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%I #7 Aug 12 2019 22:32:24
%S 1,2,5,24,1267
%N Number of unlabeled set-systems on n vertices whose dual is a weak antichain.
%C A set-system is a finite set of finite nonempty sets. The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. A weak antichain is a multiset of sets, none of which is a proper subset of any other.
%e Non-isomorphic representatives of the a(0) = 1 through a(3) = 24 set-systems:
%e {} {} {} {}
%e {{1}} {{1}} {{1}}
%e {{1,2}} {{1,2}}
%e {{1},{2}} {{1},{2}}
%e {{1},{2},{1,2}} {{1,2,3}}
%e {{1},{2,3}}
%e {{1},{2},{3}}
%e {{1},{2},{1,2}}
%e {{1,2},{1,3},{2,3}}
%e {{1},{2,3},{1,2,3}}
%e {{1},{2},{3},{2,3}}
%e {{1},{2},{1,3},{2,3}}
%e {{1},{2},{3},{1,2,3}}
%e {{3},{1,2},{1,3},{2,3}}
%e {{1},{2},{3},{1,3},{2,3}}
%e {{1,2},{1,3},{2,3},{1,2,3}}
%e {{1},{2},{3},{2,3},{1,2,3}}
%e {{2},{3},{1,2},{1,3},{2,3}}
%e {{1},{2},{1,3},{2,3},{1,2,3}}
%e {{1},{2},{3},{1,2},{1,3},{2,3}}
%e {{3},{1,2},{1,3},{2,3},{1,2,3}}
%e {{1},{2},{3},{1,3},{2,3},{1,2,3}}
%e {{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
%e {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
%Y Unlabeled set-systems are A000612.
%Y Unlabeled set-systems whose dual is strict are A326946.
%Y The labeled version is A326968.
%Y The version with empty edges allowed is A326969.
%Y The T_0 case (with strict dual) is A326972.
%Y The covering case is A326973 (first differences).
%Y Cf. A319559, A326951, A326965, A326966, A326970, A326974, A326975, A326978.
%K nonn,more
%O 0,2
%A _Gus Wiseman_, Aug 10 2019