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allocated for Gus WisemanNumber of non-isomorphic self-dual connected antichains of multisets of weight n.

1, 1, 1, 1, 2, 2, 3, 5, 10, 18, 30

0,5

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.

Non-isomorphic representatives of the a(1) = 1 through a(9) = 18 antichains:

{{1}} {{11}} {{111}} {{1111}} {{11111}} {{111111}}

{{12}{12}} {{11}{122}} {{112}{122}}

{{12}{13}{23}}

.

{{1111111}} {{11111111}} {{111111111}}

{{111}{1222}} {{111}{11222}} {{1111}{12222}}

{{112}{1222}} {{1112}{1222}} {{1112}{11222}}

{{11}{12}{233}} {{112}{12222}} {{1112}{12222}}

{{12}{13}{233}} {{1122}{1122}} {{112}{122222}}

{{11}{122}{233}} {{11}{11}{12233}}

{{12}{13}{2333}} {{11}{122}{1233}}

{{13}{112}{233}} {{112}{123}{233}}

{{13}{122}{233}} {{113}{122}{233}}

{{12}{13}{24}{34}} {{12}{111}{2333}}

{{12}{13}{23333}}

{{12}{133}{2233}}

{{123}{123}{123}}

{{13}{112}{2333}}

{{22}{113}{2333}}

{{12}{13}{14}{234}}

{{12}{13}{22}{344}}

{{12}{13}{24}{344}}

Cf. A006126, A007716, A007718, A286520, A293993, A293994, A304867, A316983, A318099, A319719, A319721, A322111, A322112.

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Gus Wiseman, Nov 26 2018

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editing

allocated for Gus Wiseman

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