OFFSET
0,5
COMMENTS
The weight of a multiset partition 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(6) = 16 set multipartitions:
{{1,2}} {{1,2,3}} {{1,2,3,4}} {{1,2,3,4,5}} {{1,2,3,4,5,6}}
{{1,2},{1,2}} {{1,2},{3,4,5}} {{1,2,3},{1,2,3}}
{{1,2},{3,4}} {{1,4},{2,3,4}} {{1,2},{3,4,5,6}}
{{1,3},{2,3}} {{2,3},{1,2,3}} {{1,2,3},{4,5,6}}
{{1,2,5},{3,4,5}}
{{1,3,4},{2,3,4}}
{{1,5},{2,3,4,5}}
{{3,4},{1,2,3,4}}
{{1,2},{1,2},{1,2}}
{{1,2},{1,3},{2,3}}
{{1,2},{3,4},{3,4}}
{{1,2},{3,4},{5,6}}
{{1,2},{3,5},{4,5}}
{{1,3},{2,3},{2,3}}
{{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), n)); s/n!)} \\ Andrew Howroyd, Jan 16 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 16 2018
EXTENSIONS
Terms a(11) and beyond from Andrew Howroyd, Sep 01 2019
STATUS
approved