A002499 Number of self-converse digraphs with n nodes.

1, 3, 10, 70, 708, 15224, 544152, 39576432, 5074417616, 1296033011648, 604178966756320, 556052774253161600, 954895322019762585664, 3224152068625567826724224, 20610090531322819956330186112

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

References

• F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 155, Table 6.6.1 (but the last entry is wrong).
• R. W. Robinson, personal communication.
• R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1980.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..50 (terms 1..28 from R. W. Robinson)
• Alastair Farrugia, Self-complementary graphs and generalizations: a comprehensive reference, M.Sc. Thesis, University of Malta, August 1999.
• F. Harary and E. M. Palmer, Enumeration of self-converse digraphs, Mathematika, 13 (1966), 151-157.

Crossrefs

Cf. A002500.

Programs

Mathematica

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]]]*If[Mod[v[[i]] v[[j]], 2]==0, 2, 1], {j, 1, i-1}], {i, 2, Length[v]}]+Sum[Quotient[v[[i]], 2] + If[Mod[v[[i]], 2]==0, Quotient[v[[i]]-2, 4]*2+1, 0], {i, 1, Length[v]}];
a[n_] := Module[{s=0}, Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!];
Array[a, 15] (* Jean-François Alcover, Aug 16 2019, after Andrew Howroyd *)

PARI

permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i],v[j])*if(v[i]*v[j]%2==0, 2, 1))) + sum(i=1, #v, v[i]\2 + if(v[i]%2==0, (v[i]-2)\4*2+1))}
a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*2^edges(p)); s/n!} \\ Andrew Howroyd, Sep 18 2018

Formula

Asymptotics (R. W. Robinson): a(n) ~ 2^((n^2 - 1)/2) * exp(sqrt(n/2) - n/2 - 1/8) * n^(n/2) / n!, (Farrugia, formula 7.28, p. 199). - Vaclav Kotesovec, Dec 31 2020

Extensions

More terms from Vladeta Jovovic, Apr 17 2000