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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.

%I A354607 #17 Jul 12 2022 17:42:34
%S A354607 1,0,1,0,0,2,0,2,0,6,0,24,16,0,24,0,544,240,120,0,120,0,22320,6608,
%T A354607 2160,960,0,720,0,1677488,315840,70224,20160,8400,0,5040,0,236522496,
%U A354607 27001984,3830400,758016,201600,80640,0,40320,0,64026088576,4268194560,366729600,46448640,8628480,2177280,846720,0,362880
%N 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.
%H A354607 N. J. A. Sloane, <a href="/A000568/a000568_1.pdf">Illustration of first 5 terms</a>
%H A354607 Peter Steinbach, <a href="/A000664/a000664_11.pdf">Field Guide to Simple Graphs, Volume 4</a>, Part 11 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
%F A354607 E.g.f.: 1/(1-y*(1-1/A(x))) where A(x) is the e.g.f. for A006125.
%e A354607 Triangle T(n,k) begins:
%e A354607   1;
%e A354607   0,     1;
%e A354607   0,     0,    2;
%e A354607   0,     2,    0,    6;
%e A354607   0,    24,   16,    0,  24;
%e A354607   0,   544,  240,  120,   0, 120;
%e A354607   0, 22320, 6608, 2160, 960,   0, 720;
%e A354607   ...
%t A354607 nn = 10; G[x_] := Sum[2^Binomial[n, 2] x^n/n!, {n, 0, nn}]; Table[
%t A354607 Take[(Range[0, nn]! CoefficientList[Series[1/(1 - y (1 - 1/ G[x])), {x, 0, nn}], {x, y}])[[i]], i], {i, 1, nn}]
%Y A354607 Cf. A006125 (row sums), A054946 (column k=1), A000142 (main diagonal).
%K A354607 nonn,tabl
%O A354607 0,6
%A A354607 _Geoffrey Critzer_, Jul 08 2022