A327020 - OEIS

%I #6 Aug 18 2019 11:27:51

%S 1,1,1,2,17,1451,3741198

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

%C 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.

%F Inverse binomial transform of A327057.

%e The a(0) = 1 through a(4) = 17 antichains:

%e   {}  {{1}}  {{12}}  {{123}}         {{1234}}

%e                      {{12}{13}{23}}  {{12}{134}{234}}

%e                                      {{13}{124}{234}}

%e                                      {{14}{123}{234}}

%e                                      {{23}{124}{134}}

%e                                      {{24}{123}{134}}

%e                                      {{34}{123}{124}}

%e                                      {{123}{124}{134}}

%e                                      {{123}{124}{234}}

%e                                      {{123}{134}{234}}

%e                                      {{124}{134}{234}}

%e                                      {{12}{13}{14}{234}}

%e                                      {{12}{23}{24}{134}}

%e                                      {{13}{23}{34}{124}}

%e                                      {{14}{24}{34}{123}}

%e                                      {{123}{124}{134}{234}}

%e                                      {{12}{13}{14}{23}{24}{34}}

%t dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];

%t 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]]]];

%t stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];

%t Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ],Union@@#==Range[n]&&stableQ[dual[#],Intersection[#1,#2]=={}&]&]],{n,0,4}]

%Y Covering, intersecting antichains are A305844.

%Y Covering, T1 antichains are A319639.

%Y Cointersecting set-systems are A327039.

%Y Covering, cointersecting set-systems are A327040.

%Y Covering, cointersecting set-systems are A327051.

%Y The non-covering version is A327057.

%Y Covering, intersecting, T1 set-systems are A327058.

%Y Unlabeled cointersecting antichains of multisets are A327060.

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

%K nonn,more

%O 0,4

%A _Gus Wiseman_, Aug 17 2019