cp's OEIS Frontend

A327038 Number of pairwise intersecting set-systems covering a subset of {1..n} where every two covered vertices appear together in some edge (cointersecting).

%I A327038 #9 Aug 18 2024 19:04:46
%S A327038 1,2,6,34,1020,1188106,909149847892,291200434288840793135801
%N A327038 Number of pairwise intersecting set-systems covering a subset of {1..n} where every two covered vertices appear together in some edge (cointersecting).
%C A327038 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}}. This sequence counts pairwise intersecting set-systems that are cointersecting, meaning their dual is pairwise intersecting.
%F A327038 Binomial transform of A327037.
%e A327038 The a(0) = 1 through a(2) = 6 set-systems:
%e A327038   {}  {}     {}
%e A327038       {{1}}  {{1}}
%e A327038              {{2}}
%e A327038              {{1,2}}
%e A327038              {{1},{1,2}}
%e A327038              {{2},{1,2}}
%e A327038 The a(3) = 34 set-systems:
%e A327038   {}  {{1}}    {{1}{12}}    {{1}{12}{123}}   {{1}{12}{13}{123}}
%e A327038       {{2}}    {{1}{13}}    {{1}{13}{123}}   {{2}{12}{23}{123}}
%e A327038       {{3}}    {{2}{12}}    {{12}{13}{23}}   {{3}{13}{23}{123}}
%e A327038       {{12}}   {{2}{23}}    {{2}{12}{123}}   {{12}{13}{23}{123}}
%e A327038       {{13}}   {{3}{13}}    {{2}{23}{123}}
%e A327038       {{23}}   {{3}{23}}    {{3}{13}{123}}
%e A327038       {{123}}  {{1}{123}}   {{3}{23}{123}}
%e A327038                {{2}{123}}   {{12}{13}{123}}
%e A327038                {{3}{123}}   {{12}{23}{123}}
%e A327038                {{12}{123}}  {{13}{23}{123}}
%e A327038                {{13}{123}}
%e A327038                {{23}{123}}
%t A327038 dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
%t A327038 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 A327038 stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t A327038 Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],Intersection[#1,#2]=={}&],stableQ[dual[#],Intersection[#1,#2]=={}&]&]],{n,0,4}]
%Y A327038 Intersecting set-systems are A051185.
%Y A327038 The unlabeled multiset partition version is A319765.
%Y A327038 The BII-numbers of these set-systems are A326912.
%Y A327038 The covering case is A327037.
%Y A327038 Cointersecting set-systems are A327039.
%Y A327038 The case where the dual is strict is A327040.
%Y A327038 Cf. A058891, A319767, A319774, A326854, A327052.
%K A327038 nonn,more
%O A327038 0,2
%A A327038 _Gus Wiseman_, Aug 17 2019
%E A327038 a(6)-a(7) from _Christian Sievers_, Aug 18 2024