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COMMENTS
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The multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices.
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}}. A multiset partition is self-dual if it is isomorphic to its dual. For example, {{1,1},{1,2,2},{2,3,3}} is self-dual, as it is isomorphic to its dual {{1,1,2},{2,2,3},{3,3}}.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
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EXAMPLE
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Non-isomorphic representatives of the a(2) = 1 through a(10) = 9 multiset partitions:
{{11}} {{111}} {{1111}} {{11111}} {{111111}} {{1111111}}
{{11}{122}} {{22}{1122}} {{111}{1222}}
{{22}{11222}}
{{11}{12}{233}}
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{{11111111}} {{111111111}} {{1111111111}}
{{111}{11222}} {{1111}{12222}} {{1111}{112222}}
{{22}{112222}} {{22}{1122222}} {{22}{11222222}}
{{11}{122}{233}} {{222}{111222}} {{222}{1112222}}
{{11}{11}{12233}} {{111}{122}{2333}}
{{11}{113}{2233}} {{22}{113}{23333}}
{{12}{111}{2333}} {{22}{1133}{2233}}
{{22}{113}{2333}} {{33}{33}{112233}}
{{12}{13}{22}{344}} {{11}{14}{223}{344}}
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