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%I #13 Aug 14 2019 18:22:32
%S 1,0,0,1,23,1834,1367903,229745722873,423295077919493525420
%N Number of intersecting antichains with empty intersection (meaning there is no vertex in common to all the edges) covering n vertices.
%C Covering means there are no isolated vertices. A set system (set of sets) is an antichain if no part is a subset of any other, and is intersecting if no two parts are disjoint.
%e The a(4) = 23 intersecting antichains with empty intersection:
%e {{1,2},{1,3},{2,3,4}}
%e {{1,2},{1,4},{2,3,4}}
%e {{1,2},{2,3},{1,3,4}}
%e {{1,2},{2,4},{1,3,4}}
%e {{1,3},{1,4},{2,3,4}}
%e {{1,3},{2,3},{1,2,4}}
%e {{1,3},{3,4},{1,2,4}}
%e {{1,4},{2,4},{1,2,3}}
%e {{1,4},{3,4},{1,2,3}}
%e {{2,3},{2,4},{1,3,4}}
%e {{2,3},{3,4},{1,2,4}}
%e {{2,4},{3,4},{1,2,3}}
%e {{1,2},{1,3,4},{2,3,4}}
%e {{1,3},{1,2,4},{2,3,4}}
%e {{1,4},{1,2,3},{2,3,4}}
%e {{2,3},{1,2,4},{1,3,4}}
%e {{2,4},{1,2,3},{1,3,4}}
%e {{3,4},{1,2,3},{1,2,4}}
%e {{1,2},{1,3},{1,4},{2,3,4}}
%e {{1,2},{2,3},{2,4},{1,3,4}}
%e {{1,3},{2,3},{3,4},{1,2,4}}
%e {{1,4},{2,4},{3,4},{1,2,3}}
%e {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
%t 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 Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],Or[Intersection[#1,#2]=={},SubsetQ[#1,#2]]&],And[Union@@#==Range[n],#=={}||Intersection@@#=={}]&]],{n,0,4}]
%Y Intersecting antichain covers are A305844.
%Y Intersecting covers with empty intersection are A326364.
%Y Antichain covers with empty intersection are A305001.
%Y The binomial transform is the non-covering case A326366.
%Y Covering, intersecting antichains with empty intersection are A326365.
%Y Cf. A006126, A007363, A014466, A051185, A058891, A305843, A307249, A318128, A318129, A326361, A326362, A326363.
%K nonn,more
%O 0,5
%A _Gus Wiseman_, Jul 01 2019
%E a(7)-a(8) from _Andrew Howroyd_, Aug 14 2019
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