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).

1, 2, 5, 1, 16, 8, 2, 67, 56, 30, 8, 1, 374, 436, 358, 188, 68, 16, 2, 2825, 4143, 4508, 3460, 2032, 924, 320, 80, 13, 1, 29212, 50460, 66976, 66092, 52412, 34280, 18630, 8376, 3072, 892, 194, 28, 2, 417199, 811790, 1246486, 1471358, 1436404, 1195166, 859650, 537750, 292880, 138280, 56048, 19168, 5382, 1188, 192, 20, 1

Offset: 0

Author

Geoffrey Critzer, Feb 12 2025

Keywords

• nonn
• tabf

Comments

A descent in a labeled directed graph is an edge s->t such that s>t.

T(n,0) = A006116(n).

Examples

			    1;
    2;
    5,    1;
   16,    8,    2;
   67,   56,   30,    8,    1;
  374,  436,  358,  188,   68,  16,   2;
 2825, 4143, 4508, 3460, 2032, 924, 320, 80, 13, 1;
 ...
		

Links

• Kassie Archer, Ira M. Gessel, Christina Graves, and Xuming Liang, Counting acyclic and strong digraphs by descents, arXiv:1909.01550 [math.CO], 20 Mar 2020.

Crossrefs

Cf. A228890, A047863 (row sums), A006116, A111636, A381102, A381192.

Programs

Mathematica

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