A004102 Number of signed graphs with n nodes. Also number of 2-multigraphs on n nodes.

1, 1, 3, 10, 66, 792, 25506, 2302938, 591901884, 420784762014, 819833163057369, 4382639993148435207, 64588133532185722290294, 2638572375815762804156666529, 300400208094064113266621946833097, 95776892467035669509813163910815022152

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

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).

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

• Andrew Howroyd, Table of n, a(n) for n = 0..50 (terms 1..22 from R. W. Robinson)
• M. Adamaszek, The smallest nonevasive graph property, Disc. Mathem. Graph Theory 34 (2014) 857
• Edward A. Bender and E. Rodney Canfield, Enumeration of connected invariant graphs, Journal of Combinatorial Theory, Series B 34.3 (1983): 268-278. See p. 273.
• J. Cummings, D. Kral, F. Pfender, K. Sperfeld et al., Monochromatic triangles in three-coloured graphs, arXiv preprint arXiv:1206.1987 [math.CO]. 2012. - From _N. J. A. Sloane_, Nov 25 2012
• Harald Fripertinger, The cycle type of the induced action on 2-subsets
• Harary, Frank; Palmer, Edgar M.; Robinson, Robert W.; Schwenk, Allen J.; Enumeration of graphs with signed points and lines, J. Graph Theory 1 (1977), no. 4, 295-308.
• Vladeta Jovovic, Formulae for the number T(n,k) of n-multigraphs on k nodes
• R. W. Robinson, Notes - "A Present for Neil Sloane"
• R. W. Robinson, Notes - computer printout
• R. W. Robinson & N. J. A. Sloane, Correspondence, 1970-1980

Crossrefs

A column of A063841.

Cf. A053465.

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]]], {j, 1, i-1}], {i, 2, Length[v]}] + Sum[Quotient[v[[i]], 2], {i, 1, Length[v]}];
a[n_] := Module[{s = 0}, Do[s += permcount[p]*3^edges[p], {p, IntegerPartitions[n]}]; s/n!];
Array[a, 16, 0] (* Jean-François Alcover, Aug 17 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]))) + sum(i=1, #v, v[i]\2)}
a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*3^edges(p)); s/n!} \\ Andrew Howroyd, Sep 25 2018

Python

from itertools import combinations
from math import prod, gcd, factorial
from fractions import Fraction
from sympy.utilities.iterables import partitions
def A004102(n): return int(sum(Fraction(3**(sum(p[r]*p[s]*gcd(r,s) for r,s in combinations(p.keys(),2))+sum((q>>1)*r+(q*r*(r-1)>>1) 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 09 2024

Formula

Euler transform of A053465. - Andrew Howroyd, Sep 25 2018

Extensions

More terms from Vladeta Jovovic, Jan 06 2000

a(0)=1 prepended and a(15) added by Andrew Howroyd, Sep 25 2018