A326972 Number of unlabeled set-systems on n vertices whose dual is a (strict) antichain, also called unlabeled T_1 set-systems.

1, 2, 4, 20, 1232

Offset: 0

Author

Gus Wiseman, Aug 11 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}}. An antichain is a set of sets, none of which is a subset of any other.

Examples

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 20 set-systems:
  {}  {}     {}               {}
      {{1}}  {{1}}            {{1}}
             {{1},{2}}        {{1},{2}}
             {{1},{2},{1,2}}  {{1},{2},{3}}
                              {{1},{2},{1,2}}
                              {{1,2},{1,3},{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 version with empty edges allowed is A326951.

The labeled version is A326965.

The version where the dual is not required to be strict is A326971.

The covering version is A326974 (first differences).

Cf. A059523, A319559, A319637, A326973, A326976, A326977, A326979.