A319769 - OEIS

%I #6 Sep 28 2018 15:23:13

%S 1,1,2,3,5,7,12,16,26,38,61

%N Number of non-isomorphic intersecting set multipartitions (multisets of sets) of weight n whose dual is also an intersecting set multipartition.

%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}}.

%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.

%C A multiset partition is intersecting iff no two parts are disjoint. The dual of a multiset partition is intersecting iff every pair of distinct vertices appear together in some part.

%e Non-isomorphic representatives of the a(1) = 1 through a(5) = 7 set multipartitions:

%e 1: {{1}}

%e 2: {{1,2}}

%e    {{1},{1}}

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

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

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

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

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

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

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

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

%e 5: {{1,2,3,4,5}}

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

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

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

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

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

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

%Y Cf. A007716, A281116, A283877, A305854, A306006,  A316980, A316983, A317757, A319616.

%Y Cf. A319752, A319765, A319766, A319767, A319768, A319773, A319774.

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Sep 27 2018