%I #6 Aug 12 2019 22:32:39

%S 1,1,3,19,1243

%N Number of unlabeled set-systems covering 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) = 19 set-systems:

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

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

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

%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 covering set-systems are A055621.

%Y The labeled version is A326970.

%Y The non-covering case is A326971 (partial sums).

%Y The case that is also T_0 is the T_1 case A326974.

%Y Cf. A000612, A059523, A319637, A326966, A326968, A326972, A326975, A326978.

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Aug 11 2019