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A321413
Number of non-isomorphic self-dual multiset partitions of weight n with no singletons and relatively prime part sizes.
2
1, 0, 0, 0, 0, 3, 0, 14, 13, 50, 65
OFFSET
0,6
COMMENTS
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) 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) = 3, a(7) = 14, and a(8) = 13 multiset partitions:
{{11}{122}} {{111}{1222}} {{111}{11222}}
{{11}{222}} {{111}{2222}} {{111}{22222}}
{{12}{122}} {{112}{1222}} {{112}{12222}}
{{11}{22222}} {{122}{11222}}
{{12}{12222}} {{11}{122}{233}}
{{122}{1122}} {{11}{122}{333}}
{{22}{11222}} {{11}{222}{333}}
{{11}{12}{233}} {{11}{223}{233}}
{{11}{22}{233}} {{12}{122}{333}}
{{11}{22}{333}} {{12}{123}{233}}
{{11}{23}{233}} {{13}{112}{233}}
{{12}{12}{333}} {{13}{122}{233}}
{{12}{13}{233}} {{23}{123}{123}}
{{13}{23}{123}}
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Nov 16 2018
STATUS
approved