A381192 -id:A381192 - cp's OEIS frontend

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

Geoffrey Critzer, Feb 12 2025

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

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

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.

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

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

A381102 Irregular triangle read by rows. For each j, 1<=j<=n properly color the vertices of a labeled graph on [n] using each of the first j colors in the color set {c1=0, 0<=k<=binomial(n,2).

Original entry on oeis.org

1, 1, 4, 1, 36, 27, 9, 1, 696, 983, 731, 330, 93, 15, 1, 27808, 60615, 72662, 59113, 35197, 15731, 5269, 1287, 216, 22, 1, 2257888, 6803655, 11412586, 13504721, 12316799, 9026017, 5427090, 2700863, 1112555, 376459, 103002, 22203, 3619, 417, 30, 1

Offset: 0

Geoffrey Critzer, Feb 16 2025

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

T(n,0) = A289545(n).

			     1;
     1;
     4,     1;
    36,    27,     9,     1;
   696,   983,   731,   330,    93,    15,    1;
 27808, 60615, 72662, 59113, 35197, 15731, 5269, 1287, 216, 22, 1;
 ...

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.

Cf. A334282(row sums), A289545, A381058, A381192.

nn = 5; 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[1/(1 - (e[z] - 1)), {z, 0, nn}], z] /. y -> 1] // Grid

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