OFFSET
0,9
COMMENTS
A set-system is a finite set of finite nonempty set of positive integers. A singleton is an edge of size 1. 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(7) = 1 through a(10) = 16 set-systems:
{12}{13}{123} {12}{134}{234} {12}{134}{1234} {12}{1345}{2345}
{12}{34}{1234} {123}{124}{134} {123}{124}{1234}
{12}{13}{24}{34} {12}{13}{14}{234} {123}{145}{2345}
{12}{13}{23}{123} {12}{345}{12345}
{12}{13}{24}{134} {12}{13}{124}{134}
{12}{13}{124}{234}
{12}{13}{14}{1234}
{12}{13}{24}{1234}
{12}{13}{245}{345}
{12}{13}{45}{2345}
{12}{34}{123}{124}
{12}{34}{125}{345}
{12}{34}{135}{245}
{13}{24}{123}{124}
{12}{13}{14}{23}{24}
{12}{13}{24}{35}{45}
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=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)}
S(q, t, k)={(x-x^2)*sum(j=1, #q, if(t%q[j]==0, q[j])) + O(x*x^k)}
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)-S(q, t, n\t), x, x^t)/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