A326363 Number of maximal intersecting antichains of subsets of {1..n}.

1, 2, 4, 6, 21, 169, 11749, 12160648

Offset

0,2

Comments

A set system (set of sets) is an antichain if no element is a subset of any other, and is intersecting if no two element are disjoint.

Links

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

Formula

For n > 1, a(n) = A007363(n + 1) + 1 = A326362(n) + n + 1.

Example

The a(1) = 1 through a(4) = 21 maximal intersecting antichains:
  {}   {}    {}            {}
  {1}  {1}   {1}           {1}
       {2}   {2}           {2}
       {12}  {3}           {3}
             {123}         {4}
             {12}{13}{23}  {1234}
                           {12}{13}{23}
                           {12}{14}{24}
                           {13}{14}{34}
                           {23}{24}{34}
                           {12}{134}{234}
                           {13}{124}{234}
                           {14}{123}{234}
                           {23}{124}{134}
                           {24}{123}{134}
                           {34}{123}{124}
                           {12}{13}{14}{234}
                           {12}{23}{24}{134}
                           {13}{23}{34}{124}
                           {14}{24}{34}{123}
                           {123}{124}{134}{234}

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], {0, n}], Or[Intersection[#1, #2]=={}, SubsetQ[#1, #2]]&]]], {n, 0, 5}]
(* 2nd program *)
n = 2^6; g = CompleteGraph[n]; i = 0;
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]]]];
sets = FindClique[g, Infinity, All];
Length[sets] (* Elijah Beregovsky, May 06 2020 *)

Crossrefs

The case with nonempty, non-singleton edges is A326362.

Antichains of nonempty, non-singleton sets are A307249.

Minimal covering antichains are A046165.

Maximal intersecting antichains are A007363.

Maximal antichains of nonempty sets are A326359.

Cf. A000372, A003182, A006126, A006602, A014466, A051185, A058891, A261005, A305000, A305844, A326358, A326360, A326361.

Sequence in context: A333992 A176652 A251724 * A273522 A227626 A152482

Adjacent sequences: A326360 A326361 A326362 * A326364 A326365 A326366

Keyword

nonn,more

Author

Gus Wiseman, Jul 01 2019

Extensions

a(7) from Elijah Beregovsky, May 06 2020

Status

approved