A326971 Number of unlabeled set-systems on n vertices whose dual is a weak antichain.

1, 2, 5, 24, 1267

Offset: 0

Author

Gus Wiseman, Aug 10 2019

Keywords

• nonn
• more

Comments

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.

Examples

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 24 set-systems:
  {}  {}     {}               {}
      {{1}}  {{1}}            {{1}}
             {{1,2}}          {{1,2}}
             {{1},{2}}        {{1},{2}}
             {{1},{2},{1,2}}  {{1,2,3}}
                              {{1},{2,3}}
                              {{1},{2},{3}}
                              {{1},{2},{1,2}}
                              {{1,2},{1,3},{2,3}}
                              {{1},{2,3},{1,2,3}}
                              {{1},{2},{3},{2,3}}
                              {{1},{2},{1,3},{2,3}}
                              {{1},{2},{3},{1,2,3}}
                              {{3},{1,2},{1,3},{2,3}}
                              {{1},{2},{3},{1,3},{2,3}}
                              {{1,2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{2,3},{1,2,3}}
                              {{2},{3},{1,2},{1,3},{2,3}}
                              {{1},{2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{1,2},{1,3},{2,3}}
                              {{3},{1,2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{1,3},{2,3},{1,2,3}}
                              {{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
		

Crossrefs

Unlabeled set-systems are A000612.

Unlabeled set-systems whose dual is strict are A326946.

The labeled version is A326968.

The version with empty edges allowed is A326969.

The T_0 case (with strict dual) is A326972.

The covering case is A326973 (first differences).

Cf. A319559, A326951, A326965, A326966, A326970, A326974, A326975, A326978.