OFFSET
0,3
COMMENTS
Total number of blocks of size one in all set partitions of set {1..n}. - Wouter Meeussen, Jul 28 2003
With offset 1, number of permutations beginning with 12 and avoiding 12-3.
a(n) = number of partitions of {1...n+1} containing exactly one pair of consecutive integers, counted within a block. With offset t-1, number of partitions of {1...N} containing one string of t consecutive integers, where N=n+j, t=2+j, j = 0,1,2,.... - Augustine O. Munagi, Apr 10 2005
LINKS
Robert Israel, Table of n, a(n) for n = 0..575
Adam M. Goyt and Lara K. Pudwell, Avoiding colored partitions of two elements in the pattern sense, arXiv preprint arXiv:1203.3786 [math.CO], 2012. - From N. J. A. Sloane, Sep 17 2012
Jia Huang and Erkko Lehtonen, Associative-commutative spectra for some varieties of groupoids, arXiv:2401.15786 [math.CO], 2024. See p. 13.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 865
Sergey Kitaev, Generalized pattern avoidance with additional restrictions, Sem. Lothar. Combinat. B48e (2003); arXiv:math/0205215 [math.CO].
Sergey Kitaev and Toufik Mansour, Simultaneous avoidance of generalized patterns, arXiv:math/0205182 [math.CO], 2002.
Augustine O. Munagi, Set Partitions with Successions and Separations, IJMMS 2005:3 (2005),451-463.
FORMULA
E.g.f.: x*exp(exp(x)-1).
a(n) = n*A000110(n-1). - Vladeta Jovovic, Sep 14 2003
Let A be the upper Hessenberg matrix of order n defined by: A[i,i-1]=-1, A[i,j]=binomial(j-1,i-1), (i<=j), and A[i,j]=0 otherwise. Then, for n>=1, a(n)=(-1)^(n-1)*coeff(charpoly(A,x),x). - Milan Janjic, Jul 08 2010
EXAMPLE
a(3) = 6 because the partitions of {1, 2, 3, 4} containing a pair of consecutive integers are 124/3, 134/2, 14/23, 12/3/4, 1/23/4, 1/2/34.
MAPLE
spec := [S, {B=Set(C), C=Set(Z, 1 <= card), S=Prod(Z, B)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
Explanation of above combstruct commands using generating functions, from Mitch Harris, Jul 28 2003:
Z = an atom (each atom used is labeled), gf: Z(x) = x
C = Set(Z, card <= 1) is the set of positive integers; gf: C(x) = e^(Z(x)) - 1 = e^x - 1 (the -1 removes the empty set); [x^n]C = 1 means there is exactly one set with n atoms since each atom is labeled
B = Set(C) the set of (ordered) sets of integers = ordered set partitions; gf: B(x) = e^C(x) = e^(e^x - 1)
S = Prod(Z, B) pairs of an atom (Z) and an ordered set partition = an ordered set partition with an adjoining single atom. The adjoining atom corresponds to choosing a "root" in the partition; gf: S(x) = x B(x) = x*e^(e^x-1)
A052889 := n -> `if`(n=0, 0, n*combinat[bell](n-1)):
seq(A052889(n), n=0..20); # Peter Luschny, Apr 19 2011
MATHEMATICA
Range[0, 20]! CoefficientList[Series[ x Exp[Exp[x]-1], {x, 0, 20}], x] (* Geoffrey Critzer, Nov 25 2011 *)
Table[If[n==0, 0, n*BellB[n-1]], {n, 0, 30}] (* G. C. Greubel, May 11 2024 *)
PROG
(Magma) [n eq 0 select 0 else n*Bell(n-1): n in [0..30]]; // G. C. Greubel, May 11 2024
(SageMath) [0]+[n*bell_number(n-1) for n in range(1, 31)] # G. C. Greubel, May 11 2024
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
STATUS
approved