OFFSET
0,7
COMMENTS
A set-system is a finite set of finite nonempty set of positive integers. An endpoint is a vertex appearing only once (degree 1). The weight of a set-system is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..50
Wikipedia, Degree (graph theory)
EXAMPLE
Non-isomorphic representatives of the a(0) = 1 through a(8) = 16 multiset partitions (empty columns not shown):
0 {1}{2}{12} {12}{13}{23} {13}{23}{123} {12}{134}{234}
{1}{23}{123} {1}{3}{23}{123} {1}{234}{1234}
{1}{2}{13}{23} {3}{12}{13}{23} {12}{34}{1234}
{1}{2}{3}{123} {1}{2}{3}{13}{23} {1}{12}{34}{234}
{12}{13}{24}{34}
{1}{2}{134}{234}
{1}{2}{34}{1234}
{2}{13}{14}{234}
{2}{13}{23}{123}
{3}{13}{23}{123}
{1}{2}{13}{24}{34}
{1}{2}{3}{14}{234}
{1}{2}{3}{23}{123}
{1}{2}{3}{4}{1234}
{2}{3}{12}{13}{23}
{1}{2}{3}{4}{12}{34}
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)={my(g=1+x*Ser(WeighT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k)))); (1-x)*g - subst(g, x, x^2)}
a(n)={if(n==0, 1, my(s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(sum(t=1, n, subst(K(q, t, n\t)/t, x, x^t) )), n)); s/n!)} \\ Andrew Howroyd, Jan 27 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 30 2019
EXTENSIONS
a(11) onwards from Andrew Howroyd, Jan 27 2024
STATUS
approved