cp's OEIS Frontend

A326359 Number of maximal antichains of nonempty subsets of {1..n}.

%I A326359 #18 Aug 31 2023 10:02:30
%S A326359 1,1,2,6,28,375,31745,123805913
%N A326359 Number of maximal antichains of nonempty subsets of {1..n}.
%C A326359 A set system (set of sets) is an antichain if no element is a subset of any other.
%H A326359 Dmitry I. Ignatov, <a href="https://doi.org/10.1134/S1995080223010158">On the Number of Maximal Antichains in Boolean Lattices for n up to 7</a>. Lobachevskii J. Math., 44 (2023), 137-146.
%F A326359 For n > 0, a(n) = A326358(n) - 1.
%e A326359 The a(0) = 1 through a(4) = 28 antichains:
%e A326359   {}   {1}    {12}      {123}           {1234}
%e A326359               {1}{2}    {1}{23}         {1}{234}
%e A326359                         {2}{13}         {2}{134}
%e A326359                         {3}{12}         {3}{124}
%e A326359                         {1}{2}{3}       {4}{123}
%e A326359                         {12}{13}{23}    {1}{2}{34}
%e A326359                                         {1}{3}{24}
%e A326359                                         {1}{4}{23}
%e A326359                                         {2}{3}{14}
%e A326359                                         {2}{4}{13}
%e A326359                                         {3}{4}{12}
%e A326359                                         {1}{2}{3}{4}
%e A326359                                         {12}{134}{234}
%e A326359                                         {13}{124}{234}
%e A326359                                         {14}{123}{234}
%e A326359                                         {23}{124}{134}
%e A326359                                         {24}{123}{134}
%e A326359                                         {34}{123}{124}
%e A326359                                         {1}{23}{24}{34}
%e A326359                                         {2}{13}{14}{34}
%e A326359                                         {3}{12}{14}{24}
%e A326359                                         {4}{12}{13}{23}
%e A326359                                         {12}{13}{14}{234}
%e A326359                                         {12}{23}{24}{134}
%e A326359                                         {13}{23}{34}{124}
%e A326359                                         {14}{24}{34}{123}
%e A326359                                         {123}{124}{134}{234}
%e A326359                                         {12}{13}{14}{23}{24}{34}
%t A326359 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 A326359 fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
%t A326359 Table[Length[fasmax[stableSets[Subsets[Range[n],{1,n}],SubsetQ]]],{n,0,5}]
%Y A326359 Antichains of nonempty sets are A014466.
%Y A326359 Minimal covering antichains are A046165.
%Y A326359 Maximal intersecting antichains are A007363.
%Y A326359 Maximal antichains of sets are A326358.
%Y A326359 Cf. A000372, A003182, A006126, A006602, A014466, A058891, A261005, A305000, A305844, A307249, A326360, A326362, A326363.
%K A326359 nonn,more
%O A326359 0,3
%A A326359 _Gus Wiseman_, Jul 01 2019
%E A326359 a(6) from _Andrew Howroyd_, Aug 14 2019
%E A326359 a(7) from _Dmitry I. Ignatov_, Oct 12 2021