A004105 Number of point-self-dual nets with 2n nodes. Also number of directed 2-multigraphs with loops on n nodes. (Formerly M3153)

1, 3, 45, 3411, 1809459, 7071729867, 208517974495911, 47481903377454219975, 85161307642554753639601848, 1221965550839348597865127102714827, 142024245093355901785105779901319683262778, 135056692539998733060710198802224149631056479068139

Offset

0,2

Comments

A 2-multigraph is similar to an ordinary graph except there are 0, 1 or 2 edges between any two nodes (self-loops are not allowed).

Also nonisomorphic relations on 3-state logic.

References

F. Harary and R. W. Robinson, Exposition of the enumeration of point-line-signed graphs, pp. 19 - 33 of Proc. Second Caribbean Conference Combinatorics and Computing (Bridgetown, 1977). Ed. R. C. Read and C. C. Cadogan. University of the West Indies, Cave Hill Campus, Barbados, 1977. vii+223 pp.

R. W. Robinson, personal communication.

R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

R. W. Robinson and Alois P. Heinz, Table of n, a(n) for n = 0..40 (terms n = 1..13 from R. W. Robinson)

Frank Harary, Edgar M. Palmer, Robert W. Robinson, Allen J. Schwenk, Enumeration of graphs with signed points and lines, J. Graph Theory 1 (1977), no. 4, 295-308.

R. W. Robinson, Notes - "A Present for Neil Sloane"

R. W. Robinson, Notes - computer printout

Formula

a(n) = Sum_{1*s_1+2*s_2+...=n} (fixA[s_1, s_2, ...]/ (1^s_1*s_1!*2^s_2*s_2!*...)) where fixA[s_1, s_2, ...] = 3^Sum_{i, j>=1} (gcd(i,j)*s_i*s_j).

Mathematica

Prepend[Table[CycleIndex[Join[PairGroup[SymmetricGroup[n], Ordered], Permutations[Range[n^2-n+1, n^2]], 2], s]/.Table[s[i]->3, {i, 1, n^2-n}], {n, 2, 7}], 1] (* Geoffrey Critzer, Oct 20 2012 *)
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[2*GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[v];
a[n_] := (s=0; Do[s += permcount[p]*3^edges[p], {p, IntegerPartitions[n]}]; s/n!);
Array[a, 15, 0] (* Jean-François Alcover, Jul 08 2018, 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, 2*gcd(v[i], v[j]))) + sum(i=1, #v, v[i])}
a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*3^edges(p)); s/n!} \\ Andrew Howroyd, Oct 22 2017
(Python)
from itertools import combinations
from math import prod, gcd, factorial
from fractions import Fraction
from sympy.utilities.iterables import partitions
def A004105(n): return int(sum(Fraction(3**((sum(p[r]*p[s]*gcd(r, s) for r, s in combinations(p.keys(), 2))<<1)+sum(q*r**2 for q, r in p.items())), prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n))) # Chai Wah Wu, Jul 10 2024

Crossrefs

Cf. A000595, A001374, A053467, A053516.

Sequence in context: A198952 A099168 A227379 * A060336 A268196 A057863

Adjacent sequences: A004102 A004103 A004104 * A004106 A004107 A004108

Keyword

easy,nonn

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Jan 14 2000

Formula from Christian G. Bower, Jan 06 2004

Status

approved