A326971 - OEIS

%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