A053468 Number of directed 3-multigraphs on n nodes.
1, 10, 720, 703760, 9168331776, 1601371799340544, 3837878966366932639744, 128777257564337108286016980992, 61454877497308462618188532330410573824, 422314689395950135433730499958070655419345928192
Offset: 1
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..40
- Rebekka Willenberg, Self-Complementary Graphs and Digraphs. In search of a natural bijection, 2017.
Programs
-
Mathematica
Table[CoefficientList[PairGroupIndex[SymmetricGroup[n],s,Ordered]/.Table[s[i]->4,{i,1,2 Binomial[n,2]}],x],{n,1,8}] (* 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 - 1]; a[n_] := (s=0; Do[s += permcount[p]*4^edges[p], {p, IntegerPartitions[n]}]; s/n!); Array[a, 15] (* Jean-François Alcover, Jul 08 2018, 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, 2*gcd(v[i],v[j]))) + sum(i=1, #v, v[i]-1)} a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*4^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 A053468(n): return int(sum(Fraction(1<<((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())-s<<1),prod(q**r*factorial(r) for q, r in p.items())) for s, p in partitions(n,size=True))) # Chai Wah Wu, Jul 10 2024