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A007363 Maximal self-dual antichains on n points.

%I A007363 M2505 #31 May 26 2020 21:58:16
%S A007363 0,1,3,5,20,168,11748,12160647
%N A007363 Maximal self-dual antichains on n points.
%C A007363 From _Gus Wiseman_, Jul 02 2019: (Start)
%C A007363 If self-dual means (pairwise) intersecting, then a(n) is the number of maximal intersecting antichains of nonempty subsets of {1..(n - 1)}. A set of sets is an antichain if no part is a subset of any other, and is intersecting if no two parts are disjoint. For example, the a(2) = 1 through a(5) = 20 maximal intersecting antichains are:
%C A007363    {1}    {1}     {1}             {1}
%C A007363           {2}     {2}             {2}
%C A007363           {12}    {3}             {3}
%C A007363                   {123}           {4}
%C A007363                   {12}{13}{23}    {1234}
%C A007363                                   {12}{13}{23}
%C A007363                                   {12}{14}{24}
%C A007363                                   {13}{14}{34}
%C A007363                                   {23}{24}{34}
%C A007363                                   {12}{134}{234}
%C A007363                                   {13}{124}{234}
%C A007363                                   {14}{123}{234}
%C A007363                                   {23}{124}{134}
%C A007363                                   {24}{123}{134}
%C A007363                                   {34}{123}{124}
%C A007363                                   {12}{13}{14}{234}
%C A007363                                   {12}{23}{24}{134}
%C A007363                                   {13}{23}{34}{124}
%C A007363                                   {14}{24}{34}{123}
%C A007363                                   {123}{124}{134}{234}
%C A007363 (End)
%D A007363 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007363 Daniel E. Loeb, <a href="http://www.labri.u-bordeaux.fr/~loeb/vote.html">On Games, Voting Schemes and Distributive Lattices</a>. LaBRI Report 625-93, University of Bordeaux I, 1993. [Broken link]
%F A007363 For n > 0, a(n) = A326363(n - 1) - 1 = A326362(n - 1) + n - 1. - _Gus Wiseman_, Jul 03 2019
%t A007363 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 A007363 fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
%t A007363 Table[Length[fasmax[stableSets[Subsets[Range[n],{1,n}],Or[Intersection[#1,#2]=={},SubsetQ[#1,#2]]&]]],{n,0,5}] (* _Gus Wiseman_, Jul 02 2019 *)
%t A007363 (* 2nd program *)
%t A007363 n = 2^6; g = CompleteGraph[n]; i = 0;
%t A007363 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 A007363 sets = FindClique[g, Infinity, All];
%t A007363 Length[sets]-1 (* _Elijah Beregovsky_, May 06 2020 *)
%Y A007363 Intersecting antichains are A326372.
%Y A007363 Intersecting antichains of nonempty sets are A001206.
%Y A007363 Unlabeled intersecting antichains are A305857.
%Y A007363 Maximal antichains of nonempty sets are A326359.
%Y A007363 The case with empty edges allowed is A326363.
%Y A007363 Cf. A000372, A305844, A306007, A326358, A326361, A326362, A326366.
%K A007363 nonn,more
%O A007363 1,3
%A A007363 _N. J. A. Sloane_
%E A007363 a(8) from _Elijah Beregovsky_, May 06 2020