OFFSET
0,8
COMMENTS
A multiset is aperiodic if its multiplicities are relatively prime.
Also the number of nonnegative integer symmetric matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, with relatively prime row sums (or column sums) and no row or column having a common divisor > 1 or summing to 1.
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(5) = 1 through a(9) = 16 multiset partitions:
{{12}{122}} {{112}{1222}} {{112}{12222}} {{1112}{11222}}
{{12}{12222}} {{122}{11222}} {{1112}{12222}}
{{12}{13}{233}} {{12}{123}{233}} {{12}{1222222}}
{{13}{23}{123}} {{13}{112}{233}} {{12}{123}{2333}}
{{13}{122}{233}} {{12}{13}{23333}}
{{23}{123}{123}} {{12}{223}{1233}}
{{13}{112}{2333}}
{{13}{223}{1233}}
{{13}{23}{12333}}
{{23}{122}{1233}}
{{23}{123}{1233}}
{{12}{12}{34}{234}}
{{12}{12}{34}{344}}
{{12}{13}{14}{234}}
{{12}{13}{24}{344}}
{{12}{14}{34}{234}}
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Nov 16 2018
STATUS
approved