A305844 -id:A305844 - cp's OEIS frontend

A007363 Maximal self-dual antichains on n points.

0, 1, 3, 5, 20, 168, 11748, 12160647

Offset: 1

From Gus Wiseman, Jul 02 2019: (Start)
If self-dual means (pairwise) intersecting, then a(n) is the number of maximal intersecting antichains of nonempty subsets of {1..(n - 1)}. A set of sets is an antichain if no part is a subset of any other, and is intersecting if no two parts are disjoint. For example, the a(2) = 1 through a(5) = 20 maximal intersecting antichains are:
   {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}
(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Daniel E. Loeb, On Games, Voting Schemes and Distributive Lattices. LaBRI Report 625-93, University of Bordeaux I, 1993. [Broken link]

Intersecting antichains are A326372.
Intersecting antichains of nonempty sets are A001206.
Unlabeled intersecting antichains are A305857.
Maximal antichains of nonempty sets are A326359.
The case with empty edges allowed is A326363.
Cf. A000372, A305844, A306007, A326358, A326361, A326362, A326366.

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}],Or[Intersection[#1,#2]=={},SubsetQ[#1,#2]]&]]],{n,0,5}] (* Gus Wiseman, Jul 02 2019 *)
(* 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]-1 (* Elijah Beregovsky, May 06 2020 *)

For n > 0, a(n) = A326363(n - 1) - 1 = A326362(n - 1) + n - 1. - Gus Wiseman, Jul 03 2019

a(8) from Elijah Beregovsky, May 06 2020

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

Original entry on oeis.org

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

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, and is intersecting if no two element are disjoint.

			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}

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.

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 *)

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

a(7) from Elijah Beregovsky, May 06 2020

A319774 Number of intersecting set systems spanning n vertices whose dual is also an intersecting set system.

Original entry on oeis.org

1, 1, 2, 14, 814, 1174774, 909125058112, 291200434263385001951232

Offset: 0

Gus Wiseman, Sep 27 2018

The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.

A multiset partition is intersecting iff no two parts are disjoint. The dual of a multiset partition is intersecting iff every pair of distinct vertices appear together in some part.

			The a(3) = 14 set systems:
   {{1},{1,2},{1,2,3}}
   {{1},{1,3},{1,2,3}}
   {{2},{1,2},{1,2,3}}
   {{2},{2,3},{1,2,3}}
   {{3},{1,3},{1,2,3}}
   {{3},{2,3},{1,2,3}}
   {{1,2},{1,3},{2,3}}
   {{1,2},{1,3},{1,2,3}}
   {{1,2},{2,3},{1,2,3}}
   {{1,3},{2,3},{1,2,3}}
   {{1},{1,2},{1,3},{1,2,3}}
   {{2},{1,2},{2,3},{1,2,3}}
   {{3},{1,3},{2,3},{1,2,3}}
   {{1,2},{1,3},{2,3},{1,2,3}}

Cf. A007716, A281116, A283877, A305854, A306006,  A316980, A316983, A317757, A319616.
Cf. A319752, A319765, A319766, A319767, A319768, A319769.
Intersecting set-systems are A051185.
The unlabeled multiset partition version is A319773.
The covering case is A327037.
The version without strict dual is A327038.
Cointersecting set-systems are A327039.
The BII-numbers of these set-systems are A327061.
Cf. A003465, A305843, A305844, A326854, A327020, A327040, A327052.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&UnsameQ@@dual[#]&&stableQ[#,Intersection[#1,#2]=={}&]&&stableQ[dual[#],Intersection[#1,#2]=={}&]&]],{n,0,3}] (* Gus Wiseman, Aug 19 2019 *)

a(6)-a(7) from Christian Sievers, Aug 18 2024

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

Original entry on oeis.org

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

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(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}

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

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.

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}]

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

a(6) from Andrew Howroyd, Aug 14 2019

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

A326361 Number of maximal intersecting antichains of sets covering n vertices with no singletons.

Original entry on oeis.org

1, 1, 1, 2, 12, 133, 11386, 12143511

Offset: 0

Gus Wiseman, Jul 01 2019

Covering means there are no isolated vertices. A set system (set of sets) is an antichain if no part is a subset of any other, and is intersecting if no two parts are disjoint.

			The a(4) = 12 antichains:
  {{1,2,3,4}}
  {{1,2},{1,3,4},{2,3,4}}
  {{1,3},{1,2,4},{2,3,4}}
  {{1,4},{1,2,3},{2,3,4}}
  {{2,3},{1,2,4},{1,3,4}}
  {{2,4},{1,2,3},{1,3,4}}
  {{3,4},{1,2,3},{1,2,4}}
  {{1,2},{1,3},{1,4},{2,3,4}}
  {{1,2},{2,3},{2,4},{1,3,4}}
  {{1,3},{2,3},{3,4},{1,2,4}}
  {{1,4},{2,4},{3,4},{1,2,3}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}

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, 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[Select[stableSets[Subsets[Range[n]],Or[Intersection[#1,#2]=={},SubsetQ[#1,#2]]&],Union@@#==Range[n]&]]],{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 = Select[FindClique[g, Infinity, All], BitOr @@ # == n - 1 &];
Length[sets] (* Elijah Beregovsky, May 05 2020 *)

a(6)-a(7) from Elijah Beregovsky, May 05 2020

A326362 Number of maximal intersecting antichains of nonempty, non-singleton subsets of {1..n}.

Original entry on oeis.org

1, 1, 1, 2, 16, 163, 11742, 12160640

Offset: 0

Gus Wiseman, Jul 01 2019

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

			The a(4) = 16 maximal intersecting antichains:
  {{1,2,3,4}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,4},{2,4}}
  {{1,3},{1,4},{3,4}}
  {{2,3},{2,4},{3,4}}
  {{1,2},{1,3,4},{2,3,4}}
  {{1,3},{1,2,4},{2,3,4}}
  {{1,4},{1,2,3},{2,3,4}}
  {{2,3},{1,2,4},{1,3,4}}
  {{2,4},{1,2,3},{1,3,4}}
  {{3,4},{1,2,3},{1,2,4}}
  {{1,2},{1,3},{1,4},{2,3,4}}
  {{1,2},{2,3},{2,4},{1,3,4}}
  {{1,3},{2,3},{3,4},{1,2,4}}
  {{1,4},{2,4},{3,4},{1,2,3}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}

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, 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}],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]-Log[2,n]-1 (* Elijah Beregovsky, May 06 2020 *)

For n > 1, a(n) = A326363(n) - n - 1 = A007363(n + 1) - n.

a(7) from Elijah Beregovsky, May 06 2020

A327020 Number of antichains covering n vertices where every two vertices appear together in some edge (cointersecting).

Original entry on oeis.org

1, 1, 1, 2, 17, 1451, 3741198

Offset: 0

Gus Wiseman, Aug 17 2019

A set-system is a finite set of finite nonempty sets. Its elements are sometimes called edges, The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. An antichain is a set of sets, none of which is a subset of any other. This sequence counts antichains with union {1..n} whose dual is pairwise intersecting.

			The a(0) = 1 through a(4) = 17 antichains:
  {}  {{1}}  {{12}}  {{123}}         {{1234}}
                     {{12}{13}{23}}  {{12}{134}{234}}
                                     {{13}{124}{234}}
                                     {{14}{123}{234}}
                                     {{23}{124}{134}}
                                     {{24}{123}{134}}
                                     {{34}{123}{124}}
                                     {{123}{124}{134}}
                                     {{123}{124}{234}}
                                     {{123}{134}{234}}
                                     {{124}{134}{234}}
                                     {{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}}

Covering, intersecting antichains are A305844.
Covering, T1 antichains are A319639.
Cointersecting set-systems are A327039.
Covering, cointersecting set-systems are A327040.
Covering, cointersecting set-systems are A327051.
The non-covering version is A327057.
Covering, intersecting, T1 set-systems are A327058.
Unlabeled cointersecting antichains of multisets are A327060.
Cf. A003465, A305843, A319765, A326853, A327037, A327038, A327053.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
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]]]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ],Union@@#==Range[n]&&stableQ[dual[#],Intersection[#1,#2]=={}&]&]],{n,0,4}]

Inverse binomial transform of A327057.

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

Original entry on oeis.org

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

A327057 Number of antichains covering a subset of {1..n} where every two covered vertices appear together in some edge (cointersecting).

Original entry on oeis.org

1, 2, 4, 9, 36, 1572, 3750221

Offset: 0

Gus Wiseman, Aug 18 2019

A set-system is a finite set of finite nonempty sets. Its elements are sometimes called edges. The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. An antichain is a set of sets, none of which is a subset of any other. This sequence counts antichains whose dual is pairwise intersecting.

			The a(0) = 1 through a(3) = 9 antichains:
  {}  {}     {}       {}
      {{1}}  {{1}}    {{1}}
             {{2}}    {{2}}
             {{1,2}}  {{3}}
                      {{1,2}}
                      {{1,3}}
                      {{2,3}}
                      {{1,2,3}}
                      {{1,2},{1,3},{2,3}}

Antichains are A000372.
The BII-numbers of these set-systems are the intersection of A326704 and A326853.
The covering case is A327020.
Cointersecting set-systems are A327039.
Cf. A006126, A051185, A245567, A305844, A326950, A327038, A327052.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
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]]]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ],stableQ[dual[#],Intersection[#1,#2]=={}&]&]],{n,0,5}]

Binomial transform of A327020.

A304999 Number of labeled antichains of finite sets spanning n vertices with singleton edges allowed.

Original entry on oeis.org

1, 1, 5, 53, 1577, 212137, 496946349, 309068823607069, 14369391923126237496803793, 146629927766168786109802623629262590838145873

Offset: 0

Gus Wiseman, May 23 2018

Only the non-singleton edges are required to form an antichain.

			The a(2) = 5 antichains:
  {{1,2}}
  {{1},{2}}
  {{1},{1,2}}
  {{2},{1,2}}
  {{1},{2},{1,2}}

Aniruddha Biswas and Palash Sarkar, Counting unate and balanced monotone Boolean functions, arXiv:2304.14069 [math.CO], 2023.
Aniruddha Biswas and Palash Sarkar, Counting Unate and Monotone Boolean Functions Under Restrictions of Balancedness and Non-Degeneracy, J. Int. Seq. (2025) Vol. 28, Art. No. 25.3.4. See p. 4.
Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.

Cf. A006126, A014466, A261005, A304996, A304997, A304998, A304999, A305000, A305001.

Cf. A000372, A003182, A304985, A305844, A306505, A319721, A321679, A324167.

Exponential transform of A304985.

Inverse binomial transform of A305000. - Aniruddha Biswas, May 12 2024

a(5)-a(8) from Gus Wiseman, May 31 2018

a(9) from Aniruddha Biswas, May 12 2024

cp's OEIS Frontend

A007363 Maximal self-dual antichains on n points.

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

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A319774 Number of intersecting set systems spanning n vertices whose dual is also an intersecting set system.

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

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A326361 Number of maximal intersecting antichains of sets covering n vertices with no singletons.

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A326362 Number of maximal intersecting antichains of nonempty, non-singleton subsets of {1..n}.

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A327020 Number of antichains covering n vertices where every two vertices appear together in some edge (cointersecting).

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

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