A322113 Number of non-isomorphic self-dual connected antichains of multisets of weight n.

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

Offset: 0

Author

Gus Wiseman, Nov 26 2018

Keywords

• nonn
• more

Comments

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.

Examples

			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}}
		

Crossrefs

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