cp's OEIS Frontend

A327059 Number of pairwise intersecting set-systems covering a subset of {1..n} whose dual is a weak antichain.

%I A327059 #5 Aug 19 2019 08:50:33
%S A327059 1,2,4,10,178
%N A327059 Number of pairwise intersecting set-systems covering a subset of {1..n} whose dual is a weak antichain.
%C A327059 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}}. A weak antichain is a multiset of sets, none of which is a proper subset of any other.
%F A327059 Binomial transform of A327058.
%e A327059 The a(0) = 1 through a(3) = 10 set-systems:
%e A327059   {}  {}     {}      {}
%e A327059       {{1}}  {{1}}   {{1}}
%e A327059              {{2}}   {{2}}
%e A327059              {{12}}  {{3}}
%e A327059                      {{12}}
%e A327059                      {{13}}
%e A327059                      {{23}}
%e A327059                      {{123}}
%e A327059                      {{12}{13}{23}}
%e A327059                      {{12}{13}{23}{123}}
%t A327059 dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
%t A327059 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 A327059 stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t A327059 Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],Intersection[#1,#2]=={}&],stableQ[dual[#],SubsetQ]&]],{n,0,3}]
%Y A327059 Intersecting set-systems are A051185.
%Y A327059 The BII-numbers of these set-systems are the intersection of A326910 and A326966.
%Y A327059 Set-systems whose dual is a weak antichain are A326968.
%Y A327059 The covering version is A327058.
%Y A327059 The unlabeled multiset partition version is A327060.
%Y A327059 Cf. A000372, A305844, A326951, A326965, A326970, A327057, A327062.
%K A327059 nonn,more
%O A327059 0,2
%A A327059 _Gus Wiseman_, Aug 18 2019