A322113 - OEIS

%I #4 Nov 27 2018 16:18:20

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

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

%C 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}}.

%C The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

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

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

%e                           {{12}{12}}  {{11}{122}}  {{112}{122}}

%e                                                    {{12}{13}{23}}

%e .

%e   {{1111111}}      {{11111111}}        {{111111111}}

%e   {{111}{1222}}    {{111}{11222}}      {{1111}{12222}}

%e   {{112}{1222}}    {{1112}{1222}}      {{1112}{11222}}

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

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

%e                    {{11}{122}{233}}    {{11}{11}{12233}}

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

%e                    {{13}{112}{233}}    {{112}{123}{233}}

%e                    {{13}{122}{233}}    {{113}{122}{233}}

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

%e                                        {{12}{13}{23333}}

%e                                        {{12}{133}{2233}}

%e                                        {{123}{123}{123}}

%e                                        {{13}{112}{2333}}

%e                                        {{22}{113}{2333}}

%e                                        {{12}{13}{14}{234}}

%e                                        {{12}{13}{22}{344}}

%e                                        {{12}{13}{24}{344}}

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

%K nonn,more

%O 0,5

%A _Gus Wiseman_, Nov 26 2018