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%I #18 May 26 2020 21:59:02
%S 1,2,4,6,21,169,11749,12160648
%N Number of maximal intersecting antichains of subsets of {1..n}.
%C A set system (set of sets) is an antichain if no element is a subset of any other, and is intersecting if no two element are disjoint.
%F For n > 1, a(n) = A007363(n + 1) + 1 = A326362(n) + n + 1.
%e The a(1) = 1 through a(4) = 21 maximal intersecting antichains:
%e {} {} {} {}
%e {1} {1} {1} {1}
%e {2} {2} {2}
%e {12} {3} {3}
%e {123} {4}
%e {12}{13}{23} {1234}
%e {12}{13}{23}
%e {12}{14}{24}
%e {13}{14}{34}
%e {23}{24}{34}
%e {12}{134}{234}
%e {13}{124}{234}
%e {14}{123}{234}
%e {23}{124}{134}
%e {24}{123}{134}
%e {34}{123}{124}
%e {12}{13}{14}{234}
%e {12}{23}{24}{134}
%e {13}{23}{34}{124}
%e {14}{24}{34}{123}
%e {123}{124}{134}{234}
%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 fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
%t Table[Length[fasmax[stableSets[Subsets[Range[n],{0,n}],Or[Intersection[#1,#2]=={},SubsetQ[#1,#2]]&]]],{n,0,5}]
%t (* 2nd program *)
%t n = 2^6; g = CompleteGraph[n]; i = 0;
%t While[i < n, i++; j = i; While[j < n, j++; If[BitAnd[i, j] == 0 || BitAnd[i, j] == i || BitAnd[i, j] == j, g = EdgeDelete[g, i <-> j]]]];
%t sets = FindClique[g, Infinity, All];
%t Length[sets] (* _Elijah Beregovsky_, May 06 2020 *)
%Y The case with nonempty, non-singleton edges is A326362.
%Y Antichains of nonempty, non-singleton sets are A307249.
%Y Minimal covering antichains are A046165.
%Y Maximal intersecting antichains are A007363.
%Y Maximal antichains of nonempty sets are A326359.
%Y Cf. A000372, A003182, A006126, A006602, A014466, A051185, A058891, A261005, A305000, A305844, A326358, A326360, A326361.
%K nonn,more
%O 0,2
%A _Gus Wiseman_, Jul 01 2019
%E a(7) from _Elijah Beregovsky_, May 06 2020
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