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 (1) the positive entries in each row and column are relatively prime and (2) the row sums and column sums are relatively prime.
The last condition (aperiodicity of the multiset union of the dual) is equivalent to the parts having relatively prime sizes.
A multiset is aperiodic if its multiplicities are relatively prime.
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}}.
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
Non-isomorphic representatives of the a(1) = 1 through a(4) = 10 multiset partitions:
{{1}} {{1},{2}} {{1},{2,3}} {{1},{2,3,4}}
{{2},{1,2}} {{2},{1,2,2}}
{{1},{2},{2}} {{3},{1,2,3}}
{{1},{2},{3}} {{1},{1},{2,3}}
{{1},{2},{3,4}}
{{1},{3},{2,3}}
{{2},{2},{1,2}}
{{1},{2},{2},{2}}
{{1},{2},{3},{3}}
{{1},{2},{3},{4}}
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Nov 07 2018
STATUS
approved