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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}.
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%I #13 Feb 06 2024 11:33:42

%S 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 20181,118125,84035,5103,7,1,1016,111132,1668296,3543750,941192,20412,

%U 8,1,2295,580644,21003948,116363646,95681250,9882516,78732,9

%N 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 The rank of a poset is the number of cover relations in a maximal chain.

%C Equivalently, T(n,k) is the number of labeled posets P on [n] of rank at most one such that |image(P)| = k.

%F E.g.f.: Sum_{n>=0} x^n/n!*exp(y*x)^(2^n-1).

%F T(n,1) = A058877(n).

%e Triangle begins

%e 1;

%e 1;

%e 1, 2;

%e 1, 9, 3;

%e 1, 28, 54, 4;

%e 1, 75, 490, 270, 5;

%e 1, 186, 3375, 6860, 1215, 6;

%e ...

%t 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 Cf. A001831 (row sums), A058877, A263859, A369921.

%K nonn,tabl

%O 0,4

%A _Geoffrey Critzer_, Feb 05 2024