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%I #5 Sep 28 2018 15:22:37
%S 1,1,3,6,15,31,74,156,358,792,1821
%N Number of non-isomorphic intersecting multiset partitions of weight n whose dual is also an intersecting multiset partition.
%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 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.
%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(4) = 15 multiset partitions:
%e 1: {{1}}
%e 2: {{1,1}}
%e {{1,2}}
%e {{1},{1}}
%e 3: {{1,1,1}}
%e {{1,2,2}}
%e {{1,2,3}}
%e {{1},{1,1}}
%e {{2},{1,2}}
%e {{1},{1},{1}}
%e 4: {{1,1,1,1}}
%e {{1,1,2,2}}
%e {{1,2,2,2}}
%e {{1,2,3,3}}
%e {{1,2,3,4}}
%e {{1},{1,1,1}}
%e {{1},{1,2,2}}
%e {{2},{1,2,2}}
%e {{3},{1,2,3}}
%e {{1,1},{1,1}}
%e {{1,2},{1,2}}
%e {{1,2},{2,2}}
%e {{1},{1},{1,1}}
%e {{2},{2},{1,2}}
%e {{1},{1},{1},{1}}
%Y Cf. A007716, A281116, A283877, A305854, A306006, A316980, A316983, A317757, A319616.
%Y Cf. A319752, A319766, A319767, A319768, A319769, A319773, A319774.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Sep 27 2018
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