Abstract
Three new methods for proving log-convexity of combinatorial sequences are presented. Their implementation is demonstrated and their performance is compared with four more familiar approaches in the context of sequences that enumerate various classes of lattice paths.
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References
Aigner, M.: Motzkin numbers. Eur. J. Comb. 19, 663–675 (1998)
André, D.: Solution directe du problème résolu par M. Bertrand. C.R. Math. Acad. Sci. Paris 105, 436–437 (1887)
Angulo, J.C., Dehesa, J.S.: Atomic-charge log-convexity and radial expectation values. J. Phys. B 24, L299–L306 (1991)
Artin, E.: The Gamma Function. Holt, Rinehart and Winston, New York (1964)
Asai, N., Kubo, I., Kuo, H.-H.: Bell numbers, log-concavity, and log-convexity. Acta Appl. Math. 63, 79–87 (2000)
Asai, N., Kubo, I., Kuo, H.-H.: Roles of log-concavity, log-convexity, and growth order in white noise analysis. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4, 59–84 (2001)
Banderier, C., Flajolet, P.: Basic analytic combinatorics of directed lattice paths. Theor. Comput. Sci. 281, 37–80 (2002)
Banderier, C., Schwer, S.: Why Delannoy numbers. J. Stat. Plan. Inference 135, 40–54 (2005)
Barcucci, E., Pinzani, R., Sprugnoli, R.: The Motzkin family. Pure Math. Appl. Ser. A 2, 249–279 (1991)
Baryshnikov, Yu.: IT security investment and Gordon-Loeb’s 1/e rule. Preprint
Bender, E.A., Canfield, E.R.: Log-concavity and related properties of the cycle index polynomials. J. Comb. Theory A 74, 56–70 (1996)
Boche, H., Stanczak, S.: Log-convexity of the minimum total power in CDMA systems with certain quality-of-service guaranteed. IEEE Trans. Inf. Theory 51, 374–381 (2005)
Bohr, H., Mollerup, J.: Lærebog i Matematisk Analyse, vol. III. Jul. Gjellerups Forlag, Copenhagen (1922)
Bóna, M., Ehrenborg, R.: A combinatorial proof of the log-concavity of the numbers of permutations with k runs. J. Comb. Theory A 90, 293–303 (2000)
Brenti, F.: Unimodal, Log-Concave and Pólya Frequency Sequences in Combinatorics. Am. Math. Soc., Providence (1989)
Callan, D.: Notes on Motzkin and Schröder numbers. Preprint (2000)
Catalan, E.: Note sur une équation aux différences finies. J. Math. Pures Appl. 3, 508–516 (1838)
Catalan, E.: Sur les nombres de Segner. Rend. Circ. Mat. Palermo 1, 190–201 (1887)
Deutsch, E.: Dyck path enumeration. Discrete Math. 204, 167–202 (1999)
Došlić, T.: Log-balanced combinatorial sequences. Int. J. Math. Math. Sci. 2005(4), 507–522 (2005)
Došlić, T., Veljan, D.: Calculus proofs of some combinatorial inequalities. Math. Inequal. Appl. 6, 197–210 (2003)
Došlić, T., Veljan, D.: Logarithmic behavior of some combinatorial sequences. Discrete Math. 308, 2182–2212 (2008)
Došlić, T., Svrtan, D., Veljan, D.: Enumerative aspects of secondary structures. Discrete Math. 285, 67–82 (2004)
Dulucq, S., Penaud, J.-G.: Interprétation bijective d’une récurrence des nombres de Motzkin. Discrete Math. 256, 671–676 (2002)
Klauder, J.R., Penson, K.A., Sixdeniers, J.M.: Constructing coherent states through solutions of Stieltjes and Hausdorff moment problems. Phys. Rev. A 64, 013817–013835 (2001)
Koshy, T.: Catalan Numbers with Applications. Oxford University Press, Oxford (2008)
Kuo, H.H.: White Noise Distribution Theory. Probability and Stochastic Series. CRC Press, Boca Raton (1996)
Kurtz, D.C.: A note on concavity properties of triangular arrays of numbers. J. Comb. Theory A 13, 135–159 (1972)
Liu, L.L., Wang, Y.: On the log-convexity of combinatorial sequences. Adv. Appl. Math. 39, 453–476 (2007)
Milenkovic, O., Compton, K.J.: Probabilistic transforms for combinatorial urn models. Comb. Probab. Comput. 13, 645–675 (2004)
Mohanty, S.G.: Lattice Path Counting and Applications. Academic Press, San Diego (1979)
Motzkin, T.: Relation between hypersurface cross ratios, and a combinatorial formula for partitions of a polygon, for permanental preponderance and for non-associative products. Bull. Am. Math. Soc. 54, 352–360 (1948)
Munarini, E., Zagaglia Salvi, N.: Binary strings without zigzags. In: Seminaire Lotharingien de Combinatoire, B49h (2004), 15 pp.
Narayana, T.V.: Lattice Path Combinatorics with Statistical Applications. Toronto Univ. Press, Toronto (1979)
Netto, E.: Kombinatorik. In: Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen. Teubner, Leipzig (1898)
Penson, K.A., Solomon, A.I.: Coherent states from combinatorial sequences. In: Quantum Theory and Symmetries (Krakow 2001), pp. 527–530. World Scientific, Singapore (2002)
Prelec, D.: Decreasing impatience: a criterion for non-stationary time preference and ‘hyperbolic’ discounting. Scand. J. Econ. 106, 511–532 (2004)
Riordan, J.: Percy Alexander MacMahon: collected papers, book review. Bull. Am. Math. Soc. New Ser. 2, 239–241 (1980)
Sagan, B.: Inductive and injective proofs of log-concavity results. Discrete Math. 68, 281–292 (1988)
Schröder, E.: Vier kombinatorische Probleme. Z. Math. Phys. 15, 361–376 (1870)
Schuster, P., Stadler, P.F.: Discrete models of biopolymers. In: Crabbe, J., Konopka, A., Drew, M. (eds.) Handbook of Computational Chemistry and Biology, pp. 187–221. Dekker, New York (2004)
Schwer, S.R., Autebert, J.-M.: Henri-Auguste Delannoy, une biographie. Math. Soc. Sci. 43, 25–67 (2006)
Shapiro, L.W., Sulanke, R.A.: Bijections for Schröder numbers. Math. Mag. 73, 369–376 (2000)
Sloane, N.J.A.: The on-line encyclopedia of integer sequences. Published electronically at http://www.research.att.com/~njas/sequences/index.html
Stanley, R.P.: Log-concave and unimodal sequences in algebra, combinatorics and geometry. Ann. N.Y. Acad. Sci. 576, 500–535 (1989)
Stanley, R.P.: Hipparchus, Plutarch, Schröder, and Hough. Am. Math. Mon. 104, 344–350 (1997)
Stanley, R.P.: Enumerative Combinatorics, vol. 2. Cambridge Univ. Press, Cambridge (1999)
Stanley, R.P.: Catalan addendum. Available at http://www-math.mit.edu/~rstan/ec/catadd.pdf
Stein, P.R., Waterman, M.S.: On some new sequences generalizing the Catalan and Motzkin numbers. Discrete Math. 26, 261–272 (1979)
Sulanke, R.A.: Objects counted by the central Delannoy numbers. J. Integer Seq. 3, 00.2.1 (2000)
Szegö, G.: Orthogonal Polynomials. AMS, New York (1959)
Takacs, L.: On the ballot theorems. In: Balakrishnan, N. (ed.) Advances in Combinatorial Methods and Applications to Probability and Statistics, pp. 98–114. Birkhäuser, Boston (1979)
Warde, W.D., Katti, S.K.: Infinite divisibility of discrete distributions II. Ann. Math. Stat. 42, 1088–1090 (1964)
Weiss, M.: Theorems on log-convex disposition curves in drug and tracer kinetics. J. Theor. Biol. 116, 355–368 (1985)
Weiss, M.: Generalizations in linear pharmacokinetics using properties of certain classes of residence time distributions. I. Log-convex drug disposition curves. J. Pharmacokinet. Biopharm. 14, 635–657 (1986)
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Došlić, T. Seven (Lattice) Paths to Log-Convexity. Acta Appl Math 110, 1373–1392 (2010). https://doi.org/10.1007/s10440-009-9515-4
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DOI: https://doi.org/10.1007/s10440-009-9515-4
Keywords
- Log-convexity
- Integer sequences
- Recurrences
- Motzkin numbers
- Catalan numbers
- Schröder numbers
- Delannoy numbers
- Lattice paths