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Number of non-isomorphic intersecting set systems of weight n whose dual is also an intersecting set system.
6

%I #5 Sep 28 2018 15:23:24

%S 1,1,0,1,0,0,2,1,2,4,5

%N Number of non-isomorphic intersecting set systems of weight n whose dual is also an intersecting set system.

%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(10) = 5 set systems:

%e 1: {{1}}

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

%e 6: {{3},{2,3},{1,2,3}}

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

%e 7: {{1,3},{2,3},{1,2,3}}

%e 8: {{2,4},{3,4},{1,2,3,4}}

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

%e 9: {{1,2,4},{1,3,4},{2,3,4}}

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

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

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

%e 10: {{4},{3,4},{2,3,4},{1,2,3,4}}

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

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

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

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

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

%Y Cf. A319752, A319765, A319766, A319767, A319768, A319769, A319774.

%K nonn,more

%O 0,7

%A _Gus Wiseman_, Sep 27 2018