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%I #5 Sep 28 2018 15:23:57
%S 1,0,0,0,1,4,20,66,226,696,2156
%N Number of intersecting multiset partitions of weight n whose dual is not 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 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(4) = 1 through a(6) = 20 multiset partitions:
%e 4: {{1,3},{2,3}}
%e 5: {{1,2},{2,3,3}}
%e {{1,3},{2,3,3}}
%e {{1,4},{2,3,4}}
%e {{3},{1,3},{2,3}}
%e 6: {{1,2},{2,3,3,3}}
%e {{1,3},{2,2,3,3}}
%e {{1,3},{2,3,3,3}}
%e {{1,3},{2,3,4,4}}
%e {{1,4},{2,3,4,4}}
%e {{1,5},{2,3,4,5}}
%e {{1,1,2},{2,3,3}}
%e {{1,2,2},{2,3,3}}
%e {{1,2,3},{3,4,4}}
%e {{1,2,4},{3,4,4}}
%e {{1,2,5},{3,4,5}}
%e {{1,3,3},{2,3,3}}
%e {{1,3,4},{2,3,4}}
%e {{2},{1,2},{2,3,3}}
%e {{3},{1,3},{2,3,3}}
%e {{4},{1,4},{2,3,4}}
%e {{1,3},{2,3},{2,3}}
%e {{1,3},{2,3},{3,3}}
%e {{1,4},{2,4},{3,4}}
%e {{3},{3},{1,3},{2,3}}
%Y Cf. A007716, A281116, A283877, A305854, A306006, A316980, A316983, A317757, A319616.
%Y Cf. A319775, A319778, A319781, A319783.
%K nonn,more
%O 0,6
%A _Gus Wiseman_, Sep 27 2018
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