A327020 Number of antichains covering n vertices where every two vertices appear together in some edge (cointersecting).

1, 1, 1, 2, 17, 1451, 3741198

Offset: 0

Author

Gus Wiseman, Aug 17 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 with union {1..n} whose dual is pairwise intersecting.

Examples

			The a(0) = 1 through a(4) = 17 antichains:
  {}  {{1}}  {{12}}  {{123}}         {{1234}}
                     {{12}{13}{23}}  {{12}{134}{234}}
                                     {{13}{124}{234}}
                                     {{14}{123}{234}}
                                     {{23}{124}{134}}
                                     {{24}{123}{134}}
                                     {{34}{123}{124}}
                                     {{123}{124}{134}}
                                     {{123}{124}{234}}
                                     {{123}{134}{234}}
                                     {{124}{134}{234}}
                                     {{12}{13}{14}{234}}
                                     {{12}{23}{24}{134}}
                                     {{13}{23}{34}{124}}
                                     {{14}{24}{34}{123}}
                                     {{123}{124}{134}{234}}
                                     {{12}{13}{14}{23}{24}{34}}
		

Crossrefs

Covering, intersecting antichains are A305844.

Covering, T1 antichains are A319639.

Cointersecting set-systems are A327039.

Covering, cointersecting set-systems are A327040.

Covering, cointersecting set-systems are A327051.

The non-covering version is A327057.

Covering, intersecting, T1 set-systems are A327058.

Unlabeled cointersecting antichains of multisets are A327060.

Cf. A003465, A305843, A319765, A326853, A327037, A327038, A327053.

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],Union@@#==Range[n]&&stableQ[dual[#],Intersection[#1,#2]=={}&]&]],{n,0,4}]

Formula

Inverse binomial transform of A327057.