A326968 Number of set-systems on n vertices whose dual is a weak antichain.

1, 2, 6, 56, 19446

Offset: 0

Author

Gus Wiseman, Aug 10 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}}. A weak antichain is a multiset of sets, none of which is a proper subset of any other.

Examples

			The a(0) = 1 through a(2) = 6 set-systems:
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1,2}}
             {{1},{2}}
             {{1},{2},{1,2}}
		

Crossrefs

The case with strict dual is A326965.

The BII-numbers of these set-systems are A326966.

The version with empty edges allowed is A326969.

The covering case is A326970.

The unlabeled version is A326971.

Cf. A058891, A059523, A326940, A326972, A326973, A326975, A326978, A326979.

Programs

Mathematica

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],stableQ[dual[#],SubsetQ]&]],{n,0,3}]

Formula

a(n) = A326969(n)/2.

Binomial transform of A326970.