cp's OEIS Frontend

A326366 Number of intersecting antichains of nonempty subsets of {1..n} with empty intersection (meaning there is no vertex in common to all the edges).

%I A326366 #16 Aug 14 2019 18:22:54
%S A326366 1,1,1,2,28,1960,1379273,229755337549,423295079757497714059
%N A326366 Number of intersecting antichains of nonempty subsets of {1..n} with empty intersection (meaning there is no vertex in common to all the edges).
%C A326366 A set system (set of sets) is an antichain if no edge is a subset of any other, and is intersecting if no two edges are disjoint.
%F A326366 a(n) = A326375(n) - 1.
%F A326366 a(n) = A001206(n+1) + A307249(n) - A014466(n). - _Andrew Howroyd_, Aug 14 2019
%e A326366 The a(0) = 1 through a(4) = 28 intersecting antichains with empty intersection:
%e A326366   {}  {}  {}  {}              {}
%e A326366               {{12}{13}{23}}  {{12}{13}{23}}
%e A326366                               {{12}{14}{24}}
%e A326366                               {{13}{14}{34}}
%e A326366                               {{23}{24}{34}}
%e A326366                               {{12}{13}{234}}
%e A326366                               {{12}{14}{234}}
%e A326366                               {{12}{23}{134}}
%e A326366                               {{12}{24}{134}}
%e A326366                               {{13}{14}{234}}
%e A326366                               {{13}{23}{124}}
%e A326366                               {{13}{34}{124}}
%e A326366                               {{14}{24}{123}}
%e A326366                               {{14}{34}{123}}
%e A326366                               {{23}{24}{134}}
%e A326366                               {{23}{34}{124}}
%e A326366                               {{24}{34}{123}}
%e A326366                               {{12}{134}{234}}
%e A326366                               {{13}{124}{234}}
%e A326366                               {{14}{123}{234}}
%e A326366                               {{23}{124}{134}}
%e A326366                               {{24}{123}{134}}
%e A326366                               {{34}{123}{124}}
%e A326366                               {{12}{13}{14}{234}}
%e A326366                               {{12}{23}{24}{134}}
%e A326366                               {{13}{23}{34}{124}}
%e A326366                               {{14}{24}{34}{123}}
%e A326366                               {{123}{124}{134}{234}}
%t A326366 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 A326366 Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],Or[Intersection[#1,#2]=={},SubsetQ[#1,#2]]&],#=={}||Intersection@@#=={}&]],{n,0,4}]
%Y A326366 The case with empty edges allowed is A326375.
%Y A326366 Intersecting antichains of nonempty sets are A001206.
%Y A326366 Intersecting set systems with empty intersection are A326373.
%Y A326366 Antichains of nonempty sets with empty intersection are A006126 or A307249.
%Y A326366 The inverse binomial transform is the covering case A326365.
%Y A326366 Cf. A007363, A014466, A051185, A058891, A305001, A305843, A305844, A318128, A318129, A326361, A326362, A326363, A326364.
%K A326366 nonn,more
%O A326366 0,4
%A A326366 _Gus Wiseman_, Jul 01 2019
%E A326366 a(7)-a(8) from _Andrew Howroyd_, Aug 14 2019