A051337 Number of strongly connected tournaments on n nodes.

1, 1, 0, 1, 1, 6, 35, 353, 6008, 178133, 9355949, 884464590, 152310149735, 48234782263293, 28304491788158056, 30964247546702883729, 63468402142317299907481, 244785748571033855024746438, 1782909084196274276970660380187, 24602074618353524534591008760307017

Offset: 0

Author

Vladeta Jovovic

Keywords

• nonn
• nice
• easy

Comments

A tournament is strongly connected (or strong) if there is a directed path between any pair of points.

References

• F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 127, Eq. (5.2.4);
• J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 523.

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..50
• John W. Moon, Topics on tournaments, Holt, Rinehard and Winston (1968)
• 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.)
• Raphael Yuster, Vector clique decompositions, Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms (2019), 1221-1238.

Crossrefs

Cf. A000568, A054946.

Programs

Mathematica

m = 20;
permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
edges[v_] := Sum[Sum[GCD[v[[i]], v[[j]]], {j, 1, i - 1}], {i, 2, Length[v]}] + Sum[Quotient[v[[i]], 2], {i, 1, Length[v]}];
oddp[v_] := (For[i = 1, i <= Length[v], i++, If[BitAnd[v[[i]], 1] == 0, Return[0]]]; 1);
b[n_] := b[n] = (s = 0; Do[If[oddp[p] == 1, s += permcount[p]*2^edges[p]], {p, IntegerPartitions[n]}]; s/n!);
B[x_] = Sum[b[k] x^k, {k, 0, m}];
A[x_] = 2 - 1/B[x];
A[x] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Sep 12 2019, after Andrew Howroyd in A000568 *)

Formula

G.f.: = 2 - 1/B(x) where B(x) = g.f. for A000568.

Extensions

a(0)=1 prepended and a(18)-a(19) from Andrew Howroyd, Sep 10 2018