A002499 Number of self-converse digraphs with n nodes. (Formerly M2875 N1156)

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

Offset

1,2

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.

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

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 *)

Prog

(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

Crossrefs

Cf. A002500.

Sequence in context: A342629 A232213 A143083 * A047833 A047834 A208999

Adjacent sequences: A002496 A002497 A002498 * A002500 A002501 A002502

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Apr 17 2000

Status

approved