OFFSET
0,5
COMMENTS
A set-system is a finite set of finite nonempty sets (edges). The weight is the sum of cardinalities of the edges. Weight is generally not the same as number of vertices.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..50
FORMULA
EXAMPLE
Non-isomorphic representatives of the a(6) = 12 set-systems:
{{1,2,3,4,5,6}}
{{1,2},{3,4,5,6}}
{{1,5},{2,3,4,5}}
{{3,4},{1,2,3,4}}
{{1,2,3},{4,5,6}}
{{1,2,5},{3,4,5}}
{{1,3,4},{2,3,4}}
{{1,2},{1,3},{2,3}}
{{1,2},{3,4},{5,6}}
{{1,2},{3,5},{4,5}}
{{1,3},{2,4},{3,4}}
{{1,4},{2,4},{3,4}}
PROG
(PARI)
WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
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}
K(q, t, k)={WeighT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k)) - Vec(sum(j=1, #q, if(t%q[j]==0, q[j])) + O(x*x^k), -k)}
a(n)={if(n==0, 1, my(s=0); forpart(q=n, my(g=sum(t=1, n, subst(x*Ser(K(q, t, n\t)/t), x, x^t) )); s+=permcount(q)*polcoef(exp(g - subst(g, x, x^2)), n)); s/n!)} \\ Andrew Howroyd, Jan 16 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 16 2018
EXTENSIONS
Terms a(11) and beyond from Andrew Howroyd, Sep 01 2019
STATUS
approved