OFFSET
0,4
COMMENTS
Also the number of nonnegative integer matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, in which every row has at least two nonzero entries.
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(5) = 13 multiset partitions:
{{1,2}} {{1,2,2}} {{1,1,2,2}} {{1,1,2,2,2}}
{{1,2,3}} {{1,2,2,2}} {{1,2,2,2,2}}
{{1,2,3,3}} {{1,2,2,3,3}}
{{1,2,3,4}} {{1,2,3,3,3}}
{{1,2},{1,2}} {{1,2,3,4,4}}
{{1,2},{3,4}} {{1,2,3,4,5}}
{{1,3},{2,3}} {{1,2},{1,2,2}}
{{1,2},{2,3,3}}
{{1,2},{3,4,4}}
{{1,2},{3,4,5}}
{{1,3},{2,3,3}}
{{1,4},{2,3,4}}
{{2,3},{1,2,3}}
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, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
S(q, t, k)={sum(j=1, #q, if(t%q[j]==0, q[j]))*vector(k, i, 1)}
a(n)={if(n==0, 1, my(s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(sum(t=1, n, subst(x*Ser(K(q, t, n\t)-S(q, t, n\t))/t, x, x^t) )), n)); s/n!)} \\ Andrew Howroyd, Jan 17 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 29 2018
EXTENSIONS
Terms a(11) and beyond from Andrew Howroyd, Jan 17 2023
STATUS
approved