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 (1) the row sums are all > 1 and (2) the positive entries in each row are relatively prime.

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.