A326967 Number of sets of subsets of {1..n} where every covered vertex is the unique common element of some subset of the edges.

2, 4, 10, 92, 38362, 4020654364, 18438434849260080818, 340282363593610212050791236025945013956, 115792089237316195072053288318104625957065868613454666314675263144628100544274

Offset: 0

Author

Gus Wiseman, Aug 10 2019

Keywords

• nonn

Comments

Alternatively, these are sets of subsets of {1..n} whose dual is a (strict) antichain, also called T_1 sets of subsets. 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 sets, none of which is a subset of any other.

Examples

			The a(0) = 2 through a(2) = 10 sets of subsets:
  {}    {}        {}
  {{}}  {{}}      {{}}
        {{1}}     {{1}}
        {{},{1}}  {{2}}
                  {{},{1}}
                  {{},{2}}
                  {{1},{2}}
                  {{},{1},{2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}
		

Crossrefs

Sets of subsets are A001146.

The unlabeled version is A326951.

The covering version is A326960.

The case without empty edges is A326965.

Sets of subsets whose dual is a weak antichain are A326969.

Cf. A059052, A059523, A326941, A326966, A326972, A326976, A326977, A326979.

Programs

Mathematica

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

Formula

a(n) = 2 * A326965(n).

Binomial transform of A326960.