A381058 - OEIS

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

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OFFSET

0,2

COMMENTS

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

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

LINKS

Table of n, a(n) for n=0..58.

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.

EXAMPLE

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;

...

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

CROSSREFS

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

Sequence in context: A216121 A104546 A121632 * A186361 A197365 A121579

Adjacent sequences: A381055 A381056 A381057 * A381059 A381060 A381061

KEYWORD

nonn,tabf

AUTHOR

Geoffrey Critzer, Feb 12 2025

STATUS

approved