A327057 Number of antichains covering a subset of {1..n} where every two covered vertices appear together in some edge (cointersecting).

1, 2, 4, 9, 36, 1572, 3750221

Offset: 0

Author

Gus Wiseman, Aug 18 2019

Keywords

• nonn
• more

Comments

A set-system is a finite set of finite nonempty sets. Its elements are sometimes called edges. 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 of sets, none of which is a subset of any other. This sequence counts antichains whose dual is pairwise intersecting.

Examples

			The a(0) = 1 through a(3) = 9 antichains:
  {}  {}     {}       {}
      {{1}}  {{1}}    {{1}}
             {{2}}    {{2}}
             {{1,2}}  {{3}}
                      {{1,2}}
                      {{1,3}}
                      {{2,3}}
                      {{1,2,3}}
                      {{1,2},{1,3},{2,3}}
		

Crossrefs

Antichains are A000372.

The BII-numbers of these set-systems are the intersection of A326704 and A326853.

The covering case is A327020.

Cointersecting set-systems are A327039.

Cf. A006126, A051185, A245567, A305844, A326950, A327038, A327052.

Programs

Mathematica

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ],stableQ[dual[#],Intersection[#1,#2]=={}&]&]],{n,0,5}]

Formula

Binomial transform of A327020.