OFFSET
0,4
COMMENTS
A set system is a finite set of finite nonempty sets. Its weight 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
EXAMPLE
Non-isomorphic representatives of the a(2) = 1 through a(5) = 13 set systems:
2: {{1},{2}}
3: {{1},{2,3}}
{{1},{2},{3}}
4: {{1},{2,3,4}}
{{1,2},{3,4}}
{{1},{2},{1,2}}
{{1},{2},{3,4}}
{{1},{3},{2,3}}
{{1},{2},{3},{4}}
5: {{1},{2,3,4,5}}
{{1,2},{3,4,5}}
{{1},{2},{3,4,5}}
{{1},{4},{2,3,4}}
{{1},{2,3},{4,5}}
{{1},{2,4},{3,4}}
{{2},{3},{1,2,3}}
{{2},{1,3},{2,3}}
{{4},{1,2},{3,4}}
{{1},{2},{3},{2,3}}
{{1},{2},{3},{4,5}}
{{1},{2},{4},{3,4}}
{{1},{2},{3},{4},{5}}
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, gcd(t, q[j])*x^lcm(t, q[j])) + O(x*x^k), -k))}
R(q, n)={vector(n, t, x*Ser(K(q, t, n)/t))}
a(n)={if(n==0, 1, my(s=0); forpart(q=n, my(u=R(q, n)); s+=permcount(q)*polcoef(exp(sum(t=1, n, u[t]-subst(u[t], x, x^2), O(x*x^n))) - exp(sum(t=1, n\2, x^t*u[t] - subst(x^t*u[t], x, x^2), O(x*x^n)))*(1+x), n)); s/n!)} \\ Andrew Howroyd, May 30 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 27 2018
EXTENSIONS
Terms a(11) and beyond from Andrew Howroyd, May 30 2023
STATUS
approved