A322111 - OEIS

%I #4 Nov 27 2018 16:18:05

%S 1,1,1,2,2,5,5,13,13,37,37

%N Number of non-isomorphic self-dual connected multiset partitions of weight n with 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(1) = 1 through a(8) = 13 multiset partitions:

%e   {{1}}                    {{1,1}}

%e .

%e   {{1,1,1}}                {{1,1,1,1}}

%e   {{2},{1,2}}              {{2},{1,2,2}}

%e .

%e   {{1,1,1,1,1}}            {{1,1,1,1,1,1}}

%e   {{1,1},{1,2,2}}          {{2},{1,2,2,2,2}}

%e   {{2},{1,2,2,2}}          {{2,2},{1,1,2,2}}

%e   {{2},{1,3},{2,3}}        {{2},{1,3},{2,3,3}}

%e   {{3},{3},{1,2,3}}        {{3},{3},{1,2,3,3}}

%e .

%e   {{1,1,1,1,1,1,1}}        {{1,1,1,1,1,1,1,1}}

%e   {{1,1,1},{1,2,2,2}}      {{1,1,1},{1,1,2,2,2}}

%e   {{2},{1,2,2,2,2,2}}      {{2},{1,2,2,2,2,2,2}}

%e   {{2,2},{1,1,2,2,2}}      {{2,2},{1,1,2,2,2,2}}

%e   {{1,1},{1,2},{2,3,3}}    {{1,1},{1,2,2},{2,3,3}}

%e   {{2},{1,3},{2,3,3,3}}    {{2},{1,3},{2,3,3,3,3}}

%e   {{2},{2,2},{1,2,3,3}}    {{2},{1,3,3},{2,2,3,3}}

%e   {{3},{1,2,2},{2,3,3}}    {{3},{3},{1,2,3,3,3,3}}

%e   {{3},{3},{1,2,3,3,3}}    {{3},{3,3},{1,2,2,3,3}}

%e   {{1},{1},{1,4},{2,3,4}}  {{2},{1,3},{2,4},{3,4,4}}

%e   {{2},{1,3},{2,4},{3,4}}  {{3},{3},{1,2,4},{3,4,4}}

%e   {{3},{4},{1,4},{2,3,4}}  {{3},{4},{1,4},{2,3,4,4}}

%e   {{4},{4},{4},{1,2,3,4}}  {{4},{4},{4},{1,2,3,4,4}}

%Y Cf. A000272, A007716, A007718, A030019, A052888, A134954, A304867, A304887, A316983, A318697, A319616, A321155, A321255.

%K nonn,more

%O 0,4

%A _Gus Wiseman_, Nov 26 2018