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A326373 Number of intersecting set systems with empty intersection (meaning there is no vertex in common to all the edges) on n vertices.

%I A326373 #9 Aug 12 2019 23:04:32
%S A326373 1,1,1,3,435,989555,887050136795,291072121058024908202443,
%T A326373 14704019422368226413236661148207899662350666147,
%U A326373 12553242487939461785560846872353486129110194529637343578112251094358919036718815137721635299
%N A326373 Number of intersecting set systems with empty intersection (meaning there is no vertex in common to all the edges) on n vertices.
%C A326373 A set system (set of sets) is intersecting if no two edges are disjoint.
%F A326373 a(n) = A051185(n) - 1 - Sum_{k=1..n-1} binomial(n,k)*A000371(k). - _Andrew Howroyd_, Aug 12 2019
%e A326373 The a(3) = 3 intersecting set systems with empty intersection:
%e A326373   {}
%e A326373   {{1,2},{1,3},{2,3}}
%e A326373   {{1,2},{1,3},{2,3},{1,2,3}}
%t A326373 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 A326373 Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],Intersection[#1,#2]=={}&],And[#=={}||Intersection@@#=={}]&]],{n,0,4}]
%Y A326373 The inverse binomial transform is the covering case A326364.
%Y A326373 Set systems with empty intersection are A318129.
%Y A326373 Intersecting set systems are A051185.
%Y A326373 Intersecting antichains with empty intersection are A326366.
%Y A326373 Cf. A000371, A006126, A007363, A014466, A058891, A305844, A307249, A318128, A326361, A326362, A326363, A326365.
%K A326373 nonn
%O A326373 0,4
%A A326373 _Gus Wiseman_, Jul 01 2019
%E A326373 a(6)-a(9) from _Andrew Howroyd_, Aug 12 2019