A326965 Number of set-systems on n vertices where every covered vertex is the unique common element of some subset of the edges.

1, 2, 5, 46, 19181, 2010327182, 9219217424630040409, 170141181796805106025395618012972506978, 57896044618658097536026644159052312978532934306727333157337631572314050272137

Offset: 0

Author

Gus Wiseman, Aug 10 2019

Keywords

• nonn

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-system where no edge is a subset of any other. This sequence counts set-systems whose dual is a (strict) antichain, also called T_1 set-systems.

Examples

			The a(0) = 1 through a(2) = 5 set-systems:
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1},{2}}
             {{1},{2},{1,2}}
		

Crossrefs

Set-systems are A058891.

T_0 set-systems are A326940.

The covering case is A326961.

The version with empty edges allowed is A326967.

Set-systems whose dual is a weak antichain are A326968.

The unlabeled version is A326972.

The BII_numbers of these set-systems are A326979.

Cf. A059052, A326951, A326966, A326970, A326971, A326976, A326977.

Programs

Mathematica

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

Formula

Binomial transform of A326961.

a(n) = A326967(n)/2.