A322112 - OEIS

A322112

Number of non-isomorphic self-dual connected multiset partitions of weight n with no singletons and multiset density -1.

1, 0, 1, 1, 1, 2, 2, 4, 4, 9, 9

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OFFSET

0,6

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..10.

EXAMPLE

Non-isomorphic representatives of the a(2) = 1 through a(10) = 9 multiset partitions:

{{11}{122}} {{22}{1122}} {{111}{1222}}

{{22}{11222}}

{{11}{12}{233}}

{{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}}