A109062 Triangle read by rows: number of atomic set compositions of size n and length k (see description in A095989) 1 <= k <= n.

1, 1, 1, 1, 4, 3, 1, 11, 23, 13, 1, 26, 112, 158, 71, 1, 57, 446, 1170, 1241, 461, 1, 120, 1593, 6880, 12871, 10912, 3447, 1, 247, 5337, 35503, 103887, 150413, 106031, 29093, 1, 502, 17190, 168982, 724148, 1589266, 1872286, 1128218, 273343, 1, 1013, 54008

Offset: 1

Author

Mike Zabrocki, Aug 24 2005

Keywords

• nonn
• tabl

Comments

Also the number of free generators and primitives of the quasi-symmetric functions in non-commuting variables. - Mike Zabrocki, Aug 06 2006

Triangle given by [1,0,2,0,3,0,4,0,5,...] DELTA [1,2,2,3,3,4,4,5,5,6,6,7,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 01 2007

Apparently, the alternating sums vanish for n > 1. - F. Chapoton, Sep 05 2023

Examples

			Atomic set compositions a(1,1)=1: [{1}]; a(2,1)=1, a(2,2)=1: [{12}], [{2},{1}]; a(3,1)=1, a(3,2)=4, a(3,3)=3: [{123}], [{2},{13}], [{3}, {12}], [{23}, {1}], [{13},{2}], [{2},{3},{1}], [{3},{1},{2}], [{3},{2},{1}].
Triangle begins:
  1;
  1,  1;
  1,  4,   3;
  1, 11,  23,  13;
  1, 26, 112, 158, 71;
  ...
		

Links

• N. Bergeron and M. Zabrocki, The Hopf algebras of symmetric functions and quasisymmetric functions in non-commutative variables are free and cofree, arXiv:math/0509265 [math.CO], 2005.
• Ming-Jian Ding and Bao-Xuan Zhu, Some results related to Hurwitz stability of combinatorial polynomials, Advances in Applied Mathematics, Volume 152, (2024), 102591. See p. 7.

Crossrefs

Row sums are equal to A095989, a(n,n) = A003319, a(n,2) = A000295.

Cf. A095989, A059438, A074664, A087903, A008277, A019538.

Programs

Maple

f:=(n,k)->coeff(coeff(series(1-1/(1+add(add(q^m*t^i*
    Stirling2(m,i)*i!,i=1..m),m=1..n)),q,n+1),q,n),t,k):
seq(seq(f(n,k), k=1..n), n=1..10);

Formula

G.f.: 1-1/(1+Sum_{n>=1} Sum_{k=1..n} q^n*t^k*Stirling2(n,k)*k!).