A326969 Number of sets of subsets of {1..n} whose dual is a weak antichain.

2, 4, 12, 112, 38892

Offset: 0

Author

Gus Wiseman, Aug 10 2019

Keywords

• nonn
• more

Comments

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

Examples

			The a(0) = 2 through a(2) = 12 sets of subsets:
  {}    {}        {}
  {{}}  {{}}      {{}}
        {{1}}     {{1}}
        {{},{1}}  {{2}}
                  {{1,2}}
                  {{},{1}}
                  {{},{2}}
                  {{1},{2}}
                  {{},{1,2}}
                  {{},{1},{2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}
		

Crossrefs

Sets of subsets whose dual is strict are A326941.

The BII-numbers of set-systems whose dual is a weak antichain are A326966.

Sets of subsets whose dual is a (strict) antichain are A326967.

The case without empty edges is A326968.

Cf. A001146, A059052, A326951, A326970, A326971, A326975, A326978.

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]]],stableQ[dual[#],SubsetQ]&]],{n,0,3}]

Formula

a(n) = 2 * A326968(n).

a(n) = 2 * Sum_{k = 0..n} binomial(n, k) * A326970(k).