A354607 Triangular array read by rows: T(n,k) is the number of labeled tournaments on [n] that have exactly k irreducible (strongly connected) components, n >= 0, 0 <= k <= n.

1, 0, 1, 0, 0, 2, 0, 2, 0, 6, 0, 24, 16, 0, 24, 0, 544, 240, 120, 0, 120, 0, 22320, 6608, 2160, 960, 0, 720, 0, 1677488, 315840, 70224, 20160, 8400, 0, 5040, 0, 236522496, 27001984, 3830400, 758016, 201600, 80640, 0, 40320, 0, 64026088576, 4268194560, 366729600, 46448640, 8628480, 2177280, 846720, 0, 362880

Offset: 0

Author

Geoffrey Critzer, Jul 08 2022

Keywords

• nonn
• tabl

Examples

			Triangle T(n,k) begins:
  1;
  0,     1;
  0,     0,    2;
  0,     2,    0,    6;
  0,    24,   16,    0,  24;
  0,   544,  240,  120,   0, 120;
  0, 22320, 6608, 2160, 960,   0, 720;
  ...
		

Links

• N. J. A. Sloane, Illustration of first 5 terms
• Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 11 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)

Crossrefs

Cf. A006125 (row sums), A054946 (column k=1), A000142 (main diagonal).

Programs

Mathematica

nn = 10; G[x_] := Sum[2^Binomial[n, 2] x^n/n!, {n, 0, nn}]; Table[
Take[(Range[0, nn]! CoefficientList[Series[1/(1 - y (1 - 1/ G[x])), {x, 0, nn}], {x, y}])[[i]], i], {i, 1, nn}]

Formula

E.g.f.: 1/(1-y*(1-1/A(x))) where A(x) is the e.g.f. for A006125.