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%I #6 Oct 26 2018 12:50:18
%S 1,1,1,1,1,1,2,2,3,4,7
%N Number of non-isomorphic weight-n antichains of distinct sets whose dual is also an antichain of distinct sets.
%C The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks 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.
%F Euler transform of A319625.
%e Non-isomorphic representatives of the a(1) = 1 through a(10) = 7 antichains:
%e 1: {{1}}
%e 2: {{1},{2}}
%e 3: {{1},{2},{3}}
%e 4: {{1},{2},{3},{4}}
%e 5: {{1},{2},{3},{4},{5}}
%e 6: {{1,2},{1,3},{2,3}}
%e {{1},{2},{3},{4},{5},{6}}
%e 7: {{1},{2,3},{2,4},{3,4}}
%e {{1},{2},{3},{4},{5},{6},{7}}
%e 8: {{1,2},{1,3},{2,4},{3,4}}
%e {{1},{2},{3,4},{3,5},{4,5}}
%e {{1},{2},{3},{4},{5},{6},{7},{8}}
%e 9: {{1,2},{1,3},{1,4},{2,3,4}}
%e {{1},{2,3},{2,4},{3,5},{4,5}}
%e {{1},{2},{3},{4,5},{4,6},{5,6}}
%e {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
%e 10: {{1,3},{2,4},{1,2,5},{3,4,5}}
%e {{1},{2,3},{2,4},{2,5},{3,4,5}}
%e {{1,2},{1,3},{2,4},{3,5},{4,5}}
%e {{1,3},{1,4},{2,3},{2,4},{3,4}}
%e {{1},{2},{3,4},{3,5},{4,6},{5,6}}
%e {{1},{2},{3},{4},{5,6},{5,7},{6,7}}
%e {{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
%Y Cf. A006126, A007716, A049311, A059201, A283877, A293606, A316980, A316983, A318099, A319558, A319616-A319646, A300913.
%K nonn,more
%O 0,7
%A _Gus Wiseman_, Sep 25 2018
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