A326965 -id:A326965 - cp's OEIS frontend

A326971 Number of unlabeled set-systems on n vertices whose dual is a weak antichain.

1, 2, 5, 24, 1267

Offset: 0

Gus Wiseman, Aug 10 2019

A set-system is a finite set of finite nonempty sets. 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}}. A weak antichain is a multiset of sets, none of which is a proper subset of any other.

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 24 set-systems:
  {}  {}     {}               {}
      {{1}}  {{1}}            {{1}}
             {{1,2}}          {{1,2}}
             {{1},{2}}        {{1},{2}}
             {{1},{2},{1,2}}  {{1,2,3}}
                              {{1},{2,3}}
                              {{1},{2},{3}}
                              {{1},{2},{1,2}}
                              {{1,2},{1,3},{2,3}}
                              {{1},{2,3},{1,2,3}}
                              {{1},{2},{3},{2,3}}
                              {{1},{2},{1,3},{2,3}}
                              {{1},{2},{3},{1,2,3}}
                              {{3},{1,2},{1,3},{2,3}}
                              {{1},{2},{3},{1,3},{2,3}}
                              {{1,2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{2,3},{1,2,3}}
                              {{2},{3},{1,2},{1,3},{2,3}}
                              {{1},{2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{1,2},{1,3},{2,3}}
                              {{3},{1,2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{1,3},{2,3},{1,2,3}}
                              {{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Unlabeled set-systems are A000612.
Unlabeled set-systems whose dual is strict are A326946.
The labeled version is A326968.
The version with empty edges allowed is A326969.
The T_0 case (with strict dual) is A326972.
The covering case is A326973 (first differences).
Cf. A319559, A326951, A326965, A326966, A326970, A326974, A326975, A326978.

A326975 Number of factorizations of n into factors > 1 whose dual is a weak antichain.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 2, 1, 2, 2, 5, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 3, 2, 1, 5, 1, 7, 2, 2, 2, 9, 1, 2, 2, 3, 1, 5, 1, 2, 2, 2, 1, 5, 2, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 4, 1, 2, 2, 11, 2, 5, 1, 2, 2, 5, 1, 12, 1, 2, 2, 2, 2, 5, 1, 5, 5, 2, 1, 4, 2, 2

Offset: 1

Gus Wiseman, Aug 13 2019

nonn

The dual of a multiset system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The dual of a factorization is the dual of the multiset partition obtained by replacing each factor with its multiset of prime indices.

A weak antichain is a multiset of multisets, none of which is a proper submultiset of any other.

			The a(36) = 9 factorizations:
  (36)
  (4*9)
  (6*6)
  (2*18)
  (3*12)
  (2*2*9)
  (2*3*6)
  (3*3*4)
  (2*2*3*3)

The T_0 case (where the dual is strict) is A316978.
Set-systems whose dual is a weak antichain are A326968.
Partitions whose dual is a weak antichain are A326978.
The T_1 case (where the dual is a strict antichain) is A327012.
Cf. A001055, A059523, A316978, A319639, A326965, A326966, A326969, A326970, A326971, A326976, A326977.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
submultQ[cap_,fat_]:=And@@Function[i,Count[fat,i]>=Count[cap,i]]/@Union[List@@cap];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[facs[n],stableQ[dual[primeMS/@#],submultQ]&]],{n,100}]

A326960 Number of sets of subsets of {1..n} covering all n vertices whose dual is a (strict) antichain, also called covering T_1 sets of subsets.

Original entry on oeis.org

2, 2, 4, 72, 38040, 4020463392, 18438434825136728352, 340282363593610211921722192165556850240, 115792089237316195072053288318104625954343609704705784618785209431974668731584

Offset: 0

Gus Wiseman, Aug 13 2019

nonn

Same as A059052 except with a(1) = 2 instead of 4.
The dual of a set of subsets 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 subsets where no edge is a subset of any other.
Alternatively, these are sets of subsets of {1..n} covering all n vertices where every vertex is the unique common element of some subset of the edges.

			The a(0) = 2 through a(2) = 4 sets of subsets:
  {}    {{1}}     {{1},{2}}
  {{}}  {{},{1}}  {{},{1},{2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}

Covering sets of subsets are A000371.
Covering T_0 sets of subsets are A326939.
The case without empty edges is A326961.
The non-covering version is A326967.
Cf. A003181, A059052, A059523, A319639, A326951, A326965, A326974, A326976, A326977, A326979.

Table[Length[Select[Subsets[Subsets[Range[n]]],Length[Union[Select[Intersection@@@Rest[Subsets[#]],Length[#]==1&]]]==n&]],{n,0,3}]

Binomial transform of A326967.

A326967 Number of sets of subsets of {1..n} where every covered vertex is the unique common element of some subset of the edges.

Original entry on oeis.org

2, 4, 10, 92, 38362, 4020654364, 18438434849260080818, 340282363593610212050791236025945013956, 115792089237316195072053288318104625957065868613454666314675263144628100544274

Offset: 0

Gus Wiseman, Aug 10 2019

nonn

Alternatively, these are sets of subsets of {1..n} whose dual is a (strict) antichain, also called T_1 sets of subsets. The dual of a set of subsets 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.

			The a(0) = 2 through a(2) = 10 sets of subsets:
  {}    {}        {}
  {{}}  {{}}      {{}}
        {{1}}     {{1}}
        {{},{1}}  {{2}}
                  {{},{1}}
                  {{},{2}}
                  {{1},{2}}
                  {{},{1},{2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}

Sets of subsets are A001146.
The unlabeled version is A326951.
The covering version is A326960.
The case without empty edges is A326965.
Sets of subsets whose dual is a weak antichain are A326969.
Cf. A059052, A059523, A326941, A326966, A326972, A326976, A326977, A326979.

tmQ[eds_]:=Union@@Select[Intersection@@@Rest[Subsets[eds]],Length[#]==1&]==Union@@eds;
Table[Length[Select[Subsets[Subsets[Range[n]]],tmQ[#]&]],{n,0,3}]

a(n) = 2 * A326965(n).

Binomial transform of A326960.

A327012 Number of factorizations of n into factors > 1 whose dual is a (strict) antichain.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 7, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 2, 2, 1, 1, 1, 5, 5, 1, 1, 2, 1, 1

Offset: 1

Gus Wiseman, Aug 13 2019

nonn

Differs from A322453 at 36, 72, 100, ...
The dual of a multiset system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The dual of a factorization is the dual of the multiset partition obtained by replacing each factor with its multiset of prime indices.
An antichain is a set of multisets, none of which is a submultiset of any other.

			The a(72) = 12 factorizations:
  (8*9)
  (3*24)
  (4*18)
  (2*4*9)
  (3*3*8)
  (3*4*6)
  (2*2*18)
  (2*3*12)
  (2*2*2*9)
  (2*2*3*6)
  (2*3*3*4)
  (2*2*2*3*3)

Set-systems whose dual is a (strict) antichain are A326965.
The version where the dual is a weak antichain is A326975.
Partitions whose dual is a (strict) antichain are A326977.
Cf. A001055, A316978, A326961, A326974, A326976, A326979.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
submultQ[cap_,fat_]:=And@@Function[i,Count[fat,i]>=Count[cap,i]]/@Union[List@@cap];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[facs[n],UnsameQ@@dual[primeMS/@#]&&stableQ[dual[primeMS/@#],submultQ]&]],{n,100}]

A327058 Number of pairwise intersecting set-systems covering n vertices whose dual is a weak antichain.

Original entry on oeis.org

1, 1, 1, 3, 155

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}}. A weak antichain is a multiset of sets, none of which is a proper subset of any other.

			The a(0) = 1 through a(3) = 3 set-systems:
  {}  {{1}}  {{12}}  {{123}}
                     {{12}{13}{23}}
                     {{12}{13}{23}{123}}

Covering intersecting set-systems are A305843.
The BII-numbers of these set-systems are the intersection of A326910 and A326966.
Covering coantichains are A326970.
The non-covering version is A327059.
The unlabeled multiset partition version is A327060.
Cf. A006126, A051185, A059523, A305844, A319639, A326961, A326965, A326968, A327020, A327057.

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}],Intersection[#1,#2]=={}&],Union@@#==Range[n]&&stableQ[dual[#],SubsetQ]&]],{n,0,3}]

Inverse binomial transform of A327059.

A327060 Number of non-isomorphic weight-n weak antichains of multisets where every two vertices appear together in some edge (cointersecting).

Original entry on oeis.org

1, 1, 3, 4, 9, 11, 30, 42, 103, 194, 443

Offset: 0

Gus Wiseman, Aug 18 2019

A multiset partition is a finite multiset of finite nonempty multisets. It is a weak antichain if no part is a proper submultiset of any other.

			Non-isomorphic representatives of the a(0) = 1 through a(5) = 11 multiset partitions:
  {}  {{1}}  {{11}}    {{111}}      {{1111}}        {{11111}}
             {{12}}    {{122}}      {{1122}}        {{11222}}
             {{1}{1}}  {{123}}      {{1222}}        {{12222}}
                       {{1}{1}{1}}  {{1233}}        {{12233}}
                                    {{1234}}        {{12333}}
                                    {{11}{11}}      {{12344}}
                                    {{12}{12}}      {{12345}}
                                    {{12}{22}}      {{11}{122}}
                                    {{1}{1}{1}{1}}  {{12}{222}}
                                                    {{33}{123}}
                                                    {{1}{1}{1}{1}{1}}

Antichains are A000372.
The BII-numbers of these set-systems are the intersection of A326853 and A326704.
Cointersecting set-systems are A327039.
The set-system version is A327057, with covering case A327058.
Cf. A006126, A007716, A051185, A305844, A326965, A327020, A327059, A327062.

A327059 Number of pairwise intersecting set-systems covering a subset of {1..n} whose dual is a weak antichain.

Original entry on oeis.org

1, 2, 4, 10, 178

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}}. A weak antichain is a multiset of sets, none of which is a proper subset of any other.

			The a(0) = 1 through a(3) = 10 set-systems:
  {}  {}     {}      {}
      {{1}}  {{1}}   {{1}}
             {{2}}   {{2}}
             {{12}}  {{3}}
                     {{12}}
                     {{13}}
                     {{23}}
                     {{123}}
                     {{12}{13}{23}}
                     {{12}{13}{23}{123}}

Intersecting set-systems are A051185.
The BII-numbers of these set-systems are the intersection of A326910 and A326966.
Set-systems whose dual is a weak antichain are A326968.
The covering version is A327058.
The unlabeled multiset partition version is A327060.
Cf. A000372, A305844, A326951, A326965, A326970, A327057, A327062.

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}],Intersection[#1,#2]=={}&],stableQ[dual[#],SubsetQ]&]],{n,0,3}]

Binomial transform of A327058.

A327017 Number of non-isomorphic multiset partitions of weight n where every vertex, as a multiset of weight 1, is the multiset-meet of some subset of the edges.

Original entry on oeis.org

1, 1, 2, 4, 9, 19, 49, 115, 310, 830, 2383

Offset: 0

Gus Wiseman, Aug 15 2019

The multiset-meet of a collection of multisets has as underlying set the intersection of their underlying sets and as multiplicities the minima of their multiplicities.

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 19 multiset partitions:
    {1}  {1}{1}  {1}{11}    {1}{111}      {1}{1111}
         {1}{2}  {1}{1}{1}  {1}{1}{11}    {1}{1}{111}
                 {1}{2}{2}  {1}{2}{12}    {1}{11}{11}
                 {1}{2}{3}  {1}{2}{22}    {1}{12}{22}
                            {1}{1}{1}{1}  {1}{2}{122}
                            {1}{1}{2}{2}  {1}{2}{222}
                            {1}{2}{2}{2}  {1}{1}{1}{11}
                            {1}{2}{3}{3}  {1}{1}{2}{22}
                            {1}{2}{3}{4}  {1}{2}{2}{12}
                                          {1}{2}{2}{22}
                                          {1}{2}{3}{23}
                                          {1}{2}{3}{33}
                                          {1}{1}{1}{1}{1}
                                          {1}{1}{2}{2}{2}
                                          {1}{2}{2}{2}{2}
                                          {1}{2}{2}{3}{3}
                                          {1}{2}{3}{3}{3}
                                          {1}{2}{3}{4}{4}
                                          {1}{2}{3}{4}{5}

Cf. A007716, A059523, A326961, A326965, A326967, A326972, A326974, A326976, A326977, A326979, A327012.

A327019 Number of non-isomorphic set-systems of weight n whose dual is a (strict) antichain.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 5, 7, 15, 26, 61

Offset: 0

Gus Wiseman, Aug 15 2019

Also the number of non-isomorphic set-systems where every vertex is the unique common element of some subset of the edges, also called non-isomorphic T_1 set-systems.
A set-system is a finite set of finite nonempty sets. 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.

			Non-isomorphic representatives of the a(1) = 1 through a(8) = 15 multiset partitions:
  {1}  {1}{2}  {1}{2}{3}  {1}{2}{12}    {1}{2}{3}{23}    {12}{13}{23}
                          {1}{2}{3}{4}  {1}{2}{3}{4}{5}  {1}{2}{13}{23}
                                                         {1}{2}{3}{123}
                                                         {1}{2}{3}{4}{34}
                                                         {1}{2}{3}{4}{5}{6}
.
  {1}{23}{24}{34}        {12}{13}{24}{34}
  {3}{12}{13}{23}        {2}{13}{14}{234}
  {1}{2}{3}{13}{23}      {1}{2}{13}{24}{34}
  {1}{2}{3}{24}{34}      {1}{2}{3}{14}{234}
  {1}{2}{3}{4}{234}      {1}{2}{3}{23}{123}
  {1}{2}{3}{4}{5}{45}    {1}{2}{3}{4}{1234}
  {1}{2}{3}{4}{5}{6}{7}  {1}{2}{34}{35}{45}
                         {1}{4}{23}{24}{34}
                         {2}{3}{12}{13}{23}
                         {1}{2}{3}{4}{12}{34}
                         {1}{2}{3}{4}{24}{34}
                         {1}{2}{3}{4}{35}{45}
                         {1}{2}{3}{4}{5}{345}
                         {1}{2}{3}{4}{5}{6}{56}
                         {1}{2}{3}{4}{5}{6}{7}{8}

Cf. A007716, A059523, A283877, A293993, A326961, A326965, A326974, A326976, A326977, A326979, A327012, A327018.

cp's OEIS Frontend

A326971 Number of unlabeled set-systems on n vertices whose dual is a weak antichain.

Views

Author

Keywords

Comments

Examples

Crossrefs

A326975 Number of factorizations of n into factors > 1 whose dual is a weak antichain.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A326960 Number of sets of subsets of {1..n} covering all n vertices whose dual is a (strict) antichain, also called covering T_1 sets of subsets.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A326967 Number of sets of subsets of {1..n} where every covered vertex is the unique common element of some subset of the edges.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A327012 Number of factorizations of n into factors > 1 whose dual is a (strict) antichain.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A327058 Number of pairwise intersecting set-systems covering n vertices whose dual is a weak antichain.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A327060 Number of non-isomorphic weight-n weak antichains of multisets where every two vertices appear together in some edge (cointersecting).

Views

Author

Keywords

Comments

Examples

Crossrefs

A327059 Number of pairwise intersecting set-systems covering a subset of {1..n} whose dual is a weak antichain.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A327017 Number of non-isomorphic multiset partitions of weight n where every vertex, as a multiset of weight 1, is the multiset-meet of some subset of the edges.

Views

Author

Keywords

Comments

Examples

Crossrefs