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A369919 Triangular array read by rows. T(n,k) is the number of labeled posets on [n] of rank at most one with exactly k elements of positive indegree, n >= 0, 0 <= k <= max{0,n-1}.

%I A369919 #13 Feb 06 2024 11:33:42
%S A369919 1,1,1,2,1,9,3,1,28,54,4,1,75,490,270,5,1,186,3375,6860,1215,6,1,441,
%T A369919 20181,118125,84035,5103,7,1,1016,111132,1668296,3543750,941192,20412,
%U A369919 8,1,2295,580644,21003948,116363646,95681250,9882516,78732,9
%N A369919 Triangular array read by rows.  T(n,k) is the number of labeled posets on [n] of rank at most one with exactly k elements of positive indegree, n >= 0, 0 <= k <= max{0,n-1}.
%C A369919 The rank of a poset is the number of cover relations in a maximal chain.
%C A369919 Equivalently, T(n,k) is the number of labeled posets P on [n] of rank at most one such that |image(P)| = k.
%F A369919 E.g.f.: Sum_{n>=0} x^n/n!*exp(y*x)^(2^n-1).
%F A369919 T(n,1) = A058877(n).
%e A369919 Triangle begins
%e A369919   1;
%e A369919   1;
%e A369919   1,   2;
%e A369919   1,   9,    3;
%e A369919   1,  28,   54,    4;
%e A369919   1,  75,  490,  270,    5;
%e A369919   1, 186, 3375, 6860, 1215, 6;
%e A369919   ...
%t A369919 nn = 9; Map[Select[#, # > 0 &] &,Table[n!, {n, 0, nn}] CoefficientList[Series[ Sum[ Exp[y  x]^(2^n - 1)  x^n/n!, {n, 0, nn}], {x, 0, nn}], {x, y}]] // Grid
%Y A369919 Cf. A001831 (row sums), A058877, A263859, A369921.
%K A369919 nonn,tabl
%O A369919 0,4
%A A369919 _Geoffrey Critzer_, Feb 05 2024