for a given list assignment $L=\{L(v): v∈V\}$, there exists a proper coloring $c$ of $G$ such that $c(v)∈L(v)$ for all $v∈V$. If $G$ is $L$-list colorable for every list assignment with $|L(v)|≥k$ for all $v∈V$, then $G$ is said $k$-choosable. A graph is said to be lineary $k$-choosable if the coloring obtained is linear. In this paper, we investigate the linear choosability of graphs for some families of graphs: graphs with small maximum degree, with given maximum average degree, planar graphs... Moreover, we prove that determining whether a bipartite subcubic planar graph is lineary 3-colorable is an NP-complete problem.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3434 https://doi.org/10.46298/dmtcs.3434 Esperet, Louis Montassier, Mickael Raspaud, André Esperet, Louis Montassier, Mickael Raspaud, André <![CDATA[A proper vertex coloring of a non oriented graph $G=(V,E)$ is linear if the graph induced by the vertices of two color classes is a forest of paths. A graph $G$ is $L$-list colorable if for a given list assignment $L=\{L(v): v∈V\}$, there exists a proper coloring $c$ of $G$ such that $c(v)∈L(v)$ for all $v∈V$. If $G$ is $L$-list colorable for every list assignment with $|L(v)|≥k$ for all $v∈V$, then $G$ is said $k$-choosable. A graph is said to be lineary $k$-choosable if the coloring obtained is linear. In this paper, we investigate the linear choosability of graphs for some families of graphs: graphs with small maximum degree, with given maximum average degree, planar graphs... Moreover, we prove that determining whether a bipartite subcubic planar graph is lineary 3-colorable is an NP-complete problem.]]> 0 <![CDATA[The minor crossing number of a graph $G$, $rmmcr(G)$, is defined as the minimum crossing number of all graphs that contain $G$ as a minor. We present some basic properties of this new minor-monotone graph invariant. We give estimates on mmcr for some important graph families using the topological structure of graphs satisfying \$mcr(G) ≤k$.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3433 https://doi.org/10.46298/dmtcs.3433 Bokal, Drago Fijavž, Gašper Mohar, Bojan Bokal, Drago Fijavž, Gašper Mohar, Bojan <![CDATA[The minor crossing number of a graph $G$, $rmmcr(G)$, is defined as the minimum crossing number of all graphs that contain $G$ as a minor. We present some basic properties of this new minor-monotone graph invariant. We give estimates on mmcr for some important graph families using the topological structure of graphs satisfying \$mcr(G) ≤k$.]]> 0 <![CDATA[The present paper connects sharpenings of Sauer's bound on forbidden configurations with color critical hypergraphs. We define a matrix to be \emphsimple if it is a $(0,1)-matrix$ with no repeated columns. Let $F$ be $a k× l (0,1)-matrix$ (the forbidden configuration). Assume $A$ is an $m× n$ simple matrix which has no submatrix which is a row and column permutation of $F$. We define $forb(m,F)$ as the best possible upper bound on n, for such a matrix $A$, which depends on m and $F$. It is known that $forb(m,F)=O(m^k)$ for any $F$, and Sauer's bond states that $forb(m,F)=O(m^k-1)$ fore simple $F$. We give sufficient condition for non-simple $F$ to have the same bound using linear algebra methods to prove a generalization of a result of Lovász on color critical hypergraphs.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3432 https://doi.org/10.46298/dmtcs.3432 Anstee, Richard Fleming, Balin Füredi, Zoltán Sali, Attila Anstee, Richard Fleming, Balin Füredi, Zoltán Sali, Attila <![CDATA[The present paper connects sharpenings of Sauer's bound on forbidden configurations with color critical hypergraphs. We define a matrix to be \emphsimple if it is a $(0,1)-matrix$ with no repeated columns. Let $F$ be $a k× l (0,1)-matrix$ (the forbidden configuration). Assume $A$ is an $m× n$ simple matrix which has no submatrix which is a row and column permutation of $F$. We define $forb(m,F)$ as the best possible upper bound on n, for such a matrix $A$, which depends on m and $F$. It is known that $forb(m,F)=O(m^k)$ for any $F$, and Sauer's bond states that $forb(m,F)=O(m^k-1)$ fore simple $F$. We give sufficient condition for non-simple $F$ to have the same bound using linear algebra methods to prove a generalization of a result of Lovász on color critical hypergraphs.]]> 0 <![CDATA[In this paper we concern ourself with the question, whether there exists a fix-free code for a given sequence of codeword lengths. We focus mostly on results which shows the $\frac{3 }{ 4}$-conjecture for special kinds of lengths sequences.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3431 https://doi.org/10.46298/dmtcs.3431 Deppe, Christian Schnettler, Holger Deppe, Christian Schnettler, Holger <![CDATA[In this paper we concern ourself with the question, whether there exists a fix-free code for a given sequence of codeword lengths. We focus mostly on results which shows the $\frac{3 }{ 4}$-conjecture for special kinds of lengths sequences.]]> 0 <![CDATA[An edge in a drawing of a graph is called $\textit{even}$ if it intersects every other edge of the graph an even number of times. Pach and Tóth proved that a graph can always be redrawn such that its even edges are not involved in any intersections. We give a new, and significantly simpler, proof of a slightly stronger statement. We show two applications of this strengthened result: an easy proof of a theorem of Hanani and Tutte (not using Kuratowski's theorem), and the result that the odd crossing number of a graph equals the crossing number of the graph for values of at most $3$. We begin with a disarmingly simple proof of a weak (but standard) version of the theorem by Hanani and Tutte.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3430 https://doi.org/10.46298/dmtcs.3430 Pelsmajer, Michael J. Schaefer, Marcus Štefankovič, Daniel Pelsmajer, Michael J. Schaefer, Marcus Štefankovič, Daniel <![CDATA[An edge in a drawing of a graph is called $\textit{even}$ if it intersects every other edge of the graph an even number of times. Pach and Tóth proved that a graph can always be redrawn such that its even edges are not involved in any intersections. We give a new, and significantly simpler, proof of a slightly stronger statement. We show two applications of this strengthened result: an easy proof of a theorem of Hanani and Tutte (not using Kuratowski's theorem), and the result that the odd crossing number of a graph equals the crossing number of the graph for values of at most $3$. We begin with a disarmingly simple proof of a weak (but standard) version of the theorem by Hanani and Tutte.]]> 0 <![CDATA[We identify the class of directed one-trees and prove the so-called min-max theorem for them. As a consequence, we establish the equality of directed tree-width and a new measure, $d$-width, on this class of graphs. In addition, we prove a property of all directed one-trees and use this property to create an $O(n^2)$ recognition algorithm and an $O(n^2)$ algorithm for solving the Hamiltonian cycle problem on directed one-trees.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3429 https://doi.org/10.46298/dmtcs.3429 Evans, William Safari, Mohammad Ali Evans, William Safari, Mohammad Ali <![CDATA[We identify the class of directed one-trees and prove the so-called min-max theorem for them. As a consequence, we establish the equality of directed tree-width and a new measure, $d$-width, on this class of graphs. In addition, we prove a property of all directed one-trees and use this property to create an $O(n^2)$ recognition algorithm and an $O(n^2)$ algorithm for solving the Hamiltonian cycle problem on directed one-trees.]]> 0 <![CDATA[The problem of finding the convex hull of the intersection points of random lines was studied in Devroye and Toussaint, 1993 and Langerman, Golin and Steiger, 2002, and algorithms with expected linear time were found. We improve the previous results of the model in Devroye and Toussaint, 1993 by giving a universal algorithm for a wider range of distributions.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3364 https://doi.org/10.46298/dmtcs.3364 Berend, Daniel Braverman, Vladimir Berend, Daniel Braverman, Vladimir <![CDATA[The problem of finding the convex hull of the intersection points of random lines was studied in Devroye and Toussaint, 1993 and Langerman, Golin and Steiger, 2002, and algorithms with expected linear time were found. We improve the previous results of the model in Devroye and Toussaint, 1993 by giving a universal algorithm for a wider range of distributions.]]> 0 <![CDATA[In this paper, we are concerned with random sampling of an n dimensional integral point on an $(n-1)$ dimensional simplex according to a multivariate discrete distribution. We employ sampling via Markov chain and propose two "hit-and-run'' chains, one is for approximate sampling and the other is for perfect sampling. We introduce an idea of alternating inequalities and show that a logarithmic separable concave function satisfies the alternating inequalities. If a probability function satisfies alternating inequalities, then our chain for approximate sampling mixes in $\textit{O}(n^2 \textit{ln}(Kɛ^{-1}))$, namely $(1/2)n(n-1) \textit{ln}(K ɛ^{-1})$, where $K$ is the side length of the simplex and $ɛ (0<ɛ<1)$ is an error rate. On the same condition, we design another chain and a perfect sampler based on monotone CFTP (Coupling from the Past). We discuss a condition that the expected number of total transitions of the chain in the perfect sampler is bounded by $\textit{O}(n^3 \textit{ln}(Kn))$.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3374 https://doi.org/10.46298/dmtcs.3374 Kijima, Shuji Matsui, Tomomi Kijima, Shuji Matsui, Tomomi <![CDATA[In this paper, we are concerned with random sampling of an n dimensional integral point on an $(n-1)$ dimensional simplex according to a multivariate discrete distribution. We employ sampling via Markov chain and propose two "hit-and-run'' chains, one is for approximate sampling and the other is for perfect sampling. We introduce an idea of alternating inequalities and show that a logarithmic separable concave function satisfies the alternating inequalities. If a probability function satisfies alternating inequalities, then our chain for approximate sampling mixes in $\textit{O}(n^2 \textit{ln}(Kɛ^{-1}))$, namely $(1/2)n(n-1) \textit{ln}(K ɛ^{-1})$, where $K$ is the side length of the simplex and $ɛ (0<ɛ<1)$ is an error rate. On the same condition, we design another chain and a perfect sampler based on monotone CFTP (Coupling from the Past). We discuss a condition that the expected number of total transitions of the chain in the perfect sampler is bounded by $\textit{O}(n^3 \textit{ln}(Kn))$.]]> 0 <![CDATA[We investigate distances between pairs of nodes in digital trees (digital search trees (DST), and tries). By analytic techniques, such as the Mellin Transform and poissonization, we describe a program to determine the moments of these distances. The program is illustrated on the mean and variance. One encounters delayed Mellin transform equations, which we solve by inspection. Interestingly, the unbiased case gives a bounded variance, whereas the biased case gives a variance growing with the number of keys. It is therefore possible in the biased case to show that an appropriately normalized version of the distance converges to a limit. The complexity of moment calculation increases substantially with each higher moment; A shortcut to the limit is needed via a method that avoids the computation of all moments. Toward this end, we utilize the contraction method to show that in biased digital search trees the distribution of a suitably normalized version of the distances approaches a limit that is the fixed-point solution (in the Wasserstein space) of a distributional equation. An explicit solution to the fixed-point equation is readily demonstrated to be Gaussian.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3373 https://doi.org/10.46298/dmtcs.3373 Aguech, Rafik Lasmar, Nabil Mahmoud, Hosam Aguech, Rafik Lasmar, Nabil Mahmoud, Hosam <![CDATA[We investigate distances between pairs of nodes in digital trees (digital search trees (DST), and tries). By analytic techniques, such as the Mellin Transform and poissonization, we describe a program to determine the moments of these distances. The program is illustrated on the mean and variance. One encounters delayed Mellin transform equations, which we solve by inspection. Interestingly, the unbiased case gives a bounded variance, whereas the biased case gives a variance growing with the number of keys. It is therefore possible in the biased case to show that an appropriately normalized version of the distance converges to a limit. The complexity of moment calculation increases substantially with each higher moment; A shortcut to the limit is needed via a method that avoids the computation of all moments. Toward this end, we utilize the contraction method to show that in biased digital search trees the distribution of a suitably normalized version of the distances approaches a limit that is the fixed-point solution (in the Wasserstein space) of a distributional equation. An explicit solution to the fixed-point equation is readily demonstrated to be Gaussian.]]> 0 <![CDATA[We study a gcd algorithm directed by Least Significant Bits, the so―called LSB algorithm, and provide a precise average―case analysis of its main parameters [number of iterations, number of shifts, etc...]. This analysis is based on a precise study of the dynamical systems which provide a continuous extension of the algorithm, and, here, it is proved convenient to use both a 2―adic extension and a real one. This leads to the framework of products of random matrices, and our results thus involve a constant $γ$ which is the Lyapunov exponent of the set of matrices relative to the algorithm. The algorithm can be viewed as a race between a dyadic hare with a speed of 2 bits by step and a "real'' tortoise with a speed equal to $γ /\textit{log} \ 2 \sim 0.05$ bits by step. Even if the tortoise starts before the hare, the hare easily catches up with the tortoise [unlike in Aesop's fable [Ae]\ldots], and the algorithm terminates.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3372 https://doi.org/10.46298/dmtcs.3372 Daireaux, Benoît Maume-Deschamps, Véronique Vallée, Brigitte Daireaux, Benoît Maume-Deschamps, Véronique Vallée, Brigitte <![CDATA[We study a gcd algorithm directed by Least Significant Bits, the so―called LSB algorithm, and provide a precise average―case analysis of its main parameters [number of iterations, number of shifts, etc...]. This analysis is based on a precise study of the dynamical systems which provide a continuous extension of the algorithm, and, here, it is proved convenient to use both a 2―adic extension and a real one. This leads to the framework of products of random matrices, and our results thus involve a constant $γ$ which is the Lyapunov exponent of the set of matrices relative to the algorithm. The algorithm can be viewed as a race between a dyadic hare with a speed of 2 bits by step and a "real'' tortoise with a speed equal to $γ /\textit{log} \ 2 \sim 0.05$ bits by step. Even if the tortoise starts before the hare, the hare easily catches up with the tortoise [unlike in Aesop's fable [Ae]\ldots], and the algorithm terminates.]]> 0 <![CDATA[In this report, we prove that under a Markovian model of order one, the average depth of suffix trees of index n is asymptotically similar to the average depth of tries (a.k.a. digital trees) built on n independent strings. This leads to an asymptotic behavior of $(\log{n})/h + C$ for the average of the depth of the suffix tree, where $h$ is the entropy of the Markov model and $C$ is constant. Our proof compares the generating functions for the average depth in tries and in suffix trees; the difference between these generating functions is shown to be asymptotically small. We conclude by using the asymptotic behavior of the average depth in a trie under the Markov model found by Jacquet and Szpankowski ([JaSz91]).]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3371 https://doi.org/10.46298/dmtcs.3371 Fayolle, Julien Ward, Mark Daniel Fayolle, Julien Ward, Mark Daniel <![CDATA[In this report, we prove that under a Markovian model of order one, the average depth of suffix trees of index n is asymptotically similar to the average depth of tries (a.k.a. digital trees) built on n independent strings. This leads to an asymptotic behavior of $(\log{n})/h + C$ for the average of the depth of the suffix tree, where $h$ is the entropy of the Markov model and $C$ is constant. Our proof compares the generating functions for the average depth in tries and in suffix trees; the difference between these generating functions is shown to be asymptotically small. We conclude by using the asymptotic behavior of the average depth in a trie under the Markov model found by Jacquet and Szpankowski ([JaSz91]).]]> 0 <![CDATA[Using recent results on singularity analysis for Hadamard products of generating functions, we obtain the limiting distributions for additive functionals on $m$-ary search trees on $n$ keys with toll sequence $(i) n^α$ with $α ≥ 0 (α =0$ and $α =1$ correspond roughly to the space requirement and total path length, respectively); $(ii) ln \binom{n} {m-1}$, which corresponds to the so-called shape functional; and $(iii) $$1$$_{n=m-1}$, which corresponds to the number of leaves.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3370 https://doi.org/10.46298/dmtcs.3370 fill, james Allen Kapur, Nevin fill, james Allen Kapur, Nevin <![CDATA[Using recent results on singularity analysis for Hadamard products of generating functions, we obtain the limiting distributions for additive functionals on $m$-ary search trees on $n$ keys with toll sequence $(i) n^α$ with $α ≥ 0 (α =0$ and $α =1$ correspond roughly to the space requirement and total path length, respectively); $(ii) ln \binom{n} {m-1}$, which corresponds to the so-called shape functional; and $(iii) $$1$$_{n=m-1}$, which corresponds to the number of leaves.]]> 0 <![CDATA[Ordinary generating series of multiple harmonic sums admit a full singular expansion in the basis of functions $\{(1-z)^α \log^β (1-z)\}_{α ∈ℤ, β ∈ℕ}$, near the singularity $z=1$. A constructive proof of this result is given, and, by combinatoric aspects, an explicit evaluation of Taylor coefficients of functions in some polylogarithmic algebra is obtained. In particular, the asymptotic expansion of multiple harmonic sums is easily deduced.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3369 https://doi.org/10.46298/dmtcs.3369 Costermans, Christian Enjalbert, Jean-Yves Minh, Hoang Ngoc Costermans, Christian Enjalbert, Jean-Yves Minh, Hoang Ngoc <![CDATA[Ordinary generating series of multiple harmonic sums admit a full singular expansion in the basis of functions $\{(1-z)^α \log^β (1-z)\}_{α ∈ℤ, β ∈ℕ}$, near the singularity $z=1$. A constructive proof of this result is given, and, by combinatoric aspects, an explicit evaluation of Taylor coefficients of functions in some polylogarithmic algebra is obtained. In particular, the asymptotic expansion of multiple harmonic sums is easily deduced.]]> 0 <![CDATA[New cache-oblivious and cache-aware algorithms for simple dynamic programming based on Valiant's context-free language recognition algorithm are designed, implemented, analyzed, and empirically evaluated with timing studies and cache simulations. The studies show that for large inputs the cache-oblivious and cache-aware dynamic programming algorithms are significantly faster than the standard dynamic programming algorithm.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3368 https://doi.org/10.46298/dmtcs.3368 Cherng, Cary Ladner, Richard E. Cherng, Cary Ladner, Richard E. <![CDATA[New cache-oblivious and cache-aware algorithms for simple dynamic programming based on Valiant's context-free language recognition algorithm are designed, implemented, analyzed, and empirically evaluated with timing studies and cache simulations. The studies show that for large inputs the cache-oblivious and cache-aware dynamic programming algorithms are significantly faster than the standard dynamic programming algorithm.]]> 0 <![CDATA[As a sequel to [arch04], the position of the maximum in a geometrically distributed sample is investigated. Samples of length n are considered, where the maximum is required to be in the first d positions. The probability that the maximum occurs in the first $d$ positions is sought for $d$ dependent on n (as opposed to d fixed in [arch04]). Two scenarios are discussed. The first is when $d=αn$ for $0 < α ≤ 1$, where Mellin transforms are used to obtain the asymptotic results. The second is when $1 ≤ d = o(n)$.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3367 https://doi.org/10.46298/dmtcs.3367 Archibald, Margaret Archibald, Margaret <![CDATA[As a sequel to [arch04], the position of the maximum in a geometrically distributed sample is investigated. Samples of length n are considered, where the maximum is required to be in the first d positions. The probability that the maximum occurs in the first $d$ positions is sought for $d$ dependent on n (as opposed to d fixed in [arch04]). Two scenarios are discussed. The first is when $d=αn$ for $0 < α ≤ 1$, where Mellin transforms are used to obtain the asymptotic results. The second is when $1 ≤ d = o(n)$.]]> 0 <![CDATA[Consider a set $S$ of points in the plane in convex position, where each point has an integer label from $\{0,1,\ldots,n-1\}$. This naturally induces a labeling of the edges: each edge $(i,j)$ is assigned label $i+j$, modulo $n$. We propose the algorithms for finding large non―crossing $\textit{harmonic}$ matchings or paths, i. e. the matchings or paths in which no two edges have the same label. When the point labels are chosen uniformly at random, and independently of each other, our matching algorithm with high probability (w.h.p.) delivers a nearly―perfect matching, a matching of size $n/2 - O(n^{1/3}\ln n)$.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3366 https://doi.org/10.46298/dmtcs.3366 Balogh, József Pittel, Boris Salazar, Gelasio Balogh, József Pittel, Boris Salazar, Gelasio <![CDATA[Consider a set $S$ of points in the plane in convex position, where each point has an integer label from $\{0,1,\ldots,n-1\}$. This naturally induces a labeling of the edges: each edge $(i,j)$ is assigned label $i+j$, modulo $n$. We propose the algorithms for finding large non―crossing $\textit{harmonic}$ matchings or paths, i. e. the matchings or paths in which no two edges have the same label. When the point labels are chosen uniformly at random, and independently of each other, our matching algorithm with high probability (w.h.p.) delivers a nearly―perfect matching, a matching of size $n/2 - O(n^{1/3}\ln n)$.]]> 0 <![CDATA[The Additive-Increase-Multiplicative Decrease (AIMD) algorithm is an effective technique for controlling competitive access to a shared resource. Let $N$ be the number of users and let $x_i(t)$ be the amount of the resource in possession of the $i$-th user. The allocations $x_i(t)$ increase linearly until the aggregate demand $\sum_i x_i(t)$ exceeds a given nominal capacity, at which point a user is selected at a random time and its allocation reduced from $x_i(t)$ to $x_i(t)/ \gamma$ , for some given parameter $\gamma >1$. In our new, generalized version of AIMD, the choice of users to have their allocations cut is determined by a selection rule whereby the probabilities of selection are proportional to $x_i^{\alpha} (t)/ \sum_j x_j^{\alpha}$, with $\alpha$ a parameter of the policy. Variations of parameters allows one to adjust fairness under AIMD (as measured for example by the variance of $x_i(t)$) as well as to provide for differentiated service. The primary contribution here is an asymptotic, large-$N$ analysis of the above nonlinear AIMD algorithm within a baseline mathematical model that leads to explicit formulas for the density function governing the allocations $x_i(t)$ in statistical equilibrium. The analysis yields explicit formulas for measures of fairness and several techniques for supplying differentiated service via AIMD.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3365 https://doi.org/10.46298/dmtcs.3365 Baryshnikov, Y. Coffman, E. Feng, J. Momčilović, P. Baryshnikov, Y. Coffman, E. Feng, J. Momčilović, P. <![CDATA[The Additive-Increase-Multiplicative Decrease (AIMD) algorithm is an effective technique for controlling competitive access to a shared resource. Let $N$ be the number of users and let $x_i(t)$ be the amount of the resource in possession of the $i$-th user. The allocations $x_i(t)$ increase linearly until the aggregate demand $\sum_i x_i(t)$ exceeds a given nominal capacity, at which point a user is selected at a random time and its allocation reduced from $x_i(t)$ to $x_i(t)/ \gamma$ , for some given parameter $\gamma >1$. In our new, generalized version of AIMD, the choice of users to have their allocations cut is determined by a selection rule whereby the probabilities of selection are proportional to $x_i^{\alpha} (t)/ \sum_j x_j^{\alpha}$, with $\alpha$ a parameter of the policy. Variations of parameters allows one to adjust fairness under AIMD (as measured for example by the variance of $x_i(t)$) as well as to provide for differentiated service. The primary contribution here is an asymptotic, large-$N$ analysis of the above nonlinear AIMD algorithm within a baseline mathematical model that leads to explicit formulas for the density function governing the allocations $x_i(t)$ in statistical equilibrium. The analysis yields explicit formulas for measures of fairness and several techniques for supplying differentiated service via AIMD.]]> 0 <![CDATA[We introduce a new class of algorithms to estimate the cardinality of very large multisets using constant memory and doing only one pass on the data. It is based on order statistics rather that on bit patterns in binary representations of numbers. We analyse three families of estimators. They attain a standard error of $\frac{1}{\sqrt{M}}$ using $M$ units of storage, which places them in the same class as the best known algorithms so far. They have a very simple internal loop, which gives them an advantage in term of processing speed. The algorithms are validated on internet traffic traces.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3353 https://doi.org/10.46298/dmtcs.3353 Giroire, Frédéric Giroire, Frédéric <![CDATA[We introduce a new class of algorithms to estimate the cardinality of very large multisets using constant memory and doing only one pass on the data. It is based on order statistics rather that on bit patterns in binary representations of numbers. We analyse three families of estimators. They attain a standard error of $\frac{1}{\sqrt{M}}$ using $M$ units of storage, which places them in the same class as the best known algorithms so far. They have a very simple internal loop, which gives them an advantage in term of processing speed. The algorithms are validated on internet traffic traces.]]> 0 <![CDATA[On modern computers memory access patterns and cache utilization are as important, if not more important, than operation count in obtaining high-performance implementations of algorithms. In this work, the memory behavior of a large family of algorithms for computing the Walsh-Hadamard transform, an important signal processing transform related to the fast Fourier transform, is investigated. Empirical evidence shows that the family of algorithms exhibit a wide range of performance, despite the fact that all algorithms perform the same number of arithmetic operations. Different algorithms, while having the same number of memory operations, access memory in different patterns and consequently have different numbers of cache misses. A recurrence relation is derived for the number of cache misses and is used to determine the distribution of cache misses over the space of WHT algorithms.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3363 https://doi.org/10.46298/dmtcs.3363 Furis, Mihai Hitczenko, Paweł Johnson, Jeremy Furis, Mihai Hitczenko, Paweł Johnson, Jeremy <![CDATA[On modern computers memory access patterns and cache utilization are as important, if not more important, than operation count in obtaining high-performance implementations of algorithms. In this work, the memory behavior of a large family of algorithms for computing the Walsh-Hadamard transform, an important signal processing transform related to the fast Fourier transform, is investigated. Empirical evidence shows that the family of algorithms exhibit a wide range of performance, despite the fact that all algorithms perform the same number of arithmetic operations. Different algorithms, while having the same number of memory operations, access memory in different patterns and consequently have different numbers of cache misses. A recurrence relation is derived for the number of cache misses and is used to determine the distribution of cache misses over the space of WHT algorithms.]]> 0 <![CDATA[This extended abstract introduces a new algorithm for the random generation of labelled planar graphs. Its principles rely on Boltzmann samplers as recently developed by Duchon, Flajolet, Louchard, and Schaeffer. It combines the Boltzmann framework, a judicious use of rejection, a new combinatorial bijection found by Fusy, Poulalhon and Schaeffer, as well as a precise analytic description of the generating functions counting planar graphs, which was recently obtained by Giménez and Noy. This gives rise to an extremely efficient algorithm for the random generation of planar graphs. There is a preprocessing step of some fixed small cost. Then, for each generation, the time complexity is quadratic for exact-size uniform sampling and linear for approximate-size sampling. This greatly improves on the best previously known time complexity for exact-size uniform sampling of planar graphs with $n$ vertices, which was a little over $\mathcal{O}(n^7)$.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3362 https://doi.org/10.46298/dmtcs.3362 Fusy, Eric Fusy, Eric <![CDATA[This extended abstract introduces a new algorithm for the random generation of labelled planar graphs. Its principles rely on Boltzmann samplers as recently developed by Duchon, Flajolet, Louchard, and Schaeffer. It combines the Boltzmann framework, a judicious use of rejection, a new combinatorial bijection found by Fusy, Poulalhon and Schaeffer, as well as a precise analytic description of the generating functions counting planar graphs, which was recently obtained by Giménez and Noy. This gives rise to an extremely efficient algorithm for the random generation of planar graphs. There is a preprocessing step of some fixed small cost. Then, for each generation, the time complexity is quadratic for exact-size uniform sampling and linear for approximate-size sampling. This greatly improves on the best previously known time complexity for exact-size uniform sampling of planar graphs with $n$ vertices, which was a little over $\mathcal{O}(n^7)$.]]> 0 <![CDATA[We summarize several limit results for the profile of random plane-oriented recursive trees. These include the limit distribution of the normalized profile, asymptotic bimodality of the variance, asymptotic approximations of the expected width and the correlation coefficients of two level sizes. We also unveil an unexpected connection between the profile of plane-oriented recursive trees (with logarithmic height) and that of random binary trees (with height proportional to the square root of tree size).]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3361 https://doi.org/10.46298/dmtcs.3361 Hwang, Hsien-Kuei Hwang, Hsien-Kuei <![CDATA[We summarize several limit results for the profile of random plane-oriented recursive trees. These include the limit distribution of the normalized profile, asymptotic bimodality of the variance, asymptotic approximations of the expected width and the correlation coefficients of two level sizes. We also unveil an unexpected connection between the profile of plane-oriented recursive trees (with logarithmic height) and that of random binary trees (with height proportional to the square root of tree size).]]> 0 <![CDATA[Renewed interest in caching techniques stems from their application to improving the performance of the World Wide Web, where storing popular documents in proxy caches closer to end-users can significantly reduce the document download latency and overall network congestion. Rules used to update the collection of frequently accessed documents inside a cache are referred to as cache replacement algorithms. Due to many different factors that influence the Web performance, the most desirable attributes of a cache replacement scheme are low complexity and high adaptability to variability in Web access patterns. These properties are primarily the reason why most of the practical Web caching algorithms are based on the easily implemented Least-Recently-Used (LRU) cache replacement heuristic. In our recent paperJelenković and Radovanović (2004c), we introduce a new algorithm, termed Persistent Access Caching (PAC), that, in addition to desirable low complexity and adaptability, somewhat surprisingly achieves nearly optimal performance for the independent reference model and generalized Zipf's law request probabilities. Two drawbacks of the PAC algorithm are its dependence on the request arrival times and variable storage requirements. In this paper, we resolve these problems by introducing a discrete version of the PAC policy (DPAC) that, after a cache miss, places the requested document in the cache only if it is requested at least $k$ times among the last $m$, $m \geq k$, requests. However, from a mathematical perspective, due to the inherent coupling of the replacement decisions for different documents, the DPAC algorithm is considerably harder to analyze than the original PAC policy. In this regard, we develop a new analytical technique for estimating the performance of the DPAC rule. Using our analysis, we show that this algorithm is close to optimal even for small values of $k$ and $m$, and, therefore, adds negligible additional storage and processing complexity in comparison to the ordinary LRU policy.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3360 https://doi.org/10.46298/dmtcs.3360 Jelenković, Predrag R. Kang, Xiaozhu Radovanović, Ana Jelenković, Predrag R. Kang, Xiaozhu Radovanović, Ana <![CDATA[Renewed interest in caching techniques stems from their application to improving the performance of the World Wide Web, where storing popular documents in proxy caches closer to end-users can significantly reduce the document download latency and overall network congestion. Rules used to update the collection of frequently accessed documents inside a cache are referred to as cache replacement algorithms. Due to many different factors that influence the Web performance, the most desirable attributes of a cache replacement scheme are low complexity and high adaptability to variability in Web access patterns. These properties are primarily the reason why most of the practical Web caching algorithms are based on the easily implemented Least-Recently-Used (LRU) cache replacement heuristic. In our recent paperJelenković and Radovanović (2004c), we introduce a new algorithm, termed Persistent Access Caching (PAC), that, in addition to desirable low complexity and adaptability, somewhat surprisingly achieves nearly optimal performance for the independent reference model and generalized Zipf's law request probabilities. Two drawbacks of the PAC algorithm are its dependence on the request arrival times and variable storage requirements. In this paper, we resolve these problems by introducing a discrete version of the PAC policy (DPAC) that, after a cache miss, places the requested document in the cache only if it is requested at least $k$ times among the last $m$, $m \geq k$, requests. However, from a mathematical perspective, due to the inherent coupling of the replacement decisions for different documents, the DPAC algorithm is considerably harder to analyze than the original PAC policy. In this regard, we develop a new analytical technique for estimating the performance of the DPAC rule. Using our analysis, we show that this algorithm is close to optimal even for small values of $k$ and $m$, and, therefore, adds negligible additional storage and processing complexity in comparison to the ordinary LRU policy.]]> 0 <![CDATA[Let $\mathcal{T}_n$ denote the set of unrooted unlabeled trees of size $n$ and let $\mathcal{M}$ be a particular (finite) tree. Assuming that every tree of $\mathcal{T}_n$ is equally likely, it is shown that the number of occurrences $X_n$ of $\mathcal{M}$ as an induced sub-tree satisfies $\mathbf{E} X_n \sim \mu n$ and $\mathbf{V}ar X_n \sim \sigma^2 n$ for some (computable) constants $\mu > 0$ and $\sigma \geq 0$. Furthermore, if $\sigma > 0$ then $(X_n - \mathbf{E} X_n) / \sqrt{\mathbf{V}ar X_n}$ converges to a limiting distribution with density $(A+Bt^2)e^{-Ct^2}$ for some constants $A,B,C$. However, in all cases in which we were able to calculate these constants, we obtained $B=0$ and thus a normal distribution. Further, if we consider planted or rooted trees instead of $T_n$ then the limiting distribution is always normal. Similar results can be proved for planar, labeled and simply generated trees.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3359 https://doi.org/10.46298/dmtcs.3359 Kok, Gerard Kok, Gerard <![CDATA[Let $\mathcal{T}_n$ denote the set of unrooted unlabeled trees of size $n$ and let $\mathcal{M}$ be a particular (finite) tree. Assuming that every tree of $\mathcal{T}_n$ is equally likely, it is shown that the number of occurrences $X_n$ of $\mathcal{M}$ as an induced sub-tree satisfies $\mathbf{E} X_n \sim \mu n$ and $\mathbf{V}ar X_n \sim \sigma^2 n$ for some (computable) constants $\mu > 0$ and $\sigma \geq 0$. Furthermore, if $\sigma > 0$ then $(X_n - \mathbf{E} X_n) / \sqrt{\mathbf{V}ar X_n}$ converges to a limiting distribution with density $(A+Bt^2)e^{-Ct^2}$ for some constants $A,B,C$. However, in all cases in which we were able to calculate these constants, we obtained $B=0$ and thus a normal distribution. Further, if we consider planted or rooted trees instead of $T_n$ then the limiting distribution is always normal. Similar results can be proved for planar, labeled and simply generated trees.]]> 0 <![CDATA[We consider a sequence of $n$ geometric random variables and interpret the outcome as an urn model. For a given parameter $m$, we treat several parameters like what is the largest urn containing at least (or exactly) $m$ balls, or how many urns contain at least $m$ balls, etc. Many of these questions have their origin in some computer science problems. Identifying the underlying distributions as (variations of) the extreme value distribution, we are able to derive asymptotic equivalents for all (centered or uncentered) moments in a fairly automatic way.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3358 https://doi.org/10.46298/dmtcs.3358 Louchard, Guy Prodinger, Helmut Ward, Mark Daniel Louchard, Guy Prodinger, Helmut Ward, Mark Daniel <![CDATA[We consider a sequence of $n$ geometric random variables and interpret the outcome as an urn model. For a given parameter $m$, we treat several parameters like what is the largest urn containing at least (or exactly) $m$ balls, or how many urns contain at least $m$ balls, etc. Many of these questions have their origin in some computer science problems. Identifying the underlying distributions as (variations of) the extreme value distribution, we are able to derive asymptotic equivalents for all (centered or uncentered) moments in a fairly automatic way.]]> 0 <![CDATA[We give several examples for Poisson approximation of quantities of interest in the analysis of algorithms: the distribution of node depth in a binary search tree, the distribution of the number of losers in an election algorithm and the discounted profile of a binary search tree. A simple and well-known upper bound for the total variation distance between the distribution of a sum of independent Bernoulli variables and the Poisson distribution with the same mean turns out to be very useful in all three cases.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3357 https://doi.org/10.46298/dmtcs.3357 Grübel, Rudolf Grübel, Rudolf <![CDATA[We give several examples for Poisson approximation of quantities of interest in the analysis of algorithms: the distribution of node depth in a binary search tree, the distribution of the number of losers in an election algorithm and the discounted profile of a binary search tree. A simple and well-known upper bound for the total variation distance between the distribution of a sum of independent Bernoulli variables and the Poisson distribution with the same mean turns out to be very useful in all three cases.]]> 0 <![CDATA[We consider simply generated trees, where the nodes are equipped with weakly monotone labellings with elements of $\{1, 2, \ldots, r\}$, for $r$ fixed. These tree families were introduced in Prodinger and Urbanek (1983) and studied further in Kirschenhofer (1984), Blieberger (1987), and Morris and Prodinger (2005). Here we give distributional results for several tree statistics (the depth of a random node, the ancestor-tree size and the Steiner-distance of $p$ randomly chosen nodes, the height of the $j$-st leaf, and the number of nodes with label $l$), which extend the existing results and also contain the corresponding results for unlabelled simply generated trees as the special case $r=1$.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3356 https://doi.org/10.46298/dmtcs.3356 Gittenberger, Bernhard Panholzer, Alois Gittenberger, Bernhard Panholzer, Alois <![CDATA[We consider simply generated trees, where the nodes are equipped with weakly monotone labellings with elements of $\{1, 2, \ldots, r\}$, for $r$ fixed. These tree families were introduced in Prodinger and Urbanek (1983) and studied further in Kirschenhofer (1984), Blieberger (1987), and Morris and Prodinger (2005). Here we give distributional results for several tree statistics (the depth of a random node, the ancestor-tree size and the Steiner-distance of $p$ randomly chosen nodes, the height of the $j$-st leaf, and the number of nodes with label $l$), which extend the existing results and also contain the corresponding results for unlabelled simply generated trees as the special case $r=1$.]]> 0 <![CDATA[We consider two probability distributions on Boolean functions defined in terms of their representations by $\texttt{and/or}$ trees (or formulas). The relationships between them, and connections with the complexity of the function, are studied. New and improved bounds on these probabilities are given for a wide class of functions, with special attention being paid to the constant function $\textit{True}$ and read-once functions in a fixed number of variables.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3355 https://doi.org/10.46298/dmtcs.3355 Gardy, Danièle Woods, Alan Gardy, Danièle Woods, Alan <![CDATA[We consider two probability distributions on Boolean functions defined in terms of their representations by $\texttt{and/or}$ trees (or formulas). The relationships between them, and connections with the complexity of the function, are studied. New and improved bounds on these probabilities are given for a wide class of functions, with special attention being paid to the constant function $\textit{True}$ and read-once functions in a fixed number of variables.]]> 0 <![CDATA[We show an asymptotic estimate for the number of labelled planar graphs on $n$ vertices. We also find limit laws for the number of edges, the number of connected components, and other parameters in random planar graphs.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3354 https://doi.org/10.46298/dmtcs.3354 Gimenez, Omer Noy, Marc Gimenez, Omer Noy, Marc <![CDATA[We show an asymptotic estimate for the number of labelled planar graphs on $n$ vertices. We also find limit laws for the number of edges, the number of connected components, and other parameters in random planar graphs.]]> 0 <![CDATA[An $\textit{anticoloring}$ of a graph is a coloring of some of the vertices, such that no two adjacent vertices are colored in distinct colors. We deal with the anticoloring problem with two colors for planar graphs, and, using Lipton and Tarjan's separation algorithm, provide an algorithm with some bound on the error. In the particular cases of graphs which are strong products of two paths or two cycles, we provide an explicit optimal solution.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3388 https://doi.org/10.46298/dmtcs.3388 Berend, Daniel Korach, Ephraim Zucker, Shira Berend, Daniel Korach, Ephraim Zucker, Shira <![CDATA[An $\textit{anticoloring}$ of a graph is a coloring of some of the vertices, such that no two adjacent vertices are colored in distinct colors. We deal with the anticoloring problem with two colors for planar graphs, and, using Lipton and Tarjan's separation algorithm, provide an algorithm with some bound on the error. In the particular cases of graphs which are strong products of two paths or two cycles, we provide an explicit optimal solution.]]> 0 <![CDATA[Rhombus tilings are tilings of zonotopes with rhombohedra. We study a class of \emphlexicographic rhombus tilings of zonotopes, which are deduced from higher Bruhat orders relaxing the unitarity condition. Precisely, we fix a sequence $(v_1, v_2,\dots, v_D)$ of vectors of $ℝ^d$ and a sequence $(m_1, m_2,\dots, m_D)$ of positive integers. We assume (lexicographic hypothesis) that for each subsequence $(v_{i1}, v_{i2},\dots, v_{id})$ of length $d$, we have $det(v_{i1}, v_{i2},\dots, v_{id}) > 0$. The zonotope $Z$ is the set $\{ Σα _iv_i 0 ≤α _i ≤m_i \}$. Each prototile used in a tiling of $Z$ is a rhombohedron constructed from a subsequence of d vectors. We prove that the space of tilings of $Z$ is a graded poset, with minimal and maximal element.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3400 https://doi.org/10.46298/dmtcs.3400 Rémila, Éric Rémila, Éric <![CDATA[Rhombus tilings are tilings of zonotopes with rhombohedra. We study a class of \emphlexicographic rhombus tilings of zonotopes, which are deduced from higher Bruhat orders relaxing the unitarity condition. Precisely, we fix a sequence $(v_1, v_2,\dots, v_D)$ of vectors of $ℝ^d$ and a sequence $(m_1, m_2,\dots, m_D)$ of positive integers. We assume (lexicographic hypothesis) that for each subsequence $(v_{i1}, v_{i2},\dots, v_{id})$ of length $d$, we have $det(v_{i1}, v_{i2},\dots, v_{id}) > 0$. The zonotope $Z$ is the set $\{ Σα _iv_i 0 ≤α _i ≤m_i \}$. Each prototile used in a tiling of $Z$ is a rhombohedron constructed from a subsequence of d vectors. We prove that the space of tilings of $Z$ is a graded poset, with minimal and maximal element.]]> 0 <![CDATA[In this paper we present an algorithmic approach to packing A-paths. It is regarded as a generalization of Edmonds' matching algorithm, however there is the significant difference that here we do not build up any kind of alternating tree. Instead we use the so-called 3-way lemma, which either provides augmentation, or a dual, or a subgraph which can be used for contraction. The method works in the general setting of packing non-returning A-paths. It also implies an ear-decomposition of criticals, as a generalization of the odd ear-decomposition of factor-critical graph.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3399 https://doi.org/10.46298/dmtcs.3399 Pap, Gyula Pap, Gyula <![CDATA[In this paper we present an algorithmic approach to packing A-paths. It is regarded as a generalization of Edmonds' matching algorithm, however there is the significant difference that here we do not build up any kind of alternating tree. Instead we use the so-called 3-way lemma, which either provides augmentation, or a dual, or a subgraph which can be used for contraction. The method works in the general setting of packing non-returning A-paths. It also implies an ear-decomposition of criticals, as a generalization of the odd ear-decomposition of factor-critical graph.]]> 0 <![CDATA[A graph is called (matching-)immune if it has no edge cut that is also a matching. Farley and Proskurowski proved that for all immune graphs $G=(V,E)$, $|E|≥\lceil 3(|V|-1)/2\rceil$ , and constructed a large class of immune graphs that attain this lower bound for every value of $|V(G)|$, called $ABC$ graphs. They conjectured that every immune graph that attains this lower bound is an $ABC$ graph. We present a proof of this conjecture.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3398 https://doi.org/10.46298/dmtcs.3398 Bonsma, Paul Bonsma, Paul <![CDATA[A graph is called (matching-)immune if it has no edge cut that is also a matching. Farley and Proskurowski proved that for all immune graphs $G=(V,E)$, $|E|≥\lceil 3(|V|-1)/2\rceil$ , and constructed a large class of immune graphs that attain this lower bound for every value of $|V(G)|$, called $ABC$ graphs. They conjectured that every immune graph that attains this lower bound is an $ABC$ graph. We present a proof of this conjecture.]]> 0 <![CDATA[A hypergraph $\mathscr{H}$ is $τ$ -critical if $τ (\mathscr{H}-E) < τ (\mathscr{H})$ for every edge $E ∈\mathscr{H}$, where $τ (\mathscr{H})$ denotes the transversal number of $\mathscr{H}$. It can be shown that a connected $τ$ -critical hypergraph $\mathscr{H}$ has at least $2τ (\mathscr{H})-1$ edges; this generalises a classical theorem of Gallai on $χ$ -vertex-critical graphs with connected complements. In this paper we study connected $τ$ -critical hypergraphs $\mathscr{H}$ with exactly $2τ (\mathscr{H)}-1$ edges. We prove that such hypergraphs have at least $2τ (\mathscr{H})-1$ vertices, and characterise those with $2τ (\mathscr{H})-1$ vertices using a directed odd ear decomposition of an associated digraph. Using Seymour's characterisation of $χ$ -critical 3-chromatic square hypergraphs, we also show that a connected square hypergraph $\mathscr{H}$ with fewer than $2τ (\mathscr{H})$ edges is $τ$ -critical if and only if it is $χ$ -critical 3-chromatic. Finally, we deduce some new results on $χ$ -vertex-critical graphs with connected complements.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3397 https://doi.org/10.46298/dmtcs.3397 Stehlík, Matěj Stehlík, Matěj <![CDATA[A hypergraph $\mathscr{H}$ is $τ$ -critical if $τ (\mathscr{H}-E) < τ (\mathscr{H})$ for every edge $E ∈\mathscr{H}$, where $τ (\mathscr{H})$ denotes the transversal number of $\mathscr{H}$. It can be shown that a connected $τ$ -critical hypergraph $\mathscr{H}$ has at least $2τ (\mathscr{H})-1$ edges; this generalises a classical theorem of Gallai on $χ$ -vertex-critical graphs with connected complements. In this paper we study connected $τ$ -critical hypergraphs $\mathscr{H}$ with exactly $2τ (\mathscr{H)}-1$ edges. We prove that such hypergraphs have at least $2τ (\mathscr{H})-1$ vertices, and characterise those with $2τ (\mathscr{H})-1$ vertices using a directed odd ear decomposition of an associated digraph. Using Seymour's characterisation of $χ$ -critical 3-chromatic square hypergraphs, we also show that a connected square hypergraph $\mathscr{H}$ with fewer than $2τ (\mathscr{H})$ edges is $τ$ -critical if and only if it is $χ$ -critical 3-chromatic. Finally, we deduce some new results on $χ$ -vertex-critical graphs with connected complements.]]> 0 <![CDATA[The windy postman problem is the NP-hard problem of finding the minimum cost of a tour traversing all edges of an undirected graph, where the cost of traversal of an edge depends on the direction. Given an undirected graph $G$, we consider the polyhedron $O(G)$ induced by the linear programming relaxation of a well-known integer programming formulation of the problem. We say that $G$ is windy postman perfect if $O(G)$ is integral. There exists a polynomial-time algorithm, based on the ellipsoid method, to solve the windy postman problem for the class of windy postman perfect graphs. Eulerian graphs and trees are windy postman perfect. By considering a family of polyhedra related to $O(G)$, we prove that series-parallel graphs are windy postman perfect, therefore solving a conjecture of [Win1987a].]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3396 https://doi.org/10.46298/dmtcs.3396 Martínez, Francisco Javier Zaragoza Martínez, Francisco Javier Zaragoza <![CDATA[The windy postman problem is the NP-hard problem of finding the minimum cost of a tour traversing all edges of an undirected graph, where the cost of traversal of an edge depends on the direction. Given an undirected graph $G$, we consider the polyhedron $O(G)$ induced by the linear programming relaxation of a well-known integer programming formulation of the problem. We say that $G$ is windy postman perfect if $O(G)$ is integral. There exists a polynomial-time algorithm, based on the ellipsoid method, to solve the windy postman problem for the class of windy postman perfect graphs. Eulerian graphs and trees are windy postman perfect. By considering a family of polyhedra related to $O(G)$, we prove that series-parallel graphs are windy postman perfect, therefore solving a conjecture of [Win1987a].]]> 0 <![CDATA[Grone and Merris [GM94] conjectured that the Laplacian spectrum of a graph is majorized by its conjugate vertex degree sequence. We prove that this conjecture holds for a class of graphs including trees. We also show that this conjecture and its generalization to graphs with Dirichlet boundary conditions are equivalent.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3395 https://doi.org/10.46298/dmtcs.3395 Stephen, Tamon Stephen, Tamon <![CDATA[Grone and Merris [GM94] conjectured that the Laplacian spectrum of a graph is majorized by its conjugate vertex degree sequence. We prove that this conjecture holds for a class of graphs including trees. We also show that this conjecture and its generalization to graphs with Dirichlet boundary conditions are equivalent.]]> 0 <![CDATA[A multi-graph $G$ on n vertices is $(k,l)$-sparse if every subset of $n'≤n$ vertices spans at most $kn'-l$ edges, $0 ≤l < 2k$. $G$ is tight if, in addition, it has exactly $kn - l$ edges. We characterize $(k,l)$-sparse graphs via a family of simple, elegant and efficient algorithms called the $(k,l)$-pebble games. As applications, we use the pebble games for computing components (maximal tight subgraphs) in sparse graphs, to obtain inductive (Henneberg) constructions, and, when $l=k$, edge-disjoint tree decompositions.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3394 https://doi.org/10.46298/dmtcs.3394 Lee, Audrey Streinu, Ileana Lee, Audrey Streinu, Ileana <![CDATA[A multi-graph $G$ on n vertices is $(k,l)$-sparse if every subset of $n'≤n$ vertices spans at most $kn'-l$ edges, $0 ≤l < 2k$. $G$ is tight if, in addition, it has exactly $kn - l$ edges. We characterize $(k,l)$-sparse graphs via a family of simple, elegant and efficient algorithms called the $(k,l)$-pebble games. As applications, we use the pebble games for computing components (maximal tight subgraphs) in sparse graphs, to obtain inductive (Henneberg) constructions, and, when $l=k$, edge-disjoint tree decompositions.]]> 0 <![CDATA[Let $H=(V,E)$ be a hypergraph and let $k≥ 1$ and$ l≥ 0$ be fixed integers. Let $\mathcal{M}$ be the matroid with ground-set $E s.t. a$ set $F⊆E$ is independent if and only if each $X⊆V$ with $k|X|-l≥ 0$ spans at most $k|X|-l$ hyperedges of $F$. We prove that if $H$ is dense enough, then $\mathcal{M}$ satisfies the double circuit property, thus the min-max formula of Dress and Lovász on the maximum matroid matching holds for $\mathcal{M}$ . Our result implies the Berge-Tutte formula on the maximum matching of graphs $(k=1, l=0)$, generalizes Lovász' graphic matroid (cycle matroid) matching formula to hypergraphs $(k=l=1)$ and gives a min-max formula for the maximum matroid matching in the 2-dimensional rigidity matroid $(k=2, l=3)$.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3393 https://doi.org/10.46298/dmtcs.3393 Makai, Márton Makai, Márton <![CDATA[Let $H=(V,E)$ be a hypergraph and let $k≥ 1$ and$ l≥ 0$ be fixed integers. Let $\mathcal{M}$ be the matroid with ground-set $E s.t. a$ set $F⊆E$ is independent if and only if each $X⊆V$ with $k|X|-l≥ 0$ spans at most $k|X|-l$ hyperedges of $F$. We prove that if $H$ is dense enough, then $\mathcal{M}$ satisfies the double circuit property, thus the min-max formula of Dress and Lovász on the maximum matroid matching holds for $\mathcal{M}$ . Our result implies the Berge-Tutte formula on the maximum matching of graphs $(k=1, l=0)$, generalizes Lovász' graphic matroid (cycle matroid) matching formula to hypergraphs $(k=l=1)$ and gives a min-max formula for the maximum matroid matching in the 2-dimensional rigidity matroid $(k=2, l=3)$.]]> 0 <![CDATA[We consider the maps $f:\mathbb{F}_{2^n} →\mathbb{F}_{2^n}$ with the property that the set $\{ f(x+a)+ f(x): x ∈F_{2^n}\}$ is a hyperplane or a complement of hyperplane for every $a ∈\mathbb{F}_{2^n}^*$. The main goal of the talk is to show that almost all maps $f(x) = Σ_{b ∈B}c_b(x+b)^d$, where $B ⊂\mathbb{F}_{2^n}$ and $Σ_{b ∈B}c_b ≠0$, are not of that type. In particular, the only such power maps have exponents $2^i+2^j$ with $gcd(n, i-j)=1$. We give also a geometrical characterization of this maps.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3392 https://doi.org/10.46298/dmtcs.3392 Kyureghyan, Gohar Kyureghyan, Gohar <![CDATA[We consider the maps $f:\mathbb{F}_{2^n} →\mathbb{F}_{2^n}$ with the property that the set $\{ f(x+a)+ f(x): x ∈F_{2^n}\}$ is a hyperplane or a complement of hyperplane for every $a ∈\mathbb{F}_{2^n}^*$. The main goal of the talk is to show that almost all maps $f(x) = Σ_{b ∈B}c_b(x+b)^d$, where $B ⊂\mathbb{F}_{2^n}$ and $Σ_{b ∈B}c_b ≠0$, are not of that type. In particular, the only such power maps have exponents $2^i+2^j$ with $gcd(n, i-j)=1$. We give also a geometrical characterization of this maps.]]> 0 <![CDATA[Let $D$ be a finite set of integers. The distance graph $G(D)$ has the set of integers as vertices and two vertices at distance $d ∈D$ are adjacent in $G(D)$. A conjecture of Xuding Zhu states that if the chromatic number of $G (D)$ achieves its maximum value $|D|+1$ then the graph has a clique of order $|D|$. We prove that the chromatic number of a distance graph with $D=\{ a,b,c,d\}$ is five if and only if either $D=\{1,2,3,4k\}$ or $D=\{ a,b,a+b,a+2b\}$ with $a \equiv 0 (mod 2)$ and $b \equiv 1 (mod 2)$. This confirms Zhu's conjecture for $|D|=4$.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3391 https://doi.org/10.46298/dmtcs.3391 Barajas, Javier Serra, Oriol Barajas, Javier Serra, Oriol <![CDATA[Let $D$ be a finite set of integers. The distance graph $G(D)$ has the set of integers as vertices and two vertices at distance $d ∈D$ are adjacent in $G(D)$. A conjecture of Xuding Zhu states that if the chromatic number of $G (D)$ achieves its maximum value $|D|+1$ then the graph has a clique of order $|D|$. We prove that the chromatic number of a distance graph with $D=\{ a,b,c,d\}$ is five if and only if either $D=\{1,2,3,4k\}$ or $D=\{ a,b,a+b,a+2b\}$ with $a \equiv 0 (mod 2)$ and $b \equiv 1 (mod 2)$. This confirms Zhu's conjecture for $|D|=4$.]]> 0 <![CDATA[We build upon previous work of Fayolle (2004) and Park and Szpankowski (2005) to study asymptotically the average internal profile of tries and of suffix-trees. The binary keys and the strings are built from a Bernoulli source $(p,q)$. We consider the average number $p_{k,\mathcal{P}}(\nu)$ of internal nodes at depth $k$ of a trie whose number of input keys follows a Poisson law of parameter $\nu$. The Mellin transform of the corresponding bivariate generating function has a major singularity at the origin, which implies a phase reversal for the saturation rate $p_{k,\mathcal{P}}(\nu)/2^k$ as $k$ reaches the value $2\log(\nu)/(\log(1/p)+\log(1/q))$. We prove that the asymptotic average profiles of random tries and suffix-trees are mostly similar, up to second order terms, a fact that has been experimentally observed in Nicodème (2003); the proof follows from comparisons to the profile of tries in the Poisson model.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3390 https://doi.org/10.46298/dmtcs.3390 Nicodème, Pierre Nicodème, Pierre <![CDATA[We build upon previous work of Fayolle (2004) and Park and Szpankowski (2005) to study asymptotically the average internal profile of tries and of suffix-trees. The binary keys and the strings are built from a Bernoulli source $(p,q)$. We consider the average number $p_{k,\mathcal{P}}(\nu)$ of internal nodes at depth $k$ of a trie whose number of input keys follows a Poisson law of parameter $\nu$. The Mellin transform of the corresponding bivariate generating function has a major singularity at the origin, which implies a phase reversal for the saturation rate $p_{k,\mathcal{P}}(\nu)/2^k$ as $k$ reaches the value $2\log(\nu)/(\log(1/p)+\log(1/q))$. We prove that the asymptotic average profiles of random tries and suffix-trees are mostly similar, up to second order terms, a fact that has been experimentally observed in Nicodème (2003); the proof follows from comparisons to the profile of tries in the Poisson model.]]> 0 <![CDATA[The machinery of Riordan arrays has been used recently by several authors. We show how meromorphic singularity analysis can be used to provide uniform bivariate asymptotic expansions, in the central regime, for a generalization of these arrays. We show how to do this systematically, for various descriptions of the array. Several examples from recent literature are given.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3389 https://doi.org/10.46298/dmtcs.3389 Wilson, Mark C. Wilson, Mark C. <![CDATA[The machinery of Riordan arrays has been used recently by several authors. We show how meromorphic singularity analysis can be used to provide uniform bivariate asymptotic expansions, in the central regime, for a generalization of these arrays. We show how to do this systematically, for various descriptions of the array. Several examples from recent literature are given.]]> 0 <![CDATA[In this paper we show that the CSMA IEEE 802.11 protocol (Wifi) provides packet access delays asymptotics in power law. This very feature allows us to specify optimal routing via polynomial algorithm while the general case is NP-hard.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3375 https://doi.org/10.46298/dmtcs.3375 Jacquet, Philippe Naimi, Amina Meraihi Rodolakis, Georgios Jacquet, Philippe Naimi, Amina Meraihi Rodolakis, Georgios <![CDATA[In this paper we show that the CSMA IEEE 802.11 protocol (Wifi) provides packet access delays asymptotics in power law. This very feature allows us to specify optimal routing via polynomial algorithm while the general case is NP-hard.]]> 0 <![CDATA[In a suffix tree, the multiplicity matching parameter (MMP) $M_n$ is the number of leaves in the subtree rooted at the branching point of the $(n+1)$st insertion. Equivalently, the MMP is the number of pointers into the database in the Lempel-Ziv '77 data compression algorithm. We prove that the MMP asymptotically follows the logarithmic series distribution plus some fluctuations. In the proof we compare the distribution of the MMP in suffix trees to its distribution in tries built over independent strings. Our results are derived by both probabilistic and analytic techniques of the analysis of algorithms. In particular, we utilize combinatorics on words, bivariate generating functions, pattern matching, recurrence relations, analytical poissonization and depoissonization, the Mellin transform, and complex analysis.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3387 https://doi.org/10.46298/dmtcs.3387 Ward, Mark Daniel Szpankowski, Wojciech Ward, Mark Daniel Szpankowski, Wojciech <![CDATA[In a suffix tree, the multiplicity matching parameter (MMP) $M_n$ is the number of leaves in the subtree rooted at the branching point of the $(n+1)$st insertion. Equivalently, the MMP is the number of pointers into the database in the Lempel-Ziv '77 data compression algorithm. We prove that the MMP asymptotically follows the logarithmic series distribution plus some fluctuations. In the proof we compare the distribution of the MMP in suffix trees to its distribution in tries built over independent strings. Our results are derived by both probabilistic and analytic techniques of the analysis of algorithms. In particular, we utilize combinatorics on words, bivariate generating functions, pattern matching, recurrence relations, analytical poissonization and depoissonization, the Mellin transform, and complex analysis.]]> 0 <![CDATA[This paper presents the first distributional analysis of a linear probing hashing scheme with buckets of size $b$. The exact distribution of the cost of successful searches for a $b \alpha$ -full table is obtained, and moments and asymptotic results are derived. With the use of the Poisson transform distributional results are also obtained for tables of size $m$ and $n$ elements. A key element in the analysis is the use of a new family of numbers that satisfies a recurrence resembling that of the Bernoulli numbers. These numbers may prove helpful in studying recurrences involving truncated generating functions, as well as in other problems related with buckets.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3386 https://doi.org/10.46298/dmtcs.3386 Viola, Alfredo Viola, Alfredo <![CDATA[This paper presents the first distributional analysis of a linear probing hashing scheme with buckets of size $b$. The exact distribution of the cost of successful searches for a $b \alpha$ -full table is obtained, and moments and asymptotic results are derived. With the use of the Poisson transform distributional results are also obtained for tables of size $m$ and $n$ elements. A key element in the analysis is the use of a new family of numbers that satisfies a recurrence resembling that of the Bernoulli numbers. These numbers may prove helpful in studying recurrences involving truncated generating functions, as well as in other problems related with buckets.]]> 0 <![CDATA[For a given matrix of size $n \times m$ over a finite alphabet $\mathcal{A}$, a bicluster is a submatrix composed of selected columns and rows satisfying a certain property. In microarrays analysis one searches for largest biclusters in which selected rows constitute the same string (pattern); in another formulation of the problem one tries to find a maximally dense submatrix. In a conceptually similar problem, namely the bipartite clique problem on graphs, one looks for the largest binary submatrix with all '1'. In this paper, we assume that the original matrix is generated by a memoryless source over a finite alphabet $\mathcal{A}$. We first consider the case where the selected biclusters are square submatrices and prove that with high probability (whp) the largest (square) bicluster having the same row-pattern is of size $\log_Q^2 n m$ where $Q^{-1}$ is the (largest) probability of a symbol. We observe, however, that when we consider $\textit{any}$ submatrices (not just $\textit{square}$ submatrices), then the largest area of a bicluster jumps to $A_n$ (whp) where $A$ is an explicitly computable constant. These findings complete some recent results concerning maximal biclusters and maximum balanced bicliques for random bipartite graphs.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3385 https://doi.org/10.46298/dmtcs.3385 Park, Gahyun Szpankowski, Wojciech Park, Gahyun Szpankowski, Wojciech <![CDATA[For a given matrix of size $n \times m$ over a finite alphabet $\mathcal{A}$, a bicluster is a submatrix composed of selected columns and rows satisfying a certain property. In microarrays analysis one searches for largest biclusters in which selected rows constitute the same string (pattern); in another formulation of the problem one tries to find a maximally dense submatrix. In a conceptually similar problem, namely the bipartite clique problem on graphs, one looks for the largest binary submatrix with all '1'. In this paper, we assume that the original matrix is generated by a memoryless source over a finite alphabet $\mathcal{A}$. We first consider the case where the selected biclusters are square submatrices and prove that with high probability (whp) the largest (square) bicluster having the same row-pattern is of size $\log_Q^2 n m$ where $Q^{-1}$ is the (largest) probability of a symbol. We observe, however, that when we consider $\textit{any}$ submatrices (not just $\textit{square}$ submatrices), then the largest area of a bicluster jumps to $A_n$ (whp) where $A$ is an explicitly computable constant. These findings complete some recent results concerning maximal biclusters and maximum balanced bicliques for random bipartite graphs.]]> 0 <![CDATA[This article deals with Pólya generalized urn models with constant balance in any dimension. It is based on the algebraic approach of Pouyanne (2005) and classifies urns having "large'' eigenvalues in five classes, depending on their almost sure asymptotics. These classes are described in terms of the spectrum of the urn's replacement matrix and examples of each case are treated. We study the cases of so-called cyclic urns in any dimension and $m$-ary search trees for $m \geq 27$.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3384 https://doi.org/10.46298/dmtcs.3384 Pouyanne, Nicolas Pouyanne, Nicolas <![CDATA[This article deals with Pólya generalized urn models with constant balance in any dimension. It is based on the algebraic approach of Pouyanne (2005) and classifies urns having "large'' eigenvalues in five classes, depending on their almost sure asymptotics. These classes are described in terms of the spectrum of the urn's replacement matrix and examples of each case are treated. We study the cases of so-called cyclic urns in any dimension and $m$-ary search trees for $m \geq 27$.]]> 0 <![CDATA[We consider the $\textit{master ring problem (MRP)}$ which often arises in optical network design. Given a network which consists of a collection of interconnected rings $R_1, \ldots, R_K$, with $n_1, \ldots, n_K$ distinct nodes, respectively, we need to find an ordering of the nodes in the network that respects the ordering of every individual ring, if one exists. Our main result is an exact algorithm for MRP whose running time approaches $Q \cdot \prod_{k=1}^K (n_k/ \sqrt{2})$ for some polynomial $Q$, as the $n_k$ values become large. For the $\textit{ring clearance problem}$, a special case of practical interest, our algorithm achieves this running time for rings of $\textit{any}$ size $n_k \geq 2$. This yields the first nontrivial improvement, by factor of $(2 \sqrt{2})^K \approx (2.82)^K$, over the running time of the naive algorithm, which exhaustively enumerates all $\prod_{k=1}^K (2n_k)$ possible solutions.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3383 https://doi.org/10.46298/dmtcs.3383 Shachnai, Hadas Zhang, Lisa Shachnai, Hadas Zhang, Lisa <![CDATA[We consider the $\textit{master ring problem (MRP)}$ which often arises in optical network design. Given a network which consists of a collection of interconnected rings $R_1, \ldots, R_K$, with $n_1, \ldots, n_K$ distinct nodes, respectively, we need to find an ordering of the nodes in the network that respects the ordering of every individual ring, if one exists. Our main result is an exact algorithm for MRP whose running time approaches $Q \cdot \prod_{k=1}^K (n_k/ \sqrt{2})$ for some polynomial $Q$, as the $n_k$ values become large. For the $\textit{ring clearance problem}$, a special case of practical interest, our algorithm achieves this running time for rings of $\textit{any}$ size $n_k \geq 2$. This yields the first nontrivial improvement, by factor of $(2 \sqrt{2})^K \approx (2.82)^K$, over the running time of the naive algorithm, which exhaustively enumerates all $\prod_{k=1}^K (2n_k)$ possible solutions.]]> 0 <![CDATA[We consider words or strings of characters $a_1a_2a_3 \ldots a_n$ of length $n$, where the letters $a_i \in \mathbb{Z}$ are independently generated with a geometric probability $\mathbb{P} \{ X=k \} = pq^{k-1}$ where $p+q=1$. Let $d$ be a fixed nonnegative integer. We say that we have an ascent of size $d$ or more if $a_{i+1} \geq a_i+d$. We determine the mean, variance and limiting distribution of the number of ascents of size $d$ or more in a random geometrically distributed word.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3382 https://doi.org/10.46298/dmtcs.3382 Brennan, Charlotte Knopfmacher, Arnold Brennan, Charlotte Knopfmacher, Arnold <![CDATA[We consider words or strings of characters $a_1a_2a_3 \ldots a_n$ of length $n$, where the letters $a_i \in \mathbb{Z}$ are independently generated with a geometric probability $\mathbb{P} \{ X=k \} = pq^{k-1}$ where $p+q=1$. Let $d$ be a fixed nonnegative integer. We say that we have an ascent of size $d$ or more if $a_{i+1} \geq a_i+d$. We determine the mean, variance and limiting distribution of the number of ascents of size $d$ or more in a random geometrically distributed word.]]> 0 <![CDATA[Given a set $\mathcal{S}$ with real-valued members, associated with each member one of two possible types; a multi-partitioning of $\mathcal{S}$ is a sequence of the members of $\mathcal{S}$ such that if $x,y \in \mathcal{S}$ have different types and $x < y$, $x$ precedes $y$ in the multi-partitioning of $\mathcal{S}$. We give two distribution-sensitive algorithms for the set multi-partitioning problem and a matching lower bound in the algebraic decision-tree model. One of the two algorithms can be made stable and can be implemented in place. We also give an output-sensitive algorithm for the problem.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3381 https://doi.org/10.46298/dmtcs.3381 Elmasry, Amr Elmasry, Amr <![CDATA[Given a set $\mathcal{S}$ with real-valued members, associated with each member one of two possible types; a multi-partitioning of $\mathcal{S}$ is a sequence of the members of $\mathcal{S}$ such that if $x,y \in \mathcal{S}$ have different types and $x < y$, $x$ precedes $y$ in the multi-partitioning of $\mathcal{S}$. We give two distribution-sensitive algorithms for the set multi-partitioning problem and a matching lower bound in the algebraic decision-tree model. One of the two algorithms can be made stable and can be implemented in place. We also give an output-sensitive algorithm for the problem.]]> 0 <![CDATA[We show that data compression methods (or universal codes) can be applied for hypotheses testing in a framework of classical mathematical statistics. Namely, we describe tests, which are based on data compression methods, for the three following problems: i) identity testing, ii) testing for independence and iii) testing of serial independence for time series. Applying our method of identity testing to pseudorandom number generators, we obtained experimental results which show that the suggested tests are quite efficient.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3380 https://doi.org/10.46298/dmtcs.3380 Ryabko, Boris Astola, Jaakko Ryabko, Boris Astola, Jaakko <![CDATA[We show that data compression methods (or universal codes) can be applied for hypotheses testing in a framework of classical mathematical statistics. Namely, we describe tests, which are based on data compression methods, for the three following problems: i) identity testing, ii) testing for independence and iii) testing of serial independence for time series. Applying our method of identity testing to pseudorandom number generators, we obtained experimental results which show that the suggested tests are quite efficient.]]> 0 <![CDATA[The aim of this paper is threefold: firstly, to explain a certain segment of ordinals in terms which are familiar to the analytic combinatorics community, secondly to state a great many of associated problems on resulting count functions and thirdly, to provide some weak asymptotic for the resulting count functions. We employ for simplicity Tauberian methods. The analytic combinatorics community is encouraged to provide (maybe in joint work) sharper results in future investigations.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3379 https://doi.org/10.46298/dmtcs.3379 Weiermann, Andreas Weiermann, Andreas <![CDATA[The aim of this paper is threefold: firstly, to explain a certain segment of ordinals in terms which are familiar to the analytic combinatorics community, secondly to state a great many of associated problems on resulting count functions and thirdly, to provide some weak asymptotic for the resulting count functions. We employ for simplicity Tauberian methods. The analytic combinatorics community is encouraged to provide (maybe in joint work) sharper results in future investigations.]]> 0 <![CDATA[A tight upper bound of the size of the antidictionary of a binary string is presented. And it is shown that the size of the antidictionary of a binary sting is always smaller than or equal to that of its dictionary. Moreover, an algorithm to reconstruct its dictionary from its antidictionary is given.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3378 https://doi.org/10.46298/dmtcs.3378 Morita, Hiroyoshi Ota, Takahiro Morita, Hiroyoshi Ota, Takahiro <![CDATA[A tight upper bound of the size of the antidictionary of a binary string is presented. And it is shown that the size of the antidictionary of a binary sting is always smaller than or equal to that of its dictionary. Moreover, an algorithm to reconstruct its dictionary from its antidictionary is given.]]> 0 <![CDATA[It has become customary to prove binomial identities by means of the method for automated proofs as developed by Petkovšek, Wilf and Zeilberger. In this paper, we wish to emphasize the role of "human'' and constructive proofs in contrast with the somewhat lazy attitude of relaying on "automated'' proofs. As a meaningful example, we consider the four formulas by Romik, related to Motzkin and central trinomial numbers. We show that a proof of these identities can be obtained by using the method of coefficients, a human method only requiring hand computations.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3377 https://doi.org/10.46298/dmtcs.3377 Merlini, D. Sprugnoli, R. Verri, M. C. Merlini, D. Sprugnoli, R. Verri, M. C. <![CDATA[It has become customary to prove binomial identities by means of the method for automated proofs as developed by Petkovšek, Wilf and Zeilberger. In this paper, we wish to emphasize the role of "human'' and constructive proofs in contrast with the somewhat lazy attitude of relaying on "automated'' proofs. As a meaningful example, we consider the four formulas by Romik, related to Motzkin and central trinomial numbers. We show that a proof of these identities can be obtained by using the method of coefficients, a human method only requiring hand computations.]]> 0 <![CDATA[For the tree algorithm introduced by [Cap79] and [TsMi78] let $L_N$ denote the expected collision resolution time given the collision multiplicity $N$. If $L(z)$ stands for the Poisson transform of $L_N$, then we show that $L_N - L(N) ≃ 1.29·10^-4 \cos (2 π \log _2 N + 0.698)$.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3376 https://doi.org/10.46298/dmtcs.3376 Gyorfi, Laszlo Gyori, Sándor Gyorfi, Laszlo Gyori, Sándor <![CDATA[For the tree algorithm introduced by [Cap79] and [TsMi78] let $L_N$ denote the expected collision resolution time given the collision multiplicity $N$. If $L(z)$ stands for the Poisson transform of $L_N$, then we show that $L_N - L(N) ≃ 1.29·10^-4 \cos (2 π \log _2 N + 0.698)$.]]> 0 <![CDATA[We consider the number of nodes in the levels of unlabeled rooted random trees and show that the joint distribution of several level sizes (where the level number is scaled by $\sqrt{n}$) weakly converges to the distribution of the local time of a Brownian excursion evaluated at the times corresponding to the level numbers. This extends existing results for simply generated trees and forests to the case of unlabeled rooted trees.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3352 https://doi.org/10.46298/dmtcs.3352 Gittenberger, Bernhard Gittenberger, Bernhard <![CDATA[We consider the number of nodes in the levels of unlabeled rooted random trees and show that the joint distribution of several level sizes (where the level number is scaled by $\sqrt{n}$) weakly converges to the distribution of the local time of a Brownian excursion evaluated at the times corresponding to the level numbers. This extends existing results for simply generated trees and forests to the case of unlabeled rooted trees.]]> 0 <![CDATA[A graph $G=(V,E)$ is said to be $\textit{magic}$ if there exists an integer labeling $f: V \cup E \to [1, |V \cup E|]$ such that $f(x)+f(y)+f(xy)$ is constant for all edges $xy \in E$. Enomoto, Masuda and Nakamigawa proved that there are magic graphs of order at most $3n^2+o(n^2)$ which contain a complete graph of order $n$. Bounds on Sidon sets show that the order of such a graph is at least $n^2+o(n^2)$. We close the gap between those two bounds by showing that, for any given graph $H$ of order $n$, there are connected magic graphs of order $n^2+o(n^2)$ containing $H$ as an induced subgraph. Moreover it can be required that the graph admits a supermagic labelling $f$, which satisfies the additional condition $f(V)=[1,|V|]$.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3414 https://doi.org/10.46298/dmtcs.3414 Lladó, Anna Lladó, Anna <![CDATA[A graph $G=(V,E)$ is said to be $\textit{magic}$ if there exists an integer labeling $f: V \cup E \to [1, |V \cup E|]$ such that $f(x)+f(y)+f(xy)$ is constant for all edges $xy \in E$. Enomoto, Masuda and Nakamigawa proved that there are magic graphs of order at most $3n^2+o(n^2)$ which contain a complete graph of order $n$. Bounds on Sidon sets show that the order of such a graph is at least $n^2+o(n^2)$. We close the gap between those two bounds by showing that, for any given graph $H$ of order $n$, there are connected magic graphs of order $n^2+o(n^2)$ containing $H$ as an induced subgraph. Moreover it can be required that the graph admits a supermagic labelling $f$, which satisfies the additional condition $f(V)=[1,|V|]$.]]> 0 <![CDATA[In this note we prove Sterboul's conjecture, that provides a sufficient condition for the bicolorability of hypergraphs.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3427 https://doi.org/10.46298/dmtcs.3427 Défossez, David Défossez, David <![CDATA[In this note we prove Sterboul's conjecture, that provides a sufficient condition for the bicolorability of hypergraphs.]]> 0 <![CDATA[Let $f_m(a,b,c,d)$ denote the maximum size of a family $\mathcal{F}$ of subsets of an $m$-element set for which there is no pair of subsets $A,B \in \mathcal{F}$ with $|A \cap B| \geq a$, $|\bar{A} \cap B| \geq b$, $|A \cap \bar{B}| \geq c$, and $|\bar{A} \cap \bar{B}| \geq d$. By symmetry we can assume $a \geq d$ and $b \geq c$. We show that $f_m(a,b,c,d)$ is $\Theta (m^{a+b-1})$ if either $b > c$ or $a,b \geq 1$. We also show that $f_m(0,b,b,0)$ is $\Theta (m^b)$ and $f_m(a,0,0,d)$ is $\Theta (m^a)$. This can be viewed as a result concerning forbidden configurations and is further evidence for a conjecture of Anstee and Sali. Our key tool is a strong stability version of the Complete Intersection Theorem of Ahlswede and Khachatrian, which is of independent interest.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3426 https://doi.org/10.46298/dmtcs.3426 Anstee, Richard P. Keevash, Peter Anstee, Richard P. Keevash, Peter <![CDATA[Let $f_m(a,b,c,d)$ denote the maximum size of a family $\mathcal{F}$ of subsets of an $m$-element set for which there is no pair of subsets $A,B \in \mathcal{F}$ with $|A \cap B| \geq a$, $|\bar{A} \cap B| \geq b$, $|A \cap \bar{B}| \geq c$, and $|\bar{A} \cap \bar{B}| \geq d$. By symmetry we can assume $a \geq d$ and $b \geq c$. We show that $f_m(a,b,c,d)$ is $\Theta (m^{a+b-1})$ if either $b > c$ or $a,b \geq 1$. We also show that $f_m(0,b,b,0)$ is $\Theta (m^b)$ and $f_m(a,0,0,d)$ is $\Theta (m^a)$. This can be viewed as a result concerning forbidden configurations and is further evidence for a conjecture of Anstee and Sali. Our key tool is a strong stability version of the Complete Intersection Theorem of Ahlswede and Khachatrian, which is of independent interest.]]> 0 <![CDATA[Let $T_t$ denote the $t$-threshold function on the $n$-cube: $T_t(x) = 1$ if $|\{i : x_i=1\}| \geq t$, and $0$ otherwise. Define the distance between Boolean functions $g$ and $h$, $d(g,h)$, to be the number of points on which $g$ and $h$ disagree. We consider the following extremal problem: Over a monotone Boolean function $g$ on the $n$-cube with $s$ zeros, what is the maximum of $d(g,T_t)$? We show that the following monotone function $p_s$ maximizes the distance: For $x \in \{0,1\}^n$, $p_s(x)=0$ if and only if $N(x) < s$, where $N(x)$ is the integer whose $n$-bit binary representation is $x$. Our result generalizes the previous work for the case $t=\lceil n/2 \rceil$ and $s=2^{n-1}$ by Blum, Burch, and Langford [BBL98-FOCS98], who considered the problem to analyze the behavior of a learning algorithm for monotone Boolean functions, and the previous work for the same $t$ and $s$ by Amano and Maruoka [AM02-ALT02].]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3425 https://doi.org/10.46298/dmtcs.3425 Amano, Kazuyuki Tarui, Jun Amano, Kazuyuki Tarui, Jun <![CDATA[Let $T_t$ denote the $t$-threshold function on the $n$-cube: $T_t(x) = 1$ if $|\{i : x_i=1\}| \geq t$, and $0$ otherwise. Define the distance between Boolean functions $g$ and $h$, $d(g,h)$, to be the number of points on which $g$ and $h$ disagree. We consider the following extremal problem: Over a monotone Boolean function $g$ on the $n$-cube with $s$ zeros, what is the maximum of $d(g,T_t)$? We show that the following monotone function $p_s$ maximizes the distance: For $x \in \{0,1\}^n$, $p_s(x)=0$ if and only if $N(x) < s$, where $N(x)$ is the integer whose $n$-bit binary representation is $x$. Our result generalizes the previous work for the case $t=\lceil n/2 \rceil$ and $s=2^{n-1}$ by Blum, Burch, and Langford [BBL98-FOCS98], who considered the problem to analyze the behavior of a learning algorithm for monotone Boolean functions, and the previous work for the same $t$ and $s$ by Amano and Maruoka [AM02-ALT02].]]> 0 <![CDATA[We study the notion of hypertree-width of hypergraphs. We prove that, up to a constant factor, hypertree-width is the same as a number of other hypergraph invariants that resemble graph invariants such as bramble-number, branch-width, linkedness, and the minimum number of cops required to win Seymour and Thomas's robber and cops game.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3424 https://doi.org/10.46298/dmtcs.3424 Adler, Isolde Gottlob, Georg Grohe, Martin Adler, Isolde Gottlob, Georg Grohe, Martin <![CDATA[We study the notion of hypertree-width of hypergraphs. We prove that, up to a constant factor, hypertree-width is the same as a number of other hypergraph invariants that resemble graph invariants such as bramble-number, branch-width, linkedness, and the minimum number of cops required to win Seymour and Thomas's robber and cops game.]]> 0 <![CDATA[Let $D(G)$ be the minimum quantifier depth of a first order sentence $\Phi$ that defines a graph $G$ up to isomorphism in terms of the adjacency and the equality relations. Let $D_0(G)$ be a variant of $D(G)$ where we do not allow quantifier alternations in $\Phi$. Using large graphs decomposable in complement-connected components by a short sequence of serial and parallel decompositions, we show examples of $G$ on $n$ vertices with $D_0(G) \leq 2 \log^{\ast}n+O(1)$. On the other hand, we prove a lower bound $D_0(G) \geq \log^{\ast}n-\log^{\ast}\log^{\ast}n-O(1)$ for all $G$. Here $\log^{\ast}n$ is equal to the minimum number of iterations of the binary logarithm needed to bring $n$ below $1$.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3423 https://doi.org/10.46298/dmtcs.3423 Pikhurko, Oleg Spencer, Joel Verbitsky, Oleg Pikhurko, Oleg Spencer, Joel Verbitsky, Oleg <![CDATA[Let $D(G)$ be the minimum quantifier depth of a first order sentence $\Phi$ that defines a graph $G$ up to isomorphism in terms of the adjacency and the equality relations. Let $D_0(G)$ be a variant of $D(G)$ where we do not allow quantifier alternations in $\Phi$. Using large graphs decomposable in complement-connected components by a short sequence of serial and parallel decompositions, we show examples of $G$ on $n$ vertices with $D_0(G) \leq 2 \log^{\ast}n+O(1)$. On the other hand, we prove a lower bound $D_0(G) \geq \log^{\ast}n-\log^{\ast}\log^{\ast}n-O(1)$ for all $G$. Here $\log^{\ast}n$ is equal to the minimum number of iterations of the binary logarithm needed to bring $n$ below $1$.]]> 0 <![CDATA[In a weak positional game, two players, Maker and Breaker, alternately claim vertices of a hypergraph until either Maker wins by getting a complete edge or all vertices are taken without this happening, a Breaker win. For the class of almost-disjoint hypergraphs of rank three (edges with up to three vertices only and edge-intersections on at most one vertex) we show how to find optimal strategies in polynomial time. Our result is based on a new type of decomposition theorem which might lead to a better understanding of weak positional games in general.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3422 https://doi.org/10.46298/dmtcs.3422 Kutz, Martin Kutz, Martin <![CDATA[In a weak positional game, two players, Maker and Breaker, alternately claim vertices of a hypergraph until either Maker wins by getting a complete edge or all vertices are taken without this happening, a Breaker win. For the class of almost-disjoint hypergraphs of rank three (edges with up to three vertices only and edge-intersections on at most one vertex) we show how to find optimal strategies in polynomial time. Our result is based on a new type of decomposition theorem which might lead to a better understanding of weak positional games in general.]]> 0 <![CDATA[We consider permutations of $1,2,...,n^2$ whose longest monotone subsequence is of length $n$ and are therefore extremal for the Erdős-Szekeres Theorem. Such permutations correspond via the Robinson-Schensted correspondence to pairs of square $n \times n$ Young tableaux. We show that all the bumping sequences are constant and therefore these permutations have a simple description in terms of the pair of square tableaux. We deduce a limit shape result for the plot of values of the typical such permutation, which in particular implies that the first value taken by such a permutation is with high probability $(1+o(1))n^2/2$.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3421 https://doi.org/10.46298/dmtcs.3421 Romik, Dan Romik, Dan <![CDATA[We consider permutations of $1,2,...,n^2$ whose longest monotone subsequence is of length $n$ and are therefore extremal for the Erdős-Szekeres Theorem. Such permutations correspond via the Robinson-Schensted correspondence to pairs of square $n \times n$ Young tableaux. We show that all the bumping sequences are constant and therefore these permutations have a simple description in terms of the pair of square tableaux. We deduce a limit shape result for the plot of values of the typical such permutation, which in particular implies that the first value taken by such a permutation is with high probability $(1+o(1))n^2/2$.]]> 0 <![CDATA[In the maximum constraint satisfaction problem ($\mathrm{Max \; CSP}$), one is given a finite collection of (possibly weighted) constraints on overlapping sets of variables, and the goal is to assign values from a given finite domain to the variables so as to maximise the number (or the total weight) of satisfied constraints. This problem is $\mathrm{NP}$-hard in general so it is natural to study how restricting the allowed types of constraints affects the complexity of the problem. In this paper, we show that any $\mathrm{Max \; CSP}$ problem with a finite set of allowed constraint types, which includes all constants (i.e. constraints of the form $x=a$), is either solvable in polynomial time or is $\mathrm{NP}$-complete. Moreover, we present a simple description of all polynomial-time solvable cases of our problem. This description uses the well-known combinatorial property of supermodularity.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3420 https://doi.org/10.46298/dmtcs.3420 Deineko, Vladimir Jonsson, Peter Klasson, Mikael Krokhin, Andrei Deineko, Vladimir Jonsson, Peter Klasson, Mikael Krokhin, Andrei <![CDATA[In the maximum constraint satisfaction problem ($\mathrm{Max \; CSP}$), one is given a finite collection of (possibly weighted) constraints on overlapping sets of variables, and the goal is to assign values from a given finite domain to the variables so as to maximise the number (or the total weight) of satisfied constraints. This problem is $\mathrm{NP}$-hard in general so it is natural to study how restricting the allowed types of constraints affects the complexity of the problem. In this paper, we show that any $\mathrm{Max \; CSP}$ problem with a finite set of allowed constraint types, which includes all constants (i.e. constraints of the form $x=a$), is either solvable in polynomial time or is $\mathrm{NP}$-complete. Moreover, we present a simple description of all polynomial-time solvable cases of our problem. This description uses the well-known combinatorial property of supermodularity.]]> 0 <![CDATA[Let $G$ be an $n$-vertex $m$-edge graph with weighted vertices. A pair of vertex sets $A,B \subseteq V(G)$ is a $\frac{2}{3} - \textit{separation}$ of $\textit{order}$ $|A \cap B|$ if $A \cup B = V(G)$, there is no edge between $A \backslash B$ and $B \backslash A$, and both $A \backslash B$ and $B \backslash A$ have weight at most $\frac{2}{3}$ the total weight of $G$. Let $\ell \in \mathbb{Z}^+$ be fixed. Alon, Seymour and Thomas [$\textit{J. Amer. Math. Soc.}$ 1990] presented an algorithm that in $\mathcal{O}(n^{1/2}m)$ time, either outputs a $K_\ell$-minor of $G$, or a separation of $G$ of order $\mathcal{O}(n^{1/2})$. Whether there is a $\mathcal{O}(n+m)$ time algorithm for this theorem was left as open problem. In this paper, we obtain a $\mathcal{O}(n+m)$ time algorithm at the expense of $\mathcal{O}(n^{2/3})$ separator. Moreover, our algorithm exhibits a tradeoff between running time and the order of the separator. In particular, for any given $\epsilon \in [0,\frac{1}{2}]$, our algorithm either outputs a $K_\ell$-minor of $G$, or a separation of $G$ with order $\mathcal{O}(n^{(2-\epsilon )/3})$ in $\mathcal{O}(n^{1+\epsilon} +m)$ time.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3419 https://doi.org/10.46298/dmtcs.3419 Reed, Bruce Wood, David R. Reed, Bruce Wood, David R. <![CDATA[Let $G$ be an $n$-vertex $m$-edge graph with weighted vertices. A pair of vertex sets $A,B \subseteq V(G)$ is a $\frac{2}{3} - \textit{separation}$ of $\textit{order}$ $|A \cap B|$ if $A \cup B = V(G)$, there is no edge between $A \backslash B$ and $B \backslash A$, and both $A \backslash B$ and $B \backslash A$ have weight at most $\frac{2}{3}$ the total weight of $G$. Let $\ell \in \mathbb{Z}^+$ be fixed. Alon, Seymour and Thomas [$\textit{J. Amer. Math. Soc.}$ 1990] presented an algorithm that in $\mathcal{O}(n^{1/2}m)$ time, either outputs a $K_\ell$-minor of $G$, or a separation of $G$ of order $\mathcal{O}(n^{1/2})$. Whether there is a $\mathcal{O}(n+m)$ time algorithm for this theorem was left as open problem. In this paper, we obtain a $\mathcal{O}(n+m)$ time algorithm at the expense of $\mathcal{O}(n^{2/3})$ separator. Moreover, our algorithm exhibits a tradeoff between running time and the order of the separator. In particular, for any given $\epsilon \in [0,\frac{1}{2}]$, our algorithm either outputs a $K_\ell$-minor of $G$, or a separation of $G$ with order $\mathcal{O}(n^{(2-\epsilon )/3})$ in $\mathcal{O}(n^{1+\epsilon} +m)$ time.]]> 0 <![CDATA[Let $\mathcal{F}\subseteq 2^{[n]}$ be a intersecting Sperner family (i.e. $A \not\subset B, A \cap B \neq \emptyset$ for all $A,B \in \mathcal{F}$) with profile vector $(f_i)_{i=0 \ldots n}$ (i.e. $f_i=|\mathcal{F} \cap \binom{[n]}{i}|$). We present quadratic inequalities in the $f_i$'s which sharpen the previously known linear $\mathrm{LYM}$-type inequalities.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3418 https://doi.org/10.46298/dmtcs.3418 Bey, Christian Bey, Christian <![CDATA[Let $\mathcal{F}\subseteq 2^{[n]}$ be a intersecting Sperner family (i.e. $A \not\subset B, A \cap B \neq \emptyset$ for all $A,B \in \mathcal{F}$) with profile vector $(f_i)_{i=0 \ldots n}$ (i.e. $f_i=|\mathcal{F} \cap \binom{[n]}{i}|$). We present quadratic inequalities in the $f_i$'s which sharpen the previously known linear $\mathrm{LYM}$-type inequalities.]]> 0 <![CDATA[An $L(2,1)$-labeling of a graph is a mapping $c:V(G) \to \{0,\ldots,K\}$ such that the labels assigned to neighboring vertices differ by at least $2$ and the labels of vertices at distance two are different. Griggs and Yeh [SIAM J. Discrete Math. 5 (1992), 586―595] conjectured that every graph $G$ with maximum degree $\Delta$ has an $L(2,1)$-labeling with $K \leq \Delta^2$. We verify the conjecture for planar graphs with maximum degree $\Delta \neq 3$.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3417 https://doi.org/10.46298/dmtcs.3417 Bella, Peter Král, Daniel Mohar, Bojan Quittnerová, Katarina Bella, Peter Král, Daniel Mohar, Bojan Quittnerová, Katarina <![CDATA[An $L(2,1)$-labeling of a graph is a mapping $c:V(G) \to \{0,\ldots,K\}$ such that the labels assigned to neighboring vertices differ by at least $2$ and the labels of vertices at distance two are different. Griggs and Yeh [SIAM J. Discrete Math. 5 (1992), 586―595] conjectured that every graph $G$ with maximum degree $\Delta$ has an $L(2,1)$-labeling with $K \leq \Delta^2$. We verify the conjecture for planar graphs with maximum degree $\Delta \neq 3$.]]> 0 <![CDATA[A double $2$-$(v,k,2 \lambda)$ design is a design which is reducible into two $2$-$(v,k,\lambda)$ designs. It is called uniquely reducible if it has, up to equivalence, only one reduction. We present properties of uniquely reducible double designs which show that their total number can be determined if only the designs with non-trivial automorphisms are classified with respect to their automorphism group. As an application, after proving that a reducible $2$-$(21,5,2)$ design is uniquely reducible, we establish that the number of all reducible $2$-$(21,5,2)$ designs is $1 746 461 307$.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3416 https://doi.org/10.46298/dmtcs.3416 Fack, Veerle Topalova, Svetlana Winne, Joost Fack, Veerle Topalova, Svetlana Winne, Joost <![CDATA[A double $2$-$(v,k,2 \lambda)$ design is a design which is reducible into two $2$-$(v,k,\lambda)$ designs. It is called uniquely reducible if it has, up to equivalence, only one reduction. We present properties of uniquely reducible double designs which show that their total number can be determined if only the designs with non-trivial automorphisms are classified with respect to their automorphism group. As an application, after proving that a reducible $2$-$(21,5,2)$ design is uniquely reducible, we establish that the number of all reducible $2$-$(21,5,2)$ designs is $1 746 461 307$.]]> 0 <![CDATA[A vertex coloring of a graph $G$ is $k \textit{-nonrepetitive}$ if one cannot find a periodic sequence with $k$ blocks on any simple path of $G$. The minimum number of colors needed for such coloring is denoted by $\pi _k(G)$ . This idea combines graph colorings with Thue sequences introduced at the beginning of 20th century. In particular Thue proved that if $G$ is a simple path of any length greater than $4$ then $\pi _2(G)=3$ and $\pi_3(G)=2$. We investigate $\pi_k(G)$ for other classes of graphs. Particularly interesting open problem is to decide if there is, possibly huge, $k$ such that $\pi_k(G)$ is bounded for planar graphs.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3415 https://doi.org/10.46298/dmtcs.3415 Alon, Noga Grytczuk, Jaroslaw Alon, Noga Grytczuk, Jaroslaw <![CDATA[A vertex coloring of a graph $G$ is $k \textit{-nonrepetitive}$ if one cannot find a periodic sequence with $k$ blocks on any simple path of $G$. The minimum number of colors needed for such coloring is denoted by $\pi _k(G)$ . This idea combines graph colorings with Thue sequences introduced at the beginning of 20th century. In particular Thue proved that if $G$ is a simple path of any length greater than $4$ then $\pi _2(G)=3$ and $\pi_3(G)=2$. We investigate $\pi_k(G)$ for other classes of graphs. Particularly interesting open problem is to decide if there is, possibly huge, $k$ such that $\pi_k(G)$ is bounded for planar graphs.]]> 0 <![CDATA[It was conjectured by Reed [reed98conjecture] that for any graph $G$, the graph's chromatic number $χ (G)$ is bounded above by $\lceil Δ (G) +1 + ω (G) / 2\rceil$ , where $Δ (G)$ and $ω (G)$ are the maximum degree and clique number of $G$, respectively. In this paper we prove that this bound holds if $G$ is the line graph of a multigraph. The proof yields a polynomial time algorithm that takes a line graph $G$ and produces a colouring that achieves our bound.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3401 https://doi.org/10.46298/dmtcs.3401 King, Andrew D. Reed, Bruce A. Vetta, Adrian R. King, Andrew D. Reed, Bruce A. Vetta, Adrian R. <![CDATA[It was conjectured by Reed [reed98conjecture] that for any graph $G$, the graph's chromatic number $χ (G)$ is bounded above by $\lceil Δ (G) +1 + ω (G) / 2\rceil$ , where $Δ (G)$ and $ω (G)$ are the maximum degree and clique number of $G$, respectively. In this paper we prove that this bound holds if $G$ is the line graph of a multigraph. The proof yields a polynomial time algorithm that takes a line graph $G$ and produces a colouring that achieves our bound.]]> 0 <![CDATA[In our paper we consider the $P_3$-packing problem in subcubic graphs of different connectivity, improving earlier results of Kelmans and Mubayi. We show that there exists a $P_3$-packing of at least $\lceil 3n/4\rceil$ vertices in any connected subcubic graph of order $n>5$ and minimum vertex degree $\delta \geq 2$, and that this bound is tight. The proof is constructive and implied by a linear-time algorithm. We use this result to show that any $2$-connected cubic graph of order $n>8$ has a $P_3$-packing of at least $\lceil 7n/9 \rceil$ vertices.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3413 https://doi.org/10.46298/dmtcs.3413 Kosowski, Adrian Malafiejski, Michal Zyliński, Pawel Kosowski, Adrian Malafiejski, Michal Zyliński, Pawel <![CDATA[In our paper we consider the $P_3$-packing problem in subcubic graphs of different connectivity, improving earlier results of Kelmans and Mubayi. We show that there exists a $P_3$-packing of at least $\lceil 3n/4\rceil$ vertices in any connected subcubic graph of order $n>5$ and minimum vertex degree $\delta \geq 2$, and that this bound is tight. The proof is constructive and implied by a linear-time algorithm. We use this result to show that any $2$-connected cubic graph of order $n>8$ has a $P_3$-packing of at least $\lceil 7n/9 \rceil$ vertices.]]> 0 <![CDATA[We introduce a notion of a $\textit{broken circuit}$ and an $\textit{NBC complex}$ for an (abstract) convex geometry. Based on these definitions, we shall show the analogues of the Whitney-Rota's formula and Brylawski's decomposition theorem for broken circuit complexes on matroids for convex geometries. We also present an Orlik-Solomon type algebra on a convex geometry, and show the NBC generating theorem.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3412 https://doi.org/10.46298/dmtcs.3412 Kashiwabara, Kenji Nakamura, Masataka Kashiwabara, Kenji Nakamura, Masataka <![CDATA[We introduce a notion of a $\textit{broken circuit}$ and an $\textit{NBC complex}$ for an (abstract) convex geometry. Based on these definitions, we shall show the analogues of the Whitney-Rota's formula and Brylawski's decomposition theorem for broken circuit complexes on matroids for convex geometries. We also present an Orlik-Solomon type algebra on a convex geometry, and show the NBC generating theorem.]]> 0 <![CDATA[A strong stable set in a graph $G$ is a stable set that contains a vertex of every maximal clique of $G$. A Meyniel obstruction is an odd circuit with at least five vertices and at most one chord. Given a graph $G$ and a vertex $v$ of $G$, we give a polytime algorithm to find either a strong stable set containing $v$ or a Meyniel obstruction in $G$. This can then be used to find in any graph, a clique and colouring of the same size or a Meyniel obstruction.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3411 https://doi.org/10.46298/dmtcs.3411 Cameron, Kathie Edmonds, Jack Cameron, Kathie Edmonds, Jack <![CDATA[A strong stable set in a graph $G$ is a stable set that contains a vertex of every maximal clique of $G$. A Meyniel obstruction is an odd circuit with at least five vertices and at most one chord. Given a graph $G$ and a vertex $v$ of $G$, we give a polytime algorithm to find either a strong stable set containing $v$ or a Meyniel obstruction in $G$. This can then be used to find in any graph, a clique and colouring of the same size or a Meyniel obstruction.]]> 0 <![CDATA[We study infinite limits of graphs generated by the duplication model for biological networks. We prove that with probability 1, the sole nontrivial connected component of the limits is unique up to isomorphism. We describe certain infinite deterministic graphs which arise naturally from the model. We characterize the isomorphism type and induced subgraph structure of these infinite graphs using the notion of dismantlability from the theory of vertex pursuit games, and graph homomorphisms.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3410 https://doi.org/10.46298/dmtcs.3410 Bonato, Anthony Janssen, Jeannette Bonato, Anthony Janssen, Jeannette <![CDATA[We study infinite limits of graphs generated by the duplication model for biological networks. We prove that with probability 1, the sole nontrivial connected component of the limits is unique up to isomorphism. We describe certain infinite deterministic graphs which arise naturally from the model. We characterize the isomorphism type and induced subgraph structure of these infinite graphs using the notion of dismantlability from the theory of vertex pursuit games, and graph homomorphisms.]]> 0 <![CDATA[We are looking for the maximum number of subsets of an n-element set not containing 4 distinct subsets satisfying $A ⊂B, C ⊂B, C ⊂D$. It is proved that this number is at least the number of the $\lfloor \frac{n }{ 2}\rfloor$ -element sets times $1+\frac{2}{ n}$, on the other hand an upper bound is given with 4 replaced by the value 2.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3409 https://doi.org/10.46298/dmtcs.3409 Katona, Gyula O.H. Katona, Gyula O.H. <![CDATA[We are looking for the maximum number of subsets of an n-element set not containing 4 distinct subsets satisfying $A ⊂B, C ⊂B, C ⊂D$. It is proved that this number is at least the number of the $\lfloor \frac{n }{ 2}\rfloor$ -element sets times $1+\frac{2}{ n}$, on the other hand an upper bound is given with 4 replaced by the value 2.]]> 0 <![CDATA[We consider the problems of finding the maximum number of vertex-disjoint triangles (VTP) and edge-disjoint triangles (ETP) in a simple graph. Both problems are NP-hard. The algorithm with the best approximation guarantee known so far for these problems has ratio $3/2 + ɛ$, a result that follows from a more general algorithm for set packing obtained by Hurkens and Schrijver in 1989. We present improvements on the approximation ratio for restricted cases of VTP and ETP that are known to be APX-hard: we give an approximation algorithm for VTP on graphs with maximum degree 4 with ratio slightly less than 1.2, and for ETP on graphs with maximum degree 5 with ratio 4/3. We also present an exact linear-time algorithm for VTP on the class of indifference graphs.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3408 https://doi.org/10.46298/dmtcs.3408 Manić, Gordana Wakabayashi, Yoshiko Manić, Gordana Wakabayashi, Yoshiko <![CDATA[We consider the problems of finding the maximum number of vertex-disjoint triangles (VTP) and edge-disjoint triangles (ETP) in a simple graph. Both problems are NP-hard. The algorithm with the best approximation guarantee known so far for these problems has ratio $3/2 + ɛ$, a result that follows from a more general algorithm for set packing obtained by Hurkens and Schrijver in 1989. We present improvements on the approximation ratio for restricted cases of VTP and ETP that are known to be APX-hard: we give an approximation algorithm for VTP on graphs with maximum degree 4 with ratio slightly less than 1.2, and for ETP on graphs with maximum degree 5 with ratio 4/3. We also present an exact linear-time algorithm for VTP on the class of indifference graphs.]]> 0 <![CDATA[A class of graphs $\mathcal{C}$ ordered by the homomorphism relation is universal if every countable partial order can be embedded in $\mathcal{C}$. It was shown in [ZH] that the class $\mathcal{C_k}$ of $k$-colorable graphs, for any fixed $k≥3$, induces a universal partial order. In [HN1], a surprisingly small subclass of $\mathcal{C_3}$ which is a proper subclass of $K_4$-minor-free graphs $(\mathcal{G/K_4)}$ is shown to be universal. In another direction, a density result was given in [PZ], that for each rational number $a/b ∈[2,8/3]∪ \{3\}$, there is a $K_4$-minor-free graph with circular chromatic number equal to $a/b$. In this note we show for each rational number $a/b$ within this interval the class $\mathcal{K_{a/b}}$ of $0K_4$-minor-free graphs with circular chromatic number $a/b$ is universal if and only if $a/b ≠2$, $5/2$ or $3$. This shows yet another surprising richness of the $K_4$-minor-free class that it contains universal classes as dense as the rational numbers.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3407 https://doi.org/10.46298/dmtcs.3407 Nešetřil, Jaroslav Nigussie, Yared Nešetřil, Jaroslav Nigussie, Yared <![CDATA[A class of graphs $\mathcal{C}$ ordered by the homomorphism relation is universal if every countable partial order can be embedded in $\mathcal{C}$. It was shown in [ZH] that the class $\mathcal{C_k}$ of $k$-colorable graphs, for any fixed $k≥3$, induces a universal partial order. In [HN1], a surprisingly small subclass of $\mathcal{C_3}$ which is a proper subclass of $K_4$-minor-free graphs $(\mathcal{G/K_4)}$ is shown to be universal. In another direction, a density result was given in [PZ], that for each rational number $a/b ∈[2,8/3]∪ \{3\}$, there is a $K_4$-minor-free graph with circular chromatic number equal to $a/b$. In this note we show for each rational number $a/b$ within this interval the class $\mathcal{K_{a/b}}$ of $0K_4$-minor-free graphs with circular chromatic number $a/b$ is universal if and only if $a/b ≠2$, $5/2$ or $3$. This shows yet another surprising richness of the $K_4$-minor-free class that it contains universal classes as dense as the rational numbers.]]> 0 <![CDATA[Barabási and Albert [1] suggested modeling scale-free networks by the following random graph process: one node is added at a time and is connected to an earlier node chosen with probability proportional to its degree. A recent empirical study of Newman [5] demonstrates existence of degree-correlation between degrees of adjacent nodes in real-world networks. Here we define the \textitdegree correlation―-correlation of the degrees in a pair of adjacent nodes―-for a random graph process. We determine asymptotically the joint probability distribution for node-degrees, $d$ and $d'$, of adjacent nodes for every $0≤d≤ d'≤n^1 / 5$, and use this result to show that the model of Barabási and Albert does not generate degree-correlation. Our theorem confirms the result in [KR01], obtained by using the mean-field heuristic approach.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3406 https://doi.org/10.46298/dmtcs.3406 Nikoloski, Zoran Deo, Narsingh Kucera, Ludek Nikoloski, Zoran Deo, Narsingh Kucera, Ludek <![CDATA[Barabási and Albert [1] suggested modeling scale-free networks by the following random graph process: one node is added at a time and is connected to an earlier node chosen with probability proportional to its degree. A recent empirical study of Newman [5] demonstrates existence of degree-correlation between degrees of adjacent nodes in real-world networks. Here we define the \textitdegree correlation―-correlation of the degrees in a pair of adjacent nodes―-for a random graph process. We determine asymptotically the joint probability distribution for node-degrees, $d$ and $d'$, of adjacent nodes for every $0≤d≤ d'≤n^1 / 5$, and use this result to show that the model of Barabási and Albert does not generate degree-correlation. Our theorem confirms the result in [KR01], obtained by using the mean-field heuristic approach.]]> 0 <![CDATA[Due to some intractability considerations, reasonable formulation of necessary and sufficient conditions for decomposability of a general multigraph G into a fixed connected multigraph H, is probably not feasible if the underlying simple graph of H has three or more edges. We study the case where H consists of two underlying edges. We present necessary and sufficient conditions for H-decomposability of G, which hold when certain size parameters of G lies within some bounds which depends on the multiplicities of the two edges of H. We also show this result to be "tight" in the sense that even a slight deviation of these size parameters from the given bounds results intractability of the corresponding decision problem.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3405 https://doi.org/10.46298/dmtcs.3405 Priesler, Miri Tarsi, Michael Priesler, Miri Tarsi, Michael <![CDATA[Due to some intractability considerations, reasonable formulation of necessary and sufficient conditions for decomposability of a general multigraph G into a fixed connected multigraph H, is probably not feasible if the underlying simple graph of H has three or more edges. We study the case where H consists of two underlying edges. We present necessary and sufficient conditions for H-decomposability of G, which hold when certain size parameters of G lies within some bounds which depends on the multiplicities of the two edges of H. We also show this result to be "tight" in the sense that even a slight deviation of these size parameters from the given bounds results intractability of the corresponding decision problem.]]> 0 <![CDATA[The deep theorem of Mader concerning the number of internally disjoint H-paths is a very powerfull tool. Nevertheless its use is very difficult, because one has to deal with a very reach family of separators. This paper shows several ways to strengthen Mader's theorem by certain additional restrictions of the appearing separators.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3404 https://doi.org/10.46298/dmtcs.3404 Göring, Frank Göring, Frank <![CDATA[The deep theorem of Mader concerning the number of internally disjoint H-paths is a very powerfull tool. Nevertheless its use is very difficult, because one has to deal with a very reach family of separators. This paper shows several ways to strengthen Mader's theorem by certain additional restrictions of the appearing separators.]]> 0 <![CDATA[Let $K_ℓ^-$ denote the graph obtained from $K_ℓ$ by deleting one edge. We show that for every $γ >0$ and every integer $ℓ≥4$ there exists an integer $n_0=n_0(γ ,ℓ)$ such that every graph $G$ whose order $n≥n_0$ is divisible by $ℓ$ and whose minimum degree is at least $(\frac{ℓ^2-3ℓ+1}{/ ℓ(ℓ-2)}+γ )n$ contains a $K_ℓ^-$-factor, i.e. a collection of disjoint copies of $K_ℓ^-$ which covers all vertices of $G$. This is best possible up to the error term $γn$ and yields an approximate solution to a conjecture of Kawarabayashi.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3403 https://doi.org/10.46298/dmtcs.3403 Kühn, Daniela Osthus, Deryk Kühn, Daniela Osthus, Deryk <![CDATA[Let $K_ℓ^-$ denote the graph obtained from $K_ℓ$ by deleting one edge. We show that for every $γ >0$ and every integer $ℓ≥4$ there exists an integer $n_0=n_0(γ ,ℓ)$ such that every graph $G$ whose order $n≥n_0$ is divisible by $ℓ$ and whose minimum degree is at least $(\frac{ℓ^2-3ℓ+1}{/ ℓ(ℓ-2)}+γ )n$ contains a $K_ℓ^-$-factor, i.e. a collection of disjoint copies of $K_ℓ^-$ which covers all vertices of $G$. This is best possible up to the error term $γn$ and yields an approximate solution to a conjecture of Kawarabayashi.]]> 0 <![CDATA[For any graph $G$, the $k$-improper chromatic number $χ ^k(G)$ is the smallest number of colours used in a colouring of $G$ such that each colour class induces a subgraph of maximum degree $k$. We investigate the ratio of the $k$-improper chromatic number to the clique number for unit disk graphs and random unit disk graphs to extend results of [McRe99, McD03] (where they considered only proper colouring).]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3402 https://doi.org/10.46298/dmtcs.3402 Kang, Ross J. Müller, Tobias Sereni, Jean-Sébastien Kang, Ross J. Müller, Tobias Sereni, Jean-Sébastien <![CDATA[For any graph $G$, the $k$-improper chromatic number $χ ^k(G)$ is the smallest number of colours used in a colouring of $G$ such that each colour class induces a subgraph of maximum degree $k$. We investigate the ratio of the $k$-improper chromatic number to the clique number for unit disk graphs and random unit disk graphs to extend results of [McRe99, McD03] (where they considered only proper colouring).]]> 0 <![CDATA[Given $k$ natural numbers $\{a_1, \ldots ,a_k\} \subset \mathbb{N}$ with $1 \leq a_1 < a_2 < \ldots < a_k$ and $\mathrm{gcd} (a_1, \ldots ,a_k)=1$, let be $R(a_1, \ldots ,a_k) = \{ \lambda_1 a_1+ \cdots + \lambda_k a_k | \space \lambda_i \in \mathbb{N}, i=1 \div k\}$ and $\overline{R}(a_1, \ldots ,a_k) = \mathbb{N} \backslash R (a_1, \ldots ,a_k)$. It is easy to see that $| \overline{R}(a_1, \ldots ,a_k)| < \infty$. The $\textit{Frobenius Problem}$ related to the set $\{a_1, \ldots ,a_k\}$ consists on the computation of $f(a_1, \ldots ,a_k)=\max \overline{R} (a_1, \ldots ,a_k)$, also called the $\textit{Frobenius number}$, and the cardinal $| \overline{R}(a_1, \ldots ,a_k)|$. The solution of the Frobenius Problem is the explicit computation of the set $\overline{R} (a_1,\ldots ,a_k)$. In some cases it is known a sharp upper bound for the Frobenius number. When $k=3$ this bound is known to be $$F(N)=\max\limits_{\substack{0 \lt a \lt b \lt N \\ \mathrm{gcd}(a,b,N)=1}} f(a,b,N)= \begin{cases} 2(\lfloor N/2 \rfloor -1)^2-1 & \textrm{if } N \equiv 0 (\mod 2),\\ 2 \lfloor N/2 \rfloor (\lfloor N/2 \rfloor -1) -1 & \textrm{if } N \equiv 1 (\mod 2).\\ \end{cases}$$ This bound is given in [Dixmier1990]. In this work we give a geometrical proof of this bound which allows us to give the solution of the Frobenius problem for all the sets $\{\alpha ,\beta ,N\}$ such that $f(\alpha ,\beta ,N)=F(N)$.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3462 https://doi.org/10.46298/dmtcs.3462 Aguiló, Francesc Miralles, Alícia Aguiló, Francesc Miralles, Alícia <![CDATA[Given $k$ natural numbers $\{a_1, \ldots ,a_k\} \subset \mathbb{N}$ with $1 \leq a_1 < a_2 < \ldots < a_k$ and $\mathrm{gcd} (a_1, \ldots ,a_k)=1$, let be $R(a_1, \ldots ,a_k) = \{ \lambda_1 a_1+ \cdots + \lambda_k a_k | \space \lambda_i \in \mathbb{N}, i=1 \div k\}$ and $\overline{R}(a_1, \ldots ,a_k) = \mathbb{N} \backslash R (a_1, \ldots ,a_k)$. It is easy to see that $| \overline{R}(a_1, \ldots ,a_k)| < \infty$. The $\textit{Frobenius Problem}$ related to the set $\{a_1, \ldots ,a_k\}$ consists on the computation of $f(a_1, \ldots ,a_k)=\max \overline{R} (a_1, \ldots ,a_k)$, also called the $\textit{Frobenius number}$, and the cardinal $| \overline{R}(a_1, \ldots ,a_k)|$. The solution of the Frobenius Problem is the explicit computation of the set $\overline{R} (a_1,\ldots ,a_k)$. In some cases it is known a sharp upper bound for the Frobenius number. When $k=3$ this bound is known to be $$F(N)=\max\limits_{\substack{0 \lt a \lt b \lt N \\ \mathrm{gcd}(a,b,N)=1}} f(a,b,N)= \begin{cases} 2(\lfloor N/2 \rfloor -1)^2-1 & \textrm{if } N \equiv 0 (\mod 2),\\ 2 \lfloor N/2 \rfloor (\lfloor N/2 \rfloor -1) -1 & \textrm{if } N \equiv 1 (\mod 2).\\ \end{cases}$$ This bound is given in [Dixmier1990]. In this work we give a geometrical proof of this bound which allows us to give the solution of the Frobenius problem for all the sets $\{\alpha ,\beta ,N\}$ such that $f(\alpha ,\beta ,N)=F(N)$.]]> 0 <![CDATA[We consider the problem of solving a system of polynomial equations over fixed algebra $A$ which we call MPolSat($A$). We restrict ourselves to unary algebras and give a partial characterization of complexity of MPolSat($A$). We isolate a preorder $P(A)$ to show that when $A$ has at most 3 elements then MPolSat($A$) is in $P$ when width of $P(A)$ is at most 2 and is NP-complete otherwise. We show also that if $P ≠ NP$ then the class of unary algebras solvable in polynomial time is not closed under homomorphic images.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3474 https://doi.org/10.46298/dmtcs.3474 Broniek, Przemyslaw Broniek, Przemyslaw <![CDATA[We consider the problem of solving a system of polynomial equations over fixed algebra $A$ which we call MPolSat($A$). We restrict ourselves to unary algebras and give a partial characterization of complexity of MPolSat($A$). We isolate a preorder $P(A)$ to show that when $A$ has at most 3 elements then MPolSat($A$) is in $P$ when width of $P(A)$ is at most 2 and is NP-complete otherwise. We show also that if $P ≠ NP$ then the class of unary algebras solvable in polynomial time is not closed under homomorphic images.]]> 0 <![CDATA[We analyze on-line chain partitioning problem and its variants as a two-person game. One person (Spoiler) builds an on-line poset presenting one point at time. The other one (Algorithm) assigns new point to a chain. Kierstead gave a strategy for Algorithm showing that width w posets can be on-line chain partitioned into $\frac{{5}^{w-1}}{4}$ chains. Felsner proved that if Spoiler presents an upgrowing poset, i.e., each new point is maximal at the moment of its arrival then there is a strategy for Algorithm using at most $\binom{w+1}{2}$ chains and it is best possible. An adaptive variant of this problem allows Algorithm to assign to the new point a set of chains and than to remove some of them (but not all) while covering next points. Felsner stated a hypothesis that in on-line adaptive chain covering of upgrowing posets Algorithm may use smaller number of chains than in non-adaptive version. In this paper we provide an argument suggesting that it is true. We present a class of upgrowing posets in which Spoiler has a strategy forcing Algorithm to use at least $\binom{w+1}{2}$ chains (in non-adaptive version) and Algorithm has a strategy using at most $O(w\sqrt{w})$ chains in adaptive version.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3473 https://doi.org/10.46298/dmtcs.3473 Bosek, Bartłomiej Micek, Piotr Bosek, Bartłomiej Micek, Piotr <![CDATA[We analyze on-line chain partitioning problem and its variants as a two-person game. One person (Spoiler) builds an on-line poset presenting one point at time. The other one (Algorithm) assigns new point to a chain. Kierstead gave a strategy for Algorithm showing that width w posets can be on-line chain partitioned into $\frac{{5}^{w-1}}{4}$ chains. Felsner proved that if Spoiler presents an upgrowing poset, i.e., each new point is maximal at the moment of its arrival then there is a strategy for Algorithm using at most $\binom{w+1}{2}$ chains and it is best possible. An adaptive variant of this problem allows Algorithm to assign to the new point a set of chains and than to remove some of them (but not all) while covering next points. Felsner stated a hypothesis that in on-line adaptive chain covering of upgrowing posets Algorithm may use smaller number of chains than in non-adaptive version. In this paper we provide an argument suggesting that it is true. We present a class of upgrowing posets in which Spoiler has a strategy forcing Algorithm to use at least $\binom{w+1}{2}$ chains (in non-adaptive version) and Algorithm has a strategy using at most $O(w\sqrt{w})$ chains in adaptive version.]]> 0 <![CDATA[An on-line vertex coloring algorithm receives vertices of a graph in some externally determined order. Each new vertex is presented together with a set of the edges connecting it to the previously presented vertices. As a vertex is presented, the algorithm assigns it a color which cannot be changed afterwards. The on-line coloring problem was addressed for many different classes of graphs defined in terms of forbidden structures. We analyze the class of $I_s$-free graphs, i.e., graphs in which the maximal size of an independent set is at most $s-1$. An old Szemerédi's result implies that for each on-line algorithm A there exists an on-line presentation of an $I_s$-free graph $G$ forcing A to use at least $\frac{s}{2}χ ^{(G)}$ colors. We prove that any greedy algorithm uses at most $\frac{s}{2}χ^{(G)}$ colors for any on-line presentation of any $I_s$-free graph $G$. Since the class of co-planar graphs is a subclass of $I_5$-free graphs all greedy algorithms use at most $\frac{5}{2}χ (G)$ colors for co-planar $G$'s. We prove that, even in a smaller class, this is an almost tight bound.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3472 https://doi.org/10.46298/dmtcs.3472 Cieslik, Iwona Kozik, Marcin Micek, Piotr Cieslik, Iwona Kozik, Marcin Micek, Piotr <![CDATA[An on-line vertex coloring algorithm receives vertices of a graph in some externally determined order. Each new vertex is presented together with a set of the edges connecting it to the previously presented vertices. As a vertex is presented, the algorithm assigns it a color which cannot be changed afterwards. The on-line coloring problem was addressed for many different classes of graphs defined in terms of forbidden structures. We analyze the class of $I_s$-free graphs, i.e., graphs in which the maximal size of an independent set is at most $s-1$. An old Szemerédi's result implies that for each on-line algorithm A there exists an on-line presentation of an $I_s$-free graph $G$ forcing A to use at least $\frac{s}{2}χ ^{(G)}$ colors. We prove that any greedy algorithm uses at most $\frac{s}{2}χ^{(G)}$ colors for any on-line presentation of any $I_s$-free graph $G$. Since the class of co-planar graphs is a subclass of $I_5$-free graphs all greedy algorithms use at most $\frac{5}{2}χ (G)$ colors for co-planar $G$'s. We prove that, even in a smaller class, this is an almost tight bound.]]> 0 <![CDATA[In this paper we prove that the question whether a language presented by a context free grammar has density, is undecidable. Moreover we show that there is no algorithm which, given two unambiguous context free grammars on input, decides whether the language defined by the first grammar has a relative density in the language defined by the second one. Our techniques can be extended to show that this problem is undecidable even for languages given by grammars from $LL(k)$ (for sufficiently large fixed $k ∈ \mathbb{N} )$.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3471 https://doi.org/10.46298/dmtcs.3471 Kozik, Jakub Kozik, Jakub <![CDATA[In this paper we prove that the question whether a language presented by a context free grammar has density, is undecidable. Moreover we show that there is no algorithm which, given two unambiguous context free grammars on input, decides whether the language defined by the first grammar has a relative density in the language defined by the second one. Our techniques can be extended to show that this problem is undecidable even for languages given by grammars from $LL(k)$ (for sufficiently large fixed $k ∈ \mathbb{N} )$.]]> 0 <![CDATA[In this paper, we introduce the $λ μ ^{∧∨}$ - call-by-value calculus and we give a proof of the Church-Rosser property of this system. This proof is an adaptation of that of Andou (2003) which uses an extended parallel reduction method and complete development.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3470 https://doi.org/10.46298/dmtcs.3470 Nour, Karim Saber, Khelifa Nour, Karim Saber, Khelifa <![CDATA[In this paper, we introduce the $λ μ ^{∧∨}$ - call-by-value calculus and we give a proof of the Church-Rosser property of this system. This proof is an adaptation of that of Andou (2003) which uses an extended parallel reduction method and complete development.]]> 0 <![CDATA[Combinatory logic shows that bound variables can be eliminated without loss of expressiveness. It has applications both in the foundations of mathematics and in the implementation of functional programming languages. The original combinatory calculus corresponds to minimal implicative logic written in a system "à la Hilbert''. We present in this paper a combinatory logic which corresponds to propositional classical logic. This system is equivalent to the system $λ ^{Sym}_{Prop}$ of Barbanera and Berardi.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3469 https://doi.org/10.46298/dmtcs.3469 Nour, Karim Nour, Karim <![CDATA[Combinatory logic shows that bound variables can be eliminated without loss of expressiveness. It has applications both in the foundations of mathematics and in the implementation of functional programming languages. The original combinatory calculus corresponds to minimal implicative logic written in a system "à la Hilbert''. We present in this paper a combinatory logic which corresponds to propositional classical logic. This system is equivalent to the system $λ ^{Sym}_{Prop}$ of Barbanera and Berardi.]]> 0 <![CDATA[In sequential games of traditional game theory, backward induction guarantees existence of Nash equilibrium by yielding a sub-game perfect equilibrium. But if payoffs range over a partially ordered set instead of the reals, then the backward induction predicate does no longer imply the Nash equilibrium predicate. Non-determinism is a solution: a suitable non-deterministic backward induction function returns a non-deterministic strategy profile which is a non-deterministic Nash equilibrium. The main notions and results in this article are constructive, conceptually simple and formalised in the proof assistant Coq.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3468 https://doi.org/10.46298/dmtcs.3468 Le Roux, Stéphane Le Roux, Stéphane <![CDATA[In sequential games of traditional game theory, backward induction guarantees existence of Nash equilibrium by yielding a sub-game perfect equilibrium. But if payoffs range over a partially ordered set instead of the reals, then the backward induction predicate does no longer imply the Nash equilibrium predicate. Non-determinism is a solution: a suitable non-deterministic backward induction function returns a non-deterministic strategy profile which is a non-deterministic Nash equilibrium. The main notions and results in this article are constructive, conceptually simple and formalised in the proof assistant Coq.]]> 0 <![CDATA[A kernel $N$ of a digraph $D$ is an independent set of vertices of $D$ such that for every $w \in V(D)-N$ there exists an arc from $w$ to $N$. If every induced subdigraph of $D$ has a kernel, $D$ is said to be a kernel perfect digraph. Minimal non-kernel perfect digraph are called critical kernel imperfect digraph. If $F$ is a set of arcs of $D$, a semikernel modulo $F$, $S$ of $D$ is an independent set of vertices of $D$ such that for every $z \in V(D)- S$ for which there exists an $Sz-$arc of $D-F$, there also exists an $zS-$arc in $D$. In this talk some structural results concerning critical kernel imperfect and sufficient conditions for a digraph to be a critical kernel imperfect digraph are presented.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3467 https://doi.org/10.46298/dmtcs.3467 Galeana-Sánchez, Hortensia Guevara, Mucuy-Kak Galeana-Sánchez, Hortensia Guevara, Mucuy-Kak <![CDATA[A kernel $N$ of a digraph $D$ is an independent set of vertices of $D$ such that for every $w \in V(D)-N$ there exists an arc from $w$ to $N$. If every induced subdigraph of $D$ has a kernel, $D$ is said to be a kernel perfect digraph. Minimal non-kernel perfect digraph are called critical kernel imperfect digraph. If $F$ is a set of arcs of $D$, a semikernel modulo $F$, $S$ of $D$ is an independent set of vertices of $D$ such that for every $z \in V(D)- S$ for which there exists an $Sz-$arc of $D-F$, there also exists an $zS-$arc in $D$. In this talk some structural results concerning critical kernel imperfect and sufficient conditions for a digraph to be a critical kernel imperfect digraph are presented.]]> 0 <![CDATA[The generalized quadrangle $Q(4,q)$ arising from the parabolic quadric in $PG(4,q)$ always has an ovoid. It is not known whether a minimal blocking set of size smaller than $q^2 + q$ (which is not an ovoid) exists in $Q(4,q)$, $q$ odd. We present results on smallest blocking sets in $Q(4,q)$, $q$ odd, obtained by a computer search. For $q = 5,7,9,11$ we found minimal blocking sets of size $q^2 + q - 2$ and we discuss their structure. By an exhaustive search we excluded the existence of a minimal blocking set of size $q^2 + 3$ in $Q(4,7)$.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3466 https://doi.org/10.46298/dmtcs.3466 Cimráková, Miroslava Fack, Veerle Cimráková, Miroslava Fack, Veerle <![CDATA[The generalized quadrangle $Q(4,q)$ arising from the parabolic quadric in $PG(4,q)$ always has an ovoid. It is not known whether a minimal blocking set of size smaller than $q^2 + q$ (which is not an ovoid) exists in $Q(4,q)$, $q$ odd. We present results on smallest blocking sets in $Q(4,q)$, $q$ odd, obtained by a computer search. For $q = 5,7,9,11$ we found minimal blocking sets of size $q^2 + q - 2$ and we discuss their structure. By an exhaustive search we excluded the existence of a minimal blocking set of size $q^2 + 3$ in $Q(4,7)$.]]> 0 <![CDATA[We prove that every cubic bridgeless graph $G$ contains a $2$-factor which intersects all (minimal) edge-cuts of size $3$ or $4$. This generalizes an earlier result of the authors, namely that such a $2$-factor exists provided that $G$ is planar. As a further extension, we show that every graph contains a cycle (a union of edge-disjoint circuits) that intersects all edge-cuts of size $3$ or $4$. Motivated by this result, we introduce the concept of a coverable set of integers and discuss a number of questions, some of which are related to classical problems of graph theory such as Tutte's $4$-flow conjecture or the Dominating circuit conjecture.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3465 https://doi.org/10.46298/dmtcs.3465 Kaiser, Tomáš Škrekovski, Riste Kaiser, Tomáš Škrekovski, Riste <![CDATA[We prove that every cubic bridgeless graph $G$ contains a $2$-factor which intersects all (minimal) edge-cuts of size $3$ or $4$. This generalizes an earlier result of the authors, namely that such a $2$-factor exists provided that $G$ is planar. As a further extension, we show that every graph contains a cycle (a union of edge-disjoint circuits) that intersects all edge-cuts of size $3$ or $4$. Motivated by this result, we introduce the concept of a coverable set of integers and discuss a number of questions, some of which are related to classical problems of graph theory such as Tutte's $4$-flow conjecture or the Dominating circuit conjecture.]]> 0 <![CDATA[We prove the existence of many complete graphs in almost all sufficiently dense partitions obtained by an application of Szemerédi's Regularity Lemma. More precisely, we consider the number of complete graphs $K_{\ell}$ on $\ell$ vertices in $\ell$-partite graphs where each partition class consists of $n$ vertices and there is an $\varepsilon$-regular graph on $m$ edges between any two partition classes. We show that for all $\beta > $0, at most a $\beta^m$-fraction of graphs in this family contain less than the expected number of copies of $K_{\ell}$ provided $\varepsilon$ is sufficiently small and $m \geq Cn^{2-1/(\ell-1)}$ for a constant $C > 0$ and $n$ sufficiently large. This result is a counting version of a restricted version of a conjecture by Kohayakawa, Łuczak and Rödl and has several implications for random graphs.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3464 https://doi.org/10.46298/dmtcs.3464 Gerke, Stefanie Marciniszyn, Martin Steger, Angelika Gerke, Stefanie Marciniszyn, Martin Steger, Angelika <![CDATA[We prove the existence of many complete graphs in almost all sufficiently dense partitions obtained by an application of Szemerédi's Regularity Lemma. More precisely, we consider the number of complete graphs $K_{\ell}$ on $\ell$ vertices in $\ell$-partite graphs where each partition class consists of $n$ vertices and there is an $\varepsilon$-regular graph on $m$ edges between any two partition classes. We show that for all $\beta > $0, at most a $\beta^m$-fraction of graphs in this family contain less than the expected number of copies of $K_{\ell}$ provided $\varepsilon$ is sufficiently small and $m \geq Cn^{2-1/(\ell-1)}$ for a constant $C > 0$ and $n$ sufficiently large. This result is a counting version of a restricted version of a conjecture by Kohayakawa, Łuczak and Rödl and has several implications for random graphs.]]> 0 <![CDATA[For a hypergraph $\mathcal{H} = (V,\mathcal{E})$, its $d$―fold symmetric product is $\Delta^d \mathcal{H} = (V^d,\{ E^d | E \in \mathcal{E} \})$. We give several upper and lower bounds for the $c$-color discrepancy of such products. In particular, we show that the bound $\textrm{disc}(\Delta^d \mathcal{H},2) \leq \textrm{disc}(\mathcal{H},2)$ proven for all $d$ in [B. Doerr, A. Srivastav, and P. Wehr, Discrepancy of Cartesian products of arithmetic progressions, Electron. J. Combin. 11(2004), Research Paper 5, 16 pp.] cannot be extended to more than $c = 2$ colors. In fact, for any $c$ and $d$ such that $c$ does not divide $d!$, there are hypergraphs having arbitrary large discrepancy and $\textrm{disc}(\Delta^d \mathcal{H},c) = \Omega_d(\textrm{disc}(\mathcal{H},c)^d)$. Apart from constant factors (depending on $c$ and $d$), in these cases the symmetric product behaves no better than the general direct product $\mathcal{H}^d$, which satisfies $\textrm{disc}(\mathcal{H}^d,c) = O_{c,d}(\textrm{disc}(\mathcal{H},c)^d)$.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3463 https://doi.org/10.46298/dmtcs.3463 Doerr, Benjamin Gnewuch, Michael Hebbinghaus, Nils Doerr, Benjamin Gnewuch, Michael Hebbinghaus, Nils <![CDATA[For a hypergraph $\mathcal{H} = (V,\mathcal{E})$, its $d$―fold symmetric product is $\Delta^d \mathcal{H} = (V^d,\{ E^d | E \in \mathcal{E} \})$. We give several upper and lower bounds for the $c$-color discrepancy of such products. In particular, we show that the bound $\textrm{disc}(\Delta^d \mathcal{H},2) \leq \textrm{disc}(\mathcal{H},2)$ proven for all $d$ in [B. Doerr, A. Srivastav, and P. Wehr, Discrepancy of Cartesian products of arithmetic progressions, Electron. J. Combin. 11(2004), Research Paper 5, 16 pp.] cannot be extended to more than $c = 2$ colors. In fact, for any $c$ and $d$ such that $c$ does not divide $d!$, there are hypergraphs having arbitrary large discrepancy and $\textrm{disc}(\Delta^d \mathcal{H},c) = \Omega_d(\textrm{disc}(\mathcal{H},c)^d)$. Apart from constant factors (depending on $c$ and $d$), in these cases the symmetric product behaves no better than the general direct product $\mathcal{H}^d$, which satisfies $\textrm{disc}(\mathcal{H}^d,c) = O_{c,d}(\textrm{disc}(\mathcal{H},c)^d)$.]]> 0 <![CDATA[An acyclic coloring of a graph $G$ is a coloring of its vertices such that: (i) no two neighbors in $G$ are assigned the same color and (ii) no bicolored cycle can exist in $G$. The acyclic chromatic number of $G$ is the least number of colors necessary to acyclically color $G$, and is denoted by $a(G)$. We show that any graph of maximum degree $\Delta$ has acyclic chromatic number at most $\frac{\Delta (\Delta -1) }{ 2}$ for any $\Delta \geq 5$, and we give an $O(n \Delta^2)$ algorithm to acyclically color any graph of maximum degree $\Delta$ with the above mentioned number of colors. This result is roughly two times better than the best general upper bound known so far, yielding $a(G) \leq \Delta (\Delta -1) +2$. By a deeper study of the case $\Delta =5$, we also show that any graph of maximum degree $5$ can be acyclically colored with at most $9$ colors, and give a linear time algorithm to achieve this bound.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3450 https://doi.org/10.46298/dmtcs.3450 Fertin, Guillaume Raspaud, André Fertin, Guillaume Raspaud, André <![CDATA[An acyclic coloring of a graph $G$ is a coloring of its vertices such that: (i) no two neighbors in $G$ are assigned the same color and (ii) no bicolored cycle can exist in $G$. The acyclic chromatic number of $G$ is the least number of colors necessary to acyclically color $G$, and is denoted by $a(G)$. We show that any graph of maximum degree $\Delta$ has acyclic chromatic number at most $\frac{\Delta (\Delta -1) }{ 2}$ for any $\Delta \geq 5$, and we give an $O(n \Delta^2)$ algorithm to acyclically color any graph of maximum degree $\Delta$ with the above mentioned number of colors. This result is roughly two times better than the best general upper bound known so far, yielding $a(G) \leq \Delta (\Delta -1) +2$. By a deeper study of the case $\Delta =5$, we also show that any graph of maximum degree $5$ can be acyclically colored with at most $9$ colors, and give a linear time algorithm to achieve this bound.]]> 0 <![CDATA[We consider an extremal problem on labelled directed trees and applications to database theory. Among others, we will show explicit keysystems on an underlying set of size $n$, that cannot be represented by a database of less than $2^{n(1-c\cdot \log \log n / \log n)}$ rows.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3461 https://doi.org/10.46298/dmtcs.3461 Katona, Gyula O.H. Tichler, Krisztián Katona, Gyula O.H. Tichler, Krisztián <![CDATA[We consider an extremal problem on labelled directed trees and applications to database theory. Among others, we will show explicit keysystems on an underlying set of size $n$, that cannot be represented by a database of less than $2^{n(1-c\cdot \log \log n / \log n)}$ rows.]]> 0 <![CDATA[It is proved that any graph of order $14n/3 + O(1)$ contains a family of n induced subgraphs of order $3$ such that they are vertex-disjoint and equivalent to each other.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3460 https://doi.org/10.46298/dmtcs.3460 Nakamigawa, Tomoki Nakamigawa, Tomoki <![CDATA[It is proved that any graph of order $14n/3 + O(1)$ contains a family of n induced subgraphs of order $3$ such that they are vertex-disjoint and equivalent to each other.]]> 0 <![CDATA[For convex bodies $K$ with $\mathcal{C}^2$ boundary in $\mathbb{R}^d$, we provide results on the volume of random polytopes with vertices chosen along the boundary of $K$ which we call $\textit{random inscribing polytopes}$. In particular, we prove results concerning the variance and higher moments of the volume, as well as show that the random inscribing polytopes generated by the Poisson process satisfy central limit theorem.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3459 https://doi.org/10.46298/dmtcs.3459 Richardson, Ross M. Vu, Van H. Wu, Lei Richardson, Ross M. Vu, Van H. Wu, Lei <![CDATA[For convex bodies $K$ with $\mathcal{C}^2$ boundary in $\mathbb{R}^d$, we provide results on the volume of random polytopes with vertices chosen along the boundary of $K$ which we call $\textit{random inscribing polytopes}$. In particular, we prove results concerning the variance and higher moments of the volume, as well as show that the random inscribing polytopes generated by the Poisson process satisfy central limit theorem.]]> 0 <![CDATA[We define an infinite permutation as a sequence of reals taken up to the order, or, equivalently, as a linear ordering of a finite or countable set. Then we introduce and characterize periodic permutations; surprisingly, for each period $t$ there is an infinite number of distinct $t$-periodic permutations. At last, we introduce a complexity notion for permutations analogous to subword complexity for words, and consider the problem of minimal complexity of non-periodic permutations. Its answer is different for the right infinite and the bi-infinite case.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3458 https://doi.org/10.46298/dmtcs.3458 Fon-Der-Flaass, Dmitri G. Frid, Anna E. Fon-Der-Flaass, Dmitri G. Frid, Anna E. <![CDATA[We define an infinite permutation as a sequence of reals taken up to the order, or, equivalently, as a linear ordering of a finite or countable set. Then we introduce and characterize periodic permutations; surprisingly, for each period $t$ there is an infinite number of distinct $t$-periodic permutations. At last, we introduce a complexity notion for permutations analogous to subword complexity for words, and consider the problem of minimal complexity of non-periodic permutations. Its answer is different for the right infinite and the bi-infinite case.]]> 0 <![CDATA[It is well known that every bipartite graph with vertex classes of size $n$ whose minimum degree is at least $n/2$ contains a perfect matching. We prove an analogue of this result for uniform hypergraphs. We also provide an analogue of Dirac's theorem on Hamilton cycles for $3$-uniform hypergraphs: We say that a $3$-uniform hypergraph has a Hamilton cycle if there is a cyclic ordering of its vertices such that every pair of consecutive vertices lies in a hyperedge which consists of three consecutive vertices. We prove that for every $\varepsilon > 0$ there is an $n_0$ such that every $3$-uniform hypergraph of order $n \geq n_0$ whose minimum degree is at least $n/4+ \varepsilon n$ contains a Hamilton cycle. Our bounds on the minimum degree are essentially best possible.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3457 https://doi.org/10.46298/dmtcs.3457 Kühn, Daniela Osthus, Deryk Kühn, Daniela Osthus, Deryk <![CDATA[It is well known that every bipartite graph with vertex classes of size $n$ whose minimum degree is at least $n/2$ contains a perfect matching. We prove an analogue of this result for uniform hypergraphs. We also provide an analogue of Dirac's theorem on Hamilton cycles for $3$-uniform hypergraphs: We say that a $3$-uniform hypergraph has a Hamilton cycle if there is a cyclic ordering of its vertices such that every pair of consecutive vertices lies in a hyperedge which consists of three consecutive vertices. We prove that for every $\varepsilon > 0$ there is an $n_0$ such that every $3$-uniform hypergraph of order $n \geq n_0$ whose minimum degree is at least $n/4+ \varepsilon n$ contains a Hamilton cycle. Our bounds on the minimum degree are essentially best possible.]]> 0 <![CDATA[Given positive integers $q$, $n$ and $d$, denote by $A_q(n,d)$ the maximum size of a $q$-ary code of length $n$ and minimum distance $d$. The famous Gilbert-Varshamov bound asserts that $A_q(n,d+1) \geq q^n / V_q(n,d)$, where $V_q(n,d)=\sum_{i=0}^d \binom{n}{i}(q-1)^i$ is the volume of a $q$-ary sphere of radius $d$. Extending a recent work of Jiang and Vardy on binary codes, we show that for any positive constant $\alpha$ less than $(q-1)/q$ there is a positive constant $c$ such that for $d \leq \alpha n, A_q(n,d+1) \geq c \frac{q^n}{ V_q(n,d)}n$. This confirms a conjecture by Jiang and Vardy.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3456 https://doi.org/10.46298/dmtcs.3456 Vu, Van H. Wu, Lei Vu, Van H. Wu, Lei <![CDATA[Given positive integers $q$, $n$ and $d$, denote by $A_q(n,d)$ the maximum size of a $q$-ary code of length $n$ and minimum distance $d$. The famous Gilbert-Varshamov bound asserts that $A_q(n,d+1) \geq q^n / V_q(n,d)$, where $V_q(n,d)=\sum_{i=0}^d \binom{n}{i}(q-1)^i$ is the volume of a $q$-ary sphere of radius $d$. Extending a recent work of Jiang and Vardy on binary codes, we show that for any positive constant $\alpha$ less than $(q-1)/q$ there is a positive constant $c$ such that for $d \leq \alpha n, A_q(n,d+1) \geq c \frac{q^n}{ V_q(n,d)}n$. This confirms a conjecture by Jiang and Vardy.]]> 0 <![CDATA[The Road Coloring Conjecture is an old and classical conjecture e posed in Adler and Weiss (1970); Adler et al. (1977). Let $G$ be a strongly connected digraph with uniform out-degree $2$. The Road Coloring Conjecture states that, under a natural (necessary) condition that $G$ is "aperiodic'', the edges of $G$ can be colored red and blue such that "universal driving directions'' can be given for each vertex. More precisely, each vertex has one red and one blue edge leaving it, and for any vertex $v$ there exists a sequence $s_v$ of reds and blues such that following the sequence from $\textit{any}$ starting vertex in $G$ ends precisely at the vertex $v$. We first generalize the conjecture to a min-max conjecture for all strongly connected digraphs. We then generalize the notion of coloring itself. Instead of assigning exactly one color to each edge we allow multiple colors to each edge. Under this relaxed notion of coloring we prove our generalized Min-Max theorem. Using the Prime Number Theorem (PNT) we further show that the number of colors needed for each edge is bounded above by $O(\log n / \log \log n)$, where $n$ is the number of vertices in the digraph.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3455 https://doi.org/10.46298/dmtcs.3455 Hegde, Rajneesh Jain, Kamal Hegde, Rajneesh Jain, Kamal <![CDATA[The Road Coloring Conjecture is an old and classical conjecture e posed in Adler and Weiss (1970); Adler et al. (1977). Let $G$ be a strongly connected digraph with uniform out-degree $2$. The Road Coloring Conjecture states that, under a natural (necessary) condition that $G$ is "aperiodic'', the edges of $G$ can be colored red and blue such that "universal driving directions'' can be given for each vertex. More precisely, each vertex has one red and one blue edge leaving it, and for any vertex $v$ there exists a sequence $s_v$ of reds and blues such that following the sequence from $\textit{any}$ starting vertex in $G$ ends precisely at the vertex $v$. We first generalize the conjecture to a min-max conjecture for all strongly connected digraphs. We then generalize the notion of coloring itself. Instead of assigning exactly one color to each edge we allow multiple colors to each edge. Under this relaxed notion of coloring we prove our generalized Min-Max theorem. Using the Prime Number Theorem (PNT) we further show that the number of colors needed for each edge is bounded above by $O(\log n / \log \log n)$, where $n$ is the number of vertices in the digraph.]]> 0 <![CDATA[We introduce and study balanced online graph avoidance games on the random graph process. The game is played by a player we call Painter. Edges of the complete graph with $n$ vertices are revealed two at a time in a random order. In each move, Painter immediately and irrevocably decides on a balanced coloring of the new edge pair: either the first edge is colored red and the second one blue or vice versa. His goal is to avoid a monochromatic copy of a given fixed graph $H$ in both colors for as long as possible. The game ends as soon as the first monochromatic copy of $H$ has appeared. We show that the duration of the game is determined by a threshold function $m_H = m_H(n)$. More precisely, Painter will asymptotically almost surely (a.a.s.) lose the game after $m = \omega (m_H)$ edge pairs in the process. On the other hand, there is an essentially optimal strategy, that is, if the game lasts for $m = o(m_H)$ moves, then Painter will a.a.s. successfully avoid monochromatic copies of H using this strategy. Our attempt is to determine the threshold function for certain graph-theoretic structures, e.g., cycles.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3454 https://doi.org/10.46298/dmtcs.3454 Marciniszyn, Martin Mitsche, Dieter Stojaković, Miloš Marciniszyn, Martin Mitsche, Dieter Stojaković, Miloš <![CDATA[We introduce and study balanced online graph avoidance games on the random graph process. The game is played by a player we call Painter. Edges of the complete graph with $n$ vertices are revealed two at a time in a random order. In each move, Painter immediately and irrevocably decides on a balanced coloring of the new edge pair: either the first edge is colored red and the second one blue or vice versa. His goal is to avoid a monochromatic copy of a given fixed graph $H$ in both colors for as long as possible. The game ends as soon as the first monochromatic copy of $H$ has appeared. We show that the duration of the game is determined by a threshold function $m_H = m_H(n)$. More precisely, Painter will asymptotically almost surely (a.a.s.) lose the game after $m = \omega (m_H)$ edge pairs in the process. On the other hand, there is an essentially optimal strategy, that is, if the game lasts for $m = o(m_H)$ moves, then Painter will a.a.s. successfully avoid monochromatic copies of H using this strategy. Our attempt is to determine the threshold function for certain graph-theoretic structures, e.g., cycles.]]> 0 <![CDATA[We find the formula for the cardinality of maximal set of integers from $[1,\ldots,n]$ which does not contain $k+1$ pairwise coprimes and has divisors from a specified set of primes. This formula is defined by the set of multiples of the generating set, which does not depend on $n$.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3453 https://doi.org/10.46298/dmtcs.3453 Blinovsky, Vladimir Blinovsky, Vladimir <![CDATA[We find the formula for the cardinality of maximal set of integers from $[1,\ldots,n]$ which does not contain $k+1$ pairwise coprimes and has divisors from a specified set of primes. This formula is defined by the set of multiples of the generating set, which does not depend on $n$.]]> 0 <![CDATA[Thomassen conjectured that every $4$-connected line graph is hamiltonian. A vertex cut $X$ of $G$ is essential if $G-X$ has at least two nontrivial components. We prove that every $3$-connected, essentially $11$-connected line graph is hamiltonian. Using Ryjáček's line graph closure, it follows that every $3$-connected, essentially $11$-connected claw-free graph is hamiltonian.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3452 https://doi.org/10.46298/dmtcs.3452 Lai, Hong-Jian Shao, Yehong Zhou, Ju Wu, Hehui Lai, Hong-Jian Shao, Yehong Zhou, Ju Wu, Hehui <![CDATA[Thomassen conjectured that every $4$-connected line graph is hamiltonian. A vertex cut $X$ of $G$ is essential if $G-X$ has at least two nontrivial components. We prove that every $3$-connected, essentially $11$-connected line graph is hamiltonian. Using Ryjáček's line graph closure, it follows that every $3$-connected, essentially $11$-connected claw-free graph is hamiltonian.]]> 0 <![CDATA[We show that the number $g_n$ of labelled series-parallel graphs on $n$ vertices is asymptotically $g_n \sim g \cdot n^{-5/2} \gamma^n n!$, where $\gamma$ and $g$ are explicit computable constants. We show that the number of edges in random series-parallel graphs is asymptotically normal with linear mean and variance, and that the number of edges is sharply concentrated around its expected value. Similar results are proved for labelled outerplanar graphs.]]> Fri, 31 Dec 2004 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3451 https://doi.org/10.46298/dmtcs.3451 Bodirsky, Manuel Gimenez, Omer Kang, Mihyun Noy, Marc Bodirsky, Manuel Gimenez, Omer Kang, Mihyun Noy, Marc <![CDATA[We show that the number $g_n$ of labelled series-parallel graphs on $n$ vertices is asymptotically $g_n \sim g \cdot n^{-5/2} \gamma^n n!$, where $\gamma$ and $g$ are explicit computable constants. We show that the number of edges in random series-parallel graphs is asymptotically normal with linear mean and variance, and that the number of edges is sharply concentrated around its expected value. Similar results are proved for labelled outerplanar graphs.]]> 0 <![CDATA[In \textitDynamical sources in information theory: fundamental intervals and word prefixes, B. Vallée studies statistical properties of words generated by dynamical sources. This is done using generalized Ruelle operators. The aim of this article is to generalize sources for which the results hold. First, we avoid the use of Grotendieck theory and Fredholm determinants, this allows dynamical sources that cannot be extended to a complex disk or that are not analytic. Second, we consider Markov sources: the language generated by the source over an alphabet \textbfM is not necessarily \textbfM^*.]]> Thu, 01 Jan 2004 07:00:00 +0000 https://doi.org/10.46298/dmtcs.329 https://doi.org/10.46298/dmtcs.329 Chazal, Frédéric Maume-Deschamps, Véronique Chazal, Frédéric Maume-Deschamps, Véronique <![CDATA[In \textitDynamical sources in information theory: fundamental intervals and word prefixes, B. Vallée studies statistical properties of words generated by dynamical sources. This is done using generalized Ruelle operators. The aim of this article is to generalize sources for which the results hold. First, we avoid the use of Grotendieck theory and Fredholm determinants, this allows dynamical sources that cannot be extended to a complex disk or that are not analytic. Second, we consider Markov sources: the language generated by the source over an alphabet \textbfM is not necessarily \textbfM^*.]]> 0 <![CDATA[We discuss Ray-Chaudhari and Wilson inequality for a 0-design and give simple proof of the result '\emphFor fixed block size k, there exist finitely many parametrically feasible t-designs with t intersection numbers and λ > 1'.]]> Thu, 01 Jan 2004 07:00:00 +0000 https://doi.org/10.46298/dmtcs.332 https://doi.org/10.46298/dmtcs.332 Pawale, Rajendra M. Pawale, Rajendra M. <![CDATA[We discuss Ray-Chaudhari and Wilson inequality for a 0-design and give simple proof of the result '\emphFor fixed block size k, there exist finitely many parametrically feasible t-designs with t intersection numbers and λ > 1'.]]> 0 <![CDATA[Observational theories are a generalization of first-order theories where two objects are observationally equal if they cannot be distinguished by experiments with observable results. Such experiments, called contexts, are usually infinite. Therfore, we consider a special finite set of contexts, called cover-contexts, ''\emphcovering'' all the observable contexts. Then, we show that to prove that two objects are observationally equal, it is sufficient to prove that they are equal (in the classical sense) under these cover-contexts. We give methods based on rewriting techniques, for constructing such cover-contexts for interesting classes of observational specifications.]]> Thu, 01 Jan 2004 07:00:00 +0000 https://doi.org/10.46298/dmtcs.331 https://doi.org/10.46298/dmtcs.331 Berregeb, Narjes Robbana, Riadh Tiwari, Ashish Berregeb, Narjes Robbana, Riadh Tiwari, Ashish <![CDATA[Observational theories are a generalization of first-order theories where two objects are observationally equal if they cannot be distinguished by experiments with observable results. Such experiments, called contexts, are usually infinite. Therfore, we consider a special finite set of contexts, called cover-contexts, ''\emphcovering'' all the observable contexts. Then, we show that to prove that two objects are observationally equal, it is sufficient to prove that they are equal (in the classical sense) under these cover-contexts. We give methods based on rewriting techniques, for constructing such cover-contexts for interesting classes of observational specifications.]]> 0 <![CDATA[An implication system (IS) on a finite set S is a set of rules called Σ -implications of the kind A →_Σ B, with A,B ⊆ S. A subset X ⊆ S satisfies A →_Σ B when ''A ⊆ X implies B ⊆ X'' holds, so ISs can be used to describe constraints on sets of elements, such as dependency or causality. ISs are formally closely linked to the well known notions of closure operators and Moore families. This paper focuses on their algorithmic aspects. A number of problems issued from an IS Σ (e.g. is it minimal, is a given implication entailed by the system) can be reduced to the computation of closures φ _Σ (X), where φ _Σ is the closure operator associated to Σ . We propose a new approach to compute such closures, based on the characterization of the direct-optimal IS Σ _do which has the following properties: \beginenumerate ıtemit is equivalent to Σ ıtemφ _Σ _do(X) (thus φ _Σ (X)) can be computed by a single scanning of Σ _do-implications ıtemit is of minimal size with respect to ISs satisfying 1. and 2. \endenumerate We give algorithms that compute Σ _do, and from Σ _do closures φ _Σ (X) and the Moore family associated to φ _Σ .]]> Thu, 01 Jan 2004 07:00:00 +0000 https://doi.org/10.46298/dmtcs.330 https://doi.org/10.46298/dmtcs.330 Bertet, Karell Nebut, Mirabelle Bertet, Karell Nebut, Mirabelle <![CDATA[An implication system (IS) on a finite set S is a set of rules called Σ -implications of the kind A →_Σ B, with A,B ⊆ S. A subset X ⊆ S satisfies A →_Σ B when ''A ⊆ X implies B ⊆ X'' holds, so ISs can be used to describe constraints on sets of elements, such as dependency or causality. ISs are formally closely linked to the well known notions of closure operators and Moore families. This paper focuses on their algorithmic aspects. A number of problems issued from an IS Σ (e.g. is it minimal, is a given implication entailed by the system) can be reduced to the computation of closures φ _Σ (X), where φ _Σ is the closure operator associated to Σ . We propose a new approach to compute such closures, based on the characterization of the direct-optimal IS Σ _do which has the following properties: \beginenumerate ıtemit is equivalent to Σ ıtemφ _Σ _do(X) (thus φ _Σ (X)) can be computed by a single scanning of Σ _do-implications ıtemit is of minimal size with respect to ISs satisfying 1. and 2. \endenumerate We give algorithms that compute Σ _do, and from Σ _do closures φ _Σ (X) and the Moore family associated to φ _Σ .]]> 0 <![CDATA[A \emph(k,t)-track layout of a graph G consists of a (proper) vertex t-colouring of G, a total order of each vertex colour class, and a (non-proper) edge k-colouring such that between each pair of colour classes no two monochromatic edges cross. This structure has recently arisen in the study of three-dimensional graph drawings. This paper presents the beginnings of a theory of track layouts. First we determine the maximum number of edges in a (k,t)-track layout, and show how to colour the edges given fixed linear orderings of the vertex colour classes. We then describe methods for the manipulation of track layouts. For example, we show how to decrease the number of edge colours in a track layout at the expense of increasing the number of tracks, and vice versa. We then study the relationship between track layouts and other models of graph layout, namely stack and queue layouts, and geometric thickness. One of our principle results is that the queue-number and track-number of a graph are tied, in the sense that one is bounded by a function of the other. As corollaries we prove that acyclic chromatic number is bounded by both queue-number and stack-number. Finally we consider track layouts of planar graphs. While it is an open problem whether planar graphs have bounded track-number, we prove bounds on the track-number of outerplanar graphs, and give the best known lower bound on the track-number of planar graphs.\]]> Thu, 01 Jan 2004 07:00:00 +0000 https://doi.org/10.46298/dmtcs.315 https://doi.org/10.46298/dmtcs.315 Dujmović, Vida Pór, Attila Wood, David R. Dujmović, Vida Pór, Attila Wood, David R. <![CDATA[A \emph(k,t)-track layout of a graph G consists of a (proper) vertex t-colouring of G, a total order of each vertex colour class, and a (non-proper) edge k-colouring such that between each pair of colour classes no two monochromatic edges cross. This structure has recently arisen in the study of three-dimensional graph drawings. This paper presents the beginnings of a theory of track layouts. First we determine the maximum number of edges in a (k,t)-track layout, and show how to colour the edges given fixed linear orderings of the vertex colour classes. We then describe methods for the manipulation of track layouts. For example, we show how to decrease the number of edge colours in a track layout at the expense of increasing the number of tracks, and vice versa. We then study the relationship between track layouts and other models of graph layout, namely stack and queue layouts, and geometric thickness. One of our principle results is that the queue-number and track-number of a graph are tied, in the sense that one is bounded by a function of the other. As corollaries we prove that acyclic chromatic number is bounded by both queue-number and stack-number. Finally we consider track layouts of planar graphs. While it is an open problem whether planar graphs have bounded track-number, we prove bounds on the track-number of outerplanar graphs, and give the best known lower bound on the track-number of planar graphs.\]]> 0 <![CDATA[We characterise ideal threshold schemes from different approaches. Since the characteristic properties are independent to particular descriptions of threshold schemes all ideal threshold schemes can be examined by new points of view and new results on ideal threshold schemes can be discovered.]]> Thu, 01 Jan 2004 07:00:00 +0000 https://doi.org/10.46298/dmtcs.328 https://doi.org/10.46298/dmtcs.328 Pieprzyk, Josef Zhang, Xian-Mo Pieprzyk, Josef Zhang, Xian-Mo <![CDATA[We characterise ideal threshold schemes from different approaches. Since the characteristic properties are independent to particular descriptions of threshold schemes all ideal threshold schemes can be examined by new points of view and new results on ideal threshold schemes can be discovered.]]> 0 <![CDATA[Recently, Green and Losonczy~GL1,GL2 introduced \emphfreely braided permutation as a special class of restricted permutations has arisen in representation theory. The freely braided permutations were introduced and studied as the upper bound for the number of commutation classes of reduced expressions for an element of a simply laced Coxeter group is achieved if and only if when the element is freely braided. In this paper, we prove that the generating function for the number of freely braided permutations in S_n is given by \par (1-3x-2x^2+(1+x)√1-4x) / (1-4x-x^2+(1-x^2)√1-4x).\par]]> Thu, 01 Jan 2004 07:00:00 +0000 https://doi.org/10.46298/dmtcs.327 https://doi.org/10.46298/dmtcs.327 Mansour, Toufik Mansour, Toufik <![CDATA[Recently, Green and Losonczy~GL1,GL2 introduced \emphfreely braided permutation as a special class of restricted permutations has arisen in representation theory. The freely braided permutations were introduced and studied as the upper bound for the number of commutation classes of reduced expressions for an element of a simply laced Coxeter group is achieved if and only if when the element is freely braided. In this paper, we prove that the generating function for the number of freely braided permutations in S_n is given by \par (1-3x-2x^2+(1+x)√1-4x) / (1-4x-x^2+(1-x^2)√1-4x).\par]]> 0 <![CDATA[This paper deals with statistics concerning distances between randomly chosen nodes in varieties of increasing trees. Increasing trees are labelled rooted trees where labels along any branch from the root go in increasing order. Many mportant tree families that have applications in computer science or are used as probabilistic models in various applications, like \emphrecursive trees, heap-ordered trees or \emphbinary increasing trees (isomorphic to binary search trees) are members of this variety of trees. We consider the parameters \textitdepth of a randomly chosen node, \textitdistance between two randomly chosen nodes, and the generalisations where \textitp nodes are randomly chosen Under the restriction that the node-degrees are bounded, we can prove that all these parameters converge in law to the Normal distribution. This extends results obtained earlier for binary search trees and heap-ordered trees to a much larger class of structures.]]> Thu, 01 Jan 2004 07:00:00 +0000 https://doi.org/10.46298/dmtcs.326 https://doi.org/10.46298/dmtcs.326 Panholzer, Alois Prodinger, Helmut Panholzer, Alois Prodinger, Helmut <![CDATA[This paper deals with statistics concerning distances between randomly chosen nodes in varieties of increasing trees. Increasing trees are labelled rooted trees where labels along any branch from the root go in increasing order. Many mportant tree families that have applications in computer science or are used as probabilistic models in various applications, like \emphrecursive trees, heap-ordered trees or \emphbinary increasing trees (isomorphic to binary search trees) are members of this variety of trees. We consider the parameters \textitdepth of a randomly chosen node, \textitdistance between two randomly chosen nodes, and the generalisations where \textitp nodes are randomly chosen Under the restriction that the node-degrees are bounded, we can prove that all these parameters converge in law to the Normal distribution. This extends results obtained earlier for binary search trees and heap-ordered trees to a much larger class of structures.]]> 0 <![CDATA[A sequence (a_i) of integers is \emphwell-spread if the sums a_i+a_j, for i]]> Thu, 01 Jan 2004 07:00:00 +0000 https://doi.org/10.46298/dmtcs.324 https://doi.org/10.46298/dmtcs.324 Kayll, Peter Mark Kayll, Peter Mark <![CDATA[A sequence (a_i) of integers is \emphwell-spread if the sums a_i+a_j, for i]]> 0 <![CDATA[In Random Cayley Graphs and Expanders, N. Alon and Y. Roichman proved that for every ε > 0 there is a finite c(ε ) such that for any sufficiently large group G, the expected value of the second largest (in absolute value) eigenvalue of the normalized adjacency matrix of the Cayley graph with respect to c(ε ) log |G| random elements is less than ε . We reduce the number of elements to c(ε )log D(G) (for the same c), where D(G) is the sum of the dimensions of the irreducible representations of G. In sufficiently non-abelian families of groups (as measured by these dimensions), log D(G) is asymptotically (1/2)log|G|. As is well known, a small eigenvalue implies large graph expansion (and conversely); see Tanner84 and AlonMilman84-2,AlonMilman84-1. For any specified eigenvalue or expansion, therefore, random Cayley graphs (of sufficiently non-abelian groups) require only half as many edges as was previously known.]]> Thu, 01 Jan 2004 07:00:00 +0000 https://doi.org/10.46298/dmtcs.316 https://doi.org/10.46298/dmtcs.316 Loh, Po-Shen Schulman, Leonard J. Loh, Po-Shen Schulman, Leonard J. <![CDATA[In Random Cayley Graphs and Expanders, N. Alon and Y. Roichman proved that for every ε > 0 there is a finite c(ε ) such that for any sufficiently large group G, the expected value of the second largest (in absolute value) eigenvalue of the normalized adjacency matrix of the Cayley graph with respect to c(ε ) log |G| random elements is less than ε . We reduce the number of elements to c(ε )log D(G) (for the same c), where D(G) is the sum of the dimensions of the irreducible representations of G. In sufficiently non-abelian families of groups (as measured by these dimensions), log D(G) is asymptotically (1/2)log|G|. As is well known, a small eigenvalue implies large graph expansion (and conversely); see Tanner84 and AlonMilman84-2,AlonMilman84-1. For any specified eigenvalue or expansion, therefore, random Cayley graphs (of sufficiently non-abelian groups) require only half as many edges as was previously known.]]> 0 <![CDATA[In a total order of the vertices of a graph, two edges with no endpoint in common can be \emphcrossing, \emphnested, or \emphdisjoint. A \emphk-stack (respectively, \emphk-queue, \emphk-arch) \emphlayout of a graph consists of a total order of the vertices, and a partition of the edges into k sets of pairwise non-crossing (non-nested, non-disjoint) edges. Motivated by numerous applications, stack layouts (also called \emphbook embeddings) and queue layouts are widely studied in the literature, while this is the first paper to investigate arch layouts.\par Our main result is a characterisation of k-arch graphs as the \emphalmost (k+1)-colourable graphs; that is, the graphs G with a set S of at most k vertices, such that G S is (k+1)-colourable.\par In addition, we survey the following fundamental questions regarding each type of layout, and in the case of queue layouts, provide simple proofs of a number of existing results. How does one partition the edges given a fixed ordering of the vertices? What is the maximum number of edges in each type of layout? What is the maximum chromatic number of a graph admitting each type of layout? What is the computational complexity of recognising the graphs that admit each type of layout?\par A comprehensive bibliography of all known references on these topics is included. \par]]> Thu, 01 Jan 2004 07:00:00 +0000 https://doi.org/10.46298/dmtcs.317 https://doi.org/10.46298/dmtcs.317 Dujmović, Vida Wood, David R. Dujmović, Vida Wood, David R. <![CDATA[In a total order of the vertices of a graph, two edges with no endpoint in common can be \emphcrossing, \emphnested, or \emphdisjoint. A \emphk-stack (respectively, \emphk-queue, \emphk-arch) \emphlayout of a graph consists of a total order of the vertices, and a partition of the edges into k sets of pairwise non-crossing (non-nested, non-disjoint) edges. Motivated by numerous applications, stack layouts (also called \emphbook embeddings) and queue layouts are widely studied in the literature, while this is the first paper to investigate arch layouts.\par Our main result is a characterisation of k-arch graphs as the \emphalmost (k+1)-colourable graphs; that is, the graphs G with a set S of at most k vertices, such that G S is (k+1)-colourable.\par In addition, we survey the following fundamental questions regarding each type of layout, and in the case of queue layouts, provide simple proofs of a number of existing results. How does one partition the edges given a fixed ordering of the vertices? What is the maximum number of edges in each type of layout? What is the maximum chromatic number of a graph admitting each type of layout? What is the computational complexity of recognising the graphs that admit each type of layout?\par A comprehensive bibliography of all known references on these topics is included. \par]]> 0 <![CDATA[We study the structure of $m$-ary search trees generated by the van der Corput sequences. The height of the tree is calculated and a generating function approach shows that the distribution of the depths of the nodes is asymptotically normal. Additionally a local limit theorem is derived.]]> Thu, 01 Jan 2004 07:00:00 +0000 https://doi.org/10.46298/dmtcs.318 https://doi.org/10.46298/dmtcs.318 Steiner, Wolfgang Steiner, Wolfgang <![CDATA[We study the structure of $m$-ary search trees generated by the van der Corput sequences. The height of the tree is calculated and a generating function approach shows that the distribution of the depths of the nodes is asymptotically normal. Additionally a local limit theorem is derived.]]> 0 <![CDATA[The paper addresses the cheating prevention in secret sharing. We consider secret sharing with binary shares. The secret also is binary. This model allows us to use results and constructions from the well developed theory of cryptographically strong boolean functions. In particular, we prove that for given secret sharing, the average cheating probability over all cheating vectors and all original vectors, i.e., 1/n 2^n ∑ _c=1...n ∑ _α ∈V n ρ _c,α , denoted by øverlineρ , satisfies øverlineρ ≥ \frac12 , and the equality holds if and only if ρ _c,α satisfies ρ _c,α = \frac12 for every cheating vector δ _c and every original vector α . In this case the secret sharing is said to be cheating immune. We further establish a relationship between cheating-immune secret sharing and cryptographic criteria of boolean functions.This enables us to construct cheating-immune secret sharing.]]> Thu, 01 Jan 2004 07:00:00 +0000 https://doi.org/10.46298/dmtcs.321 https://doi.org/10.46298/dmtcs.321 Pieprzyk, Josef Zhang, Xian-Mo Pieprzyk, Josef Zhang, Xian-Mo <![CDATA[The paper addresses the cheating prevention in secret sharing. We consider secret sharing with binary shares. The secret also is binary. This model allows us to use results and constructions from the well developed theory of cryptographically strong boolean functions. In particular, we prove that for given secret sharing, the average cheating probability over all cheating vectors and all original vectors, i.e., 1/n 2^n ∑ _c=1...n ∑ _α ∈V n ρ _c,α , denoted by øverlineρ , satisfies øverlineρ ≥ \frac12 , and the equality holds if and only if ρ _c,α satisfies ρ _c,α = \frac12 for every cheating vector δ _c and every original vector α . In this case the secret sharing is said to be cheating immune. We further establish a relationship between cheating-immune secret sharing and cryptographic criteria of boolean functions.This enables us to construct cheating-immune secret sharing.]]> 0 <![CDATA[Locating faulty processors in a multiprocessor system gives the motivation for locating-dominating codes. We consider these codes in binary hypercubes and generalize the concept for the situation in which we want to locate more than one malfunctioning processor.]]> Thu, 01 Jan 2004 07:00:00 +0000 https://doi.org/10.46298/dmtcs.322 https://doi.org/10.46298/dmtcs.322 Honkala, Iiro Laihonen, Tero Ranto, Sanna Honkala, Iiro Laihonen, Tero Ranto, Sanna <![CDATA[Locating faulty processors in a multiprocessor system gives the motivation for locating-dominating codes. We consider these codes in binary hypercubes and generalize the concept for the situation in which we want to locate more than one malfunctioning processor.]]> 0 <![CDATA[It is proved that the moments of the width of Galton-Watson trees of size n and with offspring variance σ ^2 are asymptotically given by (σ √n)^pm_p where m_p are the moments of the maximum of the local time of a standard scaled Brownian excursion. This is done by combining a weak limit theorem and a tightness estimate. The method is quite general and we state some further applications.]]> Thu, 01 Jan 2004 07:00:00 +0000 https://doi.org/10.46298/dmtcs.323 https://doi.org/10.46298/dmtcs.323 Drmota, Michael Gittenberger, Bernhard Drmota, Michael Gittenberger, Bernhard <![CDATA[It is proved that the moments of the width of Galton-Watson trees of size n and with offspring variance σ ^2 are asymptotically given by (σ √n)^pm_p where m_p are the moments of the maximum of the local time of a standard scaled Brownian excursion. This is done by combining a weak limit theorem and a tightness estimate. The method is quite general and we state some further applications.]]> 0 <![CDATA[The 3-SAT problem consists in determining if a boolean formula with 3 literals per clause is satisfiable. When the ratio between the number of clauses and the number of variables increases, a threshold phenomenon is observed: the probability of satisfiability appears to decrease sharply from 1 to 0 in the neighbourghood of a threshold value, conjectured to be close to 4.25. Although the threshold has been proved to exist for the 2-SAT formulæ and for closely related problems like 3-XORSAT, there is still no proof for the 3-sat problem. Recent works have provided so far upper and lower bounds for the threshold's potential location. We present here a unified approach to upper bounds that is based on urn models, generating functions, and saddle-point bounds. In this way, we re-derive some of the most significant upper bounds known in a simple and uniform manner.]]> Thu, 01 Jan 2004 07:00:00 +0000 https://doi.org/10.46298/dmtcs.325 https://doi.org/10.46298/dmtcs.325 Puyhaubert, Vincent Puyhaubert, Vincent <![CDATA[The 3-SAT problem consists in determining if a boolean formula with 3 literals per clause is satisfiable. When the ratio between the number of clauses and the number of variables increases, a threshold phenomenon is observed: the probability of satisfiability appears to decrease sharply from 1 to 0 in the neighbourghood of a threshold value, conjectured to be close to 4.25. Although the threshold has been proved to exist for the 2-SAT formulæ and for closely related problems like 3-XORSAT, there is still no proof for the 3-sat problem. Recent works have provided so far upper and lower bounds for the threshold's potential location. We present here a unified approach to upper bounds that is based on urn models, generating functions, and saddle-point bounds. In this way, we re-derive some of the most significant upper bounds known in a simple and uniform manner.]]> 0 <![CDATA[Let G be a graph. A component of G is a maximal connected subgraph in G. A vertex v is a cut vertex of G if k(G-v) > k(G), where k(G) is the number of components in G. Similarly, an edge e is a bridge of G if k(G-e) > k(G). In this paper, we will propose new O(n) algorithms for finding cut vertices and bridges of a trapezoid graph, assuming the trapezoid diagram is given. Our algorithms can be easily parallelized on the EREW PRAM computational model so that cut vertices and bridges can be found in O(log n) time by using O(n / log n) processors.]]> Thu, 01 Jan 2004 07:00:00 +0000 https://doi.org/10.46298/dmtcs.314 https://doi.org/10.46298/dmtcs.314 Chen, Hon-Chan Chen, Hon-Chan <![CDATA[Let G be a graph. A component of G is a maximal connected subgraph in G. A vertex v is a cut vertex of G if k(G-v) > k(G), where k(G) is the number of components in G. Similarly, an edge e is a bridge of G if k(G-e) > k(G). In this paper, we will propose new O(n) algorithms for finding cut vertices and bridges of a trapezoid graph, assuming the trapezoid diagram is given. Our algorithms can be easily parallelized on the EREW PRAM computational model so that cut vertices and bridges can be found in O(log n) time by using O(n / log n) processors.]]> 0 <![CDATA[We give an overview of how a huge class of multisum identities can be proven and discovered with the summation package Sigma implemented in the computer algebra system Mathematica. General principles of symbolic summation are discussed. We illustrate the usage of Sigma by showing how one can find and prove a multisum identity that arose in the enumeration of rhombus tilings of a symmetric hexagon. Whereas this identity has been derived alternatively with the help of highly involved transformations of special functions, our tools enable to find and prove this identity completely automatically with the computer.]]> Thu, 01 Jan 2004 07:00:00 +0000 https://doi.org/10.46298/dmtcs.313 https://doi.org/10.46298/dmtcs.313 Schneider, Carsten Schneider, Carsten <![CDATA[We give an overview of how a huge class of multisum identities can be proven and discovered with the summation package Sigma implemented in the computer algebra system Mathematica. General principles of symbolic summation are discussed. We illustrate the usage of Sigma by showing how one can find and prove a multisum identity that arose in the enumeration of rhombus tilings of a symmetric hexagon. Whereas this identity has been derived alternatively with the help of highly involved transformations of special functions, our tools enable to find and prove this identity completely automatically with the computer.]]> 0 <![CDATA[Motivated by the Coxeter complex associated to a Coxeter system (W,S), we introduce a simplicial regular cell complex Δ (G,S) with a G-action associated to any pair (G,S) where G is a group and S is a finite set of generators for G which is minimal with respect to inclusion. We examine the topology of Δ (G,S), and in particular the representations of G on its homology groups. We look closely at the case of the symmetric group S_n minimally generated by (not necessarily adjacent) transpositions, and their type-selected subcomplexes. These include not only the Coxeter complexes of type A, but also the well-studied chessboard complexes.]]> Thu, 01 Jan 2004 07:00:00 +0000 https://doi.org/10.46298/dmtcs.312 https://doi.org/10.46298/dmtcs.312 Babson, Eric Reiner, Victor Babson, Eric Reiner, Victor <![CDATA[Motivated by the Coxeter complex associated to a Coxeter system (W,S), we introduce a simplicial regular cell complex Δ (G,S) with a G-action associated to any pair (G,S) where G is a group and S is a finite set of generators for G which is minimal with respect to inclusion. We examine the topology of Δ (G,S), and in particular the representations of G on its homology groups. We look closely at the case of the symmetric group S_n minimally generated by (not necessarily adjacent) transpositions, and their type-selected subcomplexes. These include not only the Coxeter complexes of type A, but also the well-studied chessboard complexes.]]> 0 <![CDATA[For graph classes \wp_1,...,\wp_k, Generalized Graph Coloring is the problem of deciding whether the vertex set of a given graph G can be partitioned into subsets V_1,...,V_k so that V_j induces a graph in the class \wp_j (j=1,2,...,k). If \wp_1=...=\wp_k is the class of edgeless graphs, then this problem coincides with the standard vertex k-COLORABILITY, which is known to be NP-complete for any k≥ 3. Recently, this result has been generalized by showing that if all \wp_i's are additive hereditary, then the generalized graph coloring is NP-hard, with the only exception of bipartite graphs. Clearly, a similar result follows when all the \wp_i's are co-additive.]]> Thu, 01 Jan 2004 07:00:00 +0000 https://doi.org/10.46298/dmtcs.311 https://doi.org/10.46298/dmtcs.311 Alekseev, Vladimir E. Farrugia, Alastair Lozin, Vadim V. Alekseev, Vladimir E. Farrugia, Alastair Lozin, Vadim V. <![CDATA[For graph classes \wp_1,...,\wp_k, Generalized Graph Coloring is the problem of deciding whether the vertex set of a given graph G can be partitioned into subsets V_1,...,V_k so that V_j induces a graph in the class \wp_j (j=1,2,...,k). If \wp_1=...=\wp_k is the class of edgeless graphs, then this problem coincides with the standard vertex k-COLORABILITY, which is known to be NP-complete for any k≥ 3. Recently, this result has been generalized by showing that if all \wp_i's are additive hereditary, then the generalized graph coloring is NP-hard, with the only exception of bipartite graphs. Clearly, a similar result follows when all the \wp_i's are co-additive.]]> 0 <![CDATA[Some strings -the texts- are assumed to be randomly generated, according to a probability model that is either a Bernoulli model or a Markov model. A rare event is the over or under-representation of a word or a set of words. The aim of this paper is twofold. First, a single word is given. One studies the tail distribution of the number of its occurrences. Sharp large deviation estimates are derived. Second, one assumes that a given word is overrepresented. The distribution of a second word is studied; formulae for the expectation and the variance are derived. In both cases, the formulae are accurate and actually computable. These results have applications in computational biology, where a genome is viewed as a text.]]> Thu, 01 Jan 2004 07:00:00 +0000 https://doi.org/10.46298/dmtcs.310 https://doi.org/10.46298/dmtcs.310 Régnier, Mireille Denise, Alain Régnier, Mireille Denise, Alain <![CDATA[Some strings -the texts- are assumed to be randomly generated, according to a probability model that is either a Bernoulli model or a Markov model. A rare event is the over or under-representation of a word or a set of words. The aim of this paper is twofold. First, a single word is given. One studies the tail distribution of the number of its occurrences. Sharp large deviation estimates are derived. Second, one assumes that a given word is overrepresented. The distribution of a second word is studied; formulae for the expectation and the variance are derived. In both cases, the formulae are accurate and actually computable. These results have applications in computational biology, where a genome is viewed as a text.]]> 0 <![CDATA[In this paper we study two lattices of significant particular closure systems on a finite set, namely the union stable closure systems and the convex geometries. Using the notion of (admissible) quasi-closed set and of (deletable) closed set, we determine the covering relation \prec of these lattices and the changes induced, for instance, on the irreducible elements when one goes from C to C' where C and C' are two such closure systems satisfying C \prec C'. We also do a systematic study of these lattices of closure systems, characterizing for instance their join-irreducible and their meet-irreducible elements.]]> Thu, 01 Jan 2004 07:00:00 +0000 https://doi.org/10.46298/dmtcs.309 https://doi.org/10.46298/dmtcs.309 Caspard, Nathalie Monjardet, Bernard Caspard, Nathalie Monjardet, Bernard <![CDATA[In this paper we study two lattices of significant particular closure systems on a finite set, namely the union stable closure systems and the convex geometries. Using the notion of (admissible) quasi-closed set and of (deletable) closed set, we determine the covering relation \prec of these lattices and the changes induced, for instance, on the irreducible elements when one goes from C to C' where C and C' are two such closure systems satisfying C \prec C'. We also do a systematic study of these lattices of closure systems, characterizing for instance their join-irreducible and their meet-irreducible elements.]]> 0 <![CDATA[We find generating functions the number of strings (words) containing a specified number of occurrences of certain types of order-isomorphic classes of substrings called subword patterns. In particular, we find generating functions for the number of strings containing a specified number of occurrences of a given 3-letter subword pattern.]]> Wed, 01 Jan 2003 07:00:00 +0000 https://doi.org/10.46298/dmtcs.320 https://doi.org/10.46298/dmtcs.320 Burstein, Alexander Mansour, Toufik Burstein, Alexander Mansour, Toufik <![CDATA[We find generating functions the number of strings (words) containing a specified number of occurrences of certain types of order-isomorphic classes of substrings called subword patterns. In particular, we find generating functions for the number of strings containing a specified number of occurrences of a given 3-letter subword pattern.]]> 0 <![CDATA[We study numerically a non-linear integral equation that arises in the study of binary search trees. If the tree is constructed from n elements, this integral equation describes the asymptotic (as n→∞) distribution of the height of the tree. This supplements some asymptotic results we recently obtained for the tails of the distribution. The asymptotic height distribution is shown to be unimodal with highly asymmetric tails.]]> Wed, 01 Jan 2003 07:00:00 +0000 https://doi.org/10.46298/dmtcs.319 https://doi.org/10.46298/dmtcs.319 Knessl, Charles Knessl, Charles <![CDATA[We study numerically a non-linear integral equation that arises in the study of binary search trees. If the tree is constructed from n elements, this integral equation describes the asymptotic (as n→∞) distribution of the height of the tree. This supplements some asymptotic results we recently obtained for the tails of the distribution. The asymptotic height distribution is shown to be unimodal with highly asymmetric tails.]]> 0 <![CDATA[The b-chromatic number of a graph G is defined as the maximum number k of colors that can be used to color the vertices of G, such that we obtain a proper coloring and each color i, with 1 ≤ i≤ k, has at least one representant x_i adjacent to a vertex of every color j, 1 ≤ j ≠ i ≤ k. In this paper, we discuss the b-chromatic number of some power graphs. We give the exact value of the b-chromatic number of power paths and power complete binary trees, and we bound the b-chromatic number of power cycles.]]> Wed, 01 Jan 2003 07:00:00 +0000 https://doi.org/10.46298/dmtcs.333 https://doi.org/10.46298/dmtcs.333 Effantin, Brice Kheddouci, Hamamache Effantin, Brice Kheddouci, Hamamache <![CDATA[The b-chromatic number of a graph G is defined as the maximum number k of colors that can be used to color the vertices of G, such that we obtain a proper coloring and each color i, with 1 ≤ i≤ k, has at least one representant x_i adjacent to a vertex of every color j, 1 ≤ j ≠ i ≤ k. In this paper, we discuss the b-chromatic number of some power graphs. We give the exact value of the b-chromatic number of power paths and power complete binary trees, and we bound the b-chromatic number of power cycles.]]> 0 <![CDATA[Morse code sequences are very useful to give combinatorial interpretations of various properties of Fibonacci numbers. In this note we study some algebraic and combinatorial aspects of Morse code sequences and obtain several q-analogues of Fibonacci numbers and Fibonacci polynomials and their generalizations.]]> Wed, 01 Jan 2003 07:00:00 +0000 https://doi.org/10.46298/dmtcs.334 https://doi.org/10.46298/dmtcs.334 Cigler, Johann Cigler, Johann <![CDATA[Morse code sequences are very useful to give combinatorial interpretations of various properties of Fibonacci numbers. In this note we study some algebraic and combinatorial aspects of Morse code sequences and obtain several q-analogues of Fibonacci numbers and Fibonacci polynomials and their generalizations.]]> 0 <![CDATA[We present a two-variable polynomial, which simultaneously generalizes the chromatic polynomial, the independence polynomial, and the matching polynomial of a graph. This new polynomial satisfies both an edge decomposition formula and a vertex decomposition formula. We establish two general expressions for this new polynomial: one in terms of the broken circuit complex and one in terms of the lattice of forbidden colorings. We show that the new polynomial may be considered as a specialization of Stanley's chromatic symmetric function. We finally give explicit expressions for the generalized chromatic polynomial of complete graphs, complete bipartite graphs, paths, and cycles, and show that it can be computed in polynomial time for trees and graphs of restricted pathwidth.]]> Wed, 01 Jan 2003 07:00:00 +0000 https://doi.org/10.46298/dmtcs.335 https://doi.org/10.46298/dmtcs.335 Dohmen, Klaus Poenitz, André Tittmann, Peter Dohmen, Klaus Poenitz, André Tittmann, Peter <![CDATA[We present a two-variable polynomial, which simultaneously generalizes the chromatic polynomial, the independence polynomial, and the matching polynomial of a graph. This new polynomial satisfies both an edge decomposition formula and a vertex decomposition formula. We establish two general expressions for this new polynomial: one in terms of the broken circuit complex and one in terms of the lattice of forbidden colorings. We show that the new polynomial may be considered as a specialization of Stanley's chromatic symmetric function. We finally give explicit expressions for the generalized chromatic polynomial of complete graphs, complete bipartite graphs, paths, and cycles, and show that it can be computed in polynomial time for trees and graphs of restricted pathwidth.]]> 0 <![CDATA[We solve the recursion S_n=S_n-1-q^nS_n-p, both, explicitly, and in the limit for n→∞, proving in this way a formula due to Merlini and Sprugnoli. It is also discussed how computer algebra could be applied.]]> Wed, 01 Jan 2003 07:00:00 +0000 https://doi.org/10.46298/dmtcs.336 https://doi.org/10.46298/dmtcs.336 Paule, Peter Prodinger, Helmut Paule, Peter Prodinger, Helmut <![CDATA[We solve the recursion S_n=S_n-1-q^nS_n-p, both, explicitly, and in the limit for n→∞, proving in this way a formula due to Merlini and Sprugnoli. It is also discussed how computer algebra could be applied.]]> 0 <![CDATA[We collect major known algorithms in the literature for finding the maxima of multi-dimensional points and provide a simple classification. Several new algorithms are proposed. In particular, we give a new maxima-finding algorithm with expected complexity n+O(√n\log n) when the input is a sequence of points uniformly chosen at random from general planar regions. We also give a sequential algorithm, very efficient for practical purposes.]]> Wed, 01 Jan 2003 07:00:00 +0000 https://doi.org/10.46298/dmtcs.337 https://doi.org/10.46298/dmtcs.337 Chen, Wei-Mei Hwang, Hsien-Kuei Tsai, Tsung-Hsi Chen, Wei-Mei Hwang, Hsien-Kuei Tsai, Tsung-Hsi <![CDATA[We collect major known algorithms in the literature for finding the maxima of multi-dimensional points and provide a simple classification. Several new algorithms are proposed. In particular, we give a new maxima-finding algorithm with expected complexity n+O(√n\log n) when the input is a sequence of points uniformly chosen at random from general planar regions. We also give a sequential algorithm, very efficient for practical purposes.]]> 0 <![CDATA[We study in graphs properties related to fault-tolerance in case a node fails. A graph G is k-self-repairing, where k is a non-negative integer, if after the removal of any vertex no distance in the surviving graph increases by more than k. In the design of interconnection networks such graphs guarantee good fault-tolerance properties. We give upper and lower bounds on the minimum number of edges of a k-self-repairing graph for prescribed k and n, where n is the order of the graph. We prove that the problem of finding, in a k-self-repairing graph, a spanning k-self-repairing subgraph of minimum size is NP-Hard.]]> Wed, 01 Jan 2003 07:00:00 +0000 https://doi.org/10.46298/dmtcs.338 https://doi.org/10.46298/dmtcs.338 Djelloul, Selma Kouider, Mekkia Djelloul, Selma Kouider, Mekkia <![CDATA[We study in graphs properties related to fault-tolerance in case a node fails. A graph G is k-self-repairing, where k is a non-negative integer, if after the removal of any vertex no distance in the surviving graph increases by more than k. In the design of interconnection networks such graphs guarantee good fault-tolerance properties. We give upper and lower bounds on the minimum number of edges of a k-self-repairing graph for prescribed k and n, where n is the order of the graph. We prove that the problem of finding, in a k-self-repairing graph, a spanning k-self-repairing subgraph of minimum size is NP-Hard.]]> 0 <![CDATA[The Ackermann function is a fascinating and well studied paradigm for a function which eventually dominates all primitive recursive functions. By a classical result from the theory of recursive functions it is known that the Ackermann function can be defined by an unnested or descent recursion along the segment of ordinals below ω ^ω (or equivalently along the order type of the polynomials under eventual domination). In this article we give a fine structure analysis of such a Ackermann type descent recursion in the case that the ordinals below ω ^ω are represented via a Hardy Ramanujan style coding. This paper combines number-theoretic results by Hardy and Ramanujan, Karamata's celebrated Tauberian theorem and techniques from the theory of computability in a perhaps surprising way.]]> Wed, 01 Jan 2003 07:00:00 +0000 https://doi.org/10.46298/dmtcs.339 https://doi.org/10.46298/dmtcs.339 Weiermann, Andreas Weiermann, Andreas <![CDATA[The Ackermann function is a fascinating and well studied paradigm for a function which eventually dominates all primitive recursive functions. By a classical result from the theory of recursive functions it is known that the Ackermann function can be defined by an unnested or descent recursion along the segment of ordinals below ω ^ω (or equivalently along the order type of the polynomials under eventual domination). In this article we give a fine structure analysis of such a Ackermann type descent recursion in the case that the ordinals below ω ^ω are represented via a Hardy Ramanujan style coding. This paper combines number-theoretic results by Hardy and Ramanujan, Karamata's celebrated Tauberian theorem and techniques from the theory of computability in a perhaps surprising way.]]> 0 <![CDATA[Constellations are colored planar maps that generalize different families of maps (planar maps, bipartite planar maps, bi-Eulerian planar maps, planar cacti, ...) and are strongly related to factorizations of permutations. They were recently studied by Bousquet-Mélou and Schaeffer who describe a correspondence between these maps and a family of trees, called Eulerian trees. In this paper, we derive from their result a relationship between planar constellations and another family of trees, called stellar trees. This correspondence generalizes a well known result for planar cacti, and shows that planar constellations are colored Lagrangian objects (that is objects that can be enumerated by the Good-Lagrange formula). We then deduce from this result a new formula for the number of planar constellations having a given face distribution, different from the formula one can derive from the results of Bousquet-Mélou and Schaeffer, along with systems of functional equations for the generating functions of bipartite and bi-Eulerian planar maps enumerated according to the partition of faces and vertices.]]> Wed, 01 Jan 2003 07:00:00 +0000 https://doi.org/10.46298/dmtcs.340 https://doi.org/10.46298/dmtcs.340 Chauve, Cedric Chauve, Cedric <![CDATA[Constellations are colored planar maps that generalize different families of maps (planar maps, bipartite planar maps, bi-Eulerian planar maps, planar cacti, ...) and are strongly related to factorizations of permutations. They were recently studied by Bousquet-Mélou and Schaeffer who describe a correspondence between these maps and a family of trees, called Eulerian trees. In this paper, we derive from their result a relationship between planar constellations and another family of trees, called stellar trees. This correspondence generalizes a well known result for planar cacti, and shows that planar constellations are colored Lagrangian objects (that is objects that can be enumerated by the Good-Lagrange formula). We then deduce from this result a new formula for the number of planar constellations having a given face distribution, different from the formula one can derive from the results of Bousquet-Mélou and Schaeffer, along with systems of functional equations for the generating functions of bipartite and bi-Eulerian planar maps enumerated according to the partition of faces and vertices.]]> 0 <![CDATA[\emphAlon, Kleitman, Lipton, Meshulam, Rabin and \emphSpencer (Graphs. Combin. 7 (1991), no. 2, 97-99) proved, that for any hypergraph \textbf\textitF=\F_1,F_2,\ldots, F_d(q-1)+1\, where q is a prime-power, and d denotes the maximal degree of the hypergraph, there exists an \textbf\textitF_0⊂ \textbf\textitF, such that |\bigcup_F∈\textbf\textitF_0F| ≡ 0 (q). We give a direct, alternative proof for this theorem, and we also show that an explicit construction exists for a hypergraph of degree d and size Ω (d^2) which does not contain a non-empty sub-hypergraph with a union of size 0 modulo 6, consequently, the theorem does not generalize for non-prime-power moduli.]]> Wed, 01 Jan 2003 07:00:00 +0000 https://doi.org/10.46298/dmtcs.341 https://doi.org/10.46298/dmtcs.341 Grolmusz, Vince Grolmusz, Vince <![CDATA[\emphAlon, Kleitman, Lipton, Meshulam, Rabin and \emphSpencer (Graphs. Combin. 7 (1991), no. 2, 97-99) proved, that for any hypergraph \textbf\textitF=\F_1,F_2,\ldots, F_d(q-1)+1\, where q is a prime-power, and d denotes the maximal degree of the hypergraph, there exists an \textbf\textitF_0⊂ \textbf\textitF, such that |\bigcup_F∈\textbf\textitF_0F| ≡ 0 (q). We give a direct, alternative proof for this theorem, and we also show that an explicit construction exists for a hypergraph of degree d and size Ω (d^2) which does not contain a non-empty sub-hypergraph with a union of size 0 modulo 6, consequently, the theorem does not generalize for non-prime-power moduli.]]> 0 <![CDATA[We consider random paths on a square lattice which take a left or a right turn at every vertex. The possible turns are taken with equal probability, except at a vertex which has been visited before. In such case the vertex is left via the unused edge. When the initial edge is reached the path is considered completed. We also consider families of such paths which together cover every edge of the lattice once and visit every vertex twice. Because these paths may touch but not intersect each other and themselves, we call them osculating walks. The ensemble of such families is also known as the dense $O(n=1)$ model. We consider in particular such paths in a cylindrical geometry, with the cylindrical axis parallel with one of the lattice directions. We formulate a conjecture for the probability that a face of the lattice is surrounded by m distinct osculating paths. For even system sizes we give a conjecture for the probability that a path winds round the cylinder. For odd system sizes we conjecture the probability that a point is visited by a path spanning the infinite length of the cylinder. Finally we conjecture an expression for the asymptotics of a binomial determinant]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3320 https://doi.org/10.46298/dmtcs.3320 Mitra, Saibal Nienhuis, Bernard Mitra, Saibal Nienhuis, Bernard <![CDATA[We consider random paths on a square lattice which take a left or a right turn at every vertex. The possible turns are taken with equal probability, except at a vertex which has been visited before. In such case the vertex is left via the unused edge. When the initial edge is reached the path is considered completed. We also consider families of such paths which together cover every edge of the lattice once and visit every vertex twice. Because these paths may touch but not intersect each other and themselves, we call them osculating walks. The ensemble of such families is also known as the dense $O(n=1)$ model. We consider in particular such paths in a cylindrical geometry, with the cylindrical axis parallel with one of the lattice directions. We formulate a conjecture for the probability that a face of the lattice is surrounded by m distinct osculating paths. For even system sizes we give a conjecture for the probability that a path winds round the cylinder. For odd system sizes we conjecture the probability that a point is visited by a path spanning the infinite length of the cylinder. Finally we conjecture an expression for the asymptotics of a binomial determinant]]> 0 <![CDATA[This paper gives a survey of the limit distributions of the areas of different types of random walks, namely Dyck paths, bilateral Dyck paths, meanders, and Bernoulli random walks, using the technology of generating functions only.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3321 https://doi.org/10.46298/dmtcs.3321 Nguyên Thê, Michel Nguyên Thê, Michel <![CDATA[This paper gives a survey of the limit distributions of the areas of different types of random walks, namely Dyck paths, bilateral Dyck paths, meanders, and Bernoulli random walks, using the technology of generating functions only.]]> 0 <![CDATA[Words of fixed size q are commonly referred to as $q$-grams. We consider the problem of $q$-gram filtration, a method commonly used to speed upsequence comparison. We are interested in the statistics of the number of $q$-grams common to two random texts (where multiplicities are not counted) in the non uniform Bernoulli model. In the exact and dependent model, when omitting border effects, a $q$-gramin a random sequence depends on the $q-1$ preceding $q$-grams. In an approximate and independent model, we draw randomly a $q$-gram at each position, independently of the others positions. Using ball and urn models, we analyze the independent model. Numerical simulations show that this model is an excellent first order approximationto the dependent model. We provide an algorithm to compute the moments.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3322 https://doi.org/10.46298/dmtcs.3322 Nicodème, Pierre Nicodème, Pierre <![CDATA[Words of fixed size q are commonly referred to as $q$-grams. We consider the problem of $q$-gram filtration, a method commonly used to speed upsequence comparison. We are interested in the statistics of the number of $q$-grams common to two random texts (where multiplicities are not counted) in the non uniform Bernoulli model. In the exact and dependent model, when omitting border effects, a $q$-gramin a random sequence depends on the $q-1$ preceding $q$-grams. In an approximate and independent model, we draw randomly a $q$-gram at each position, independently of the others positions. Using ball and urn models, we analyze the independent model. Numerical simulations show that this model is an excellent first order approximationto the dependent model. We provide an algorithm to compute the moments.]]> 0 <![CDATA[We study some lattice paths related to the concept ofgenerating trees. When the matrix associated to this kind of trees is a Riordan array $D=(d(t),h(t))$, we are able to find the generating function for the total area below these paths expressed in terms of the functions $d(t)$ and $h(t)$.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3323 https://doi.org/10.46298/dmtcs.3323 Merlini, Donatella Merlini, Donatella <![CDATA[We study some lattice paths related to the concept ofgenerating trees. When the matrix associated to this kind of trees is a Riordan array $D=(d(t),h(t))$, we are able to find the generating function for the total area below these paths expressed in terms of the functions $d(t)$ and $h(t)$.]]> 0 <![CDATA[We report on the asymptotic behaviour of a new model of random walk, we term the bindweed model, evolving in a random environment on an infinite multiplexed tree.The term multiplexed means that the model can be viewed as a nearest neighbours random walk on a tree whose vertices carry an internal degree of freedom from the finite set $\{1,...,d\}$, for some integer $d$. The consequence of the internal degree of freedom is an enhancement of the tree graph structure induced by the replacement of ordinary edges by multi-edges, indexed by the set $\{1,...,d\} × \{1,...,d\}.$ This indexing conveys the information on the internal degree of freedom of the vertices contiguous to each edge. The term random environment means that the jumping rates for the random walk are a family of edge-indexed random variables, independent of the natural filtration generated by the random variables entering in the definition of the random walk; their joint distribution depends on the index of each component of the multi-edges. We study the large time asymptotic behaviour of this random walk and classify it with respect to positive recurrence or transience in terms of a specific parameter of the probability distribution of the jump rates.This classifying parameter is shown to coincide with the critical value of a matrix-valued multiplicative cascade on the ordinary tree (i.e.the one without internal degrees of freedom attached to the vertices) having the same vertex set as the state space of the random walk. Only results are presented here since the detailed proofs will appear elsewhere.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3324 https://doi.org/10.46298/dmtcs.3324 Menshikov, Mikhail Petritis, Dimitri Popov, Serguei Menshikov, Mikhail Petritis, Dimitri Popov, Serguei <![CDATA[We report on the asymptotic behaviour of a new model of random walk, we term the bindweed model, evolving in a random environment on an infinite multiplexed tree.The term multiplexed means that the model can be viewed as a nearest neighbours random walk on a tree whose vertices carry an internal degree of freedom from the finite set $\{1,...,d\}$, for some integer $d$. The consequence of the internal degree of freedom is an enhancement of the tree graph structure induced by the replacement of ordinary edges by multi-edges, indexed by the set $\{1,...,d\} × \{1,...,d\}.$ This indexing conveys the information on the internal degree of freedom of the vertices contiguous to each edge. The term random environment means that the jumping rates for the random walk are a family of edge-indexed random variables, independent of the natural filtration generated by the random variables entering in the definition of the random walk; their joint distribution depends on the index of each component of the multi-edges. We study the large time asymptotic behaviour of this random walk and classify it with respect to positive recurrence or transience in terms of a specific parameter of the probability distribution of the jump rates.This classifying parameter is shown to coincide with the critical value of a matrix-valued multiplicative cascade on the ordinary tree (i.e.the one without internal degrees of freedom attached to the vertices) having the same vertex set as the state space of the random walk. Only results are presented here since the detailed proofs will appear elsewhere.]]> 0 <![CDATA[We consider themodel of broadcasting on a tree, with binary state space, on theinfinite rooted tree $T^k$ in which each node has $k$ children. The root of the tree takesa random value $0$ or $1$, and then each node passes a value independently to each of its children according to a $2x2$ transition matrix $\mathbf{P}$. We say that reconstruction is possible if the values at the dth level of the tree contain non-vanishing information about the value at the root as $d→∞$. Extending a method of Brightwell and Winkler, we obtain new conditions under which reconstruction is impossible, both in the general case and in the special case $p_11=0$. The latter case is closely related to the hard-core model from statistical physics; a corollary of our results is that, for the hard-core model on the $(k+1)$-regular tree with activity $λ =1$, the unique simple invariant Gibbs measure is extremal in the set of Gibbs measures, for any $k ≥ 2$.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3325 https://doi.org/10.46298/dmtcs.3325 Martin, James B. Martin, James B. <![CDATA[We consider themodel of broadcasting on a tree, with binary state space, on theinfinite rooted tree $T^k$ in which each node has $k$ children. The root of the tree takesa random value $0$ or $1$, and then each node passes a value independently to each of its children according to a $2x2$ transition matrix $\mathbf{P}$. We say that reconstruction is possible if the values at the dth level of the tree contain non-vanishing information about the value at the root as $d→∞$. Extending a method of Brightwell and Winkler, we obtain new conditions under which reconstruction is impossible, both in the general case and in the special case $p_11=0$. The latter case is closely related to the hard-core model from statistical physics; a corollary of our results is that, for the hard-core model on the $(k+1)$-regular tree with activity $λ =1$, the unique simple invariant Gibbs measure is extremal in the set of Gibbs measures, for any $k ≥ 2$.]]> 0 <![CDATA[We study annihilating random walks on $\mathbb{Z}$ using techniques of P.W. Kasteleyn and $R$. Kenyonon perfect matchings of planar graphs. We obtain the asymptotic of the density of remaining particles and the partition function of the underlying statistical mechanical model.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3326 https://doi.org/10.46298/dmtcs.3326 Mattera, Massimiliano Mattera, Massimiliano <![CDATA[We study annihilating random walks on $\mathbb{Z}$ using techniques of P.W. Kasteleyn and $R$. Kenyonon perfect matchings of planar graphs. We obtain the asymptotic of the density of remaining particles and the partition function of the underlying statistical mechanical model.]]> 0 <![CDATA[Here we consider two parameters for random non-crossing trees: $\textit{(i)}$ the number of random cuts to destroy a size-$n$ non-crossing tree and $\textit{(ii)}$ the spanning subtree-size of $p$ randomly chosen nodes in a size-$n$ non-crossing tree. For both quantities, we are able to characterise for $n → ∞$ the limiting distributions. Non-crossing trees are almost conditioned Galton-Watson trees, and it has been already shown, that the contour and other usually associated discrete excursions converge, suitable normalised, to the Brownian excursion. We can interpret parameter $\textit{(ii)}$ as a functional of a conditioned random walk, and although we do not have such an interpretation for parameter $\textit{(i)}$, we obtain here limiting distributions, that are also arising as limits of some functionals of conditioned random walks.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3327 https://doi.org/10.46298/dmtcs.3327 Panholzer, Alois Panholzer, Alois <![CDATA[Here we consider two parameters for random non-crossing trees: $\textit{(i)}$ the number of random cuts to destroy a size-$n$ non-crossing tree and $\textit{(ii)}$ the spanning subtree-size of $p$ randomly chosen nodes in a size-$n$ non-crossing tree. For both quantities, we are able to characterise for $n → ∞$ the limiting distributions. Non-crossing trees are almost conditioned Galton-Watson trees, and it has been already shown, that the contour and other usually associated discrete excursions converge, suitable normalised, to the Brownian excursion. We can interpret parameter $\textit{(ii)}$ as a functional of a conditioned random walk, and although we do not have such an interpretation for parameter $\textit{(i)}$, we obtain here limiting distributions, that are also arising as limits of some functionals of conditioned random walks.]]> 0 <![CDATA[We review some recent results for a system of simple random walks on graphs, known as \emphfrog model. Also, we discuss several modifications of this model, and present a few open problems. A simple version of the frog model can be described as follows: There are active and sleeping particles living on some graph. Each active particle performs a simple random walk with discrete time and at each moment it may disappear with probability 1-p. When an active particle hits a sleeping particle, the latter becomes active.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3328 https://doi.org/10.46298/dmtcs.3328 Popov, Serguei Yu. Popov, Serguei Yu. <![CDATA[We review some recent results for a system of simple random walks on graphs, known as \emphfrog model. Also, we discuss several modifications of this model, and present a few open problems. A simple version of the frog model can be described as follows: There are active and sleeping particles living on some graph. Each active particle performs a simple random walk with discrete time and at each moment it may disappear with probability 1-p. When an active particle hits a sleeping particle, the latter becomes active.]]> 0 <![CDATA[In this paper we study the identity of the Abelian Sandpile Model on a rectangular lattice.This configuration can be computed with the burning algorithm, which, starting from the empty lattice, computes a sequence of configurations, the last of which is the identity.We extend this algorithm to an infinite lattice, which allows us to prove that the first steps of the algorithm on a finite lattice are the same whatever its size.Finally we introduce a new configuration, which shares the intriguing properties of the identity, but is easier to study.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2308 https://doi.org/10.46298/dmtcs.2308 Dartois, Arnaud Magnien, Clémence Dartois, Arnaud Magnien, Clémence <![CDATA[In this paper we study the identity of the Abelian Sandpile Model on a rectangular lattice.This configuration can be computed with the burning algorithm, which, starting from the empty lattice, computes a sequence of configurations, the last of which is the identity.We extend this algorithm to an infinite lattice, which allows us to prove that the first steps of the algorithm on a finite lattice are the same whatever its size.Finally we introduce a new configuration, which shares the intriguing properties of the identity, but is easier to study.]]> 0 <![CDATA[We present the first experimental demonstration of intermittency in a granular medium. The medium consists of magnets embedded within spheres. These spheres are placed in a horizontal Petri dish where they roll by virtue of an alternating, homogenous magnetic field. Due to collisions with the wall, clustering leads to self-organization into ring pieces circulating along the wall. The intermi ttent behaviour consists of an aperiodical alternation of this circular motion with a gaslike state extended over the entire dish. Molecular dynamic simulations agree with observations]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2315 https://doi.org/10.46298/dmtcs.2315 Schmick, Malte Markus, Mario Schmick, Malte Markus, Mario <![CDATA[We present the first experimental demonstration of intermittency in a granular medium. The medium consists of magnets embedded within spheres. These spheres are placed in a horizontal Petri dish where they roll by virtue of an alternating, homogenous magnetic field. Due to collisions with the wall, clustering leads to self-organization into ring pieces circulating along the wall. The intermi ttent behaviour consists of an aperiodical alternation of this circular motion with a gaslike state extended over the entire dish. Molecular dynamic simulations agree with observations]]> 0 <![CDATA[A class of finite discrete dynamical systems, called Sequential Dynamical Systems (SDSs), was introduced in [BR99] as a formal model for analyzing simulation systems. Here, we address the complexity of two basic problems and their generalizations for SDSs.Given an SDS $\mathcal{S}$ and a configuration $\mathcal{C}$, the PREDECESSOR EXISTENCE (or PRE) problem is to determine whether there is a configuration $\mathcal{C}'$ such that $\mathcal{S}$ has a transition from $\mathcal{C}'$ to $\mathcal{C}$. Our results provide separations between efficiently solvable and computationally intractable instances of the PRE problem. For example, we show that the PRE problem can be solved efficiently for SDSs with Boolean state values when the node functions are symmetric and the underlying graph is of bounded tree width. In contrast, we show that allowing just one non-symmetric node function renders the problem $\mathbf{NP}$-complete even when the underlying graph is a star (which has a tree width of 1). Our results extend some of the earlier results by Sutner [Su95] and Green [Gr87] on the complexity of the PREDECESSOR EXISTENCE problem for 1-dimensional cellular automata.Given two configurations $\mathcal{C}$ and $\mathcal{C}'$ of a partial SDS $\mathcal{S}$, the PERMUTATION EXISTENCE (or PME) problem is to determine whether there is a permutation of nodes such that $\mathcal{S}$ has a transition from $\mathcal{C}'$ to $\mathcal{C}$ in one step. We show that the PME problem is $\mathbf{NP}$-complete even when the function associated with each node is a simple-threshold function. We also show that the problem can be solved efficiently for SDSs whose underlying graphs are of bounded degree and bounded tree width. We consider a generalized version (GEN-PME) of the PME problem and show that the problem is $\mathbf{NP}$-complete for SDSs where each node function is NOR and the underlying graph has a maximum node degree of 3. When each node computes the OR function or when each node computes the AND function, we show that the GEN-PME problem is solvable in polynomial time.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2314 https://doi.org/10.46298/dmtcs.2314 Barrett, Christopher L. Hunt, Harry, Marathe, Madhav V. Ravi, S. S. Rosenkrantz, Daniel J. Stearns, Richard E. Barrett, Christopher L. Hunt, Harry, Marathe, Madhav V. Ravi, S. S. Rosenkrantz, Daniel J. Stearns, Richard E. <![CDATA[A class of finite discrete dynamical systems, called Sequential Dynamical Systems (SDSs), was introduced in [BR99] as a formal model for analyzing simulation systems. Here, we address the complexity of two basic problems and their generalizations for SDSs.Given an SDS $\mathcal{S}$ and a configuration $\mathcal{C}$, the PREDECESSOR EXISTENCE (or PRE) problem is to determine whether there is a configuration $\mathcal{C}'$ such that $\mathcal{S}$ has a transition from $\mathcal{C}'$ to $\mathcal{C}$. Our results provide separations between efficiently solvable and computationally intractable instances of the PRE problem. For example, we show that the PRE problem can be solved efficiently for SDSs with Boolean state values when the node functions are symmetric and the underlying graph is of bounded tree width. In contrast, we show that allowing just one non-symmetric node function renders the problem $\mathbf{NP}$-complete even when the underlying graph is a star (which has a tree width of 1). Our results extend some of the earlier results by Sutner [Su95] and Green [Gr87] on the complexity of the PREDECESSOR EXISTENCE problem for 1-dimensional cellular automata.Given two configurations $\mathcal{C}$ and $\mathcal{C}'$ of a partial SDS $\mathcal{S}$, the PERMUTATION EXISTENCE (or PME) problem is to determine whether there is a permutation of nodes such that $\mathcal{S}$ has a transition from $\mathcal{C}'$ to $\mathcal{C}$ in one step. We show that the PME problem is $\mathbf{NP}$-complete even when the function associated with each node is a simple-threshold function. We also show that the problem can be solved efficiently for SDSs whose underlying graphs are of bounded degree and bounded tree width. We consider a generalized version (GEN-PME) of the PME problem and show that the problem is $\mathbf{NP}$-complete for SDSs where each node function is NOR and the underlying graph has a maximum node degree of 3. When each node computes the OR function or when each node computes the AND function, we show that the GEN-PME problem is solvable in polynomial time.]]> 0 <![CDATA[A polycube in dimension $d$ is a finite union of unit $d$-cubes whose vertices are on knots of the lattice $\mathbb{Z}^d$. We show that, for each family of polycubes $E$, there exists a finite set $F$ of bricks (parallelepiped rectangles) such that the bricks which can be tiled by $E$ are exactly the bricks which can be tiled by $F$. Consequently, if we know the set $F$, then we have an algorithm to decide in polynomial time if a brick is tilable or not by the tiles of $E$.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2313 https://doi.org/10.46298/dmtcs.2313 Bodini, Olivier Bodini, Olivier <![CDATA[A polycube in dimension $d$ is a finite union of unit $d$-cubes whose vertices are on knots of the lattice $\mathbb{Z}^d$. We show that, for each family of polycubes $E$, there exists a finite set $F$ of bricks (parallelepiped rectangles) such that the bricks which can be tiled by $E$ are exactly the bricks which can be tiled by $F$. Consequently, if we know the set $F$, then we have an algorithm to decide in polynomial time if a brick is tilable or not by the tiles of $E$.]]> 0 <![CDATA[The Langton's ant is studied from the point of view of topological dynamical systems. A new approach which associate a subshift to the system is proposed.The transition rule is generalized to the family of bi-regular graphs $\Gamma(k,d)$ and the dependence of the dynamical system on $k$ and $d$ is analyzed. A classification of the $\Gamma (k,d)$ graphs based on the dynamical properties of the subshift is established. Also a hierarchy is defined on the graphs through the subset relation of the respective subshifts. The analysis are worked out by establishing an algebraic characterization of the forbidden words of the subshift.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2312 https://doi.org/10.46298/dmtcs.2312 Gajardo, Anahi Gajardo, Anahi <![CDATA[The Langton's ant is studied from the point of view of topological dynamical systems. A new approach which associate a subshift to the system is proposed.The transition rule is generalized to the family of bi-regular graphs $\Gamma(k,d)$ and the dependence of the dynamical system on $k$ and $d$ is analyzed. A classification of the $\Gamma (k,d)$ graphs based on the dynamical properties of the subshift is established. Also a hierarchy is defined on the graphs through the subset relation of the respective subshifts. The analysis are worked out by establishing an algebraic characterization of the forbidden words of the subshift.]]> 0 <![CDATA[This paper studies a conservative transformation defined on families of finite sets. It consists in removing one element from each set and adding a new set composed of the removed elements. This transformation is conservative in the sense that the union of all sets of the family always remains the same.We study the dynamical process obtained when iterating this deterministic transformation on a family of sets and we focus on the evolution of the cardinalities of the sets of the family. This point of view allows to consider the transformation as an application defined on the set of all partitions of a fixed integer (which is the total number of elements in the sets).We show that iterating this particular transformation always leads to a heterogeneous distribution of the cardinalities, where almost all integers within an interval are represented.We also tackle some issues concerning the structure of the transition graph which sums up the whole dynamics of this process for all partitions of a fixed integer.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2311 https://doi.org/10.46298/dmtcs.2311 Phan, Thi Ha Duong Thierry, Eric Phan, Thi Ha Duong Thierry, Eric <![CDATA[This paper studies a conservative transformation defined on families of finite sets. It consists in removing one element from each set and adding a new set composed of the removed elements. This transformation is conservative in the sense that the union of all sets of the family always remains the same.We study the dynamical process obtained when iterating this deterministic transformation on a family of sets and we focus on the evolution of the cardinalities of the sets of the family. This point of view allows to consider the transformation as an application defined on the set of all partitions of a fixed integer (which is the total number of elements in the sets).We show that iterating this particular transformation always leads to a heterogeneous distribution of the cardinalities, where almost all integers within an interval are represented.We also tackle some issues concerning the structure of the transition graph which sums up the whole dynamics of this process for all partitions of a fixed integer.]]> 0 <![CDATA[It is increasingly common to encounter time-varying random fields on networks (metabolic networks, sensor arrays, distributed computing, etc.).This paper considers the problem of optimal, nonlinear prediction of these fields, showing from an information-theoretic perspective that it is formally identical to the problem of finding minimal local sufficient statistics.I derive general properties of these statistics, show that they can be composed into global predictors, and explore their recursive estimation properties.For the special case of discrete-valued fields, I describe a convergent algorithm to identify the local predictors from empirical data, with minimal prior information about the field, and no distributional assumptions.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2310 https://doi.org/10.46298/dmtcs.2310 Shalizi, Cosma Rohilla Shalizi, Cosma Rohilla <![CDATA[It is increasingly common to encounter time-varying random fields on networks (metabolic networks, sensor arrays, distributed computing, etc.).This paper considers the problem of optimal, nonlinear prediction of these fields, showing from an information-theoretic perspective that it is formally identical to the problem of finding minimal local sufficient statistics.I derive general properties of these statistics, show that they can be composed into global predictors, and explore their recursive estimation properties.For the special case of discrete-valued fields, I describe a convergent algorithm to identify the local predictors from empirical data, with minimal prior information about the field, and no distributional assumptions.]]> 0 <![CDATA[We present a lattice gas technique for simulating molecular self-assembly of amphiphilic polymers in aqueous environments. Water molecules, hydrocarbons tail-groups and amphiphilic head-groups are explicitly represented on a three dimensional discrete lattice. Molecules move on the lattice proportional to their continuous momentum. Collision rules preserve momentum and kinetic energy. Potential energy from molecular interactions are also included explicitly. Non-trivial thermodynamics of large scale and long time dynamics are studied. In this paper we specifically demonstrate how, from a random initial distribution, micelles are formed, and grow until they destabilize and divide. Eventually a steady state of growing and dividing micelles is formed.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2309 https://doi.org/10.46298/dmtcs.2309 Nilsson, Martin Rasmussen, Steen Nilsson, Martin Rasmussen, Steen <![CDATA[We present a lattice gas technique for simulating molecular self-assembly of amphiphilic polymers in aqueous environments. Water molecules, hydrocarbons tail-groups and amphiphilic head-groups are explicitly represented on a three dimensional discrete lattice. Molecules move on the lattice proportional to their continuous momentum. Collision rules preserve momentum and kinetic energy. Potential energy from molecular interactions are also included explicitly. Non-trivial thermodynamics of large scale and long time dynamics are studied. In this paper we specifically demonstrate how, from a random initial distribution, micelles are formed, and grow until they destabilize and divide. Eventually a steady state of growing and dividing micelles is formed.]]> 0 <![CDATA[ECO is a method for the enumeration of classes of combinatorial objects based on recursive constructions of such classes. In the first part of this paper we present a construction for the class of convex polyominoes based on the ECO method. Then we translate this construction into a succession rule. The final goal of the paper is to determine the generating function of convex polyominoes according to the semi-perimeter, and it is achieved by applying an idea introduced in [11].]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2307 https://doi.org/10.46298/dmtcs.2307 Del Lungo, A. Duchi, E. Frosini, A. Rinaldi, S. Del Lungo, A. Duchi, E. Frosini, A. Rinaldi, S. <![CDATA[ECO is a method for the enumeration of classes of combinatorial objects based on recursive constructions of such classes. In the first part of this paper we present a construction for the class of convex polyominoes based on the ECO method. Then we translate this construction into a succession rule. The final goal of the paper is to determine the generating function of convex polyominoes according to the semi-perimeter, and it is achieved by applying an idea introduced in [11].]]> 0 <![CDATA[This paper provides a definition of a cellular line in a cellular array that is independent of the notion of a line in $\mathfrak{R}^2$.It also presents a way of determining whether or not a cell set is a cellular line.Brief statements about existence, uniqueness, and properties of cellular lines are included.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2306 https://doi.org/10.46298/dmtcs.2306 Geer, Panama McLaughlin, Harry W. Unsworth, Keith Geer, Panama McLaughlin, Harry W. Unsworth, Keith <![CDATA[This paper provides a definition of a cellular line in a cellular array that is independent of the notion of a line in $\mathfrak{R}^2$.It also presents a way of determining whether or not a cell set is a cellular line.Brief statements about existence, uniqueness, and properties of cellular lines are included.]]> 0 <![CDATA[The problem of counting monomer-dimer coverings of a lattice is a longstanding problem in statistical mechanics.It has only been exactly solved for the special case of dimer coverings in two dimensions ([Ka61], [TF61]). In earlier work, Stanley [St85] proved a reciprocity principle governing the number $N(m,n)$ of dimer coverings of an $m$ by $n$ rectangular grid (also known as perfect matchings), where $m$ is fixed and $n$ is allowed to vary. As reinterpreted by Propp [P01], Stanley's result concerns the unique way of extending $N(m,n)$ to $n<0$ so that the resulting bi-infinite sequence, $N(m,n)$ for $n \in \mathbb{Z}$, satisfies a linear recurrence relation with constant coefficients. In particular, Stanley shows that $N(m,n)$ is always an integer satisfying the relation $N(m,-2-n) = \varepsilon_{m,n} N(m,n)$ where $\varepsilon_{m,n}=1$ unless $m \equiv 2(\mod 4)$ and $n$ is odd, in which case $\varepsilon_{m,n}=-1$. Furthermore, Propp's method was applicable to higher-dimensional cases.This paper discusses similar investigations of the numbers $M(m,n)$, of monomer-dimer coverings, or equivalently (not necessarily perfect) matchings of an $m$ by $n$ rectangular grid. We show that for each fixed $m$ there is a unique way of extending $M(m,n)$ to $n<0$ so that the resulting bi-infinite sequence, $M(m,n)$ for $n \in \mathbb{Z}$, satisfies a linear recurrence relation with constant coefficients.We show that $M(m,n)$, a priori a rational number, is always an integer, using a generalization of the combinatorial model offered by Propp. Lastly, we give a new statement of reciprocity in terms of multivariate generating functions from which Stanley's result follows.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2305 https://doi.org/10.46298/dmtcs.2305 Anzalone, Nick Baldwin, John Bronshtein, Ilya Petersen, Kyle, Anzalone, Nick Baldwin, John Bronshtein, Ilya Petersen, Kyle, <![CDATA[The problem of counting monomer-dimer coverings of a lattice is a longstanding problem in statistical mechanics.It has only been exactly solved for the special case of dimer coverings in two dimensions ([Ka61], [TF61]). In earlier work, Stanley [St85] proved a reciprocity principle governing the number $N(m,n)$ of dimer coverings of an $m$ by $n$ rectangular grid (also known as perfect matchings), where $m$ is fixed and $n$ is allowed to vary. As reinterpreted by Propp [P01], Stanley's result concerns the unique way of extending $N(m,n)$ to $n<0$ so that the resulting bi-infinite sequence, $N(m,n)$ for $n \in \mathbb{Z}$, satisfies a linear recurrence relation with constant coefficients. In particular, Stanley shows that $N(m,n)$ is always an integer satisfying the relation $N(m,-2-n) = \varepsilon_{m,n} N(m,n)$ where $\varepsilon_{m,n}=1$ unless $m \equiv 2(\mod 4)$ and $n$ is odd, in which case $\varepsilon_{m,n}=-1$. Furthermore, Propp's method was applicable to higher-dimensional cases.This paper discusses similar investigations of the numbers $M(m,n)$, of monomer-dimer coverings, or equivalently (not necessarily perfect) matchings of an $m$ by $n$ rectangular grid. We show that for each fixed $m$ there is a unique way of extending $M(m,n)$ to $n<0$ so that the resulting bi-infinite sequence, $M(m,n)$ for $n \in \mathbb{Z}$, satisfies a linear recurrence relation with constant coefficients.We show that $M(m,n)$, a priori a rational number, is always an integer, using a generalization of the combinatorial model offered by Propp. Lastly, we give a new statement of reciprocity in terms of multivariate generating functions from which Stanley's result follows.]]> 0 <![CDATA[Classifying cellular automata in order to capture the notion of chaos algorithmically is a challenging problem than can be tackled in many ways.We here give a classification based on the computation of a macroscopic parameter, the $d$-spectrum, and show how our classifying scheme can be used to separate the chaotic ECA from the non-chaotic ones.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2304 https://doi.org/10.46298/dmtcs.2304 Fatès, Nazim, Fatès, Nazim, <![CDATA[Classifying cellular automata in order to capture the notion of chaos algorithmically is a challenging problem than can be tackled in many ways.We here give a classification based on the computation of a macroscopic parameter, the $d$-spectrum, and show how our classifying scheme can be used to separate the chaotic ECA from the non-chaotic ones.]]> 0 <![CDATA[In an effort to continue the pioneering work of Harary in USA and Flament in France, we have undertaken to develop, on an experimental basis, a formalized theory of systems of beliefs and their modifications. This theory uses the psycho-social concepts of theories of cognitive consistency and of the tools of discrete mathematics, such as rewriting and intervals within graphs. The axioms and rewriting rules are elaborated from experimental data, and we demonstrate that the system we have built has the property of termination. This result is in accordance with experimental observations that show that every subject having an inconsistent system of beliefs (i.e., one containing contradictions) makes this system evolve towards consistency to reach a simple, consistent reference framework.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2303 https://doi.org/10.46298/dmtcs.2303 Lafaye de Micheaux, N. López, G. Vitiello, P. Beauvois, J. L. Lafaye de Micheaux, N. López, G. Vitiello, P. Beauvois, J. L. <![CDATA[In an effort to continue the pioneering work of Harary in USA and Flament in France, we have undertaken to develop, on an experimental basis, a formalized theory of systems of beliefs and their modifications. This theory uses the psycho-social concepts of theories of cognitive consistency and of the tools of discrete mathematics, such as rewriting and intervals within graphs. The axioms and rewriting rules are elaborated from experimental data, and we demonstrate that the system we have built has the property of termination. This result is in accordance with experimental observations that show that every subject having an inconsistent system of beliefs (i.e., one containing contradictions) makes this system evolve towards consistency to reach a simple, consistent reference framework.]]> 0 <![CDATA[Equicontinuity classification is a popular classification of cellular automata based on their dynamical behavior. In this paper we prove that most of its classes are undecidable.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2302 https://doi.org/10.46298/dmtcs.2302 Durand, Bruno Formenti, Enrico Varouchas, Georges Durand, Bruno Formenti, Enrico Varouchas, Georges <![CDATA[Equicontinuity classification is a popular classification of cellular automata based on their dynamical behavior. In this paper we prove that most of its classes are undecidable.]]> 0 <![CDATA[This paper is a survey on our recent results about number conserving cellular automata. First, we prove the linear time decidability of the property of number conservation. The sequel focuses on dynamical evolutions of number conserving cellular automata.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2301 https://doi.org/10.46298/dmtcs.2301 Durand, Bruno Formenti, Enrico Grange, Aristide Róka, Zsuzsanna Durand, Bruno Formenti, Enrico Grange, Aristide Róka, Zsuzsanna <![CDATA[This paper is a survey on our recent results about number conserving cellular automata. First, we prove the linear time decidability of the property of number conservation. The sequel focuses on dynamical evolutions of number conserving cellular automata.]]> 0 <![CDATA[Using a transfer matrix method, we present some results for directed lattice walkers in a horizontal strip of finite width. Some cases with two walkers in a small width are solved exactly, as are a couple of cases with vicious walkers in a small width; a conjecture is made for a case with three walkers. We also derive the general transfer matrix for two walkers. Lastly, we examine the dependence of the growth constant on the width and friendliness.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3341 https://doi.org/10.46298/dmtcs.3341 Chan, Yao-Ban Guttmann, Anthony J. Chan, Yao-Ban Guttmann, Anthony J. <![CDATA[Using a transfer matrix method, we present some results for directed lattice walkers in a horizontal strip of finite width. Some cases with two walkers in a small width are solved exactly, as are a couple of cases with vicious walkers in a small width; a conjecture is made for a case with three walkers. We also derive the general transfer matrix for two walkers. Lastly, we examine the dependence of the growth constant on the width and friendliness.]]> 0 <![CDATA[Our model is a generalized linear programming relaxation of a much studied random K-SAT problem. Specifically, a set of linear constraints $C$ on $K$ variables is fixed. From a pool of $n$ variables, $K$ variables are chosen uniformly at random and a constraint is chosen from $C$ also uniformly at random. This procedure is repeated $m$ times independently. We are interested in whether the resulting linear programming problem is feasible. We prove that the feasibility property experiences a linear phase transition,when $n→∞$ and $m=cn$ for a constant $c$. Namely, there exists a critical value $c^*$ such that, when $c < c^*$, the problem is feasible or is asymptotically almost feasible, as $n→∞$, but, when $c > c^*$, the "distance" to feasibility is at least a positive constant independent of $n$. Our result is obtained using the combination of a powerful local weak convergence method developed in Aldous [1992, 2000], Aldous and Steele [2003], Steele [2002] and martingale techniques. By exploiting a linear programming duality, our theorem impliesthe following result in the context of sparse random graphs $G(n, cn)$ on $n$ nodes with $cn$ edges, where edges are equipped with randomly generated weights. Let $\mathcal{M}(n,c)$ denote maximum weight matching in $G(n, cn)$. We prove that when $c$ is a constant and $n→∞$, the limit $lim_{n→∞} \mathcal{M}(n,c)/n$, exists, with high probability. We further extend this result to maximum weight b-matchings also in $G(n,cn)$.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3351 https://doi.org/10.46298/dmtcs.3351 Gamarnik, David Gamarnik, David <![CDATA[Our model is a generalized linear programming relaxation of a much studied random K-SAT problem. Specifically, a set of linear constraints $C$ on $K$ variables is fixed. From a pool of $n$ variables, $K$ variables are chosen uniformly at random and a constraint is chosen from $C$ also uniformly at random. This procedure is repeated $m$ times independently. We are interested in whether the resulting linear programming problem is feasible. We prove that the feasibility property experiences a linear phase transition,when $n→∞$ and $m=cn$ for a constant $c$. Namely, there exists a critical value $c^*$ such that, when $c < c^*$, the problem is feasible or is asymptotically almost feasible, as $n→∞$, but, when $c > c^*$, the "distance" to feasibility is at least a positive constant independent of $n$. Our result is obtained using the combination of a powerful local weak convergence method developed in Aldous [1992, 2000], Aldous and Steele [2003], Steele [2002] and martingale techniques. By exploiting a linear programming duality, our theorem impliesthe following result in the context of sparse random graphs $G(n, cn)$ on $n$ nodes with $cn$ edges, where edges are equipped with randomly generated weights. Let $\mathcal{M}(n,c)$ denote maximum weight matching in $G(n, cn)$. We prove that when $c$ is a constant and $n→∞$, the limit $lim_{n→∞} \mathcal{M}(n,c)/n$, exists, with high probability. We further extend this result to maximum weight b-matchings also in $G(n,cn)$.]]> 0 <![CDATA[We consider the motion of a discrete d-dimensional random surface interacting by exclusion with a rarefied wall. The dynamics is given by the serial harness process. We prove that the process delocalizes iff the mean number of visits to the set of sites where the wall is present by some random walk is infinite. In case where there is a delocalization, bounds on its speed are obtained.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3350 https://doi.org/10.46298/dmtcs.3350 Fontes, L. R. G. Vachkovskaia, M. Yambartsev, A. Fontes, L. R. G. Vachkovskaia, M. Yambartsev, A. <![CDATA[We consider the motion of a discrete d-dimensional random surface interacting by exclusion with a rarefied wall. The dynamics is given by the serial harness process. We prove that the process delocalizes iff the mean number of visits to the set of sites where the wall is present by some random walk is infinite. In case where there is a delocalization, bounds on its speed are obtained.]]> 0 <![CDATA[A new approach is used to determine the transient probability functions of Markov processes. This new solution method is a sample path counting approach and uses dual processes and randomization. The approach is illustrated by determining transient probability functions for a three-state Markov process. This approach also provides a way to calculate transient probability functions for Markov processes which have specific sample path characteristics.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3349 https://doi.org/10.46298/dmtcs.3349 Green, Michael L. Krinik, Alan Mortensen, Carrie Rubino, Gerardo Swift, Randall J. Green, Michael L. Krinik, Alan Mortensen, Carrie Rubino, Gerardo Swift, Randall J. <![CDATA[A new approach is used to determine the transient probability functions of Markov processes. This new solution method is a sample path counting approach and uses dual processes and randomization. The approach is illustrated by determining transient probability functions for a three-state Markov process. This approach also provides a way to calculate transient probability functions for Markov processes which have specific sample path characteristics.]]> 0 <![CDATA[We obtain a new result concerning harmonic functions on infinite Cayley graphs $X$: either every nonconstant harmonic function has infinite radial variation in a certain uniform sense, or there is a nontrivial boundary with hyperbolic properties at infinity of $X$. In the latter case, relying on a theorem of Woess, it follows that the Dirichlet problem is solvable with respect to this boundary. Certain relations to group cohomology are also discussed.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3348 https://doi.org/10.46298/dmtcs.3348 Karlsson, Anders Karlsson, Anders <![CDATA[We obtain a new result concerning harmonic functions on infinite Cayley graphs $X$: either every nonconstant harmonic function has infinite radial variation in a certain uniform sense, or there is a nontrivial boundary with hyperbolic properties at infinity of $X$. In the latter case, relying on a theorem of Woess, it follows that the Dirichlet problem is solvable with respect to this boundary. Certain relations to group cohomology are also discussed.]]> 0 <![CDATA[Random compositions of integers are used as theoretical models for many applications. The degree of distinctness of a composition is a natural and important parameter. A possible measure of distinctness is the number $X$ of distinct parts (or components). This parameter has been analyzed in several papers. In this article we consider a variant of the distinctness: the number $X(m)$ of distinct parts of multiplicity m that we call the $m$-distinctness. A firstmotivation is a question asked by Wilf for random compositions: what is the asymptotic value of the probability that a randomly chosen part size in a random composition of an integer $ν$ has multiplicity $m$. This is related to $\mathbb{E}(X(m))$, which has been analyzed by Hitczenko, Rousseau and Savage. Here, we investigate, from a probabilistic point of view, the first full part, the maximum part size and the distribution of $X(m)$. We obtain asymptotically, as $ν → ∞$, the moments and an expression for a continuous distribution $φ$ , the (discrete) distribution of $X(m,ν )$ being computable from $φ$ .]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3347 https://doi.org/10.46298/dmtcs.3347 Louchard, Guy Louchard, Guy <![CDATA[Random compositions of integers are used as theoretical models for many applications. The degree of distinctness of a composition is a natural and important parameter. A possible measure of distinctness is the number $X$ of distinct parts (or components). This parameter has been analyzed in several papers. In this article we consider a variant of the distinctness: the number $X(m)$ of distinct parts of multiplicity m that we call the $m$-distinctness. A firstmotivation is a question asked by Wilf for random compositions: what is the asymptotic value of the probability that a randomly chosen part size in a random composition of an integer $ν$ has multiplicity $m$. This is related to $\mathbb{E}(X(m))$, which has been analyzed by Hitczenko, Rousseau and Savage. Here, we investigate, from a probabilistic point of view, the first full part, the maximum part size and the distribution of $X(m)$. We obtain asymptotically, as $ν → ∞$, the moments and an expression for a continuous distribution $φ$ , the (discrete) distribution of $X(m,ν )$ being computable from $φ$ .]]> 0 <![CDATA[In these expository paper we describe the role of the rooted trees as a base for convenient tools in studies ofrandom matrices. Regarding the Wigner ensemble of random matrices, we represent main ingredients ofthis approach. Also werefine our previous result on the limit of the spectral norm of adjacency matrix of large random graphs.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3346 https://doi.org/10.46298/dmtcs.3346 Khorunzhiy, Oleksiy Khorunzhiy, Oleksiy <![CDATA[In these expository paper we describe the role of the rooted trees as a base for convenient tools in studies ofrandom matrices. Regarding the Wigner ensemble of random matrices, we represent main ingredients ofthis approach. Also werefine our previous result on the limit of the spectral norm of adjacency matrix of large random graphs.]]> 0 <![CDATA[We study the phase transition in a random graph in which vertices and edges are added at constant rates. Two recent papers in Physical Review E by Callaway, Hopcroft, Kleinberg, Newman, and Strogatz, and Dorogovstev, Mendes, and Samukhin have computed the critical value of this model, shown that the fraction of vertices in finite clusters is infinitely differentiable at the critical value, and that in the subcritical phase the cluster size distribution has a polynomial decay rate with a continuously varying power. Here we sketch rigorous proofs for the first and third results and a new estimates about connectivity probabilities at the critical value.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3345 https://doi.org/10.46298/dmtcs.3345 Durrett, Rick Durrett, Rick <![CDATA[We study the phase transition in a random graph in which vertices and edges are added at constant rates. Two recent papers in Physical Review E by Callaway, Hopcroft, Kleinberg, Newman, and Strogatz, and Dorogovstev, Mendes, and Samukhin have computed the critical value of this model, shown that the fraction of vertices in finite clusters is infinitely differentiable at the critical value, and that in the subcritical phase the cluster size distribution has a polynomial decay rate with a continuously varying power. Here we sketch rigorous proofs for the first and third results and a new estimates about connectivity probabilities at the critical value.]]> 0 <![CDATA[In this paper we consider discrete random walks on infinite graphs that are generated by copying and shifting one finite (strongly connected) graph into one direction and connecting successive copies always in the same way. With help of generating functions it is shown that there are only three types for the asymptotic behaviour of the random walk. It either converges to the stationary distribution or it can be approximated in terms of a reflected Brownian motion or by a Brownian motion. In terms of Markov chains these cases correspond to positive recurrence, to null recurrence, and to non recurrence.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3344 https://doi.org/10.46298/dmtcs.3344 Drmota, Michael Drmota, Michael <![CDATA[In this paper we consider discrete random walks on infinite graphs that are generated by copying and shifting one finite (strongly connected) graph into one direction and connecting successive copies always in the same way. With help of generating functions it is shown that there are only three types for the asymptotic behaviour of the random walk. It either converges to the stationary distribution or it can be approximated in terms of a reflected Brownian motion or by a Brownian motion. In terms of Markov chains these cases correspond to positive recurrence, to null recurrence, and to non recurrence.]]> 0 <![CDATA[Our work is motivated by Bourque-Pevzner's simulation study of the effectiveness of the parsimony method in studying genome rearrangement, and leads to a surprising result about the random transposition walk in continuous time on the group of permutations on $n$ elements starting from the identity. Let $D_t$ be the minimum number of transpositions needed to go back to the identity element from the location at time $t$. $D_t$ undergoes a phase transition: for $0 < c ≤ 1$, the distance $D_cn/2 ~ cn/2$, i.e., the distance increases linearly with time; for $c > 1$, $D_cn/2 ~ u(c)n$ where u is an explicit function satisfying $u(x) < x/2$. Moreover we describe the fluctuations of $D_{cn/2}$ about its mean at each of the three stages (subcritical, critical and supercritical). The techniques used involve viewing the cycles in the random permutation as a coagulation-fragmentation process and relating the behavior to the Erdős-Rényi random graph model.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3343 https://doi.org/10.46298/dmtcs.3343 Berestycki, Nathanael Durrett, Rick Berestycki, Nathanael Durrett, Rick <![CDATA[Our work is motivated by Bourque-Pevzner's simulation study of the effectiveness of the parsimony method in studying genome rearrangement, and leads to a surprising result about the random transposition walk in continuous time on the group of permutations on $n$ elements starting from the identity. Let $D_t$ be the minimum number of transpositions needed to go back to the identity element from the location at time $t$. $D_t$ undergoes a phase transition: for $0 < c ≤ 1$, the distance $D_cn/2 ~ cn/2$, i.e., the distance increases linearly with time; for $c > 1$, $D_cn/2 ~ u(c)n$ where u is an explicit function satisfying $u(x) < x/2$. Moreover we describe the fluctuations of $D_{cn/2}$ about its mean at each of the three stages (subcritical, critical and supercritical). The techniques used involve viewing the cycles in the random permutation as a coagulation-fragmentation process and relating the behavior to the Erdős-Rényi random graph model.]]> 0 <![CDATA[The random stirring process is a natural random walk on the set of permutations of the vertex set of a graph. The cyclic time random walk is a self interacting random walk on a graph. It is influenced by its past, in that it is constrained to repeat its past choices if it returns to a previously visited edge after a multiple of some period of time. The two models are fundamentally equivalent to each other as well as to a certain coalescence and fragmentation process.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3342 https://doi.org/10.46298/dmtcs.3342 Angel, Omer Angel, Omer <![CDATA[The random stirring process is a natural random walk on the set of permutations of the vertex set of a graph. The cyclic time random walk is a self interacting random walk on a graph. It is influenced by its past, in that it is constrained to repeat its past choices if it returns to a previously visited edge after a multiple of some period of time. The two models are fundamentally equivalent to each other as well as to a certain coalescence and fragmentation process.]]> 0 <![CDATA[The study of thermodynamic properties of classical spin models on infinite graphs naturally leads to consider the new combinatorial problems of random-walks and percolation on the average. Indeed, spinmodels with O(n) continuous symmetry present spontaneous magnetization only on transient on the average graphs, while models with discrete symmetry (Ising and Potts) are spontaneously magnetized on graphs exhibiting percolation on the average. In this paper we define the combinatorial problems on the average, showing that they give rise to classifications of graph topology which are different from the ones obtained in usual (local) random-walks and percolation. Furthermore, we illustrate the theorem proving the correspondence between Potts model and average percolation.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3329 https://doi.org/10.46298/dmtcs.3329 Vezzani, Alessandro Cassi, Davide Burioni, Raffaella Vezzani, Alessandro Cassi, Davide Burioni, Raffaella <![CDATA[The study of thermodynamic properties of classical spin models on infinite graphs naturally leads to consider the new combinatorial problems of random-walks and percolation on the average. Indeed, spinmodels with O(n) continuous symmetry present spontaneous magnetization only on transient on the average graphs, while models with discrete symmetry (Ising and Potts) are spontaneously magnetized on graphs exhibiting percolation on the average. In this paper we define the combinatorial problems on the average, showing that they give rise to classifications of graph topology which are different from the ones obtained in usual (local) random-walks and percolation. Furthermore, we illustrate the theorem proving the correspondence between Potts model and average percolation.]]> 0 <![CDATA[Benjamini, Lyons and Schramm (1999) considered properties of an infinite graph $G$, and the simple random walk on it, that are preserved by random perturbations. To address problems raised by those authors, we study simple random walk on the infinite percolation cluster in Cayley graphs of certain amenable groups known as "lamplighter groups''.We prove that zero speed for random walk on a lamplighter group implies zero speed for random walk on an infinite cluster, for any supercritical percolation parameter $p$. For $p$ large enough, we also establish the converse. We prove that if $G$ has a positive anchored expansion constant then so does every infinite cluster of independent percolation with parameter $p$ sufficiently close to 1; We also show that positivity of the anchored expansion constant is preserved under a random stretch if, and only if, the stretching law has an exponential tail.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3340 https://doi.org/10.46298/dmtcs.3340 Chen, Dayue Peres, Yuval Chen, Dayue Peres, Yuval <![CDATA[Benjamini, Lyons and Schramm (1999) considered properties of an infinite graph $G$, and the simple random walk on it, that are preserved by random perturbations. To address problems raised by those authors, we study simple random walk on the infinite percolation cluster in Cayley graphs of certain amenable groups known as "lamplighter groups''.We prove that zero speed for random walk on a lamplighter group implies zero speed for random walk on an infinite cluster, for any supercritical percolation parameter $p$. For $p$ large enough, we also establish the converse. We prove that if $G$ has a positive anchored expansion constant then so does every infinite cluster of independent percolation with parameter $p$ sufficiently close to 1; We also show that positivity of the anchored expansion constant is preserved under a random stretch if, and only if, the stretching law has an exponential tail.]]> 0 <![CDATA[Consider the single server queue with an infinite buffer and a FIFO discipline, either of type M/M/1 or Geom/Geom/1. Denote by $\mathcal{A}$ the arrival process and by $s$ the services. Assume the stability condition to be satisfied. Denote by $\mathcal{D}$ the departure process in equilibrium and by $r$ the time spent by the customers at the very back of the queue. We prove that $(\mathcal{D},r)$ has the same law as $(\mathcal{A},s)$ which is an extension of the classical Burke Theorem. In fact, $r$ can be viewed as the departures from a dual storage model. This duality between the two models also appears when studying the transient behavior of a tandem by means of the RSK algorithm: the first and last row of the resulting semi-standard Young tableau are respectively the last instant of departure in the queue and the total number of departures in the store.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3339 https://doi.org/10.46298/dmtcs.3339 Draief, Moez Mairesse, Jean O'Connell, Neil Draief, Moez Mairesse, Jean O'Connell, Neil <![CDATA[Consider the single server queue with an infinite buffer and a FIFO discipline, either of type M/M/1 or Geom/Geom/1. Denote by $\mathcal{A}$ the arrival process and by $s$ the services. Assume the stability condition to be satisfied. Denote by $\mathcal{D}$ the departure process in equilibrium and by $r$ the time spent by the customers at the very back of the queue. We prove that $(\mathcal{D},r)$ has the same law as $(\mathcal{A},s)$ which is an extension of the classical Burke Theorem. In fact, $r$ can be viewed as the departures from a dual storage model. This duality between the two models also appears when studying the transient behavior of a tandem by means of the RSK algorithm: the first and last row of the resulting semi-standard Young tableau are respectively the last instant of departure in the queue and the total number of departures in the store.]]> 0 <![CDATA[A discrete space-filling curve provides a linear traversal/indexing of a multi-dimensional grid space.This paper presents an application of random walk to the study of inter-clustering of space-filling curves and an analytical study on the inter-clustering performances of 2-dimensional Hilbert and z-order curve families.Two underlying measures are employed: the mean inter-cluster distance over all inter-cluster gaps and the mean total inter-cluster distance over all subgrids.We show how approximating the mean inter-cluster distance statistics of continuous multi-dimensional space-filling curves fits into the formalism of random walk, and derive the exact formulas for the two statistics for both curve families.The excellent agreement in the approximate and true mean inter-cluster distance statistics suggests that the random walk may furnish an effective model to develop approximations to clustering and locality statistics for space-filling curves.Based upon the analytical results, the asymptotic comparisons indicate that z-order curve family performs better than Hilbert curve family with respect to both statistics.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3338 https://doi.org/10.46298/dmtcs.3338 Dai, Ho-Kwok Su, Hung-Chi Dai, Ho-Kwok Su, Hung-Chi <![CDATA[A discrete space-filling curve provides a linear traversal/indexing of a multi-dimensional grid space.This paper presents an application of random walk to the study of inter-clustering of space-filling curves and an analytical study on the inter-clustering performances of 2-dimensional Hilbert and z-order curve families.Two underlying measures are employed: the mean inter-cluster distance over all inter-cluster gaps and the mean total inter-cluster distance over all subgrids.We show how approximating the mean inter-cluster distance statistics of continuous multi-dimensional space-filling curves fits into the formalism of random walk, and derive the exact formulas for the two statistics for both curve families.The excellent agreement in the approximate and true mean inter-cluster distance statistics suggests that the random walk may furnish an effective model to develop approximations to clustering and locality statistics for space-filling curves.Based upon the analytical results, the asymptotic comparisons indicate that z-order curve family performs better than Hilbert curve family with respect to both statistics.]]> 0 <![CDATA[Consider a simple symmetric random walk on the line. The parts of the random walk between consecutive returns to the origin are called excursions. The heights and lengths of these excursions can be arranged in decreasing order. In this paper we give the exact and limiting distributions of these ranked quantities. These results are analogues of the corresponding results of Pitman and Yor [1997, 1998, 2001] for Brownian motion.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3337 https://doi.org/10.46298/dmtcs.3337 Csáki, Endre Hu, Yueyun Csáki, Endre Hu, Yueyun <![CDATA[Consider a simple symmetric random walk on the line. The parts of the random walk between consecutive returns to the origin are called excursions. The heights and lengths of these excursions can be arranged in decreasing order. In this paper we give the exact and limiting distributions of these ranked quantities. These results are analogues of the corresponding results of Pitman and Yor [1997, 1998, 2001] for Brownian motion.]]> 0 <![CDATA[We present the main results of a study for the existence of vacant and occupied unbounded connected components in a non-homogeneous Poisson blob process. The method used in the proofs is a multi-scale percolation comparison.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3336 https://doi.org/10.46298/dmtcs.3336 Machado, Fabio P. Machado, Fabio P. <![CDATA[We present the main results of a study for the existence of vacant and occupied unbounded connected components in a non-homogeneous Poisson blob process. The method used in the proofs is a multi-scale percolation comparison.]]> 0 <![CDATA[We give an algorithm which constructs recursively a sequence of simple random walks on $\mathbb{Z}$ converging almost surely to a Brownian motion. One obtains by the same method conditional versions of the simple random walk converging to the excursion, the bridge, the meander or the normalized pseudobridge.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3335 https://doi.org/10.46298/dmtcs.3335 Marchal, Philippe Marchal, Philippe <![CDATA[We give an algorithm which constructs recursively a sequence of simple random walks on $\mathbb{Z}$ converging almost surely to a Brownian motion. One obtains by the same method conditional versions of the simple random walk converging to the excursion, the bridge, the meander or the normalized pseudobridge.]]> 0 <![CDATA[This paper presents necessary and sufficient conditions for on- and off-diagonal transition probability estimates for random walks on weighted graphs. On the integer lattice and on may fractal type graphs both the volume of a ball and the mean exit time from a ball are independent of the center, uniform in space. Here the upper estimate is given without such restriction and two-sided estimate is given if the mean exit time is independent of the center but the volume is not.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3334 https://doi.org/10.46298/dmtcs.3334 Telcs, András Telcs, András <![CDATA[This paper presents necessary and sufficient conditions for on- and off-diagonal transition probability estimates for random walks on weighted graphs. On the integer lattice and on may fractal type graphs both the volume of a ball and the mean exit time from a ball are independent of the center, uniform in space. Here the upper estimate is given without such restriction and two-sided estimate is given if the mean exit time is independent of the center but the volume is not.]]> 0 <![CDATA[Understanding reflection is one of the key competences in graphic arts industry. A very popular approach was given by Kubelka andMunk [1931] who derived a simple relationship between the scattering and absorption coefficients and the overall reflectance. This paper presents an alternative approach which describes the behavior of light in matter as a special kind of random walk.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3333 https://doi.org/10.46298/dmtcs.3333 Simon, Klaus Trachsler, Beat Simon, Klaus Trachsler, Beat <![CDATA[Understanding reflection is one of the key competences in graphic arts industry. A very popular approach was given by Kubelka andMunk [1931] who derived a simple relationship between the scattering and absorption coefficients and the overall reflectance. This paper presents an alternative approach which describes the behavior of light in matter as a special kind of random walk.]]> 0 <![CDATA[The distribution function of the integral of the absolute value of the Brownian motion was expressed by L.Takács in the form of various series. In the present paper we determine the exact tail asymptotics of this distribution function. The proposed method is applicable to a variety of other Wiener functionals as well.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3332 https://doi.org/10.46298/dmtcs.3332 Tolmatz, Leonid Tolmatz, Leonid <![CDATA[The distribution function of the integral of the absolute value of the Brownian motion was expressed by L.Takács in the form of various series. In the present paper we determine the exact tail asymptotics of this distribution function. The proposed method is applicable to a variety of other Wiener functionals as well.]]> 0 <![CDATA[A continuous time branching random walk on the lattice $\mathbb{Z}$ is considered in which individuals may produce children at the origin only. Assuming that the underlying random walk is symmetric and the offspring reproduction law is critical we prove a conditional limit theorem for the number of individuals at the origin.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3331 https://doi.org/10.46298/dmtcs.3331 Topchii, Valentin Vatutin, Vladimir Topchii, Valentin Vatutin, Vladimir <![CDATA[A continuous time branching random walk on the lattice $\mathbb{Z}$ is considered in which individuals may produce children at the origin only. Assuming that the underlying random walk is symmetric and the offspring reproduction law is critical we prove a conditional limit theorem for the number of individuals at the origin.]]> 0 <![CDATA[Given $\epsilon _i ∈ [0,1)$ for each $1 < i < n$, a particle performs the following random walk on $\{1,2,...,n\:\}$par If the particle is at $n$, it chooses a point uniformly at random (u.a.r.) from $\{1,...,n-1\}$. If the current position of the particle is $m (1 < m < n)$, with probability $\epsilon _m$ it decides to go back, in which case it chooses a point u.a.r. from $\{m+1,...,n\}$. With probability $1-\epsilon _m$ it decides to go forward, in which case it chooses a point u.a.r. from $\{1,...,m-1\}$. The particle moves to the selected point. What is the expected time taken by the particle to reach 1 if it starts the walk at $n$? Apart from being a natural variant of the classical one dimensional random walk, variants and special cases of this problemarise in Theoretical Computer Science [Linial, Fagin, Karp, Vishnoi]. In this paper we study this problem and observe interesting properties of this walk. First we show that the expected number of times the particle visits $i$ (before getting absorbed at 1) is the same when the walk is started at $j$, for all $j > i$. Then we show that for the following parameterized family of $\epsilon 's: \epsilon _i = \frac{n-i}{n-i+ α · (i-1)}$,$1 < i < n$ where $α$ does not depend on $i$, the expected number of times the particle visits $i$ is the same when the walk is started at $j$, for all $j < i$. Using these observations we obtain the expected absorption time for this family of $\epsilon 's$. As $α$ varies from infinity to 1, this time goes from $Θ (log n) to Θ (n)$. Finally we studythe behavior of the expected convergence timeas a function of $\epsilon$ . It remains an open question to determine whether this quantity increases when all $\epsilon 's$ are increased. We give some preliminary results to this effect.]]> Tue, 31 Dec 2002 23:00:00 +0000 https://doi.org/10.46298/dmtcs.3330 https://doi.org/10.46298/dmtcs.3330 Vishnoi, Nisheeth Vishnoi, Nisheeth <![CDATA[Given $\epsilon _i ∈ [0,1)$ for each $1 < i < n$, a particle performs the following random walk on $\{1,2,...,n\:\}$par If the particle is at $n$, it chooses a point uniformly at random (u.a.r.) from $\{1,...,n-1\}$. If the current position of the particle is $m (1 < m < n)$, with probability $\epsilon _m$ it decides to go back, in which case it chooses a point u.a.r. from $\{m+1,...,n\}$. With probability $1-\epsilon _m$ it decides to go forward, in which case it chooses a point u.a.r. from $\{1,...,m-1\}$. The particle moves to the selected point. What is the expected time taken by the particle to reach 1 if it starts the walk at $n$? Apart from being a natural variant of the classical one dimensional random walk, variants and special cases of this problemarise in Theoretical Computer Science [Linial, Fagin, Karp, Vishnoi]. In this paper we study this problem and observe interesting properties of this walk. First we show that the expected number of times the particle visits $i$ (before getting absorbed at 1) is the same when the walk is started at $j$, for all $j > i$. Then we show that for the following parameterized family of $\epsilon 's: \epsilon _i = \frac{n-i}{n-i+ α · (i-1)}$,$1 < i < n$ where $α$ does not depend on $i$, the expected number of times the particle visits $i$ is the same when the walk is started at $j$, for all $j < i$. Using these observations we obtain the expected absorption time for this family of $\epsilon 's$. As $α$ varies from infinity to 1, this time goes from $Θ (log n) to Θ (n)$. Finally we studythe behavior of the expected convergence timeas a function of $\epsilon$ . It remains an open question to determine whether this quantity increases when all $\epsilon 's$ are increased. We give some preliminary results to this effect.]]> 0 <![CDATA[Tree languages are powerful tools for the representation and schematization of infinite sets of terms for various purposes (unification theory, verification and specification ...). In order to extend the regular tree language framework, more complex formalisms have been developed. In this paper, we focus on Tree Synchronized Grammars and Primal Grammars which introduce specific control structures to represent non regular sets of terms. We propose a common unified framework in order to achieve the membership test for these particular languages. Thanks to a proof system, we provide a full operational framework, that allows us to transform tree grammars into Prolog programs (as it already exists for word grammars with DCG) whose goal is to recognize terms of the corresponding language.]]> Tue, 01 Jan 2002 07:00:00 +0000 https://doi.org/10.46298/dmtcs.300 https://doi.org/10.46298/dmtcs.300 Saubion, Frédéric Stéphan, Igor Saubion, Frédéric Stéphan, Igor <![CDATA[Tree languages are powerful tools for the representation and schematization of infinite sets of terms for various purposes (unification theory, verification and specification ...). In order to extend the regular tree language framework, more complex formalisms have been developed. In this paper, we focus on Tree Synchronized Grammars and Primal Grammars which introduce specific control structures to represent non regular sets of terms. We propose a common unified framework in order to achieve the membership test for these particular languages. Thanks to a proof system, we provide a full operational framework, that allows us to transform tree grammars into Prolog programs (as it already exists for word grammars with DCG) whose goal is to recognize terms of the corresponding language.]]> 0 <![CDATA[The (k,n)-perfect shuffle, a generalisation of the 2-way perfect shuffle, cuts a deck of kn cards into k equal size decks and interleaves them perfectly with the first card of the last deck at the top, the first card of the second-to-last deck as the second card, and so on. It is formally defined to be the permutation ρ _k,n: i → ki \bmod (kn+1), for 1 ≤ i ≤ kn. We uncover the cycle structure of the (k,n)-perfect shuffle permutation by a group-theoretic analysis and show how to compute representative elements from its cycles by an algorithm using O(kn) time and O((\log kn)^2) space. Consequently it is possible to realise the (k,n)-perfect shuffle via an in-place, linear-time algorithm. Algorithms that accomplish this for the 2-way shuffle have already been demonstrated.]]> Tue, 01 Jan 2002 07:00:00 +0000 https://doi.org/10.46298/dmtcs.308 https://doi.org/10.46298/dmtcs.308 Ellis, John Fan, Hongbing Shallit, Jeffrey Ellis, John Fan, Hongbing Shallit, Jeffrey <![CDATA[The (k,n)-perfect shuffle, a generalisation of the 2-way perfect shuffle, cuts a deck of kn cards into k equal size decks and interleaves them perfectly with the first card of the last deck at the top, the first card of the second-to-last deck as the second card, and so on. It is formally defined to be the permutation ρ _k,n: i → ki \bmod (kn+1), for 1 ≤ i ≤ kn. We uncover the cycle structure of the (k,n)-perfect shuffle permutation by a group-theoretic analysis and show how to compute representative elements from its cycles by an algorithm using O(kn) time and O((\log kn)^2) space. Consequently it is possible to realise the (k,n)-perfect shuffle via an in-place, linear-time algorithm. Algorithms that accomplish this for the 2-way shuffle have already been demonstrated.]]> 0 <![CDATA[A certain unimodal conjecture in matroid theory states the number of rank-r matroids on a set of size n is unimodal in r and attains its maximum at r=\lfloor n/2 \rfloor . We show that this conjecture holds up to r=3 by constructing a map from a class of rank-2 matroids into the class of loopless rank-3 matroids. Similar inequalities are proven for the number of non-isomorphic loopless matroids, loopless matroids and matroids.]]> Tue, 01 Jan 2002 07:00:00 +0000 https://doi.org/10.46298/dmtcs.307 https://doi.org/10.46298/dmtcs.307 Dukes, W. M. B. Dukes, W. M. B. <![CDATA[A certain unimodal conjecture in matroid theory states the number of rank-r matroids on a set of size n is unimodal in r and attains its maximum at r=\lfloor n/2 \rfloor . We show that this conjecture holds up to r=3 by constructing a map from a class of rank-2 matroids into the class of loopless rank-3 matroids. Similar inequalities are proven for the number of non-isomorphic loopless matroids, loopless matroids and matroids.]]> 0 <![CDATA[Can a complete graph on an even number n (>4) of vertices be properly edge-colored with n-1 colors in such a way that the edges can be partitioned into edge disjoint colorful isomorphic spanning trees? A spanning treee is colorful if all n-1 colors occur among its edges. It is proved that this is possible to accomplish whenever n is a power of two, or five times a power of two.]]> Tue, 01 Jan 2002 07:00:00 +0000 https://doi.org/10.46298/dmtcs.306 https://doi.org/10.46298/dmtcs.306 Constantine, Gregory Constantine, Gregory <![CDATA[Can a complete graph on an even number n (>4) of vertices be properly edge-colored with n-1 colors in such a way that the edges can be partitioned into edge disjoint colorful isomorphic spanning trees? A spanning treee is colorful if all n-1 colors occur among its edges. It is proved that this is possible to accomplish whenever n is a power of two, or five times a power of two.]]> 0 <![CDATA[In this paper we study the synthesis of space-time optimal systolic arrays for the Cholesky Factorization (CF). First, we discuss previous allocation methods and their application to CF. Second, stemming from a new allocation method we derive a space-time optimal array, with nearest neighbor connections, that requires 3N + Θ (1) time steps and N^2/8 + Θ (N) processors, where N is the size of the problem. The number of processors required by this new design improves the best previously known bound, N^2/6 + Θ (N), induced by previous allocation methods. This is the first contribution of the paper. The second contribution stemms from the fact that the paper also introduces a new allocation method that suggests to first perform clever index transformations on the initial dependence graph of a given system of uniform recurrent equations before applying the weakest allocation method, the projection method.]]> Tue, 01 Jan 2002 07:00:00 +0000 https://doi.org/10.46298/dmtcs.305 https://doi.org/10.46298/dmtcs.305 Tayou Djamegni, Clémentin Tayou Djamegni, Clémentin <![CDATA[In this paper we study the synthesis of space-time optimal systolic arrays for the Cholesky Factorization (CF). First, we discuss previous allocation methods and their application to CF. Second, stemming from a new allocation method we derive a space-time optimal array, with nearest neighbor connections, that requires 3N + Θ (1) time steps and N^2/8 + Θ (N) processors, where N is the size of the problem. The number of processors required by this new design improves the best previously known bound, N^2/6 + Θ (N), induced by previous allocation methods. This is the first contribution of the paper. The second contribution stemms from the fact that the paper also introduces a new allocation method that suggests to first perform clever index transformations on the initial dependence graph of a given system of uniform recurrent equations before applying the weakest allocation method, the projection method.]]> 0 <![CDATA[q-binary search trees are obtained from words, equipped with a geometric distribution instead of permutations. The average and variance of the heighth computated, based on random words of length n, as well as a Gaussian limit law.]]> Tue, 01 Jan 2002 07:00:00 +0000 https://doi.org/10.46298/dmtcs.304 https://doi.org/10.46298/dmtcs.304 Drmota, Michael Prodinger, Helmut Drmota, Michael Prodinger, Helmut <![CDATA[q-binary search trees are obtained from words, equipped with a geometric distribution instead of permutations. The average and variance of the heighth computated, based on random words of length n, as well as a Gaussian limit law.]]> 0 <![CDATA[A complementation operation on a vertex of a digraph changes all outgoing arcs into non-arcs, and outgoing non-arcs into arcs. This defines an equivalence relation where two digraphs are equivalent if one can be obtained from the other by a sequence of such operations. We show that given an adjacency-list representation of a digraph G, many fundamental graph algorithms can be carried out on any member G' of G's equivalence class in O(n+m) time, where m is the number of arcs in G, not the number of arcs in G' . This may have advantages when G' is much larger than G. We use this to generalize to digraphs a simple O(n + m log n) algorithm of McConnell and Spinrad for finding the modular decomposition of undirected graphs. A key step is finding the strongly-connected components of a digraph F in G's equivalence class, where F may have ~(m log n) arcs.]]> Tue, 01 Jan 2002 07:00:00 +0000 https://doi.org/10.46298/dmtcs.303 https://doi.org/10.46298/dmtcs.303 Dahlhaus, Elias Gustedt, Jens Mcconnell, Ross M. Dahlhaus, Elias Gustedt, Jens Mcconnell, Ross M. <![CDATA[A complementation operation on a vertex of a digraph changes all outgoing arcs into non-arcs, and outgoing non-arcs into arcs. This defines an equivalence relation where two digraphs are equivalent if one can be obtained from the other by a sequence of such operations. We show that given an adjacency-list representation of a digraph G, many fundamental graph algorithms can be carried out on any member G' of G's equivalence class in O(n+m) time, where m is the number of arcs in G, not the number of arcs in G' . This may have advantages when G' is much larger than G. We use this to generalize to digraphs a simple O(n + m log n) algorithm of McConnell and Spinrad for finding the modular decomposition of undirected graphs. A key step is finding the strongly-connected components of a digraph F in G's equivalence class, where F may have ~(m log n) arcs.]]> 0 <![CDATA[Using generating functions and limit theorems, we obtain a stochastic description of Carlitz compositions of large integer n (i.e. compositions two successive parts of which are different). We analyze: the number M of parts, the number of compositions T(m,n) with m parts, the distribution of the last part size, the correlation between two successive parts, leading to a Markov chain. We describe also the associated processes and the limiting trajectories, the width and thickness of a composition. We finally present a typical simulation. The limiting processes are characterized by Brownian Motion and some discrete distributions.]]> Tue, 01 Jan 2002 07:00:00 +0000 https://doi.org/10.46298/dmtcs.302 https://doi.org/10.46298/dmtcs.302 Louchard, Guy Prodinger, Helmut Louchard, Guy Prodinger, Helmut <![CDATA[Using generating functions and limit theorems, we obtain a stochastic description of Carlitz compositions of large integer n (i.e. compositions two successive parts of which are different). We analyze: the number M of parts, the number of compositions T(m,n) with m parts, the distribution of the last part size, the correlation between two successive parts, leading to a Markov chain. We describe also the associated processes and the limiting trajectories, the width and thickness of a composition. We finally present a typical simulation. The limiting processes are characterized by Brownian Motion and some discrete distributions.]]> 0 <![CDATA[In the present paper we consider a generalized class of extended binary trees in which leaves are distinguished in order to represent the location of a key within a trie of the same structure. We prove an exact asymptotic equivalent to the average stack-size of trees with α internal nodes and β leaves corresponding to keys; we assume that all trees with the same parameters α and β have the same probability. The assumption of that uniform model is motivated for example by the usage of tries for the compression of blockcodes. Furthermore, we will prove asymptotics for the r-th moments of the stack-size and we will show that a normalized stack-size possesses a theta distribution in the limit.]]> Tue, 01 Jan 2002 07:00:00 +0000 https://doi.org/10.46298/dmtcs.301 https://doi.org/10.46298/dmtcs.301 Nebel, Markus E. Nebel, Markus E. <![CDATA[In the present paper we consider a generalized class of extended binary trees in which leaves are distinguished in order to represent the location of a key within a trie of the same structure. We prove an exact asymptotic equivalent to the average stack-size of trees with α internal nodes and β leaves corresponding to keys; we assume that all trees with the same parameters α and β have the same probability. The assumption of that uniform model is motivated for example by the usage of tries for the compression of blockcodes. Furthermore, we will prove asymptotics for the r-th moments of the stack-size and we will show that a normalized stack-size possesses a theta distribution in the limit.]]> 0 <![CDATA[We present a full analysis of the expected number of 'rigid' 3-colourings of a sparse random graph. This shows that, if the average degree is at least 4.99, then as n → ∞ the expected number of such colourings tends to 0 and so the probability that the graph is 3-colourable tends to 0. (This result is tight, in that with average degree 4.989 the expected number tends to ∞.) This bound appears independently in Kaporis \textitet al. [Kap]. We then give a minor improvement, showing that the probability that the graph is 3-colourable tends to 0 if the average degree is at least 4.989.]]> Tue, 01 Jan 2002 07:00:00 +0000 https://doi.org/10.46298/dmtcs.299 https://doi.org/10.46298/dmtcs.299 Fountoulakis, Nikolaos McDiarmid, Colin Fountoulakis, Nikolaos McDiarmid, Colin <![CDATA[We present a full analysis of the expected number of 'rigid' 3-colourings of a sparse random graph. This shows that, if the average degree is at least 4.99, then as n → ∞ the expected number of such colourings tends to 0 and so the probability that the graph is 3-colourable tends to 0. (This result is tight, in that with average degree 4.989 the expected number tends to ∞.) This bound appears independently in Kaporis \textitet al. [Kap]. We then give a minor improvement, showing that the probability that the graph is 3-colourable tends to 0 if the average degree is at least 4.989.]]> 0 <![CDATA[A factorizing permutation of a given graph is simply a permutation of the vertices in which all decomposition sets appear to be factors. Such a concept seems to play a central role in recent papers dealing with graph decomposition. It is applied here for modular decomposition and we propose a linear algorithm that computes the whole decomposition tree when a factorizing permutation is provided. This algorithm can be seen as a common generalization of Ma and Hsu for modular decomposition of chordal graphs and Habib, Huchard and Spinrad for inheritance graphs decomposition. It also suggests many new decomposition algorithms for various notions of graph decompositions.]]> Tue, 01 Jan 2002 07:00:00 +0000 https://doi.org/10.46298/dmtcs.298 https://doi.org/10.46298/dmtcs.298 Capelle, Christian Habib, Michel Montgolfier, Fabien, Capelle, Christian Habib, Michel Montgolfier, Fabien, <![CDATA[A factorizing permutation of a given graph is simply a permutation of the vertices in which all decomposition sets appear to be factors. Such a concept seems to play a central role in recent papers dealing with graph decomposition. It is applied here for modular decomposition and we propose a linear algorithm that computes the whole decomposition tree when a factorizing permutation is provided. This algorithm can be seen as a common generalization of Ma and Hsu for modular decomposition of chordal graphs and Habib, Huchard and Spinrad for inheritance graphs decomposition. It also suggests many new decomposition algorithms for various notions of graph decompositions.]]> 0 <![CDATA[We use Conway's \emphFractran language to derive a function R:\textbfZ^+ → \textbfZ^+ of the form R(n) = r_in if n ≡ i \bmod d where d is a positive integer, 0 ≤ i < d and r_0,r_1, ... r_d-1 are rational numbers, such that the famous 3x+1 conjecture holds if and only if the R-orbit of 2^n contains 2 for all positive integers n. We then show that the R-orbit of an arbitrary positive integer is a constant multiple of an orbit that contains a power of 2. Finally we apply our main result to show that any cycle \ x_0, ... ,x_m-1 \ of positive integers for the 3x+1 function must satisfy \par ∑ _i∈ \textbfE \lfloor x_i/2 \rfloor = ∑ _i∈ \textbfO \lfloor x_i/2 \rfloor +k. \par where \textbfO=\ i : x_i is odd \ , \textbfE=\ i : x_i is even \ , and k=|\textbfO|. \par The method used illustrates a general mechanism for deriving mathematical results about the iterative dynamics of arbitrary integer functions from \emphFractran algorithms.]]> Tue, 01 Jan 2002 07:00:00 +0000 https://doi.org/10.46298/dmtcs.297 https://doi.org/10.46298/dmtcs.297 Monks, Kenneth G. Monks, Kenneth G. <![CDATA[We use Conway's \emphFractran language to derive a function R:\textbfZ^+ → \textbfZ^+ of the form R(n) = r_in if n ≡ i \bmod d where d is a positive integer, 0 ≤ i < d and r_0,r_1, ... r_d-1 are rational numbers, such that the famous 3x+1 conjecture holds if and only if the R-orbit of 2^n contains 2 for all positive integers n. We then show that the R-orbit of an arbitrary positive integer is a constant multiple of an orbit that contains a power of 2. Finally we apply our main result to show that any cycle \ x_0, ... ,x_m-1 \ of positive integers for the 3x+1 function must satisfy \par ∑ _i∈ \textbfE \lfloor x_i/2 \rfloor = ∑ _i∈ \textbfO \lfloor x_i/2 \rfloor +k. \par where \textbfO=\ i : x_i is odd \ , \textbfE=\ i : x_i is even \ , and k=|\textbfO|. \par The method used illustrates a general mechanism for deriving mathematical results about the iterative dynamics of arbitrary integer functions from \emphFractran algorithms.]]> 0 <![CDATA[The i-th symbol of the well-known infinite word of Thue on the alphabet \ 0,1\ can be characterized as the parity of the number of occurrences of the digit 1 in the binary notation of i. Generalized words of Thue are based on counting the parity of occurrences of an arbitrary word w∈\ 0,1\^+-0^* in the binary notation of i. We provide here the standard Lyndon factorization of some subclasses of this class of infinite words.]]> Tue, 01 Jan 2002 07:00:00 +0000 https://doi.org/10.46298/dmtcs.296 https://doi.org/10.46298/dmtcs.296 Černý, Anton Černý, Anton <![CDATA[The i-th symbol of the well-known infinite word of Thue on the alphabet \ 0,1\ can be characterized as the parity of the number of occurrences of the digit 1 in the binary notation of i. Generalized words of Thue are based on counting the parity of occurrences of an arbitrary word w∈\ 0,1\^+-0^* in the binary notation of i. We provide here the standard Lyndon factorization of some subclasses of this class of infinite words.]]> 0 <![CDATA[We present symmetric function theoretical methods for solving a performance analysis problem related with modulation protocols with diversity.]]> Tue, 01 Jan 2002 07:00:00 +0000 https://doi.org/10.46298/dmtcs.295 https://doi.org/10.46298/dmtcs.295 Dornstetter, Jean Louis Krob, Daniel Thibon, Jean-Yves Vassilieva, Ekaterina, Dornstetter, Jean Louis Krob, Daniel Thibon, Jean-Yves Vassilieva, Ekaterina, <![CDATA[We present symmetric function theoretical methods for solving a performance analysis problem related with modulation protocols with diversity.]]> 0 <![CDATA[The P_4-structure of a graph G is a hypergraph \textbfH on the same vertex set such that four vertices form a hyperedge in \textbfH whenever they induce a P_4 in G. We present a constructive algorithm which tests in polynomial time whether a given 4-uniform hypergraph is the P_4-structure of a claw-free graph and of (banner,chair,dart)-free graphs. The algorithm relies on new structural results for (banner,chair,dart)-free graphs which are based on the concept of p-connectedness. As a byproduct, we obtain a polynomial time criterion for perfectness for a large class of graphs properly containing claw-free graphs.]]> Tue, 01 Jan 2002 07:00:00 +0000 https://doi.org/10.46298/dmtcs.294 https://doi.org/10.46298/dmtcs.294 Babel, Luitpold Brandstädt, Andreas Le, Van Bang Babel, Luitpold Brandstädt, Andreas Le, Van Bang <![CDATA[The P_4-structure of a graph G is a hypergraph \textbfH on the same vertex set such that four vertices form a hyperedge in \textbfH whenever they induce a P_4 in G. We present a constructive algorithm which tests in polynomial time whether a given 4-uniform hypergraph is the P_4-structure of a claw-free graph and of (banner,chair,dart)-free graphs. The algorithm relies on new structural results for (banner,chair,dart)-free graphs which are based on the concept of p-connectedness. As a byproduct, we obtain a polynomial time criterion for perfectness for a large class of graphs properly containing claw-free graphs.]]> 0 <![CDATA[Fix positive integers k and l. Consider a random k-partite graph on n vertices obtained by partitioning the vertex set into V_i, (i=1, \ldots,k) each having size Ω (n) and choosing each possible edge with probability p. Consider any vertex x in any V_i and any vertex y. We show that the expected number of simple paths of even length l between x and y differ significantly depending on whether y belongs to the same V_i (as x does) or not. A similar phenomenon occurs when l is odd. This result holds even when k,l vary slowly with n. This fact has implications to coloring random graphs. The proof is based on establishing bijections between sets of paths.]]> Mon, 01 Jan 2001 07:00:00 +0000 https://doi.org/10.46298/dmtcs.286 https://doi.org/10.46298/dmtcs.286 Subramanian, C.R. Subramanian, C.R. <![CDATA[Fix positive integers k and l. Consider a random k-partite graph on n vertices obtained by partitioning the vertex set into V_i, (i=1, \ldots,k) each having size Ω (n) and choosing each possible edge with probability p. Consider any vertex x in any V_i and any vertex y. We show that the expected number of simple paths of even length l between x and y differ significantly depending on whether y belongs to the same V_i (as x does) or not. A similar phenomenon occurs when l is odd. This result holds even when k,l vary slowly with n. This fact has implications to coloring random graphs. The proof is based on establishing bijections between sets of paths.]]> 0 <![CDATA[A D0L word on an alphabet Σ =\0,1,\ldots,q-1\ is called symmetric if it is a fixed point w=\varphi(w) of a morphism \varphi:Σ ^* → Σ ^* defined by \varphi(i)=øverlinet_1 + i øverlinet_2 + i\ldots øverlinet_m + i for some word t_1t_2\ldots t_m (equal to \varphi(0)) and every i ∈ Σ ; here øverlinea means a \bmod q. We prove a result conjectured by J. Shallit: if all the symbols in \varphi(0) are distinct (i.e., if t_i ≠q t_j for i ≠q j), then the symmetric D0L word w is overlap-free, i.e., contains no factor of the form axaxa for any x ∈ Σ ^* and a ∈ Σ .]]> Mon, 01 Jan 2001 07:00:00 +0000 https://doi.org/10.46298/dmtcs.293 https://doi.org/10.46298/dmtcs.293 Frid, Anna Frid, Anna <![CDATA[A D0L word on an alphabet Σ =\0,1,\ldots,q-1\ is called symmetric if it is a fixed point w=\varphi(w) of a morphism \varphi:Σ ^* → Σ ^* defined by \varphi(i)=øverlinet_1 + i øverlinet_2 + i\ldots øverlinet_m + i for some word t_1t_2\ldots t_m (equal to \varphi(0)) and every i ∈ Σ ; here øverlinea means a \bmod q. We prove a result conjectured by J. Shallit: if all the symbols in \varphi(0) are distinct (i.e., if t_i ≠q t_j for i ≠q j), then the symmetric D0L word w is overlap-free, i.e., contains no factor of the form axaxa for any x ∈ Σ ^* and a ∈ Σ .]]> 0 <![CDATA[We show that it is impossible to compute (or even to approximate) the topological entropy of a continuous piecewise affine function in dimension four. The same result holds for saturated linear functions in unbounded dimension. We ask whether the topological entropy of a piecewise affine function is always a computable real number, and conversely whether every non-negative computable real number can be obtained as the topological entropy of a piecewise affine function. It seems that these two questions are also open for cellular automata.]]> Mon, 01 Jan 2001 07:00:00 +0000 https://doi.org/10.46298/dmtcs.292 https://doi.org/10.46298/dmtcs.292 Koiran, Pascal Koiran, Pascal <![CDATA[We show that it is impossible to compute (or even to approximate) the topological entropy of a continuous piecewise affine function in dimension four. The same result holds for saturated linear functions in unbounded dimension. We ask whether the topological entropy of a piecewise affine function is always a computable real number, and conversely whether every non-negative computable real number can be obtained as the topological entropy of a piecewise affine function. It seems that these two questions are also open for cellular automata.]]> 0 <![CDATA[Efficient algorithms for temporal reasoning are essential in knowledge-based systems. This is central in many areas of Artificial Intelligence including scheduling, planning, plan recognition, and natural language understanding. As such, scalability is a crucial consideration in temporal reasoning. While reasoning in the interval algebra is NP-complete, reasoning in the less expressive point algebra is tractable. In this paper, we explore an extension to the work of Gerevini and Schubert which is based on the point algebra. In their seminal framework, temporal relations are expressed as a directed acyclic graph partitioned into chains and supported by a \emphmetagraph data structure, where time points or events are represented by vertices, and directed edges are labelled with < or ≤ . They are interested in fast algorithms for determining the strongest relation between two events. They begin by developing fast algorithms for the case where all points lie on a chain. In this paper, we are interested in a generalization of this, namely we consider the problem of finding the maximum ''distance'' between two vertices in a \emphchain; this problem arises in real world applications such as in process control and crew scheduling. We describe an O(n) time preprocessing algorithm for the maximum distance problem on chains. It allows queries for the maximum number of < edges between two vertices to be answered in O(1) time. This matches the performance of the algorithm of Gerevini and Schubert for determining the strongest relation holding between two vertices in a chain.]]> Mon, 01 Jan 2001 07:00:00 +0000 https://doi.org/10.46298/dmtcs.291 https://doi.org/10.46298/dmtcs.291 Grün, Gabrielle Assunta Grün, Gabrielle Assunta <![CDATA[Efficient algorithms for temporal reasoning are essential in knowledge-based systems. This is central in many areas of Artificial Intelligence including scheduling, planning, plan recognition, and natural language understanding. As such, scalability is a crucial consideration in temporal reasoning. While reasoning in the interval algebra is NP-complete, reasoning in the less expressive point algebra is tractable. In this paper, we explore an extension to the work of Gerevini and Schubert which is based on the point algebra. In their seminal framework, temporal relations are expressed as a directed acyclic graph partitioned into chains and supported by a \emphmetagraph data structure, where time points or events are represented by vertices, and directed edges are labelled with < or ≤ . They are interested in fast algorithms for determining the strongest relation between two events. They begin by developing fast algorithms for the case where all points lie on a chain. In this paper, we are interested in a generalization of this, namely we consider the problem of finding the maximum ''distance'' between two vertices in a \emphchain; this problem arises in real world applications such as in process control and crew scheduling. We describe an O(n) time preprocessing algorithm for the maximum distance problem on chains. It allows queries for the maximum number of < edges between two vertices to be answered in O(1) time. This matches the performance of the algorithm of Gerevini and Schubert for determining the strongest relation holding between two vertices in a chain.]]> 0 <![CDATA[Several recent papers have shown how to approximate the difference ∑ _i|a_i-b_i| or ∑ |a_i-b_i|^2 between two functions, when the function values a_i and b_i are given in a data stream, and their order is chosen by an adversary. These algorithms use little space (much less than would be needed to store the entire stream) and little time to process each item in the stream. They approximate with small relative error. Using different techniques, we show how to approximate the L^p-difference ∑ _i|a_i-b_i|^p for any rational-valued p∈(0,2], with comparable efficiency and error. We also show how to approximate ∑ _i|a_i-b_i|^p for larger values of p but with a worse error guarantee. Our results fill in gaps left by recent work, by providing an algorithm that is precisely tunable for the application at hand. These results can be used to assess the difference between two chronologically or physically separated massive data sets, making one quick pass over each data set, without buffering the data or requiring the data source to pause. For example, one can use our techniques to judge whether the traffic on two remote network routers are similar without requiring either router to transmit a copy of its traffic. A web search engine could use such algorithms to construct a library of small ''sketches,'' one for each distinct page on the web; one can approximate the extent to which new web pages duplicate old ones by comparing the sketches of the web pages. Such techniques will become increasingly important as the enormous scale, distributional nature, and one-pass processing requirements of data sets become more commonplace.]]> Mon, 01 Jan 2001 07:00:00 +0000 https://doi.org/10.46298/dmtcs.290 https://doi.org/10.46298/dmtcs.290 Fong, Jessica H. Strauss, Martin Fong, Jessica H. Strauss, Martin <![CDATA[Several recent papers have shown how to approximate the difference ∑ _i|a_i-b_i| or ∑ |a_i-b_i|^2 between two functions, when the function values a_i and b_i are given in a data stream, and their order is chosen by an adversary. These algorithms use little space (much less than would be needed to store the entire stream) and little time to process each item in the stream. They approximate with small relative error. Using different techniques, we show how to approximate the L^p-difference ∑ _i|a_i-b_i|^p for any rational-valued p∈(0,2], with comparable efficiency and error. We also show how to approximate ∑ _i|a_i-b_i|^p for larger values of p but with a worse error guarantee. Our results fill in gaps left by recent work, by providing an algorithm that is precisely tunable for the application at hand. These results can be used to assess the difference between two chronologically or physically separated massive data sets, making one quick pass over each data set, without buffering the data or requiring the data source to pause. For example, one can use our techniques to judge whether the traffic on two remote network routers are similar without requiring either router to transmit a copy of its traffic. A web search engine could use such algorithms to construct a library of small ''sketches,'' one for each distinct page on the web; one can approximate the extent to which new web pages duplicate old ones by comparing the sketches of the web pages. Such techniques will become increasingly important as the enormous scale, distributional nature, and one-pass processing requirements of data sets become more commonplace.]]> 0 <![CDATA[Consider a (MOD_q,MOD_p) circuit, where the inputs of the bottom MOD_p gates are degree-d polynomials with integer coefficients of the input variables (p, q are different primes). Using our main tool ―- the Degree Decreasing Lemma ―- we show that this circuit can be converted to a (MOD_q,MOD_p) circuit with \emphlinear polynomials on the input-level with the price of increasing the size of the circuit. This result has numerous consequences: for the Constant Degree Hypothesis of Barrington, Straubing and Thérien, and generalizing the lower bound results of Yan and Parberry, Krause and Waack, and Krause and Pudlák. Perhaps the most important application is an exponential lower bound for the size of (MOD_q,MOD_p) circuits computing the n fan-in AND, where the input of each MOD_p gate at the bottom is an \empharbitrary integer valued function of cn variables (c<1) plus an arbitrary linear function of n input variables.]]> Mon, 01 Jan 2001 07:00:00 +0000 https://doi.org/10.46298/dmtcs.289 https://doi.org/10.46298/dmtcs.289 Grolmusz, Vince Grolmusz, Vince <![CDATA[Consider a (MOD_q,MOD_p) circuit, where the inputs of the bottom MOD_p gates are degree-d polynomials with integer coefficients of the input variables (p, q are different primes). Using our main tool ―- the Degree Decreasing Lemma ―- we show that this circuit can be converted to a (MOD_q,MOD_p) circuit with \emphlinear polynomials on the input-level with the price of increasing the size of the circuit. This result has numerous consequences: for the Constant Degree Hypothesis of Barrington, Straubing and Thérien, and generalizing the lower bound results of Yan and Parberry, Krause and Waack, and Krause and Pudlák. Perhaps the most important application is an exponential lower bound for the size of (MOD_q,MOD_p) circuits computing the n fan-in AND, where the input of each MOD_p gate at the bottom is an \empharbitrary integer valued function of cn variables (c<1) plus an arbitrary linear function of n input variables.]]> 0 <![CDATA[We have worked with the local access network design problem with two cable technologies. This is an optimization problem in graphs that consists of linking an origin node to a set of terminal nodes which have a flow demand. There are also a set of Steiner or transshipment nodes which do not have demand. Each arc of the graph has two associated costs: a variable cost depending on the flow through the arc and a fixed cost associated with the installation of the arc. Moreover, in each arc we can install one of two available technologies: optical fiber or copper (we can also use radio links with any other cable technology). Each one of these technologies has different variable and fixed costs. To be more precise, the fixed cost of the optical fiber is greater than that of the copper, but its variable cost is much smaller. The problem was modeled using a multicommodity flow formulation in which we added some structural constraints. This model was used to apply the Benders decomposition method. The structural constraints have the objective of trying to guarantee that the master problem of the Benders decomposition will yield a tree. The Benders subproblems are trivial network flow problems. The dual variables have commodity meaningfull values and are evaluated in a systematic form. The algorithm was implemented in C++ with CPLEX 3.0 callable library. We have tested the algorithm with some test instances obtained by a generator of problems that we developed.]]> Mon, 01 Jan 2001 07:00:00 +0000 https://doi.org/10.46298/dmtcs.288 https://doi.org/10.46298/dmtcs.288 Randazzo, C. D. Luna, H. P. L. Mahey, P. Randazzo, C. D. Luna, H. P. L. Mahey, P. <![CDATA[We have worked with the local access network design problem with two cable technologies. This is an optimization problem in graphs that consists of linking an origin node to a set of terminal nodes which have a flow demand. There are also a set of Steiner or transshipment nodes which do not have demand. Each arc of the graph has two associated costs: a variable cost depending on the flow through the arc and a fixed cost associated with the installation of the arc. Moreover, in each arc we can install one of two available technologies: optical fiber or copper (we can also use radio links with any other cable technology). Each one of these technologies has different variable and fixed costs. To be more precise, the fixed cost of the optical fiber is greater than that of the copper, but its variable cost is much smaller. The problem was modeled using a multicommodity flow formulation in which we added some structural constraints. This model was used to apply the Benders decomposition method. The structural constraints have the objective of trying to guarantee that the master problem of the Benders decomposition will yield a tree. The Benders subproblems are trivial network flow problems. The dual variables have commodity meaningfull values and are evaluated in a systematic form. The algorithm was implemented in C++ with CPLEX 3.0 callable library. We have tested the algorithm with some test instances obtained by a generator of problems that we developed.]]> 0 <![CDATA[We examine the power of nondeterministic finite automata with acceptance of an input word defined by a leaf language, i.e., a condition on the sequence of leaves in the automaton's computation tree. We study leaf languages either taken from one of the classes of the Chomsky hierarchy, or taken from a time- or space-bounded complexity class. We contrast the obtained results with those known for leaf languages for Turing machines and Boolean circuits.]]> Mon, 01 Jan 2001 07:00:00 +0000 https://doi.org/10.46298/dmtcs.287 https://doi.org/10.46298/dmtcs.287 Peichl, Timo Vollmer, Heribert Peichl, Timo Vollmer, Heribert <![CDATA[We examine the power of nondeterministic finite automata with acceptance of an input word defined by a leaf language, i.e., a condition on the sequence of leaves in the automaton's computation tree. We study leaf languages either taken from one of the classes of the Chomsky hierarchy, or taken from a time- or space-bounded complexity class. We contrast the obtained results with those known for leaf languages for Turing machines and Boolean circuits.]]> 0 <![CDATA[We investigate a scheduling problem in which packets, or datagrams, may be fragmented. While there are a few applications to scheduling with datagram fragmentation, our model of the problem is derived from a scheduling problem present in data over CATV networks. In the scheduling problem datagrams of variable lengths must be assigned (packed) into fixed length time slots. One of the capabilities of the system is the ability to break a datagram into several fragments. When a datagram is fragmented, extra bits are added to the original datagram to enable the reassembly of all the fragments. We convert the scheduling problem into the problem of bin packing with item fragmentation, which we define in the following way: we are asked to pack a list of items into a minimum number of unit capacity bins. Each item may be fragmented in which case overhead units are added to the size of every fragment. The cost associated with fragmentation renders the problem NP-hard, therefore an approximation algorithm is needed. We define a version of the well-known Next-Fit algorithm, capable of fragmenting items, and investigate its performance. We present both worst case and average case results and compare them to the case where fragmentation is not allowed.]]> Mon, 01 Jan 2001 07:00:00 +0000 https://doi.org/10.46298/dmtcs.277 https://doi.org/10.46298/dmtcs.277 Namman, Nir Rom, Raphaël Namman, Nir Rom, Raphaël <![CDATA[We investigate a scheduling problem in which packets, or datagrams, may be fragmented. While there are a few applications to scheduling with datagram fragmentation, our model of the problem is derived from a scheduling problem present in data over CATV networks. In the scheduling problem datagrams of variable lengths must be assigned (packed) into fixed length time slots. One of the capabilities of the system is the ability to break a datagram into several fragments. When a datagram is fragmented, extra bits are added to the original datagram to enable the reassembly of all the fragments. We convert the scheduling problem into the problem of bin packing with item fragmentation, which we define in the following way: we are asked to pack a list of items into a minimum number of unit capacity bins. Each item may be fragmented in which case overhead units are added to the size of every fragment. The cost associated with fragmentation renders the problem NP-hard, therefore an approximation algorithm is needed. We define a version of the well-known Next-Fit algorithm, capable of fragmenting items, and investigate its performance. We present both worst case and average case results and compare them to the case where fragmentation is not allowed.]]> 0 <![CDATA[A vertex partition of a graph into disjoint subsets V_is is said to be a P_4-free coloring if each color class V_i induces a subgraph without chordless path on four vertices (denoted by P_4). Examples of P_4-free 2-colorable graphs (also called P_4-bipartite graphs) include parity graphs and graphs with ''few'' P_4s like P_4-reducible and P_4-sparse graphs. We prove that, given k≥ 2, \emphP_4-Free k-Colorability is NP-complete even for comparability graphs, and for P_5-free graphs. We then discuss the recognition, perfection and the Strong Perfect Graph Conjecture (SPGC) for P_4-bipartite graphs with special P_4-structure. In particular, we show that the SPGC is true for P_4-bipartite graphs with one P_3-free color class meeting every P_4 at a midpoint.]]> Mon, 01 Jan 2001 07:00:00 +0000 https://doi.org/10.46298/dmtcs.272 https://doi.org/10.46298/dmtcs.272 Hoàng, Chinh T. Le, Van Bang Hoàng, Chinh T. Le, Van Bang <![CDATA[A vertex partition of a graph into disjoint subsets V_is is said to be a P_4-free coloring if each color class V_i induces a subgraph without chordless path on four vertices (denoted by P_4). Examples of P_4-free 2-colorable graphs (also called P_4-bipartite graphs) include parity graphs and graphs with ''few'' P_4s like P_4-reducible and P_4-sparse graphs. We prove that, given k≥ 2, \emphP_4-Free k-Colorability is NP-complete even for comparability graphs, and for P_5-free graphs. We then discuss the recognition, perfection and the Strong Perfect Graph Conjecture (SPGC) for P_4-bipartite graphs with special P_4-structure. In particular, we show that the SPGC is true for P_4-bipartite graphs with one P_3-free color class meeting every P_4 at a midpoint.]]> 0 <![CDATA[Eulerian numbers (and ''Alternate Eulerian numbers'') are often interpreted as distributions of statistics defined over the Symmetric group. The main purpose of this paper is to define a way to represent permutations that provides some other combinatorial interpretations of these numbers. This representation uses a one-to-one correspondence between permutations and the so-called \emphsubexceedant functions.]]> Mon, 01 Jan 2001 07:00:00 +0000 https://doi.org/10.46298/dmtcs.271 https://doi.org/10.46298/dmtcs.271 Mantaci, Roberto Rakotondrajao, Fanja Mantaci, Roberto Rakotondrajao, Fanja <![CDATA[Eulerian numbers (and ''Alternate Eulerian numbers'') are often interpreted as distributions of statistics defined over the Symmetric group. The main purpose of this paper is to define a way to represent permutations that provides some other combinatorial interpretations of these numbers. This representation uses a one-to-one correspondence between permutations and the so-called \emphsubexceedant functions.]]> 0 <![CDATA[A graph is a P4-indifference graph if it admits an ordering < on its vertices such that every chordless path with vertices a, b, c, d and edges ab, bc, cd has a]]> Mon, 01 Jan 2001 07:00:00 +0000 https://doi.org/10.46298/dmtcs.269 https://doi.org/10.46298/dmtcs.269 Habib, Michel Paul, Christophe Viennot, Laurent Habib, Michel Paul, Christophe Viennot, Laurent <![CDATA[A graph is a P4-indifference graph if it admits an ordering < on its vertices such that every chordless path with vertices a, b, c, d and edges ab, bc, cd has a]]> 0 <![CDATA[Many protocols need a discovery mechanism to enable a given node to locate one or several nodes involved in the same communication. However, there is no protocol ready to fulfill this service at the network-layer. Every protocol usually implements its own solution. In particular, multicast protocols often use a searching technique based on an algorithm called expanding ring search. This algorithm searches for nodes in all directions and thus uses much bandwidth. However a typical search can usually restrict its scan in a specific direction. To enable this broadcast restriction, we propose an oriented multicast routing algorithm. The algorithm's principle is to direct the multicast of packets towards a special node, involved in the communication, in order to search only in a limited area. The area must be as small as possible to reduce network flooding but still has to contain many nodes satisfying the search criteria. This new algorithm is the core part of a network-level node search framework also defined herein. A search protocol based on this framework could provide a network-level agent discovery service to current protocols. We have simulated an agent search with our algorithm on one side and with the expanding ring algorithm on the other side and we give comparative results.]]> Mon, 01 Jan 2001 07:00:00 +0000 https://doi.org/10.46298/dmtcs.268 https://doi.org/10.46298/dmtcs.268 Magoni, Damien Pansiot, Jean-Jacques Magoni, Damien Pansiot, Jean-Jacques <![CDATA[Many protocols need a discovery mechanism to enable a given node to locate one or several nodes involved in the same communication. However, there is no protocol ready to fulfill this service at the network-layer. Every protocol usually implements its own solution. In particular, multicast protocols often use a searching technique based on an algorithm called expanding ring search. This algorithm searches for nodes in all directions and thus uses much bandwidth. However a typical search can usually restrict its scan in a specific direction. To enable this broadcast restriction, we propose an oriented multicast routing algorithm. The algorithm's principle is to direct the multicast of packets towards a special node, involved in the communication, in order to search only in a limited area. The area must be as small as possible to reduce network flooding but still has to contain many nodes satisfying the search criteria. This new algorithm is the core part of a network-level node search framework also defined herein. A search protocol based on this framework could provide a network-level agent discovery service to current protocols. We have simulated an agent search with our algorithm on one side and with the expanding ring algorithm on the other side and we give comparative results.]]> 0 <![CDATA[The system FT< of ordering constraints over feature trees has been introduced as an extension of the system FT of equality constraints over feature trees. We investigate the first-order theory of FT< and its fragments in detail, both over finite trees and over possibly infinite trees. We prove that the first-order theory of FT< is undecidable, in contrast to the first-order theory of FT which is well-known to be decidable. We show that the entailment problem of FT< with existential quantification is PSPACE-complete. So far, this problem has been shown decidable, coNP-hard in case of finite trees, PSPACE-hard in case of arbitrary trees, and cubic time when restricted to quantifier-free entailment judgments. To show PSPACE-completeness, we show that the entailment problem of FT< with existential quantification is equivalent to the inclusion problem of non-deterministic finite automata. Available at http://www.ps.uni-saarland.de/Publications/documents/FTSubTheory_98.pdf]]> Mon, 01 Jan 2001 07:00:00 +0000 https://doi.org/10.46298/dmtcs.267 https://doi.org/10.46298/dmtcs.267 Müller, Martin Niehren, Joachim Treinen, Ralf Müller, Martin Niehren, Joachim Treinen, Ralf <![CDATA[The system FT< of ordering constraints over feature trees has been introduced as an extension of the system FT of equality constraints over feature trees. We investigate the first-order theory of FT< and its fragments in detail, both over finite trees and over possibly infinite trees. We prove that the first-order theory of FT< is undecidable, in contrast to the first-order theory of FT which is well-known to be decidable. We show that the entailment problem of FT< with existential quantification is PSPACE-complete. So far, this problem has been shown decidable, coNP-hard in case of finite trees, PSPACE-hard in case of arbitrary trees, and cubic time when restricted to quantifier-free entailment judgments. To show PSPACE-completeness, we show that the entailment problem of FT< with existential quantification is equivalent to the inclusion problem of non-deterministic finite automata. Available at http://www.ps.uni-saarland.de/Publications/documents/FTSubTheory_98.pdf]]> 0 <![CDATA[We consider the problem of constructing a multicast tree that connects a group of source nodes to a group of sink nodes (receivers) and minimizes the maximum end-to-end delay between any pair of source/sink nodes. This is known as the \emphminimum eccentricity multicast tree problem, and is directly related to the quality of service requirements of real multipoint applications. We deal directly with the problem in its general form, meaning that the sets of source and sink nodes need not be overlapping nor disjoint. The main contribution of this work is a polynomial algorithm for this problem on general networks which is inspired by an innovative method that uses geometric relationships on the xy-plane.]]> Mon, 01 Jan 2001 07:00:00 +0000 https://doi.org/10.46298/dmtcs.278 https://doi.org/10.46298/dmtcs.278 Krumme, David Fragopoulou, Paraskevi Krumme, David Fragopoulou, Paraskevi <![CDATA[We consider the problem of constructing a multicast tree that connects a group of source nodes to a group of sink nodes (receivers) and minimizes the maximum end-to-end delay between any pair of source/sink nodes. This is known as the \emphminimum eccentricity multicast tree problem, and is directly related to the quality of service requirements of real multipoint applications. We deal directly with the problem in its general form, meaning that the sets of source and sink nodes need not be overlapping nor disjoint. The main contribution of this work is a polynomial algorithm for this problem on general networks which is inspired by an innovative method that uses geometric relationships on the xy-plane.]]> 0 <![CDATA[Let X be a two-element set of words over a finite alphabet. If a bi-infinite word possesses two X-factorizations which are not shiftequivalent, then the primitive roots of the words in X are conjugates. Note, that this is a strict sharpening of a defect theorem for bi-infinite words stated in \emphKMP. Moreover, we prove that there is at most one bi-infinite word possessing two different X-factorizations and give a necessary and sufficient conditions on X for the existence of such a word. Finally, we prove that the family of sets X for which such a word exists is parameterizable.]]> Mon, 01 Jan 2001 07:00:00 +0000 https://doi.org/10.46298/dmtcs.279 https://doi.org/10.46298/dmtcs.279 Maňuch, Ján Maňuch, Ján <![CDATA[Let X be a two-element set of words over a finite alphabet. If a bi-infinite word possesses two X-factorizations which are not shiftequivalent, then the primitive roots of the words in X are conjugates. Note, that this is a strict sharpening of a defect theorem for bi-infinite words stated in \emphKMP. Moreover, we prove that there is at most one bi-infinite word possessing two different X-factorizations and give a necessary and sufficient conditions on X for the existence of such a word. Finally, we prove that the family of sets X for which such a word exists is parameterizable.]]> 0 <![CDATA[We exhibit an initial specification of the rational numbers equipped with addition, subtraction, multiplication, greatest integer function, and absolute value. Our specification uses only the sort of rational numbers. It uses one hidden function; that function is unary. But it does not use an error constant, or extra (hidden) sorts, or conditional equations. All of our work is elementary and self-contained.]]> Mon, 01 Jan 2001 07:00:00 +0000 https://doi.org/10.46298/dmtcs.280 https://doi.org/10.46298/dmtcs.280 Moss, Lawrence S. Moss, Lawrence S. <![CDATA[We exhibit an initial specification of the rational numbers equipped with addition, subtraction, multiplication, greatest integer function, and absolute value. Our specification uses only the sort of rational numbers. It uses one hidden function; that function is unary. But it does not use an error constant, or extra (hidden) sorts, or conditional equations. All of our work is elementary and self-contained.]]> 0 <![CDATA[We propose martingale central limit theorems as an tool to prove asymptotic normality of the costs of certain recursive algorithms which are subjected to random input data. The recursive algorithms that we have in mind are such that if input data of size N produce random costs L_N, then L_N=^D L_n+ L_N-n+R_N for N ≥ n_0≥ 2, where n follows a certain distribution P_N on the integers \0, \ldots ,N\ and L_k =^D L_k for k≥ 0. L_n, L_N-n and R_N are independent, conditional on n, and R_N are random variables, which may also depend on n, corresponding to the cost of splitting the input data of size N (into subsets of size n and N-n) and combining the results of the recursive calls to yield the overall result. We construct a martingale difference array with rows converging to Z_N:= [L_N - E L_N] / [√Var L_N]. Under certain compatibility assumptions on the sequence (P_N)_N≥ 0 we show that a pair of sufficient conditions (of Lyapunov type) for Z_N → ^DN(0,1) can be restated as a pair of conditions regarding asymptotic relations between three sequences. All these sequences satisfy the same type of linear equation, that is also the defining equation for the sequence (E L_N)_N≥ 0 and thus very likely a well studied object. In the case that the P_N are binomial distributions with the same parameter p, and for deterministic R_N, we demonstrate the power of this approach. We derive very general sufficient conditions in terms of the sequence (R_N)_N≥ 0 (and for the scale R_N=N^α a characterization of those α ) leading to asymptotic normality of Z_N.]]> Mon, 01 Jan 2001 07:00:00 +0000 https://doi.org/10.46298/dmtcs.281 https://doi.org/10.46298/dmtcs.281 Schachinger, Werner Schachinger, Werner <![CDATA[We propose martingale central limit theorems as an tool to prove asymptotic normality of the costs of certain recursive algorithms which are subjected to random input data. The recursive algorithms that we have in mind are such that if input data of size N produce random costs L_N, then L_N=^D L_n+ L_N-n+R_N for N ≥ n_0≥ 2, where n follows a certain distribution P_N on the integers \0, \ldots ,N\ and L_k =^D L_k for k≥ 0. L_n, L_N-n and R_N are independent, conditional on n, and R_N are random variables, which may also depend on n, corresponding to the cost of splitting the input data of size N (into subsets of size n and N-n) and combining the results of the recursive calls to yield the overall result. We construct a martingale difference array with rows converging to Z_N:= [L_N - E L_N] / [√Var L_N]. Under certain compatibility assumptions on the sequence (P_N)_N≥ 0 we show that a pair of sufficient conditions (of Lyapunov type) for Z_N → ^DN(0,1) can be restated as a pair of conditions regarding asymptotic relations between three sequences. All these sequences satisfy the same type of linear equation, that is also the defining equation for the sequence (E L_N)_N≥ 0 and thus very likely a well studied object. In the case that the P_N are binomial distributions with the same parameter p, and for deterministic R_N, we demonstrate the power of this approach. We derive very general sufficient conditions in terms of the sequence (R_N)_N≥ 0 (and for the scale R_N=N^α a characterization of those α ) leading to asymptotic normality of Z_N.]]> 0 <![CDATA[Any attempt to find connections between mathematical properties and complexity has a strong relevance to the field of Complexity Theory. This is due to the lack of mathematical techniques to prove lower bounds for general models of computation.\par This work represents a step in this direction: we define a combinatorial property that makes Boolean functions ''\emphhard'' to compute in constant depth and show how the harmonic analysis on the hypercube can be applied to derive new lower bounds on the size complexity of previously unclassified Boolean functions.]]> Mon, 01 Jan 2001 07:00:00 +0000 https://doi.org/10.46298/dmtcs.283 https://doi.org/10.46298/dmtcs.283 Bernasconi, Anna Bernasconi, Anna <![CDATA[Any attempt to find connections between mathematical properties and complexity has a strong relevance to the field of Complexity Theory. This is due to the lack of mathematical techniques to prove lower bounds for general models of computation.\par This work represents a step in this direction: we define a combinatorial property that makes Boolean functions ''\emphhard'' to compute in constant depth and show how the harmonic analysis on the hypercube can be applied to derive new lower bounds on the size complexity of previously unclassified Boolean functions.]]> 0 <![CDATA[We review an elementary approach to the construction of all irreducible representations of the finite Heisenberg group. Determining the number of inequivalent classes of irreducible representations by different methods leads to an identity of sums involving greatest common divisors. We show how this identity can be generalized and derive an explicit formula for the sums.]]> Mon, 01 Jan 2001 07:00:00 +0000 https://doi.org/10.46298/dmtcs.284 https://doi.org/10.46298/dmtcs.284 Grassberger, Johannes Hörmann, Günther Grassberger, Johannes Hörmann, Günther <![CDATA[We review an elementary approach to the construction of all irreducible representations of the finite Heisenberg group. Determining the number of inequivalent classes of irreducible representations by different methods leads to an identity of sums involving greatest common divisors. We show how this identity can be generalized and derive an explicit formula for the sums.]]> 0 <![CDATA[In this paper we apply Polya's Theorem to the problem of enumerating Cayley graphs on permutation groups up to isomorphisms induced by conjugacy in the symmetric group. We report the results of a search of all three-regular Cayley graphs on permutation groups of degree at most nine for small diameter graphs. We explore several methods of constructing covering graphs of these Cayley graphs. Examples of large graphs with small diameter are obtained.]]> Mon, 01 Jan 2001 07:00:00 +0000 https://doi.org/10.46298/dmtcs.285 https://doi.org/10.46298/dmtcs.285 Curtin, Eugene Curtin, Eugene <![CDATA[In this paper we apply Polya's Theorem to the problem of enumerating Cayley graphs on permutation groups up to isomorphisms induced by conjugacy in the symmetric group. We report the results of a search of all three-regular Cayley graphs on permutation groups of degree at most nine for small diameter graphs. We explore several methods of constructing covering graphs of these Cayley graphs. Examples of large graphs with small diameter are obtained.]]> 0 <![CDATA[What is the maximum number of edges of the d-dimensional hypercube, denoted by S(d,k), that can be sliced by k hyperplanes? This question on combinatorial properties of Euclidean geometry arising from linear separability considerations in the theory of Perceptrons has become an issue on its own. We use computational and combinatorial methods to obtain new bounds for S(d,k), d ≤ 8. These strengthen earlier results on hypercube cut numbers.]]> Sun, 31 Dec 2000 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2296 https://doi.org/10.46298/dmtcs.2296 Emamy-Khansary, M. Reza Ziegler, Martin Emamy-Khansary, M. Reza Ziegler, Martin <![CDATA[What is the maximum number of edges of the d-dimensional hypercube, denoted by S(d,k), that can be sliced by k hyperplanes? This question on combinatorial properties of Euclidean geometry arising from linear separability considerations in the theory of Perceptrons has become an issue on its own. We use computational and combinatorial methods to obtain new bounds for S(d,k), d ≤ 8. These strengthen earlier results on hypercube cut numbers.]]> 0 <![CDATA[A triangulation of a finite point set A in $\mathbb{R}^d$ is a geometric simplicial complex which covers the convex hull of $A$ and whose vertices are points of $A$. We study the graph of triangulations whose vertices represent the triangulations and whose edges represent geometric bistellar flips. The main result of this paper is that the graph of triangulations in three dimensions is connected when the points of $A$ are in convex position. We introduce a tree of triangulations and present an algorithm for enumerating triangulations in $O(log log n)$ time per triangulation.]]> Sun, 31 Dec 2000 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2295 https://doi.org/10.46298/dmtcs.2295 Bespamyatnikh, Sergei Bespamyatnikh, Sergei <![CDATA[A triangulation of a finite point set A in $\mathbb{R}^d$ is a geometric simplicial complex which covers the convex hull of $A$ and whose vertices are points of $A$. We study the graph of triangulations whose vertices represent the triangulations and whose edges represent geometric bistellar flips. The main result of this paper is that the graph of triangulations in three dimensions is connected when the points of $A$ are in convex position. We introduce a tree of triangulations and present an algorithm for enumerating triangulations in $O(log log n)$ time per triangulation.]]> 0 <![CDATA[A class of finite discrete dynamical systems, called Sequential Dynamical Systems (SDSs), was proposed in [BMR99,BR99] as an abstract model of computer simulations. Here, we address some questions concerning two special types of the SDS configurations, namely Garden of Eden and Fixed Point configurations. A configuration $C$ of an SDS is a Garden of Eden (GE) configuration if it cannot be reached from any configuration. A necessary and sufficient condition for the non-existence of GE configurations in SDSs whose state values are from a finite domain was provided in [MR00]. We show this condition is sufficient but not necessary for SDSs whose state values are drawn from an infinite domain. We also present results that relate the existence of GE configurations to other properties of an SDS. A configuration $C$ of an SDS is a fixed point if the transition out of $C$ is to $C$ itself. The FIXED POINT EXISTENCE (or FPE) problem is to determine whether a given SDS has a fixed point. We show thatthe FPE problem is NP-complete even for some simple classes of SDSs (e.g., SDSs in which each local transition function is from the set{NAND, XNOR}). We also identify several classes of SDSs (e.g., SDSs with linear or monotone local transition functions) for which the FPE problem can be solved efficiently.]]> Sun, 31 Dec 2000 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2294 https://doi.org/10.46298/dmtcs.2294 Barrett, Christopher , Hunt, Marry, Marathe, Madhav, Ravi, S., Rosenkrantz, Daniel, Stearns, Richard, Tosic, Predrag, Barrett, Christopher , Hunt, Marry, Marathe, Madhav, Ravi, S., Rosenkrantz, Daniel, Stearns, Richard, Tosic, Predrag, <![CDATA[A class of finite discrete dynamical systems, called Sequential Dynamical Systems (SDSs), was proposed in [BMR99,BR99] as an abstract model of computer simulations. Here, we address some questions concerning two special types of the SDS configurations, namely Garden of Eden and Fixed Point configurations. A configuration $C$ of an SDS is a Garden of Eden (GE) configuration if it cannot be reached from any configuration. A necessary and sufficient condition for the non-existence of GE configurations in SDSs whose state values are from a finite domain was provided in [MR00]. We show this condition is sufficient but not necessary for SDSs whose state values are drawn from an infinite domain. We also present results that relate the existence of GE configurations to other properties of an SDS. A configuration $C$ of an SDS is a fixed point if the transition out of $C$ is to $C$ itself. The FIXED POINT EXISTENCE (or FPE) problem is to determine whether a given SDS has a fixed point. We show thatthe FPE problem is NP-complete even for some simple classes of SDSs (e.g., SDSs in which each local transition function is from the set{NAND, XNOR}). We also identify several classes of SDSs (e.g., SDSs with linear or monotone local transition functions) for which the FPE problem can be solved efficiently.]]> 0 <![CDATA[Several classic tilings, including rhombuses and dominoes, possess height functions which allow us to 1) prove ergodicity and polynomial mixing times for Markov chains based on local moves, 2) use coupling from the past to sample perfectly random tilings, 3) map the statistics of random tilings at large scales to physical models of random surfaces, and and 4) are related to the "arctic circle"' phenomenon.However, few examples are known for which this approach works in three or more dimensions.Here we show that the rhombus tiling can be generalized to n-dimensional tiles for any $n ≥ 3$. For each $n$, we show that a certain local move is ergodic, and conjecture that it has a mixing time of $O(L^{n+2} log L)$ on regions of size $L$. For $n=3$, the tiles are rhombohedra, and the local move consists of switching between two tilings of a rhombic dodecahedron.We use coupling from the past to sample random tilings of a large rhombic dodecahedron, and show that arctic regions exist in which the tiling is frozen into a fixed state.However, unlike the two-dimensional case in which the arctic region is an inscribed circle, here it seems to be octahedral.In addition, height fluctuations between the boundary of the region and the center appear to be constant rather than growing logarithmically.We conjecture that this is because the physics of the model is in a "smooth" phase where it is rigid at large scales, rather than a "rough" phase in which it is elastic.]]> Sun, 31 Dec 2000 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2293 https://doi.org/10.46298/dmtcs.2293 Linde, Joakim Moore, Cristopher Nordahl, Mats G. Linde, Joakim Moore, Cristopher Nordahl, Mats G. <![CDATA[Several classic tilings, including rhombuses and dominoes, possess height functions which allow us to 1) prove ergodicity and polynomial mixing times for Markov chains based on local moves, 2) use coupling from the past to sample perfectly random tilings, 3) map the statistics of random tilings at large scales to physical models of random surfaces, and and 4) are related to the "arctic circle"' phenomenon.However, few examples are known for which this approach works in three or more dimensions.Here we show that the rhombus tiling can be generalized to n-dimensional tiles for any $n ≥ 3$. For each $n$, we show that a certain local move is ergodic, and conjecture that it has a mixing time of $O(L^{n+2} log L)$ on regions of size $L$. For $n=3$, the tiles are rhombohedra, and the local move consists of switching between two tilings of a rhombic dodecahedron.We use coupling from the past to sample random tilings of a large rhombic dodecahedron, and show that arctic regions exist in which the tiling is frozen into a fixed state.However, unlike the two-dimensional case in which the arctic region is an inscribed circle, here it seems to be octahedral.In addition, height fluctuations between the boundary of the region and the center appear to be constant rather than growing logarithmically.We conjecture that this is because the physics of the model is in a "smooth" phase where it is rigid at large scales, rather than a "rough" phase in which it is elastic.]]> 0 <![CDATA[I give a survey of different combinatorial forms of alternating-sign matrices, starting with the original form introduced by Mills, Robbins and Rumsey as well as corner-sum matrices, height-function matrices, three-colorings, monotone triangles, tetrahedral order ideals, square ice, gasket-and-basket tilings and full packings of loops.]]> Sun, 31 Dec 2000 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2292 https://doi.org/10.46298/dmtcs.2292 Propp, James Propp, James <![CDATA[I give a survey of different combinatorial forms of alternating-sign matrices, starting with the original form introduced by Mills, Robbins and Rumsey as well as corner-sum matrices, height-function matrices, three-colorings, monotone triangles, tetrahedral order ideals, square ice, gasket-and-basket tilings and full packings of loops.]]> 0 <![CDATA[The aim of this paper is to give an overview of recent results about tilings, discrete approximations of lines and planes, and Markov partitions for toral automorphisms.The main tool is a generalization of the notion of substitution. The simplest examples which correspond to algebraic parameters, are related to the iteration of one substitution, but we show that it is possible to treat arbitrary irrationalexamples by using multidimensional continued fractions.We give some non-trivial applications to Diophantine approximation, numeration systems and tilings, and we expose the main unsolved questions.]]> Sun, 31 Dec 2000 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2291 https://doi.org/10.46298/dmtcs.2291 Arnoux, Pierre Berthe, Valerie Ei, Hiromi Ito, Shunji Arnoux, Pierre Berthe, Valerie Ei, Hiromi Ito, Shunji <![CDATA[The aim of this paper is to give an overview of recent results about tilings, discrete approximations of lines and planes, and Markov partitions for toral automorphisms.The main tool is a generalization of the notion of substitution. The simplest examples which correspond to algebraic parameters, are related to the iteration of one substitution, but we show that it is possible to treat arbitrary irrationalexamples by using multidimensional continued fractions.We give some non-trivial applications to Diophantine approximation, numeration systems and tilings, and we expose the main unsolved questions.]]> 0 <![CDATA[We discuss certain linear cellular automata whose cells take values in a finite field. We investigate the periodic behavior of the verticals of an orbit of the cellular automaton and establish that there exists, depending on the characteristic of the field, a universal behavior for the evolution of periodic verticals.]]> Sun, 31 Dec 2000 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2290 https://doi.org/10.46298/dmtcs.2290 Barbé, André Haeseler, Fritz, Barbé, André Haeseler, Fritz, <![CDATA[We discuss certain linear cellular automata whose cells take values in a finite field. We investigate the periodic behavior of the verticals of an orbit of the cellular automaton and establish that there exists, depending on the characteristic of the field, a universal behavior for the evolution of periodic verticals.]]> 0 <![CDATA[The permutations by decimation problem is thought to be applicable to computer graphics, and raises interesting theoretical questions in combinatory theory.We present the results of some theoretical and practical investigation into this problem.We show that sequences of this form are $O(n^2)$ in length, but finding optimal solutions can be difficult.]]> Sun, 31 Dec 2000 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2289 https://doi.org/10.46298/dmtcs.2289 Erra, Robert Lygeros, Nik Stewart, Nigel Erra, Robert Lygeros, Nik Stewart, Nigel <![CDATA[The permutations by decimation problem is thought to be applicable to computer graphics, and raises interesting theoretical questions in combinatory theory.We present the results of some theoretical and practical investigation into this problem.We show that sequences of this form are $O(n^2)$ in length, but finding optimal solutions can be difficult.]]> 0 <![CDATA[We introduce the Larger than Life family of two-dimensional two-state cellular automata that generalize certain nearest neighbor outer totalistic cellular automaton rules to large neighborhoods. We describe linear and quadratic rescalings of John Conway's celebrated Game of Life to these large neighborhood cellular automaton rules and present corresponding generalizations of Life's famous gliders and spaceships. We show that, as is becoming well known for nearest neighbor cellular automaton rules, these ``digital creatures'' are ubiquitous for certain parameter values.]]> Sun, 31 Dec 2000 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2288 https://doi.org/10.46298/dmtcs.2288 Evans, Kellie M. Evans, Kellie M. <![CDATA[We introduce the Larger than Life family of two-dimensional two-state cellular automata that generalize certain nearest neighbor outer totalistic cellular automaton rules to large neighborhoods. We describe linear and quadratic rescalings of John Conway's celebrated Game of Life to these large neighborhood cellular automaton rules and present corresponding generalizations of Life's famous gliders and spaceships. We show that, as is becoming well known for nearest neighbor cellular automaton rules, these ``digital creatures'' are ubiquitous for certain parameter values.]]> 0 <![CDATA[In this article, we study the question of tilings on a hexagon mesh with balanced 3-tiles. This problem has been studied by Conway and Lagarias in [CL90], by studying the tiling groups, in fact a group containing the tiling-groups, and their Cayley graphs. We will use two different approaches. The first one is based on matchings in bipartite graphs, which in this case are in correspondance with tilings of domains by lozenges, and thus can be efficiently studied, using Thurston's algorithm (see [Thu90]). The second one is based on a color and balancing approach of Thurston's algorithm, exposed in [Fou96].]]> Sun, 31 Dec 2000 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2287 https://doi.org/10.46298/dmtcs.2287 Radenne, Gilles Radenne, Gilles <![CDATA[In this article, we study the question of tilings on a hexagon mesh with balanced 3-tiles. This problem has been studied by Conway and Lagarias in [CL90], by studying the tiling groups, in fact a group containing the tiling-groups, and their Cayley graphs. We will use two different approaches. The first one is based on matchings in bipartite graphs, which in this case are in correspondance with tilings of domains by lozenges, and thus can be efficiently studied, using Thurston's algorithm (see [Thu90]). The second one is based on a color and balancing approach of Thurston's algorithm, exposed in [Fou96].]]> 0 <![CDATA[We study a poset $\Re$ on the free monoid (X*) on a countable alphabet X.This poset is determined by the fact that its total extensions are precisely the standard term orders on X*. We also investigate the poset classifying degree-compatible standard term orders, and the poset classifying sorted term orders. For the latter poset, we give a Galois coconnection with the Young lattice.]]> Sun, 31 Dec 2000 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2286 https://doi.org/10.46298/dmtcs.2286 Snellman, Jan Snellman, Jan <![CDATA[We study a poset $\Re$ on the free monoid (X*) on a countable alphabet X.This poset is determined by the fact that its total extensions are precisely the standard term orders on X*. We also investigate the poset classifying degree-compatible standard term orders, and the poset classifying sorted term orders. For the latter poset, we give a Galois coconnection with the Young lattice.]]> 0 <![CDATA[We present a characteristic-free algorithm for computing minimal generating sets of invariant rings of permutation groups. We circumvent the main weaknesses of the usual approaches (using classical Gröbner basis inside the full polynomial ring, or pure linear algebra inside the invariant ring) by relying on the theory of SAGBI- Gröbner basis. This theory takes, in this special case, a strongly combinatorial flavor, which makes it particularly effective. Our algorithm does not require the computation of a Hironaka decomposition, nor even the computation of a system of parameters, and could be parallelized. Our implementation, as part of the library $permuvar$ for $mupad$, is in many cases much more efficient than the other existing software.]]> Sun, 31 Dec 2000 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2285 https://doi.org/10.46298/dmtcs.2285 Thiéry, Nicolas, Thiéry, Nicolas, <![CDATA[We present a characteristic-free algorithm for computing minimal generating sets of invariant rings of permutation groups. We circumvent the main weaknesses of the usual approaches (using classical Gröbner basis inside the full polynomial ring, or pure linear algebra inside the invariant ring) by relying on the theory of SAGBI- Gröbner basis. This theory takes, in this special case, a strongly combinatorial flavor, which makes it particularly effective. Our algorithm does not require the computation of a Hironaka decomposition, nor even the computation of a system of parameters, and could be parallelized. Our implementation, as part of the library $permuvar$ for $mupad$, is in many cases much more efficient than the other existing software.]]> 0 <![CDATA[We consider the usual model of hypermaps or, equivalently, bipartite maps, represented by pairs of permutations that act transitively on a set of edges E. The specific feature of our construction is the fact that the elements of E are themselves (or are labelled by) rather complicated combinatorial objects, namely, the 4-constellations, while the permutations defining the hypermap originate from an action of the Hurwitz braid group on these 4-constellations.The motivation for the whole construction is the combinatorial representation of the parameter space of the ramified coverings of the Riemann sphere having four ramification points.]]> Sun, 31 Dec 2000 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2284 https://doi.org/10.46298/dmtcs.2284 Zvonkin, Alexander Zvonkin, Alexander <![CDATA[We consider the usual model of hypermaps or, equivalently, bipartite maps, represented by pairs of permutations that act transitively on a set of edges E. The specific feature of our construction is the fact that the elements of E are themselves (or are labelled by) rather complicated combinatorial objects, namely, the 4-constellations, while the permutations defining the hypermap originate from an action of the Hurwitz braid group on these 4-constellations.The motivation for the whole construction is the combinatorial representation of the parameter space of the ramified coverings of the Riemann sphere having four ramification points.]]> 0 <![CDATA[In this paper, we provide the first study of the sand pile model SPM(0) where we assume that all the grains are numbered with a distinct integer.We obtain a lower bound on the number of terminal sand piles by establishing a bijection between a subset of these sand piles and the set of shifted Young tableaux. We then prove that this number is at least factorial.]]> Sun, 31 Dec 2000 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2283 https://doi.org/10.46298/dmtcs.2283 Novelli, Jean-Christophe Rossin, Dominique Novelli, Jean-Christophe Rossin, Dominique <![CDATA[In this paper, we provide the first study of the sand pile model SPM(0) where we assume that all the grains are numbered with a distinct integer.We obtain a lower bound on the number of terminal sand piles by establishing a bijection between a subset of these sand piles and the set of shifted Young tableaux. We then prove that this number is at least factorial.]]> 0 <![CDATA[It is known the one dimensional prototile $0,a,a+b$ and its reflection $0,b,a+b$ always tile some interval. The subject has not received a great deal of further attention, although many interesting questions exist. All the information about tilings can be encoded in a finite digraph $D_{ab}$. We present several results about cycles and other structures in this graph. A number of conjectures and open problems are given.In [Go] an elegant proof by contradiction shows that a greedy algorithm will produce an interval tiling. We show that the process of converting to a direct proof leads to much stronger results.]]> Sun, 31 Dec 2000 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2282 https://doi.org/10.46298/dmtcs.2282 Meyerowitz, Aaron Meyerowitz, Aaron <![CDATA[It is known the one dimensional prototile $0,a,a+b$ and its reflection $0,b,a+b$ always tile some interval. The subject has not received a great deal of further attention, although many interesting questions exist. All the information about tilings can be encoded in a finite digraph $D_{ab}$. We present several results about cycles and other structures in this graph. A number of conjectures and open problems are given.In [Go] an elegant proof by contradiction shows that a greedy algorithm will produce an interval tiling. We show that the process of converting to a direct proof leads to much stronger results.]]> 0 <![CDATA[The distribution for the number of searches needed to find k of n lost objects is expressed in terms of a refinement of the q-Eulerian polynomials, for which formulae are developed involving homogeneous symmetric polynomials. In the case when k=n and the find probability remains constant, relatively simple and efficient formulas are obtained.From our main theorem, we further (1) deduce the inverse absorption distribution and (2) determine the expected number of times the survivor pulls the trigger in an n-player game of Russian roulette.]]> Sun, 31 Dec 2000 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2281 https://doi.org/10.46298/dmtcs.2281 Herbranson, Travis Rawlings, Don Herbranson, Travis Rawlings, Don <![CDATA[The distribution for the number of searches needed to find k of n lost objects is expressed in terms of a refinement of the q-Eulerian polynomials, for which formulae are developed involving homogeneous symmetric polynomials. In the case when k=n and the find probability remains constant, relatively simple and efficient formulas are obtained.From our main theorem, we further (1) deduce the inverse absorption distribution and (2) determine the expected number of times the survivor pulls the trigger in an n-player game of Russian roulette.]]> 0 <![CDATA[This paper provides a combinatorial approach for analyzing the performance of demodulation methods used in GSM. We also show how to obtain combinatorially a nice specialization of an important performance evaluation formula, using its connection with a classical bijection of Knuth between pairs of Young tableaux and {0,1}-matrices.]]> Sun, 31 Dec 2000 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2280 https://doi.org/10.46298/dmtcs.2280 Krob, Daniel Vassilieva, Ekaterina A. Krob, Daniel Vassilieva, Ekaterina A. <![CDATA[This paper provides a combinatorial approach for analyzing the performance of demodulation methods used in GSM. We also show how to obtain combinatorially a nice specialization of an important performance evaluation formula, using its connection with a classical bijection of Knuth between pairs of Young tableaux and {0,1}-matrices.]]> 0 <![CDATA[In this paper, we use a simple discrete dynamical model to study partitions of integers into powers of another integer. We extend and generalize some known results about their enumeration and counting, and we give new structural results. In particular, we show that the set of these partitions can be ordered in a natural way which gives the distributive lattice structure to this set. We also give a tree structure which allow efficient and simple enumeration of the partitions of an integer.]]> Sun, 31 Dec 2000 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2279 https://doi.org/10.46298/dmtcs.2279 Latapy, Matthieu Latapy, Matthieu <![CDATA[In this paper, we use a simple discrete dynamical model to study partitions of integers into powers of another integer. We extend and generalize some known results about their enumeration and counting, and we give new structural results. In particular, we show that the set of these partitions can be ordered in a natural way which gives the distributive lattice structure to this set. We also give a tree structure which allow efficient and simple enumeration of the partitions of an integer.]]> 0 <![CDATA[In this paper we construct from a cographic matroid M, a pure multicomplex whose degree sequence is the h―vector of the the matroid complex of M. This result provesa conjecture of Richard Stanley [Sta96] in the particular case of cographic matroids. We also prove that the multicomplexes constructed are M―shellable, so proving a conjecture of Manoj Chari [Cha97] again in the case of cographic matroids. The proofs use results on a game for graphs called the chip firing game.]]> Sun, 31 Dec 2000 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2278 https://doi.org/10.46298/dmtcs.2278 Merino, Criel Merino, Criel <![CDATA[In this paper we construct from a cographic matroid M, a pure multicomplex whose degree sequence is the h―vector of the the matroid complex of M. This result provesa conjecture of Richard Stanley [Sta96] in the particular case of cographic matroids. We also prove that the multicomplexes constructed are M―shellable, so proving a conjecture of Manoj Chari [Cha97] again in the case of cographic matroids. The proofs use results on a game for graphs called the chip firing game.]]> 0 <![CDATA[The Chip Firing Game (CFG) is a discrete dynamical model used in physics, computer science and economics. It is known that the set of configurationsreachable from an initial configuration (this set is called the \textitconfiguration space) can be ordered as a lattice. We first present a structural result about this model, which allows us to introduce some useful tools for describing those lattices. Then we establish that the class of lattices that are the configuration space of a CFG is strictly between the class of distributive lattices and the class of upper locally distributive (or ULD) lattices. Finally we propose an extension of the model, the \textitcoloured Chip Firing Game, which generates exactly the class of ULD lattices.]]> Sun, 31 Dec 2000 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2277 https://doi.org/10.46298/dmtcs.2277 Magnien, Clémence Phan, Ha Duong Vuillon, Laurent Magnien, Clémence Phan, Ha Duong Vuillon, Laurent <![CDATA[The Chip Firing Game (CFG) is a discrete dynamical model used in physics, computer science and economics. It is known that the set of configurationsreachable from an initial configuration (this set is called the \textitconfiguration space) can be ordered as a lattice. We first present a structural result about this model, which allows us to introduce some useful tools for describing those lattices. Then we establish that the class of lattices that are the configuration space of a CFG is strictly between the class of distributive lattices and the class of upper locally distributive (or ULD) lattices. Finally we propose an extension of the model, the \textitcoloured Chip Firing Game, which generates exactly the class of ULD lattices.]]> 0 <![CDATA[Cellular automata are mappings over infinite lattices such that each cell is updated according tothe states around it and a unique local function.Block permutations are mappings that generalize a given permutation of blocks (finite arrays of fixed size) to a given partition of the lattice in blocks.We prove that any d-dimensional reversible cellular automaton can be exp ressed as thecomposition of d+1 block permutations.We built a simulation in linear time of reversible cellular automata by reversible block cellular automata (also known as partitioning CA and CA with the Margolus neighborhood) which is valid for both finite and infinite configurations. This proves a 1990 conjecture by Toffoli and Margolus (Physica D 45) improved by Kari in 1996 (Mathematical System Theory 29).]]> Sun, 31 Dec 2000 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2297 https://doi.org/10.46298/dmtcs.2297 Durand-Lose, Jérôme Durand-Lose, Jérôme <![CDATA[Cellular automata are mappings over infinite lattices such that each cell is updated according tothe states around it and a unique local function.Block permutations are mappings that generalize a given permutation of blocks (finite arrays of fixed size) to a given partition of the lattice in blocks.We prove that any d-dimensional reversible cellular automaton can be exp ressed as thecomposition of d+1 block permutations.We built a simulation in linear time of reversible cellular automata by reversible block cellular automata (also known as partitioning CA and CA with the Margolus neighborhood) which is valid for both finite and infinite configurations. This proves a 1990 conjecture by Toffoli and Margolus (Physica D 45) improved by Kari in 1996 (Mathematical System Theory 29).]]> 0 <![CDATA[We address the question of single flip discrete dynamics in sets of two-dimensional random rhombus tilings with fixed polygonal boundaries. Single flips are local rearrangements of tiles which enable to sample the configuration sets of tilings via Markov chains. We determine the convergence rates of these dynamical processes towards the statistical equilibrium distributions and we demonstrate that the dynamics are rapidly mixing: the ergodic times are polynomial in the number of tiles up to logarithmic corrections. We use an inherent symmetry of tiling sets which enables to decompose them into smaller subsets where a technique from probability theory, the so-called coupling technique, can be applied. We also point out an interesting occurrence in this work of extreme-value statistics, namely Gumbel distributions.]]> Sun, 31 Dec 2000 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2300 https://doi.org/10.46298/dmtcs.2300 Destainville, Nicolas Destainville, Nicolas <![CDATA[We address the question of single flip discrete dynamics in sets of two-dimensional random rhombus tilings with fixed polygonal boundaries. Single flips are local rearrangements of tiles which enable to sample the configuration sets of tilings via Markov chains. We determine the convergence rates of these dynamical processes towards the statistical equilibrium distributions and we demonstrate that the dynamics are rapidly mixing: the ergodic times are polynomial in the number of tiles up to logarithmic corrections. We use an inherent symmetry of tiling sets which enables to decompose them into smaller subsets where a technique from probability theory, the so-called coupling technique, can be applied. We also point out an interesting occurrence in this work of extreme-value statistics, namely Gumbel distributions.]]> 0 <![CDATA[In this paper, we provide the second part of the study of the pseudo-permutations. We first derive a complete analysis of the pseudo-permutations, based on hyperplane arrangements, generalizing the usual way of translating the permutations. We then study the module of the pseudo-permutations over the symmetric group and provide the characteristics of this action.]]> Sun, 31 Dec 2000 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2299 https://doi.org/10.46298/dmtcs.2299 Boulier, François Hivert, Florent Krob, Daniel Novelli, Jean-Christophe Boulier, François Hivert, Florent Krob, Daniel Novelli, Jean-Christophe <![CDATA[In this paper, we provide the second part of the study of the pseudo-permutations. We first derive a complete analysis of the pseudo-permutations, based on hyperplane arrangements, generalizing the usual way of translating the permutations. We then study the module of the pseudo-permutations over the symmetric group and provide the characteristics of this action.]]> 0 <![CDATA[In this paper we consider two classes of lattice paths on the plane which use \textitnorth, \textiteast, \textitsouth,and \textitwest unitary steps, beginningand ending at (0,0).We enumerate them according to the number ofsteps by means of bijective arguments; in particular, we apply the cycle lemma.Then, using these results, we provide a bijective proof for the number of directed-convex polyominoes having a fixed number of rows and columns.]]> Sun, 31 Dec 2000 23:00:00 +0000 https://doi.org/10.46298/dmtcs.2298 https://doi.org/10.46298/dmtcs.2298 Del Lungo, Alberto Mirolli, Massimo Pinzani, Renzo Rinaldi, Simone Del Lungo, Alberto Mirolli, Massimo Pinzani, Renzo Rinaldi, Simone <![CDATA[In this paper we consider two classes of lattice paths on the plane which use \textitnorth, \textiteast, \textitsouth,and \textitwest unitary steps, beginningand ending at (0,0).We enumerate them according to the number ofsteps by means of bijective arguments; in particular, we apply the cycle lemma.Then, using these results, we provide a bijective proof for the number of directed-convex polyominoes having a fixed number of rows and columns.]]> 0 <![CDATA[We present an algorithm for unification of higher-order patterns modulo simple syntactic equational theories as defined by Kirchner [14]. The algorithm by Miller [17] for pattern unification, refined by Nipkow [18] is first modified in order to behave as a first-order unification algorithm. Then the mutation rule for syntactic theories of Kirchner [13,14] is adapted to pattern E-unification. If the syntactic algorithm for a theory E terminates in the first-order case, then our algorithm will also terminate for pattern E-unification. The result is a DAG-solved form plus some equations of the form λ øverlinex.F(øverlinex) = λ øverlinex. F(øverlinex^π ) where øverlinex^π is a permutation of øverlinex When all function symbols are decomposable these latter equations can be discarded, otherwise the compatibility of such equations with the solved form remains open.]]> Sat, 01 Jan 2000 07:00:00 +0000 https://doi.org/10.46298/dmtcs.270 https://doi.org/10.46298/dmtcs.270 Boudet, Alexandre Boudet, Alexandre <![CDATA[We present an algorithm for unification of higher-order patterns modulo simple syntactic equational theories as defined by Kirchner [14]. The algorithm by Miller [17] for pattern unification, refined by Nipkow [18] is first modified in order to behave as a first-order unification algorithm. Then the mutation rule for syntactic theories of Kirchner [13,14] is adapted to pattern E-unification. If the syntactic algorithm for a theory E terminates in the first-order case, then our algorithm will also terminate for pattern E-unification. The result is a DAG-solved form plus some equations of the form λ øverlinex.F(øverlinex) = λ øverlinex. F(øverlinex^π ) where øverlinex^π is a permutation of øverlinex When all function symbols are decomposable these latter equations can be discarded, otherwise the compatibility of such equations with the solved form remains open.]]> 0 <![CDATA[We find an explicit expression for the generating function of the number of permutations in S_n avoiding a subgroup of S_k generated by all but one simple transpositions. The generating function turns out to be rational, and its denominator is a rook polynomial for a rectangular board.]]> Sat, 01 Jan 2000 07:00:00 +0000 https://doi.org/10.46298/dmtcs.276 https://doi.org/10.46298/dmtcs.276 Mansour, Toufik Vainshtein, Alek Mansour, Toufik Vainshtein, Alek <![CDATA[We find an explicit expression for the generating function of the number of permutations in S_n avoiding a subgroup of S_k generated by all but one simple transpositions. The generating function turns out to be rational, and its denominator is a rook polynomial for a rectangular board.]]> 0 <![CDATA[Let (A_v)_v ∈ V be a finite family of sets. We establish an improved inclusion-exclusion identity for each closure operator on the power set of V having the unique base property. The result generalizes three improvements of the inclusion-exclusion principle as well as Whitney's broken circuit theorem on the chromatic polynomial of a graph.]]> Sat, 01 Jan 2000 07:00:00 +0000 https://doi.org/10.46298/dmtcs.275 https://doi.org/10.46298/dmtcs.275 Dohmen, Klaus Dohmen, Klaus <![CDATA[Let (A_v)_v ∈ V be a finite family of sets. We establish an improved inclusion-exclusion identity for each closure operator on the power set of V having the unique base property. The result generalizes three improvements of the inclusion-exclusion principle as well as Whitney's broken circuit theorem on the chromatic polynomial of a graph.]]> 0 <![CDATA[Let s_k(n) denote the sum of the digits in the base-k representation of n. In a celebrated paper, Thue showed that the infinite word (s_2(n) \bmod 2)_n≥ 0 is \emphoverlap-free, i.e., contains no subword of the form axaxa where x is any finite word and a is a single symbol. Let k,m be integers with k>2, m≥ 1. In this paper, generalizing Thue's result, we prove that the infinite word t_k,m := (s_k(n) \bmod m)_n≥ 0 is overlap-free if and only if m≥ k. We also prove that t_k,m contains arbitrarily long squares (i.e., subwords of the form xx where x is nonempty), and contains arbitrarily long palindromes if and only if m≤ 2.]]> Sat, 01 Jan 2000 07:00:00 +0000 https://doi.org/10.46298/dmtcs.282 https://doi.org/10.46298/dmtcs.282 Allouche, Jean-Paul Shallit, Jeffrey Allouche, Jean-Paul Shallit, Jeffrey <![CDATA[Let s_k(n) denote the sum of the digits in the base-k representation of n. In a celebrated paper, Thue showed that the infinite word (s_2(n) \bmod 2)_n≥ 0 is \emphoverlap-free, i.e., contains no subword of the form axaxa where x is any finite word and a is a single symbol. Let k,m be integers with k>2, m≥ 1. In this paper, generalizing Thue's result, we prove that the infinite word t_k,m := (s_k(n) \bmod m)_n≥ 0 is overlap-free if and only if m≥ k. We also prove that t_k,m contains arbitrarily long squares (i.e., subwords of the form xx where x is nonempty), and contains arbitrarily long palindromes if and only if m≤ 2.]]> 0 <![CDATA[A transitive orientation of a graph is an orientation of the edges that produces a transitive digraph. The modular decomposition of a graph is a canonical representation of all of its modules. Finding a transitive orientation and finding the modular decomposition are in some sense dual problems. In this paper, we describe a simple O(n + m \log n) algorithm that uses this duality to find both a transitive orientation and the modular decomposition. Though the running time is not optimal, this algorithm is much simpler than any previous algorithms that are not Ω (n^2). The best known time bounds for the problems are O(n+m) but they involve sophisticated techniques.]]> Sat, 01 Jan 2000 07:00:00 +0000 https://doi.org/10.46298/dmtcs.274 https://doi.org/10.46298/dmtcs.274 Mcconnell, Ross M. Spinrad, Jeremy P. Mcconnell, Ross M. Spinrad, Jeremy P. <![CDATA[A transitive orientation of a graph is an orientation of the edges that produces a transitive digraph. The modular decomposition of a graph is a canonical representation of all of its modules. Finding a transitive orientation and finding the modular decomposition are in some sense dual problems. In this paper, we describe a simple O(n + m \log n) algorithm that uses this duality to find both a transitive orientation and the modular decomposition. Though the running time is not optimal, this algorithm is much simpler than any previous algorithms that are not Ω (n^2). The best known time bounds for the problems are O(n+m) but they involve sophisticated techniques.]]> 0 <![CDATA[A permutation π is said to be τ -avoiding if it does not contain any subsequence having all the same pairwise comparisons as τ . This paper concerns the characterization and enumeration of permutations which avoid a set F^j of subsequences increasing both in number and in length at the same time. Let F^j be the set of subsequences of the form σ (j+1)(j+2), σ being any permutation on \1,...,j\. For j=1 the only subsequence in F^1 is 123 and the 123-avoiding permutations are enumerated by the Catalan numbers; for j=2 the subsequences in F^2 are 1234 2134 and the (1234,2134)avoiding permutations are enumerated by the Schröder numbers; for each other value of j greater than 2 the subsequences in F^j are j! and their length is (j+2) the permutations avoiding these j! subsequences are enumerated by a number sequence \a_n\ such that C_n ≤ a_n ≤ n!, C_n being the nth Catalan number. For each j we determine the generating function of permutations avoiding the subsequences in F^j according to the length, to the number of left minima and of non-inversions.]]> Sat, 01 Jan 2000 07:00:00 +0000 https://doi.org/10.46298/dmtcs.273 https://doi.org/10.46298/dmtcs.273 Barcucci, Elena Del Lungo, Alberto Pergola, Elisa Pinzani, Renzo Barcucci, Elena Del Lungo, Alberto Pergola, Elisa Pinzani, Renzo <![CDATA[A permutation π is said to be τ -avoiding if it does not contain any subsequence having all the same pairwise comparisons as τ . This paper concerns the characterization and enumeration of permutations which avoid a set F^j of subsequences increasing both in number and in length at the same time. Let F^j be the set of subsequences of the form σ (j+1)(j+2), σ being any permutation on \1,...,j\. For j=1 the only subsequence in F^1 is 123 and the 123-avoiding permutations are enumerated by the Catalan numbers; for j=2 the subsequences in F^2 are 1234 2134 and the (1234,2134)avoiding permutations are enumerated by the Schröder numbers; for each other value of j greater than 2 the subsequences in F^j are j! and their length is (j+2) the permutations avoiding these j! subsequences are enumerated by a number sequence \a_n\ such that C_n ≤ a_n ≤ n!, C_n being the nth Catalan number. For each j we determine the generating function of permutations avoiding the subsequences in F^j according to the length, to the number of left minima and of non-inversions.]]> 0 <![CDATA[We describe the relationship between different parameters of the initial word and its image obtained by application of a uniform marked morphism. The functions described include the subword complexity, frequency of factors, and the recurrence function. The relations obtained for the image of a word can be used also for the image of a factorial language. Using induction, we give a full description of the involved functions of the fixed point of the morphism considered.]]> Fri, 01 Jan 1999 07:00:00 +0000 https://doi.org/10.46298/dmtcs.255 https://doi.org/10.46298/dmtcs.255 Frid, Anna Frid, Anna <![CDATA[We describe the relationship between different parameters of the initial word and its image obtained by application of a uniform marked morphism. The functions described include the subword complexity, frequency of factors, and the recurrence function. The relations obtained for the image of a word can be used also for the image of a factorial language. Using induction, we give a full description of the involved functions of the fixed point of the morphism considered.]]> 0 <![CDATA[The theory and practice of polytypic programming is intimately connected with the initial algebra semantics of datatypes. This is both a blessing and a curse. It is a blessing because the underlying theory is beautiful and well developed. It is a curse because the initial algebra semantics is restricted to so-called regular datatypes. Recent work by R. Bird and L. Meertens [3] on the semantics of non-regular or nested datatypes suggests that an extension to general datatypes is not entirely straightforward. Here we propose an alternative that extends polytypism to arbitrary datatypes, including nested datatypes and mutually recursive datatypes. The central idea is to use rational trees over a suitable set of functor symbols as type arguments for polytypic functions. Besides covering a wider range of types the approach is also simpler and technically less involving than previous ones. We present several examples of polytypic functions, among others polytypic reduction and polytypic equality. The presentation assumes some background in functional and in polytypic programming. A basic knowledge of monads is required for some of the examples.]]> Fri, 01 Jan 1999 07:00:00 +0000 https://doi.org/10.46298/dmtcs.266 https://doi.org/10.46298/dmtcs.266 Hinze, Ralf Hinze, Ralf <![CDATA[The theory and practice of polytypic programming is intimately connected with the initial algebra semantics of datatypes. This is both a blessing and a curse. It is a blessing because the underlying theory is beautiful and well developed. It is a curse because the initial algebra semantics is restricted to so-called regular datatypes. Recent work by R. Bird and L. Meertens [3] on the semantics of non-regular or nested datatypes suggests that an extension to general datatypes is not entirely straightforward. Here we propose an alternative that extends polytypism to arbitrary datatypes, including nested datatypes and mutually recursive datatypes. The central idea is to use rational trees over a suitable set of functor symbols as type arguments for polytypic functions. Besides covering a wider range of types the approach is also simpler and technically less involving than previous ones. We present several examples of polytypic functions, among others polytypic reduction and polytypic equality. The presentation assumes some background in functional and in polytypic programming. A basic knowledge of monads is required for some of the examples.]]> 0 <![CDATA[In this paper, the author presents a method, based on WZ theory, for finding rapidly converging series for universal constants. This method is analogous but different from Amdeberhan and Zeilberger's method.]]> Fri, 01 Jan 1999 07:00:00 +0000 https://doi.org/10.46298/dmtcs.265 https://doi.org/10.46298/dmtcs.265 Wilf, Herbert S. Wilf, Herbert S. <![CDATA[In this paper, the author presents a method, based on WZ theory, for finding rapidly converging series for universal constants. This method is analogous but different from Amdeberhan and Zeilberger's method.]]> 0 <![CDATA[The lattice \textbfS_n of all permutations on a n-element set has been shown to be \emphbounded [CAS], which is a strong constructive property characterized by the fact that \textbfS_n admits what we call an \emph interval doubling scheme. In this paper we characterize all interval doubling schemes of the lattice \textbfS_n, a result that gives a nice precision on the bounded nature of the lattice of permutations. This theorem is a direct corollary of two strong properties that are also given with their proofs.]]> Fri, 01 Jan 1999 07:00:00 +0000 https://doi.org/10.46298/dmtcs.264 https://doi.org/10.46298/dmtcs.264 Caspard, Nathalie Caspard, Nathalie <![CDATA[The lattice \textbfS_n of all permutations on a n-element set has been shown to be \emphbounded [CAS], which is a strong constructive property characterized by the fact that \textbfS_n admits what we call an \emph interval doubling scheme. In this paper we characterize all interval doubling schemes of the lattice \textbfS_n, a result that gives a nice precision on the bounded nature of the lattice of permutations. This theorem is a direct corollary of two strong properties that are also given with their proofs.]]> 0 <![CDATA[We introduce q-proper interval graphs as interval graphs with interval models in which no interval is properly contained in more than q other intervals, and also provide a forbidden induced subgraph characterization of this class of graphs. We initiate a graph-theoretic study of subgraphs of q-proper interval graphs with maximum clique size k+1 and give an equivalent characterization of these graphs by restricted path-decomposition. By allowing the parameter q to vary from 0 to k, we obtain a nested hierarchy of graph families, from graphs of bandwidth at most k to graphs of pathwidth at most k. Allowing both parameters to vary, we have an infinite lattice of graph classes ordered by containment.]]> Fri, 01 Jan 1999 07:00:00 +0000 https://doi.org/10.46298/dmtcs.263 https://doi.org/10.46298/dmtcs.263 Proskurowski, Andrzej Telle, Jan Arne Proskurowski, Andrzej Telle, Jan Arne <![CDATA[We introduce q-proper interval graphs as interval graphs with interval models in which no interval is properly contained in more than q other intervals, and also provide a forbidden induced subgraph characterization of this class of graphs. We initiate a graph-theoretic study of subgraphs of q-proper interval graphs with maximum clique size k+1 and give an equivalent characterization of these graphs by restricted path-decomposition. By allowing the parameter q to vary from 0 to k, we obtain a nested hierarchy of graph families, from graphs of bandwidth at most k to graphs of pathwidth at most k. Allowing both parameters to vary, we have an infinite lattice of graph classes ordered by containment.]]> 0 <![CDATA[In this paper, we consider the problem of scheduling independent parallel tasks in parallel systems with identical processors. The problem is NP-hard, since it includes the bin packing problem as a special case when all tasks have unit execution time. We propose and analyze a simple approximation algorithm called H_m, where m is a positive integer. Algorithm H_m has a moderate asymptotic worst-case performance ratio in the range [4/3 ... 31/18] for all m≥ 6; but the algorithm has a small asymptotic worst-case performance ratio in the range [1+1/(r+1)..1+1/r], when task sizes do not exceed 1/r of the total available processors, where r>1 is an integer. Furthermore, we show that if the task sizes are independent, identically distributed (i.i.d.) uniform random variables, and task execution times are i.i.d. random variables with finite mean and variance, then the average-case performance ratio of algorithm H_m is no larger than 1.2898680..., and for an exponential distribution of task sizes, it does not exceed 1.2898305.... As demonstrated by our analytical as well as numerical results, the average-case performance ratio improves significantly when tasks request for smaller numbers of processors.]]> Fri, 01 Jan 1999 07:00:00 +0000 https://doi.org/10.46298/dmtcs.262 https://doi.org/10.46298/dmtcs.262 Li, Keqin Li, Keqin <![CDATA[In this paper, we consider the problem of scheduling independent parallel tasks in parallel systems with identical processors. The problem is NP-hard, since it includes the bin packing problem as a special case when all tasks have unit execution time. We propose and analyze a simple approximation algorithm called H_m, where m is a positive integer. Algorithm H_m has a moderate asymptotic worst-case performance ratio in the range [4/3 ... 31/18] for all m≥ 6; but the algorithm has a small asymptotic worst-case performance ratio in the range [1+1/(r+1)..1+1/r], when task sizes do not exceed 1/r of the total available processors, where r>1 is an integer. Furthermore, we show that if the task sizes are independent, identically distributed (i.i.d.) uniform random variables, and task execution times are i.i.d. random variables with finite mean and variance, then the average-case performance ratio of algorithm H_m is no larger than 1.2898680..., and for an exponential distribution of task sizes, it does not exceed 1.2898305.... As demonstrated by our analytical as well as numerical results, the average-case performance ratio improves significantly when tasks request for smaller numbers of processors.]]> 0 <![CDATA[We prove that the number of permutations which avoid 132-patterns and have exactly one 123-pattern, equals $(n-2)2^{n-3}$, for $n \ge 3$. We then give a bijection onto the set of permutations which avoid 123-patterns and have exactly one 132-pattern. Finally, we show that the number of permutations which contain exactly one 123-pattern and exactly one 132-pattern is $(n-3)(n-4)2^{n-5}$, for $n \ge 5$.]]> Fri, 01 Jan 1999 07:00:00 +0000 https://doi.org/10.46298/dmtcs.261 https://doi.org/10.46298/dmtcs.261 Robertson, Aaron Robertson, Aaron <![CDATA[We prove that the number of permutations which avoid 132-patterns and have exactly one 123-pattern, equals $(n-2)2^{n-3}$, for $n \ge 3$. We then give a bijection onto the set of permutations which avoid 123-patterns and have exactly one 132-pattern. Finally, we show that the number of permutations which contain exactly one 123-pattern and exactly one 132-pattern is $(n-3)(n-4)2^{n-5}$, for $n \ge 5$.]]> 0 <![CDATA[In Valiant developed an algebraic analogue of the theory of NP-completeness for computations of polynomials over a field. We further develop this theory in the spirit of structural complexity and obtain analogues of well-known results by Baker, Gill, and Solovay, Ladner, and Schöning.\par We show that if Valiant's hypothesis is true, then there is a p-definable family, which is neither p-computable nor \textitVNP-complete. More generally, we define the posets of p-degrees and c-degrees of p-definable families and prove that any countable poset can be embedded in either of them, provided Valiant's hypothesis is true. Moreover, we establish the existence of minimal pairs for \textitVP in \textitVNP.\par Over finite fields, we give a \emphspecific example of a family of polynomials which is neither \textitVNP-complete nor p-computable, provided the polynomial hierarchy does not collapse.\par We define relativized complexity classes VP^h and VNP^h and construct complete families in these classes. Moreover, we prove that there is a p-family h satisfying VP^h = VNP^h.]]> Fri, 01 Jan 1999 07:00:00 +0000 https://doi.org/10.46298/dmtcs.260 https://doi.org/10.46298/dmtcs.260 Bürgisser, Peter Bürgisser, Peter <![CDATA[In Valiant developed an algebraic analogue of the theory of NP-completeness for computations of polynomials over a field. We further develop this theory in the spirit of structural complexity and obtain analogues of well-known results by Baker, Gill, and Solovay, Ladner, and Schöning.\par We show that if Valiant's hypothesis is true, then there is a p-definable family, which is neither p-computable nor \textitVNP-complete. More generally, we define the posets of p-degrees and c-degrees of p-definable families and prove that any countable poset can be embedded in either of them, provided Valiant's hypothesis is true. Moreover, we establish the existence of minimal pairs for \textitVP in \textitVNP.\par Over finite fields, we give a \emphspecific example of a family of polynomials which is neither \textitVNP-complete nor p-computable, provided the polynomial hierarchy does not collapse.\par We define relativized complexity classes VP^h and VNP^h and construct complete families in these classes. Moreover, we prove that there is a p-family h satisfying VP^h = VNP^h.]]> 0 <![CDATA[We prove the existence of an invariant ring \textbfC[X_1,...,X_n]^T generated by elements with a total degree of at most 2, which has no finite SAGBI basis with respect to any admissible order. Therefore, 2 is the optimal lower bound for the total degree of generators of invariant rings with such a property.]]> Fri, 01 Jan 1999 07:00:00 +0000 https://doi.org/10.46298/dmtcs.259 https://doi.org/10.46298/dmtcs.259 Göbel, Manfred Göbel, Manfred <![CDATA[We prove the existence of an invariant ring \textbfC[X_1,...,X_n]^T generated by elements with a total degree of at most 2, which has no finite SAGBI basis with respect to any admissible order. Therefore, 2 is the optimal lower bound for the total degree of generators of invariant rings with such a property.]]> 0 <![CDATA[In [DO95], Ding and Oporowski proved that for every k, and d, there exists a constant c_k,d, such that every graph with treewidth at most k and maximum degree at most d has domino treewidth at most c_k,d. This note gives a new simple proof of this fact, with a better bound for c_k,d, namely (9k+7)d(d+1) -1. It is also shown that a lower bound of Ω (kd) holds: there are graphs with domino treewidth at least 1/12 × kd-1, treewidth at most k, and maximum degree at most d, for many values k and d. The domino treewidth of a tree is at most its maximum degree.]]> Fri, 01 Jan 1999 07:00:00 +0000 https://doi.org/10.46298/dmtcs.256 https://doi.org/10.46298/dmtcs.256 Bodlaender, Hans L. Bodlaender, Hans L. <![CDATA[In [DO95], Ding and Oporowski proved that for every k, and d, there exists a constant c_k,d, such that every graph with treewidth at most k and maximum degree at most d has domino treewidth at most c_k,d. This note gives a new simple proof of this fact, with a better bound for c_k,d, namely (9k+7)d(d+1) -1. It is also shown that a lower bound of Ω (kd) holds: there are graphs with domino treewidth at least 1/12 × kd-1, treewidth at most k, and maximum degree at most d, for many values k and d. The domino treewidth of a tree is at most its maximum degree.]]> 0 <![CDATA[Building on work of Gaifman [Gai82] it is shown that every first-order formula is logically equivalent to a formula of the form ∃ x_1,...,x_l, \forall y, φ where φ is r-local around y, i.e. quantification in φ is restricted to elements of the universe of distance at most r from y. \par From this and related normal forms, variants of the Ehrenfeucht game for first-order and existential monadic second-order logic are developed that restrict the possible strategies for the spoiler, one of the two players. This makes proofs of the existence of a winning strategy for the duplicator, the other player, easier and can thus simplify inexpressibility proofs. \par As another application, automata models are defined that have, on arbitrary classes of relational structures, exactly the expressive power of first-order logic and existential monadic second-order logic, respectively.]]> Fri, 01 Jan 1999 07:00:00 +0000 https://doi.org/10.46298/dmtcs.254 https://doi.org/10.46298/dmtcs.254 Schwentick, Thomas Barthelmann, Klaus Schwentick, Thomas Barthelmann, Klaus <![CDATA[Building on work of Gaifman [Gai82] it is shown that every first-order formula is logically equivalent to a formula of the form ∃ x_1,...,x_l, \forall y, φ where φ is r-local around y, i.e. quantification in φ is restricted to elements of the universe of distance at most r from y. \par From this and related normal forms, variants of the Ehrenfeucht game for first-order and existential monadic second-order logic are developed that restrict the possible strategies for the spoiler, one of the two players. This makes proofs of the existence of a winning strategy for the duplicator, the other player, easier and can thus simplify inexpressibility proofs. \par As another application, automata models are defined that have, on arbitrary classes of relational structures, exactly the expressive power of first-order logic and existential monadic second-order logic, respectively.]]> 0 <![CDATA[When dictionaries are persistent, it is natural to introduce a transcript operation which reports the status changes for a given key over time. We discuss when and how a time and space efficient implementation of this operation can be provided.]]> Fri, 01 Jan 1999 07:00:00 +0000 https://doi.org/10.46298/dmtcs.253 https://doi.org/10.46298/dmtcs.253 Larsen, Kim S. Larsen, Kim S. <![CDATA[When dictionaries are persistent, it is natural to introduce a transcript operation which reports the status changes for a given key over time. We discuss when and how a time and space efficient implementation of this operation can be provided.]]> 0 <![CDATA[We consider the standard Quicksort algorithm that sorts n distinct keys with all possible n! orderings of keys being equally likely. Equivalently, we analyze the total path length L(n) in a randomly built \emphbinary search tree. Obtaining the limiting distribution of L(n) is still an outstanding open problem. In this paper, we establish an integral equation for the probability density of the number of comparisons L(n). Then, we investigate the large deviations of L(n). We shall show that the left tail of the limiting distribution is much ''thinner'' (i.e., double exponential) than the right tail (which is only exponential). Our results contain some constants that must be determined numerically. We use formal asymptotic methods of applied mathematics such as the WKB method and matched asymptotics.]]> Fri, 01 Jan 1999 07:00:00 +0000 https://doi.org/10.46298/dmtcs.252 https://doi.org/10.46298/dmtcs.252 Knessl, Charles Szpankowski, Wojciech Knessl, Charles Szpankowski, Wojciech <![CDATA[We consider the standard Quicksort algorithm that sorts n distinct keys with all possible n! orderings of keys being equally likely. Equivalently, we analyze the total path length L(n) in a randomly built \emphbinary search tree. Obtaining the limiting distribution of L(n) is still an outstanding open problem. In this paper, we establish an integral equation for the probability density of the number of comparisons L(n). Then, we investigate the large deviations of L(n). We shall show that the left tail of the limiting distribution is much ''thinner'' (i.e., double exponential) than the right tail (which is only exponential). Our results contain some constants that must be determined numerically. We use formal asymptotic methods of applied mathematics such as the WKB method and matched asymptotics.]]> 0 <![CDATA[We define parallelograms of base a and b in a group. They appear as minimal relators in a presentation of a subgroup with generators a and b. In a Lie group they are realized as closed polygonal lines, with sides being orbits of left-invariant vector fields. We estimate the number of sides of parallelograms in a free nilpotent group and point out a relation to the rank of rational series.]]> Fri, 01 Jan 1999 07:00:00 +0000 https://doi.org/10.46298/dmtcs.251 https://doi.org/10.46298/dmtcs.251 Falbel, Elisha Koseleff, Pierre-Vincent Falbel, Elisha Koseleff, Pierre-Vincent <![CDATA[We define parallelograms of base a and b in a group. They appear as minimal relators in a presentation of a subgroup with generators a and b. In a Lie group they are realized as closed polygonal lines, with sides being orbits of left-invariant vector fields. We estimate the number of sides of parallelograms in a free nilpotent group and point out a relation to the rank of rational series.]]> 0 <![CDATA[This paper presents a new systematic approach for the uniform random generation of combinatorial objects. The method is based on the notion of object grammars which give recursive descriptions of objects and generalize context-freegrammars. The application of particular valuations to these grammars leads to enumeration and random generation of objects according to non algebraic parameters.]]> Thu, 01 Jan 1998 07:00:00 +0000 https://doi.org/10.46298/dmtcs.250 https://doi.org/10.46298/dmtcs.250 Dutour, I. Fedou, Jean-Marc Dutour, I. Fedou, Jean-Marc <![CDATA[This paper presents a new systematic approach for the uniform random generation of combinatorial objects. The method is based on the notion of object grammars which give recursive descriptions of objects and generalize context-freegrammars. The application of particular valuations to these grammars leads to enumeration and random generation of objects according to non algebraic parameters.]]> 0 <![CDATA[In this paper we consider the problem of computing on a local memory machine the product y = Ax,where A is a random n×n sparse matrix with Θ (n) nonzero elements. To study the average case communication cost of this problem, we introduce four different probability measures on the set of sparse matrices. We prove that on most local memory machines with p processors, this computation requires Ω ((n/p) \log p) time on the average. We prove that the same lower bound also holds, in the worst case, for matrices with only 2n or 3n nonzero elements.]]> Thu, 01 Jan 1998 07:00:00 +0000 https://doi.org/10.46298/dmtcs.249 https://doi.org/10.46298/dmtcs.249 Manzini, Giovanni Manzini, Giovanni <![CDATA[In this paper we consider the problem of computing on a local memory machine the product y = Ax,where A is a random n×n sparse matrix with Θ (n) nonzero elements. To study the average case communication cost of this problem, we introduce four different probability measures on the set of sparse matrices. We prove that on most local memory machines with p processors, this computation requires Ω ((n/p) \log p) time on the average. We prove that the same lower bound also holds, in the worst case, for matrices with only 2n or 3n nonzero elements.]]> 0 <![CDATA[We prove some new results on a family of operations on binary trees, some of which are similar to addition, multiplication and exponentiation for natural numbers. The main result is that each operation in the family is right-cancellable.]]> Thu, 01 Jan 1998 07:00:00 +0000 https://doi.org/10.46298/dmtcs.248 https://doi.org/10.46298/dmtcs.248 Duchon, Philippe Duchon, Philippe <![CDATA[We prove some new results on a family of operations on binary trees, some of which are similar to addition, multiplication and exponentiation for natural numbers. The main result is that each operation in the family is right-cancellable.]]> 0 <![CDATA[We introduce a data structure called \emphSOUR graphs and present an efficient Knuth-Bendix completion procedure based on it. \emphSOUR graphs allow for a maximal structure sharing of terms in rewriting systems. The term representation is a dag representation, except that edges are labelled with equational constraints and variable renamings. The rewrite rules correspond to rewrite edges, the unification problems to unification edges. The Critical Pair and Simplification inferences are recognized as patterns in the graph and are performed as local graph transformations. Our algorithm avoids duplicating term structure while performing inferences, which causes exponential behavior in the standard procedure. This approach gives a basis to design other completion algorithms, such as goal-oriented completion, concurrent completion and group completion procedures.]]> Thu, 01 Jan 1998 07:00:00 +0000 https://doi.org/10.46298/dmtcs.247 https://doi.org/10.46298/dmtcs.247 Lynch, Christopher Strogova, Polina Lynch, Christopher Strogova, Polina <![CDATA[We introduce a data structure called \emphSOUR graphs and present an efficient Knuth-Bendix completion procedure based on it. \emphSOUR graphs allow for a maximal structure sharing of terms in rewriting systems. The term representation is a dag representation, except that edges are labelled with equational constraints and variable renamings. The rewrite rules correspond to rewrite edges, the unification problems to unification edges. The Critical Pair and Simplification inferences are recognized as patterns in the graph and are performed as local graph transformations. Our algorithm avoids duplicating term structure while performing inferences, which causes exponential behavior in the standard procedure. This approach gives a basis to design other completion algorithms, such as goal-oriented completion, concurrent completion and group completion procedures.]]> 0 <![CDATA[A bijective proof for Stanley's hook-content formula for the generating function for column-strict reverse plane partitions of a given shape is given that does not involve the involution principle of Garsia and Milne. It is based on the Hillman-Grassl algorithm and Schützenberger's \emphjeu de taquin.]]> Thu, 01 Jan 1998 07:00:00 +0000 https://doi.org/10.46298/dmtcs.258 https://doi.org/10.46298/dmtcs.258 Krattenthaler, Christian Krattenthaler, Christian <![CDATA[A bijective proof for Stanley's hook-content formula for the generating function for column-strict reverse plane partitions of a given shape is given that does not involve the involution principle of Garsia and Milne. It is based on the Hillman-Grassl algorithm and Schützenberger's \emphjeu de taquin.]]> 0 <![CDATA[For any fixed parameter t greater or equal to 1, a \emph t-spanner of a graph G is a spanning subgraph in which the distance between every pair of vertices is at most t times their distance in G. A \emph minimum t-spanner is a t-spanner with minimum total edge weight or, in unweighted graphs, minimum number of edges. In this paper, we prove the NP-hardness of finding minimum t-spanners for planar weighted graphs and digraphs if t greater or equal to 3, and for planar unweighted graphs and digraphs if t greater or equal to 5. We thus extend results on that problem to the interesting case where the instances are known to be planar. We also introduce the related problem of finding minimum \emphplanar t-spanners and establish its NP-hardness for similar fixed values of t.]]> Thu, 01 Jan 1998 07:00:00 +0000 https://doi.org/10.46298/dmtcs.257 https://doi.org/10.46298/dmtcs.257 Brandes, Ulrik Handke, Dagmar Brandes, Ulrik Handke, Dagmar <![CDATA[For any fixed parameter t greater or equal to 1, a \emph t-spanner of a graph G is a spanning subgraph in which the distance between every pair of vertices is at most t times their distance in G. A \emph minimum t-spanner is a t-spanner with minimum total edge weight or, in unweighted graphs, minimum number of edges. In this paper, we prove the NP-hardness of finding minimum t-spanners for planar weighted graphs and digraphs if t greater or equal to 3, and for planar unweighted graphs and digraphs if t greater or equal to 5. We thus extend results on that problem to the interesting case where the instances are known to be planar. We also introduce the related problem of finding minimum \emphplanar t-spanners and establish its NP-hardness for similar fixed values of t.]]> 0 <![CDATA[We consider the following problem: what is the most general Lie algebra satisfying a given set of Lie polynomial equations? The presentation of Lie algebras by a finite set of generators and defining relations is one of the most general mathematical and algorithmic schemes of their analysis. That problem is of great practical importance, covering applications ranging from mathematical physics to combinatorial algebra. Some particular applications are constructionof prolongation algebras in the Wahlquist-Estabrook method for integrability analysis of nonlinear partial differential equations and investigation of Lie algebras arising in different physical models. The finite presentations also indicate a way to q-quantize Lie algebras. To solve this problem, one should perform a large volume of algebraic transformations which is sharply increased with growth of the number of generators and relations. For this reason, in practice one needs to use a computer algebra tool. We describe here an algorithm for constructing the basis of a finitely presented Lie algebra and its commutator table, and its implementation in the C language. Some computer results illustrating our algorithmand its actual implementation are also presented.]]> Wed, 01 Jan 1997 07:00:00 +0000 https://doi.org/10.46298/dmtcs.243 https://doi.org/10.46298/dmtcs.243 Gerdt, Vladimir P. Kornyak, Vladimir V. Gerdt, Vladimir P. Kornyak, Vladimir V. <![CDATA[We consider the following problem: what is the most general Lie algebra satisfying a given set of Lie polynomial equations? The presentation of Lie algebras by a finite set of generators and defining relations is one of the most general mathematical and algorithmic schemes of their analysis. That problem is of great practical importance, covering applications ranging from mathematical physics to combinatorial algebra. Some particular applications are constructionof prolongation algebras in the Wahlquist-Estabrook method for integrability analysis of nonlinear partial differential equations and investigation of Lie algebras arising in different physical models. The finite presentations also indicate a way to q-quantize Lie algebras. To solve this problem, one should perform a large volume of algebraic transformations which is sharply increased with growth of the number of generators and relations. For this reason, in practice one needs to use a computer algebra tool. We describe here an algorithm for constructing the basis of a finitely presented Lie algebra and its commutator table, and its implementation in the C language. Some computer results illustrating our algorithmand its actual implementation are also presented.]]> 0 <![CDATA[There are three classical algorithms to visit all the nodes of a binary tree - preorder, inorder and postorder traversal. From this one gets a natural labelling of the n internal nodes of a binary tree by the numbers 1, 2, ..., n, indicating the sequence in which the nodes are visited. For given n (size of the tree) and j (a number between 1 and n), we consider the statistics number of ascendants of node j and number of descendants of node j. By appropriate trivariate generating functions, we are able to find explicit formulae for the expectation and the variance in all instances. The heavy computations that are necessary are facilitated by MAPLE and Zeilberger's algorithm. A similar problem comes fromlabelling the leaves from left to right by 1, 2, ..., n and considering the statistic number of ascendants (=height) of leaf j. For this, Kirschenhofer [1] has computed the average. With our approach, we are also able to get the variance. In the last section, a table with asymptotic equivalents is provided for the reader's convenience.]]> Wed, 01 Jan 1997 07:00:00 +0000 https://doi.org/10.46298/dmtcs.246 https://doi.org/10.46298/dmtcs.246 Panholzer, Alois Prodinger, Helmut Panholzer, Alois Prodinger, Helmut <![CDATA[There are three classical algorithms to visit all the nodes of a binary tree - preorder, inorder and postorder traversal. From this one gets a natural labelling of the n internal nodes of a binary tree by the numbers 1, 2, ..., n, indicating the sequence in which the nodes are visited. For given n (size of the tree) and j (a number between 1 and n), we consider the statistics number of ascendants of node j and number of descendants of node j. By appropriate trivariate generating functions, we are able to find explicit formulae for the expectation and the variance in all instances. The heavy computations that are necessary are facilitated by MAPLE and Zeilberger's algorithm. A similar problem comes fromlabelling the leaves from left to right by 1, 2, ..., n and considering the statistic number of ascendants (=height) of leaf j. For this, Kirschenhofer [1] has computed the average. With our approach, we are also able to get the variance. In the last section, a table with asymptotic equivalents is provided for the reader's convenience.]]> 0 <![CDATA[The aim of this paper is to present some recent results on controllability of right-invariant systems on Lie groups. From the Lie-theoretical point of view, we study conditions under which subsemigroups generated by half-planes in the Lie algebra of a Lie group coincide with the whole Lie group.]]> Wed, 01 Jan 1997 07:00:00 +0000 https://doi.org/10.46298/dmtcs.245 https://doi.org/10.46298/dmtcs.245 Sachkov, Yuri L. Sachkov, Yuri L. <![CDATA[The aim of this paper is to present some recent results on controllability of right-invariant systems on Lie groups. From the Lie-theoretical point of view, we study conditions under which subsemigroups generated by half-planes in the Lie algebra of a Lie group coincide with the whole Lie group.]]> 0 <![CDATA[We develop the bialgebraic structure based on the set of functional graphs, which generalize the case of the forests of rooted trees. We use noncommutative polynomials as generating monomials of the functional graphs, and we introduce circular and arborescent brackets in accordance with the decomposition in connected components of the graph of a mapping of \1, 2, \ldots, n\ in itself as in the frame of the discrete dynamical systems. We give applications fordifferential algebras and algebras of differential operators.]]> Wed, 01 Jan 1997 07:00:00 +0000 https://doi.org/10.46298/dmtcs.244 https://doi.org/10.46298/dmtcs.244 Ginocchio, Maurice Ginocchio, Maurice <![CDATA[We develop the bialgebraic structure based on the set of functional graphs, which generalize the case of the forests of rooted trees. We use noncommutative polynomials as generating monomials of the functional graphs, and we introduce circular and arborescent brackets in accordance with the decomposition in connected components of the graph of a mapping of \1, 2, \ldots, n\ in itself as in the frame of the discrete dynamical systems. We give applications fordifferential algebras and algebras of differential operators.]]> 0 <![CDATA[This paper discusses various deformations of free associative algebras and of their convolution algebras. Our main examples are deformations of noncommutative symmetric functions related to families of idempotents in descent algebras, and a simple q-analogue of the shuffle product, which has unexpected connections with quantum groups, hyperplane arrangements, and certain questions in mathematical physics (the quon algebra, generalized Brownian motion).]]> Wed, 01 Jan 1997 07:00:00 +0000 https://doi.org/10.46298/dmtcs.231 https://doi.org/10.46298/dmtcs.231 Duchamp, Gérard, Klyachko, Alexander Krob, Daniel Thibon, Jean-Yves Duchamp, Gérard, Klyachko, Alexander Krob, Daniel Thibon, Jean-Yves <![CDATA[This paper discusses various deformations of free associative algebras and of their convolution algebras. Our main examples are deformations of noncommutative symmetric functions related to families of idempotents in descent algebras, and a simple q-analogue of the shuffle product, which has unexpected connections with quantum groups, hyperplane arrangements, and certain questions in mathematical physics (the quon algebra, generalized Brownian motion).]]> 0 <![CDATA[This paper describes progress made in context with the construction of a general library of Lie algebra algorithms, called ELIAS (Eindhoven Lie Algebra System), within the computer algebra package GAP. A first sketch of the package can be found in Cohen and de Graaf[1]. Since then, in a collaborative effort with G. Ivanyos, the authors have continued to develop algorithms which were implemented in ELIAS by the second author. These activities are part of a bigger project, called ACELA and financed by STW, the Dutch Technology Foundation, which aims at an interactive book on Lie algebras (cf. Cohen and Meertens [2]). This paper gives a global description of the main ways in which to present Lie algebras on a computer. We focus on the transition from a Lie algebra abstractly given by an array of structure constants to a Lie algebra presented as a subalgebra of the Lie algebra of n×n matrices. We describe an algorithm typical of the structure analysis of a finite-dimensional Lie algebra: finding a Levi subalgebra of a Lie algebra.]]> Wed, 01 Jan 1997 07:00:00 +0000 https://doi.org/10.46298/dmtcs.242 https://doi.org/10.46298/dmtcs.242 Cohen, A. M. Graaf, W. A., Rónyai, L. Cohen, A. M. Graaf, W. A., Rónyai, L. <![CDATA[This paper describes progress made in context with the construction of a general library of Lie algebra algorithms, called ELIAS (Eindhoven Lie Algebra System), within the computer algebra package GAP. A first sketch of the package can be found in Cohen and de Graaf[1]. Since then, in a collaborative effort with G. Ivanyos, the authors have continued to develop algorithms which were implemented in ELIAS by the second author. These activities are part of a bigger project, called ACELA and financed by STW, the Dutch Technology Foundation, which aims at an interactive book on Lie algebras (cf. Cohen and Meertens [2]). This paper gives a global description of the main ways in which to present Lie algebras on a computer. We focus on the transition from a Lie algebra abstractly given by an array of structure constants to a Lie algebra presented as a subalgebra of the Lie algebra of n×n matrices. We describe an algorithm typical of the structure analysis of a finite-dimensional Lie algebra: finding a Levi subalgebra of a Lie algebra.]]> 0 <![CDATA[An elementary introduction to the classical version of gauge theories is made. The shortcomings of the usual gauge fixing process are pointed out. They justify the need to replace it by a global symmetry: the BRST symmetry and its associated BRST charge. The main mathematical steps required to construct it are described. The algebra of constraints is, in general, a nonlinear Poisson algebra. In the nonlinear case the computation of the BRST charge by hand is hard. Itis explained how this computation can be made algorithmic. The main features of a recently created BRST computer algebra program are described. It can handle quadratic algebras very easily. Its capability to compute the BRST charge as a formal power series in the generic case of a cubic algebra is illustrated.]]> Wed, 01 Jan 1997 07:00:00 +0000 https://doi.org/10.46298/dmtcs.241 https://doi.org/10.46298/dmtcs.241 Caprasse, H. Caprasse, H. <![CDATA[An elementary introduction to the classical version of gauge theories is made. The shortcomings of the usual gauge fixing process are pointed out. They justify the need to replace it by a global symmetry: the BRST symmetry and its associated BRST charge. The main mathematical steps required to construct it are described. The algebra of constraints is, in general, a nonlinear Poisson algebra. In the nonlinear case the computation of the BRST charge by hand is hard. Itis explained how this computation can be made algorithmic. The main features of a recently created BRST computer algebra program are described. It can handle quadratic algebras very easily. Its capability to compute the BRST charge as a formal power series in the generic case of a cubic algebra is illustrated.]]> 0 <![CDATA[The goal of this paper is both to give an E-unification procedure that always terminates, and to decide unifiability. For this, we assume that the equational theory is specified by a confluent and constructor-based rewrite system, and that four additional restrictions are satisfied. We give a procedure that represents the (possibly infinite) set of solutions thanks to a tree tuple synchronized grammar, and that can decide upon unifiability thanks to an emptiness test. Moreover, we show that if only three of the four additional restrictions are satisfied then unifiability is undecidable.]]> Wed, 01 Jan 1997 07:00:00 +0000 https://doi.org/10.46298/dmtcs.240 https://doi.org/10.46298/dmtcs.240 Limet, Sébastien Réty, Pierre Limet, Sébastien Réty, Pierre <![CDATA[The goal of this paper is both to give an E-unification procedure that always terminates, and to decide unifiability. For this, we assume that the equational theory is specified by a confluent and constructor-based rewrite system, and that four additional restrictions are satisfied. We give a procedure that represents the (possibly infinite) set of solutions thanks to a tree tuple synchronized grammar, and that can decide upon unifiability thanks to an emptiness test. Moreover, we show that if only three of the four additional restrictions are satisfied then unifiability is undecidable.]]> 0 <![CDATA[This paper presents a new proof of the hook-length formula, which computes the number of standard Young tableaux of a given shape. After recalling the basic definitions, we present two inverse algorithms giving the desired bijection. The next part of the paper presents the proof of the bijectivity of our construction. The paper concludes with some examples.]]> Wed, 01 Jan 1997 07:00:00 +0000 https://doi.org/10.46298/dmtcs.239 https://doi.org/10.46298/dmtcs.239 Novelli, Jean-Christophe Pak, Igor Stoyanovskii, Alexander V. Novelli, Jean-Christophe Pak, Igor Stoyanovskii, Alexander V. <![CDATA[This paper presents a new proof of the hook-length formula, which computes the number of standard Young tableaux of a given shape. After recalling the basic definitions, we present two inverse algorithms giving the desired bijection. The next part of the paper presents the proof of the bijectivity of our construction. The paper concludes with some examples.]]> 0 <![CDATA[It is shown that there is a non-Hamiltonian vector field that provides a Lie algebraic connection between Hamiltonian and Lagrangian optics. With the aid of this connection, geometrical optics can be formulated in such a way that all aberrations are attributed to ray transformations occurring only at lens surfaces. That is, in this formulation there are no aberrations arising from simple transit in a uniform medium. The price to be paid for this formulation is that the Lie algebra of Hamiltonian vector fields must be enlarged to include certain non-Hamiltonian vector fields. It is shown that three such vector fields are required at the level of third-order aberrations, and sufficient machinery is developed to generalize these results to higher order.]]> Wed, 01 Jan 1997 07:00:00 +0000 https://doi.org/10.46298/dmtcs.238 https://doi.org/10.46298/dmtcs.238 Dragt, Alex J. Dragt, Alex J. <![CDATA[It is shown that there is a non-Hamiltonian vector field that provides a Lie algebraic connection between Hamiltonian and Lagrangian optics. With the aid of this connection, geometrical optics can be formulated in such a way that all aberrations are attributed to ray transformations occurring only at lens surfaces. That is, in this formulation there are no aberrations arising from simple transit in a uniform medium. The price to be paid for this formulation is that the Lie algebra of Hamiltonian vector fields must be enlarged to include certain non-Hamiltonian vector fields. It is shown that three such vector fields are required at the level of third-order aberrations, and sufficient machinery is developed to generalize these results to higher order.]]> 0 <![CDATA[Noncommutative algebras, defined by the generators and relations, are considered. The definition and main results connected with the Gröbner basis, Hilbert series and Anick's resolution are formulated. Most attention is paid to universal enveloping algebras. Four main examples illustrate the main concepts and ideas. Algorithmic problems arising in the calculation of the Hilbert series are investigated. The existence of finite state automata, defining thebehaviour of the Hilbert series, is discussed. The extensions of the BERGMAN package for IBM PC compatible computers are described. A table is provided permitting a comparison of the effectiveness of the calculations in BERGMAN with the other systems.]]> Wed, 01 Jan 1997 07:00:00 +0000 https://doi.org/10.46298/dmtcs.237 https://doi.org/10.46298/dmtcs.237 Cojocaru, S. Ufnarovski, V. Cojocaru, S. Ufnarovski, V. <![CDATA[Noncommutative algebras, defined by the generators and relations, are considered. The definition and main results connected with the Gröbner basis, Hilbert series and Anick's resolution are formulated. Most attention is paid to universal enveloping algebras. Four main examples illustrate the main concepts and ideas. Algorithmic problems arising in the calculation of the Hilbert series are investigated. The existence of finite state automata, defining thebehaviour of the Hilbert series, is discussed. The extensions of the BERGMAN package for IBM PC compatible computers are described. A table is provided permitting a comparison of the effectiveness of the calculations in BERGMAN with the other systems.]]> 0 <![CDATA[We first give a fast algorithm to compute the maximal Lyndon word (with respect to lexicographic order) of \textitLy_α (A) for every given multidegree alpha in \textbfN^k. We then give an algorithm to compute all the words living in \textitLy_α (A) for any given α in \textbfN^k. The best known method for generating Lyndon words is that of Duval [1], which gives a way to go from every Lyndon word of length n to its successor (with respect to lexicographic order by length), in space and worst case time complexity O(n). Finally, we give a simple algorithm which uses Duval's method (the one above) to compute the next standard bracketing of a Lyndon word for lexicographic order by length. We can find an interesting application of this algorithm in control theory, where one wants to compute within the command Lie algebra of a dynamical system (letters are actually vector fields).]]> Wed, 01 Jan 1997 07:00:00 +0000 https://doi.org/10.46298/dmtcs.236 https://doi.org/10.46298/dmtcs.236 Andary, Philippe Andary, Philippe <![CDATA[We first give a fast algorithm to compute the maximal Lyndon word (with respect to lexicographic order) of \textitLy_α (A) for every given multidegree alpha in \textbfN^k. We then give an algorithm to compute all the words living in \textitLy_α (A) for any given α in \textbfN^k. The best known method for generating Lyndon words is that of Duval [1], which gives a way to go from every Lyndon word of length n to its successor (with respect to lexicographic order by length), in space and worst case time complexity O(n). Finally, we give a simple algorithm which uses Duval's method (the one above) to compute the next standard bracketing of a Lyndon word for lexicographic order by length. We can find an interesting application of this algorithm in control theory, where one wants to compute within the command Lie algebra of a dynamical system (letters are actually vector fields).]]> 0 <![CDATA[This special issue is an outgrowth of the MEDICIS thematic workshop on Lie Computations that was held at the Centre International de Rencontres Mathématiques in Marseilles in November 1994. It was jointly sponsored by the Groupe de Recherche MEDICIS, the CIRM (Société Mathématique de France), and the European project INTAS 93-30.]]> Wed, 01 Jan 1997 07:00:00 +0000 https://doi.org/10.46298/dmtcs.235 https://doi.org/10.46298/dmtcs.235 Jacob, Gérard Koseleff, Pierre-Vincent Jacob, Gérard Koseleff, Pierre-Vincent <![CDATA[This special issue is an outgrowth of the MEDICIS thematic workshop on Lie Computations that was held at the Centre International de Rencontres Mathématiques in Marseilles in November 1994. It was jointly sponsored by the Groupe de Recherche MEDICIS, the CIRM (Société Mathématique de France), and the European project INTAS 93-30.]]> 0 <![CDATA[A nilpotent quotient algorithm for finitely presented Lie rings over \textbfZ (and \textbfQ) is described. The paper studies the graded and non-graded cases separately. The algorithm computes the so-called nilpotent presentation for a finitely presented, nilpotent Lie ring. A nilpotent presentation consists of generators for the abelian group and the products expressed as linear combinations for pairs formed by generators. Using that presentation the word problem is decidable in L. Provided that the Lie ring L is graded, it is possible to determine the canonical presentation for a lower central factor of L. Complexity is studied and it is shown that optimising the presentation is NP-hard. Computational details are provided with examples, timing and some structure theorems obtained from computations. Implementation in C and GAP interface are available.]]> Wed, 01 Jan 1997 07:00:00 +0000 https://doi.org/10.46298/dmtcs.234 https://doi.org/10.46298/dmtcs.234 Schneider, Csaba Schneider, Csaba <![CDATA[A nilpotent quotient algorithm for finitely presented Lie rings over \textbfZ (and \textbfQ) is described. The paper studies the graded and non-graded cases separately. The algorithm computes the so-called nilpotent presentation for a finitely presented, nilpotent Lie ring. A nilpotent presentation consists of generators for the abelian group and the products expressed as linear combinations for pairs formed by generators. Using that presentation the word problem is decidable in L. Provided that the Lie ring L is graded, it is possible to determine the canonical presentation for a lower central factor of L. Complexity is studied and it is shown that optimising the presentation is NP-hard. Computational details are provided with examples, timing and some structure theorems obtained from computations. Implementation in C and GAP interface are available.]]> 0 <![CDATA[We compute the Lyndon factorization of the Thue-Morse word. We also compute the Lyndon factorization of two related sequences involving morphisms that give rise to new presentations of these sequences.]]> Wed, 01 Jan 1997 07:00:00 +0000 https://doi.org/10.46298/dmtcs.233 https://doi.org/10.46298/dmtcs.233 Ido, Augustin Melançon, Guy Ido, Augustin Melançon, Guy <![CDATA[We compute the Lyndon factorization of the Thue-Morse word. We also compute the Lyndon factorization of two related sequences involving morphisms that give rise to new presentations of these sequences.]]> 0 <![CDATA[We study the P_4-tidy graphs, a new class defined by Rusu [30] in order to illustrate the notion of P_4-domination in perfect graphs. This class strictly contains the P_4-extendible graphs and the P_4-lite graphs defined by Jamison & Olariu in [19] and [23] and we show that the P_4-tidy graphs and P_4-lite graphs are closely related. Note that the class of P_4-lite graphs is a class of brittle graphs strictly containing the P_4-sparse graphs defined by Hoang in [14]. McConnel & Spinrad [2] and independently Cournier & Habib [5] have shown that the modular decomposition tree of any graph is computable in linear time. For recognizing in linear time P_4-tidy graphs, we apply a method introduced by Giakoumakis in [9] and Giakoumakis & Fouquet in [6] using modular decomposition of graphs and we propose linear algorithms for optimization problems on such graphs, as clique number, stability number, chromatic number and scattering number. We show that the Hamiltonian Path Problem is linear for this class of graphs. Our study unifies and generalizes previous results of Jamison & Olariu ([18], [21], [22]), Hochstattler & Schindler[16], Jung [25] and Hochstattler & Tinhofer [15].]]> Wed, 01 Jan 1997 07:00:00 +0000 https://doi.org/10.46298/dmtcs.232 https://doi.org/10.46298/dmtcs.232 Giakoumakis, V. Roussel, F. Thuillier, H. Giakoumakis, V. Roussel, F. Thuillier, H. <![CDATA[We study the P_4-tidy graphs, a new class defined by Rusu [30] in order to illustrate the notion of P_4-domination in perfect graphs. This class strictly contains the P_4-extendible graphs and the P_4-lite graphs defined by Jamison & Olariu in [19] and [23] and we show that the P_4-tidy graphs and P_4-lite graphs are closely related. Note that the class of P_4-lite graphs is a class of brittle graphs strictly containing the P_4-sparse graphs defined by Hoang in [14]. McConnel & Spinrad [2] and independently Cournier & Habib [5] have shown that the modular decomposition tree of any graph is computable in linear time. For recognizing in linear time P_4-tidy graphs, we apply a method introduced by Giakoumakis in [9] and Giakoumakis & Fouquet in [6] using modular decomposition of graphs and we propose linear algorithms for optimization problems on such graphs, as clique number, stability number, chromatic number and scattering number. We show that the Hamiltonian Path Problem is linear for this class of graphs. Our study unifies and generalizes previous results of Jamison & Olariu ([18], [21], [22]), Hochstattler & Schindler[16], Jung [25] and Hochstattler & Tinhofer [15].]]> 0