A327057 - OEIS

%I #5 Aug 18 2019 11:27:05

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

%N Number of antichains covering a subset of {1..n} where every two covered 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 whose dual is pairwise intersecting.

%F Binomial transform of A327020.

%e The a(0) = 1 through a(3) = 9 antichains:

%e   {}  {}     {}       {}

%e       {{1}}  {{1}}    {{1}}

%e              {{2}}    {{2}}

%e              {{1,2}}  {{3}}

%e                       {{1,2}}

%e                       {{1,3}}

%e                       {{2,3}}

%e                       {{1,2,3}}

%e                       {{1,2},{1,3},{2,3}}

%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],stableQ[dual[#],Intersection[#1,#2]=={}&]&]],{n,0,5}]

%Y Antichains are A000372.

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

%Y The covering case is A327020.

%Y Cointersecting set-systems are A327039.

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

%K nonn,more

%O 0,2

%A _Gus Wiseman_, Aug 18 2019