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%I #18 May 26 2020 21:57:15
%S 1,1,1,2,12,133,11386,12143511
%N Number of maximal intersecting antichains of sets covering n vertices with no singletons.
%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) = 12 antichains:
%e {{1,2,3,4}}
%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 fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
%t Table[Length[fasmax[Select[stableSets[Subsets[Range[n]],Or[Intersection[#1,#2]=={},SubsetQ[#1,#2]]&],Union@@#==Range[n]&]]],{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 = Select[FindClique[g, Infinity, All], BitOr @@ # == n - 1 &];
%t Length[sets] (* _Elijah Beregovsky_, May 05 2020 *)
%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, A326362, A326363.
%K nonn,more
%O 0,4
%A _Gus Wiseman_, Jul 01 2019
%E a(6)-a(7) from _Elijah Beregovsky_, May 05 2020
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