|
%I #41 Nov 04 2023 21:41:53
%S 1,1,1,2,13,279,29820,123590767
%N Number of maximal antichains of nonempty, non-singleton subsets of {1..n}.
%C A set system (set of sets) is an antichain if no element is a subset of any other.
%H 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.
%H Dmitry I. Ignatov, <a href="https://github.com/dimachine/NonEquivMACs/">Supporting iPython code and input files for a(7) based on inequivalent maximal antichains for n=7 and related sequences</a>, Github repository, section 3
%H Dmitry I. Ignatov, <a href="/A326360/a326360.pdf">PDF version of the supporting iPython notebook for a(7)</a>
%H Dmitry I. Ignatov, <a href="/A326360/a326360.ipynb.txt">Supporting iPython notebook for a(7): A326360.ipynb</a>
%F a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*A326359(k) for n >= 2. - _Andrew Howroyd_, Nov 19 2021
%e The a(1) = 1 through a(4) = 13 maximal antichains:
%e {} {12} {123} {1234}
%e {12}{13}{23} {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}
%e {12}{13}{14}{23}{24}{34}
%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],{2,n}],SubsetQ]]],{n,0,4}]
%o (Python)
%o # see Ignatov links
%o # _Dmitry I. Ignatov_, Oct 14 2021
%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, A058891, A261005, A305000, A305844, A326358, A326361, A326362, A326363.
%K nonn,more
%O 0,4
%A _Gus Wiseman_, Jul 01 2019
%E a(6) from _Andrew Howroyd_, Aug 14 2019
%E a(7) from _Dmitry I. Ignatov_, Oct 14 2021
| 