
COMMENTS

Also the number of nonnegative integer matrices up to row and column permutations with sum of entries equal to n and no zero rows or columns, in which each row and each column has relatively prime nonzero entries.
The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
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.


EXAMPLE

Nonisomorphic representatives of the a(1) = 1 through a(4) = 17 multiset partitions:
{{1}} {{1,2}} {{1,2,3}} {{1,2,3,4}}
{{1},{1}} {{1},{2,3}} {{1,2},{1,2}}
{{1},{2}} {{2},{1,2}} {{1},{2,3,4}}
{{1},{1},{1}} {{1,2},{3,4}}
{{1},{2},{2}} {{1,3},{2,3}}
{{1},{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}}
