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

1, 1, 2, 6, 28, 375, 31745, 123805913

Offset

0,3

Comments

A set system (set of sets) is an antichain if no element is a subset of any other.

Links

Table of n, a(n) for n=0..7.

Dmitry I. Ignatov, On the Number of Maximal Antichains in Boolean Lattices for n up to 7. Lobachevskii J. Math., 44 (2023), 137-146.

Formula

For n > 0, a(n) = A326358(n) - 1.

Example

The a(0) = 1 through a(4) = 28 antichains:
  {}   {1}    {12}      {123}           {1234}
              {1}{2}    {1}{23}         {1}{234}
                        {2}{13}         {2}{134}
                        {3}{12}         {3}{124}
                        {1}{2}{3}       {4}{123}
                        {12}{13}{23}    {1}{2}{34}
                                        {1}{3}{24}
                                        {1}{4}{23}
                                        {2}{3}{14}
                                        {2}{4}{13}
                                        {3}{4}{12}
                                        {1}{2}{3}{4}
                                        {12}{134}{234}
                                        {13}{124}{234}
                                        {14}{123}{234}
                                        {23}{124}{134}
                                        {24}{123}{134}
                                        {34}{123}{124}
                                        {1}{23}{24}{34}
                                        {2}{13}{14}{34}
                                        {3}{12}{14}{24}
                                        {4}{12}{13}{23}
                                        {12}{13}{14}{234}
                                        {12}{23}{24}{134}
                                        {13}{23}{34}{124}
                                        {14}{24}{34}{123}
                                        {123}{124}{134}{234}
                                        {12}{13}{14}{23}{24}{34}

Mathematica

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]]]];
fasmax[y_]:=Complement[y, Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[stableSets[Subsets[Range[n], {1, n}], SubsetQ]]], {n, 0, 5}]

Crossrefs

Antichains of nonempty sets are A014466.

Minimal covering antichains are A046165.

Maximal intersecting antichains are A007363.

Maximal antichains of sets are A326358.

Cf. A000372, A003182, A006126, A006602, A014466, A058891, A261005, A305000, A305844, A307249, A326360, A326362, A326363.

Sequence in context: A108800 A325507 A306793 * A136639 A368003 A368143

Adjacent sequences: A326356 A326357 A326358 * A326360 A326361 A326362

Keyword

nonn,more

Author

Gus Wiseman, Jul 01 2019

Extensions

a(6) from Andrew Howroyd, Aug 14 2019

a(7) from Dmitry I. Ignatov, Oct 12 2021

Status

approved