cp's OEIS Frontend

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

%I A327057 #5 Aug 18 2019 11:27:05
%S A327057 1,2,4,9,36,1572,3750221
%N A327057 Number of antichains covering a subset of {1..n} where every two covered vertices appear together in some edge (cointersecting).
%C A327057 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 A327057 Binomial transform of A327020.
%e A327057 The a(0) = 1 through a(3) = 9 antichains:
%e A327057   {}  {}     {}       {}
%e A327057       {{1}}  {{1}}    {{1}}
%e A327057              {{2}}    {{2}}
%e A327057              {{1,2}}  {{3}}
%e A327057                       {{1,2}}
%e A327057                       {{1,3}}
%e A327057                       {{2,3}}
%e A327057                       {{1,2,3}}
%e A327057                       {{1,2},{1,3},{2,3}}
%t A327057 dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
%t A327057 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 A327057 stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t A327057 Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ],stableQ[dual[#],Intersection[#1,#2]=={}&]&]],{n,0,5}]
%Y A327057 Antichains are A000372.
%Y A327057 The BII-numbers of these set-systems are the intersection of A326704 and A326853.
%Y A327057 The covering case is A327020.
%Y A327057 Cointersecting set-systems are A327039.
%Y A327057 Cf. A006126, A051185, A245567, A305844, A326950, A327038, A327052.
%K A327057 nonn,more
%O A327057 0,2
%A A327057 _Gus Wiseman_, Aug 18 2019