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Number of maximal intersecting antichains of subsets of {1..n}.
15

%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