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A321412
Number of non-isomorphic self-dual multiset partitions of weight n with no singletons and with aperiodic parts.
1
1, 0, 0, 0, 1, 1, 3, 4, 12, 20, 42
OFFSET
0,7
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 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(8) = 12 multiset partitions:
{{12}{12}} {{12}{122}} {{112}{122}} {{112}{1222}} {{1112}{1222}}
{{12}{1222}} {{12}{12222}} {{112}{12222}}
{{12}{13}{23}} {{12}{13}{233}} {{12}{122222}}
{{13}{23}{123}} {{122}{11222}}
{{12}{123}{233}}
{{12}{13}{2333}}
{{13}{112}{233}}
{{13}{122}{233}}
{{13}{23}{1233}}
{{23}{123}{123}}
{{12}{12}{34}{34}}
{{12}{13}{24}{34}}
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Nov 16 2018
STATUS
approved