OFFSET
0,3
COMMENTS
A multiset is aperiodic if its multiplicities are relatively prime.
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(1) = 1 through a(4) = 21 multiset partitions with aperiodic parts:
{{1}} {{1,2}} {{1,2,2}} {{1,2,2,2}}
{{1},{1}} {{1,2,3}} {{1,2,3,3}}
{{1},{2}} {{1},{2,3}} {{1,2,3,4}}
{{2},{1,2}} {{1},{1,2,2}}
{{1},{1},{1}} {{1,2},{1,2}}
{{1},{2},{2}} {{1},{2,3,3}}
{{1},{2},{3}} {{1},{2,3,4}}
{{1,2},{3,4}}
{{1,3},{2,3}}
{{2},{1,2,2}}
{{3},{1,2,3}}
{{1},{1},{2,3}}
{{1},{2},{1,2}}
{{1},{2},{3,4}}
{{1},{3},{2,3}}
{{2},{2},{1,2}}
{{1},{1},{1},{1}}
{{1},{1},{2},{2}}
{{1},{2},{2},{2}}
{{1},{2},{3},{3}}
{{1},{2},{3},{4}}
PROG
(PARI)
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, 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)={EulerT(Vec(sum(j=1, #q, gcd(t, q[j])*x^lcm(t, q[j])) + O(x*x^k), -k))}
a(n)={if(n==0, 1, my(mbt=vector(n, d, moebius(d)), s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(x*Ser(dirmul(mbt, sum(t=1, n, K(q, t, n)/t)))), n)); s/n!)} \\ Andrew Howroyd, Jan 16 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 06 2018
EXTENSIONS
Terms a(11) and beyond from Andrew Howroyd, Jan 16 2023
STATUS
approved