A326359 -id:A326359 - cp's OEIS frontend

A326360 Number of maximal antichains of nonempty, non-singleton subsets of {1..n}.

1, 1, 1, 2, 13, 279, 29820, 123590767

Offset: 0

Gus Wiseman, Jul 01 2019

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

			The a(1) = 1 through a(4) = 13 maximal antichains:
  {}  {12}  {123}         {1234}
            {12}{13}{23}  {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}
                          {12}{13}{14}{23}{24}{34}

Dmitry I. Ignatov, On the Number of Maximal Antichains in Boolean Lattices for n up to 7. Lobachevskii J. Math., 44 (2023), 137-146.
Dmitry I. Ignatov, Supporting iPython code and input files for a(7) based on inequivalent maximal antichains for n=7 and related sequences, Github repository, section 3
Dmitry I. Ignatov, PDF version of the supporting iPython notebook for a(7)
Dmitry I. Ignatov, Supporting iPython notebook for a(7): A326360.ipynb

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, A058891, A261005, A305000, A305844, A326358, A326361, A326362, A326363.

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],{2,n}],SubsetQ]]],{n,0,4}]

# see Ignatov links
# Dmitry I. Ignatov, Oct 14 2021

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*A326359(k) for n >= 2. - Andrew Howroyd, Nov 19 2021

a(6) from Andrew Howroyd, Aug 14 2019

a(7) from Dmitry I. Ignatov, Oct 14 2021

A343660 Number of maximal pairwise coprime sets of at least two divisors > 1 of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 4, 0, 0, 1, 1, 1, 4, 0, 1, 1, 3, 0, 4, 0, 2, 2, 1, 0, 4, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 8, 0, 1, 2, 0, 1, 4, 0, 2, 1, 4, 0, 6, 0, 1, 2, 2, 1, 4, 0, 4, 0, 1, 0, 8, 1, 1, 1

Offset: 1

Gus Wiseman, Apr 26 2021

nonn

			The a(n) sets for n = 6, 12, 24, 30, 36, 60, 72, 96:
  {2,3}  {2,3}  {2,3}  {5,6}    {2,3}  {5,6}    {2,3}  {2,3}
         {3,4}  {3,4}  {2,15}   {2,9}  {2,15}   {2,9}  {3,4}
                {3,8}  {3,10}   {3,4}  {3,10}   {3,4}  {3,8}
                       {2,3,5}  {4,9}  {3,20}   {3,8}  {3,16}
                                       {4,15}   {4,9}  {3,32}
                                       {5,12}   {8,9}
                                       {2,3,5}
                                       {3,4,5}

The case of pairs is A089233.
The case with 1's is A343652.
The case with singletons is (also) A343652.
The non-maximal version is A343653.
The non-maximal version with 1's is A343655.
The version for subsets of {2..n} is A343659 (for n > 2).
A018892 counts coprime unordered pairs of divisors.
A051026 counts pairwise indivisible subsets of {1..n}.
A066620 counts pairwise coprime 3-sets of divisors.
A100565 counts pairwise coprime unordered triples of divisors.
Cf. A005361, A007359, A007360, A067824, A074206, A225520, A276187, A320426, A325683, A326077, A326359, A326496, A337485, A343654.

fasmax[y_]:=Complement[y,Union@@Most@*Subsets/@y];
Table[Length[fasmax[Select[Subsets[Rest[Divisors[n]]],CoprimeQ@@#&]]],{n,100}]

a(n) = A343652(n) - A005361(n).

A348260 Number of inequivalent maximal antichains of the Boolean lattice on a set of n elements.

Original entry on oeis.org

1, 2, 3, 5, 10, 30, 233, 35925

Offset: 0

Dmitry I. Ignatov, Oct 13 2021

a(n) is the number of orbits for the corresponding families of maximal antichains (see also A326358) of the powerset of {1,2,...,n} under the action of the symmetry group S_n.

			The a(0)=1 maximal antichains is {}.
The a(1)=2 maximal antichains are {}, {1}.
The a(2)=3 maximal antichains {}, {1}{2}, {12}.
Representatives of the a(3)=5 maximal antichains are: {}, {1}{2}{3}, {12}{3}, {12}{13}{23}, {123}.
Representatives of the a(4)=10 maximal antichains are:
   {},                       {1}{2}{3}{4},
   {12}{3}{4},               {12}{13}{23}{4},
   {123}{4},                 {12}{13}{24}{14}{24}{34},
   {123}{14}{24}{34},        {123}{124}{34},
   {123}{124}{134}{234},     {1234}.

Dmitry I. Ignatov, On the Number of Maximal Antichains in Boolean Lattices for n up to 7. Lobachevskii J. Math., 44 (2023), 137-146.
Dmitry I. Ignatov, Supporting iPython code and input files for generating and counting inequivalent maximal antichains for n up to 7 and related sequences, Github repository, sections 0,1,4
Dmitry I. Ignatov, Supporting iPython notebook: A348260.ipynb
Dmitry I. Ignatov, PDF of the supporting iPython notebook

Cf. A003182 (not necessarily maximal), A326358 (labeled case).

Cf. A326359, A326360.

cp's OEIS Frontend

A326360 Number of maximal antichains of nonempty, non-singleton subsets of {1..n}.

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A343660 Number of maximal pairwise coprime sets of at least two divisors > 1 of n.

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A348260 Number of inequivalent maximal antichains of the Boolean lattice on a set of n elements.

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