A326974 Number of unlabeled set-systems covering n vertices where every vertex is the unique common element of some subset of the edges, also called unlabeled covering T_1 set-systems.

1, 1, 2, 16, 1212

Offset: 0

Author

Gus Wiseman, Aug 11 2019

Keywords

• nonn
• more

Comments

Alternatively, these are unlabeled set-systems covering n vertices whose dual is a (strict) antichain. 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-system where no edge is a subset of any other.

Examples

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 16 set-systems:
  {}  {{1}}  {{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 covers are A055621.

The same with T_0 instead of T_1 is A319637.

The labeled version is A326961.

The non-covering version is A326972 (partial sums).

Unlabeled covering set-systems whose dual is a weak antichain are A326973.

Cf. A000612, A059523, A319559, A326946, A326951, A326965, A326970, A326971, A326976, A326977, A326979.

Formula

a(n > 0) = A326972(n) - A326972(n - 1).