cp's OEIS Frontend

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

%I A109062 #27 Sep 05 2023 15:02:22
%S A109062 1,1,1,1,4,3,1,11,23,13,1,26,112,158,71,1,57,446,1170,1241,461,1,120,
%T A109062 1593,6880,12871,10912,3447,1,247,5337,35503,103887,150413,106031,
%U A109062 29093,1,502,17190,168982,724148,1589266,1872286,1128218,273343,1,1013,54008
%N A109062 Triangle read by rows: number of atomic set compositions of size n and length k (see description in A095989) 1 <= k <= n.
%C A109062 Also the number of free generators and primitives of the quasi-symmetric functions in non-commuting variables. - _Mike Zabrocki_, Aug 06 2006
%C A109062 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
%C A109062 Apparently, the alternating sums vanish for n > 1. - _F. Chapoton_, Sep 05 2023
%H A109062 N. Bergeron and M. Zabrocki, <a href="https://arxiv.org/abs/math/0509265">The Hopf algebras of symmetric functions and quasisymmetric functions in non-commutative variables are free and cofree</a>, arXiv:math/0509265 [math.CO], 2005.
%H A109062 Ming-Jian Ding and Bao-Xuan Zhu, <a href="https://doi.org/10.1016/j.aam.2023.102591">Some results related to Hurwitz stability of combinatorial polynomials</a>, Advances in Applied Mathematics, Volume 152, (2024), 102591. See p. 7.
%F A109062 G.f.: 1-1/(1+Sum_{n>=1} Sum_{k=1..n} q^n*t^k*Stirling2(n,k)*k!).
%e A109062 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}].
%e A109062 Triangle begins:
%e A109062   1;
%e A109062   1,  1;
%e A109062   1,  4,   3;
%e A109062   1, 11,  23,  13;
%e A109062   1, 26, 112, 158, 71;
%e A109062   ...
%p A109062 f:=(n,k)->coeff(coeff(series(1-1/(1+add(add(q^m*t^i*
%p A109062     Stirling2(m,i)*i!,i=1..m),m=1..n)),q,n+1),q,n),t,k):
%p A109062 seq(seq(f(n,k), k=1..n), n=1..10);
%Y A109062 Row sums are equal to A095989, a(n,n) = A003319, a(n,2) = A000295.
%Y A109062 Cf. A095989, A059438, A074664, A087903, A008277, A019538.
%K A109062 nonn,tabl
%O A109062 1,5
%A A109062 _Mike Zabrocki_, Aug 24 2005