A228890 -id:A228890 - 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

A228861 Irregular triangular array read by rows. T(n,k) is the number of connected labeled bipartite graphs on n nodes with exactly k edges; n >= 1, 0 <= k <= A002620(n+1).

Original entry on oeis.org

1, 0, 1, 0, 0, 3, 0, 0, 0, 16, 3, 0, 0, 0, 0, 125, 60, 10, 0, 0, 0, 0, 0, 1296, 1140, 480, 105, 10, 0, 0, 0, 0, 0, 0, 16807, 23100, 16800, 7770, 2331, 420, 35, 0, 0, 0, 0, 0, 0, 0, 262144, 513240, 555520, 412440, 222936, 88648, 25480, 5040, 616, 35

Offset: 1

Geoffrey Critzer, Sep 05 2013

			Irregular Triangle begins:
  1;
  0, 1;
  0, 0, 3;
  0, 0, 0, 16,   3;
  0, 0, 0,  0, 125,   60,   10;
  0, 0, 0,  0,   0, 1296, 1140, 480, 105, 10;
  ...

Row sums are A001832.

Cf. A002620, A228890.

nn=8;f[x_,y_]:=Sum[Sum[Binomial[n,k](1+y)^(k(n-k)),{k,0,n}]x^n/n!,{n,0,nn}];Table[PadLeft[a=Map[Select[#,#>0&]&,Drop[Range[0,nn]!CoefficientList[Series[Log[f[x,y]]/2,{x,0,nn}],{x,y}],1]][[n]],Length[a]+n-1],{n,1,nn}]//Grid

E.g.f.: log(A(x,y))/2 where A(x,y) is the e.g.f. for A228890.

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