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A381058 Irregular triangular array read by rows. Let S_n be the set of labeled graphs G on [n] with 2-colored nodes where black nodes are only connected to white nodes and vice versa. Orient the edges in each such graph G from black to white. T(n,k) is the number of graphs in S_n containing exactly k descents, n>=0, 0<=k<=A002620(n).

%I A381058 #21 Feb 17 2025 03:22:46
%S A381058 1,2,5,1,16,8,2,67,56,30,8,1,374,436,358,188,68,16,2,2825,4143,4508,
%T A381058 3460,2032,924,320,80,13,1,29212,50460,66976,66092,52412,34280,18630,
%U A381058 8376,3072,892,194,28,2,417199,811790,1246486,1471358,1436404,1195166,859650,537750,292880,138280,56048,19168,5382,1188,192,20,1
%N A381058 Irregular triangular array read by rows. Let S_n be the set of labeled graphs G on [n] with 2-colored nodes where black nodes are only connected to white nodes and vice versa.  Orient the edges in each such graph G from black to white.  T(n,k) is the number of graphs in S_n containing exactly k descents, n>=0, 0<=k<=A002620(n).
%C A381058 A descent in a labeled directed graph is an edge s->t such that s>t.
%C A381058 T(n,0) = A006116(n).
%H A381058 Kassie Archer, Ira M. Gessel, Christina Graves, and Xuming Liang, <a href="https://arxiv.org/abs/1909.01550">Counting acyclic and strong digraphs by descents</a>, arXiv:1909.01550 [math.CO], 20 Mar 2020.
%e A381058     1;
%e A381058     2;
%e A381058     5,    1;
%e A381058    16,    8,    2;
%e A381058    67,   56,   30,    8,    1;
%e A381058   374,  436,  358,  188,   68,  16,   2;
%e A381058  2825, 4143, 4508, 3460, 2032, 924, 320, 80, 13, 1;
%e A381058  ...
%t A381058 nn = 7; B[n_] := FunctionExpand[QFactorial[n, (1 + u y)/(1 + y)]] (1+y)^Binomial[n,2]; e[z_] := Sum[z^n/B[n], {n, 0, nn}];Map[CoefficientList[#, u] &,  Table[B[n], {n, 0, nn}] CoefficientList[Series[e[z]^2, {z, 0, nn}],z] /. y -> 1] // Grid
%Y A381058 Cf. A228890, A047863 (row sums), A006116, A111636, A381102, A381192.
%K A381058 nonn,tabf
%O A381058 0,2
%A A381058 _Geoffrey Critzer_, Feb 12 2025