0,3

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,.... - A. O. Munagi (amunagi(AT)yahoo.com), Apr 10 2005

A. O. Munagi, Set Partitions with Successions and Separations, Int. J. Math and Math. Sc. 2005, no. 3 (2005), 451-463.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 865

S. Kitaev and T. Mansour, Simultaneous avoidance of generalized patterns.

A. O. Munagi, Set Partitions with Successions and Separations,IJMMS 2005:3 (2005),451-463.

S. Kitaev, Generalized pattern avoidance with additional restrictions, Sem. Lothar. Combinat. B48e (2003).

E.g.f.: exp(exp(x)-1)*x

a(n) = n*Bell(n-1). - Vladeta Jovovic (vladeta(AT)eunet.rs), Sep 14 2003

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.

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)

[seq((n+1)*bell(n), n=0..27)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 08 2006

a:=n->sum(bell(n), j=0..n): seq(a(n), n=-1..19); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 25 2007

a:=n->sum(numbcomb (n, 0)*bell(n), j=0..n): seq(a(n), n=0..21); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 25 2007

a:=n->sum(sum(stirling2(n, k), j=0..n), k=0..n): seq(a(n), n=-1..19); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 11 2007

Table[Sum[BellB[n, 1], {i, 0, n}], {n, -1, 19}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 16 2009]

Second column of triangle A033306.

Sequence in context: A150168 A145870 A134957 this_sequence A150169 A083691 A150170

Adjacent sequences: A052886 A052887 A052888 this_sequence A052890 A052891 A052892

easy,nonn

encyclopedia(AT)pommard.inria.fr, Jan 25 2000