A326974 -id:A326974 - 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.

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.

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}]

A327018 Number of non-isomorphic set-systems of weight n whose dual is a weak antichain.

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 17, 24, 51, 80, 180

Offset: 0

Gus Wiseman, Aug 15 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(1) = 1 through a(6) = 17 multiset partitions:
  {1}  {12}    {123}      {1234}        {12345}          {123456}
       {1}{2}  {1}{23}    {1}{234}      {1}{2345}        {1}{23456}
               {1}{2}{3}  {12}{34}      {12}{345}        {12}{3456}
                          {1}{2}{12}    {1}{2}{345}      {123}{456}
                          {1}{2}{34}    {1}{23}{45}      {12}{13}{23}
                          {1}{2}{3}{4}  {1}{2}{3}{23}    {1}{23}{123}
                                        {1}{2}{3}{45}    {1}{2}{3456}
                                        {1}{2}{3}{4}{5}  {1}{23}{456}
                                                         {12}{34}{56}
                                                         {1}{2}{13}{23}
                                                         {1}{2}{3}{123}
                                                         {1}{2}{3}{456}
                                                         {1}{2}{34}{56}
                                                         {3}{4}{12}{34}
                                                         {1}{2}{3}{4}{34}
                                                         {1}{2}{3}{4}{56}
                                                         {1}{2}{3}{4}{5}{6}

Cf. A007716, A283877, A293993, A319643, A319721, A326966, A326968, A326970, A326972, A326973, A326974, A326975, A326978, A327017, A327019.

A327011 Number of unlabeled sets of subsets covering n vertices where every vertex is the unique common element of some subset of the edges, also called unlabeled covering T_1 sets of subsets.

Original entry on oeis.org

2, 2, 4, 32, 2424

Offset: 0

Gus Wiseman, Aug 13 2019

Alternatively, these are unlabeled sets of subsets covering n vertices whose dual is a (strict) antichain. 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.

			Non-isomorphic representatives of the a(0) = 1 through a(2) = 4 sets of subsets:
  {}    {{1}}     {{1},{2}}
  {{}}  {{},{1}}  {{},{1},{2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}

Unlabeled covering sets of subsets are A003181.
The same with T_0 instead of T_1 is A326942.
The non-covering version is A326951 (partial sums).
The labeled version is A326960.
The case without empty edges is A326974.
Cf. A001146, A055621, A059523, A319637, A326961, A326972, A326973, A326976, A326977, A326979.

a(n) = A326974(n) / 2.

a(n > 0) = A326951(n) - A326951(n - 1).

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

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

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A327018 Number of non-isomorphic set-systems of weight n whose dual is a weak antichain.

Views

Author

Keywords

Comments

Examples

Crossrefs

A327011 Number of unlabeled sets of subsets covering n vertices where every vertex is the unique common element of some subset of the edges, also called unlabeled covering T_1 sets of subsets.

Views

Author

Keywords

Comments

Examples

Crossrefs

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

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

Views

Author

Keywords

Comments

Examples

Crossrefs