A326960 Number of sets of subsets of {1..n} covering all n vertices whose dual is a (strict) antichain, also called covering T_1 sets of subsets.

2, 2, 4, 72, 38040, 4020463392, 18438434825136728352, 340282363593610211921722192165556850240, 115792089237316195072053288318104625954343609704705784618785209431974668731584

Offset: 0

Author

Gus Wiseman, Aug 13 2019

Keywords

• nonn

Comments

Same as A059052 except with a(1) = 2 instead of 4.

The dual of a set of subsets 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 subsets where no edge is a subset of any other.

Alternatively, these are sets of subsets of {1..n} covering all n vertices where every vertex is the unique common element of some subset of the edges.

Examples

			The a(0) = 2 through a(2) = 4 sets of subsets:
  {}    {{1}}     {{1},{2}}
  {{}}  {{},{1}}  {{},{1},{2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}
		

Crossrefs

Covering sets of subsets are A000371.

Covering T_0 sets of subsets are A326939.

The case without empty edges is A326961.

The non-covering version is A326967.

Cf. A003181, A059052, A059523, A319639, A326951, A326965, A326974, A326976, A326977, A326979.

Programs

Mathematica

Table[Length[Select[Subsets[Subsets[Range[n]]],Length[Union[Select[Intersection@@@Rest[Subsets[#]],Length[#]==1&]]]==n&]],{n,0,3}]

Formula

Binomial transform of A326967.