A004103 Number of nets on n unlabeled nodes.

1, 2, 9, 56, 705, 19548, 1419237, 278474976, 148192635483, 213558945249402, 836556995284293897, 8962975658381123937708, 264404516190234685662666051, 21610417954162750247842392794292, 4921335335427778307286708119839406529, 3138313838161414849743136458064895837170596

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

A net in this context is a graph with both signed vertices and signed edges. - Andrew Howroyd, Sep 25 2018

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)
• 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

Crossrefs

Cf. A004102 (signed edges only), A000666 (signed vertices only).

Cf. A004107.

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]*2^Length[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)*2^#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 A004103(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()))<Chai Wah Wu, Jul 09 2024

Extensions

a(0)=1 prepended and a(13)-a(14) added by Andrew Howroyd, Sep 25 2018