A322112 - OEIS

%I #5 Nov 27 2018 16:18:12

%S 1,0,1,1,1,2,2,4,4,9,9

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

%C 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.

%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(2) = 1 through a(10) = 9 multiset partitions:

%e   {{11}}  {{111}}  {{1111}}  {{11111}}    {{111111}}    {{1111111}}

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

%e                                                         {{22}{11222}}

%e                                                         {{11}{12}{233}}

%e .

%e   {{11111111}}      {{111111111}}        {{1111111111}}

%e   {{111}{11222}}    {{1111}{12222}}      {{1111}{112222}}

%e   {{22}{112222}}    {{22}{1122222}}      {{22}{11222222}}

%e   {{11}{122}{233}}  {{222}{111222}}      {{222}{1112222}}

%e                     {{11}{11}{12233}}    {{111}{122}{2333}}

%e                     {{11}{113}{2233}}    {{22}{113}{23333}}

%e                     {{12}{111}{2333}}    {{22}{1133}{2233}}

%e                     {{22}{113}{2333}}    {{33}{33}{112233}}

%e                     {{12}{13}{22}{344}}  {{11}{14}{223}{344}}

%Y Cf. A000272, A007716, A007718, A030019, A052888, A134954, A304867, A304887, A316983, A321155, A321255, A322111.

%K nonn,more

%O 0,6

%A _Gus Wiseman_, Nov 26 2018