A004107 Number of self-dual nets with 2n nodes.
1, 1, 9, 165, 24651, 29522961, 286646256675, 21717897090413481, 12980536689318626076840, 62082697145168772833294318409, 2405195296608025717214293025492960466, 762399078635131851885116768114137369439908725
Offset: 0
Keywords
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..40 (terms 1..13 from R. W. Robinson)
- Frank Harary, Edgar M. Palmer, Robert W. Robinson, and 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
Programs
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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_] := 2 Sum[Sum[GCD[v[[i]], v[[j]]], {j, 1, i - 1}], {i, 2, Length[v]}] + Sum[2 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, 12, 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) = {2*sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2*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 A004107(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&-2)+q*(r-1))*r 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
Extensions
a(0)=1 prepended by Andrew Howroyd, Sep 25 2018
Comments