A293993 -id:A293993 - cp's OEIS frontend

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

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.

A327425 Number of unlabeled antichains of nonempty sets covering n vertices where every two vertices appear together in some edge (cointersecting).

Original entry on oeis.org

1, 1, 1, 2, 6, 54

Offset: 0

Gus Wiseman, Sep 11 2019

An antichain is a set of sets, none of which is a subset of any other. It is covering if there are no isolated vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 6 antichains:
    {1}  {12}  {123}         {1234}
               {12}{13}{23}  {12}{134}{234}
                             {124}{134}{234}
                             {12}{13}{14}{234}
                             {123}{124}{134}{234}
                             {12}{13}{14}{23}{24}{34}

The labeled version is A327020.
Unlabeled covering antichains are A261005.
The weighted version is A327060.
Cf. A006126, A014466, A055621, A293606, A293993, A305844, A307249, A319639, A326704, A327057, A327058, A327358, A327359.

A319643 Number of non-isomorphic weight-n strict multiset partitions whose dual is an antichain of (not necessarily distinct) multisets.

Original entry on oeis.org

1, 1, 3, 6, 15, 29, 82, 179, 504, 1302, 3822

Offset: 0

Gus Wiseman, Sep 25 2018

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
From Gus Wiseman, Aug 15 2019: (Start)
Also the number of non-isomorphic T_0 weak antichains of weight n. The T_0 condition means that the dual is strict (no repeated edges). A weak antichain is a multiset of multisets, none of which is a proper submultiset of any other. For example, non-isomorphic representatives of the a(0) = 1 through a(4) = 15 T_0 weak antichains are:
  {}  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
             {{1},{1}}  {{1,2,2}}      {{1,2,2,2}}
             {{1},{2}}  {{1},{2,2}}    {{1,1},{1,1}}
                        {{1},{1},{1}}  {{1,1},{2,2}}
                        {{1},{2},{2}}  {{1},{2,2,2}}
                        {{1},{2},{3}}  {{1,2},{2,2}}
                                       {{1},{2,3,3}}
                                       {{1,3},{2,3}}
                                       {{1},{1},{2,2}}
                                       {{1},{2},{3,3}}
                                       {{1},{1},{1},{1}}
                                       {{1},{1},{2},{2}}
                                       {{1},{2},{2},{2}}
                                       {{1},{2},{3},{3}}
                                       {{1},{2},{3},{4}}
(End)

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 15 multiset partitions:
1: {{1}}
2: {{1,1}}
   {{1,2}}
   {{1},{2}}
3: {{1,1,1}}
   {{1,2,3}}
   {{1},{1,1}}
   {{1},{2,2}}
   {{1},{2,3}}
   {{1},{2},{3}}
4: {{1,1,1,1}}
   {{1,1,2,2}}
   {{1,2,3,4}}
   {{1},{1,1,1}}
   {{1},{1,2,2}}
   {{1},{2,2,2}}
   {{1},{2,3,4}}
   {{1,1},{2,2}}
   {{1,2},{3,3}}
   {{1,2},{3,4}}
   {{1},{2},{1,2}}
   {{1},{2},{2,2}}
   {{1},{2},{3,3}}
   {{1},{2},{3,4}}
   {{1},{2},{3},{4}}

Cf. A006126, A007716, A059201, A283877, A293606, A316980, A316983, A318099, A319558, A319616-A319646.

Cf. A245567, A293993, A319721, A326704, A326950, A326973, A326978.

A321678 Number of non-isomorphic weight-n strict antichains of sets with no singletons.

Original entry on oeis.org

1, 0, 1, 1, 3, 3, 11, 13, 39, 67, 174

Offset: 0

Gus Wiseman, Nov 16 2018

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(2) = 1 through a(6) = 11 antichains:
  {{1,2}}  {{1,2,3}}  {{1,2,3,4}}    {{1,2,3,4,5}}    {{1,2,3,4,5,6}}
                      {{1,2},{3,4}}  {{1,2},{3,4,5}}  {{1,2},{3,4,5,6}}
                      {{1,3},{2,3}}  {{1,4},{2,3,4}}  {{1,2,3},{4,5,6}}
                                                      {{1,2,5},{3,4,5}}
                                                      {{1,3,4},{2,3,4}}
                                                      {{1,5},{2,3,4,5}}
                                                      {{1,2},{1,3},{2,3}}
                                                      {{1,2},{3,4},{5,6}}
                                                      {{1,2},{3,5},{4,5}}
                                                      {{1,3},{2,4},{3,4}}
                                                      {{1,4},{2,4},{3,4}}

Cf. A006126, A049311, A096827, A285572, A293993, A293994, A318099, A319719, A319721, A320799, A321679.

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.

A321185 Number of integer partitions of n that are the vertex-degrees of some strict antichain of sets with no singletons.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 5, 5, 9, 11, 17, 20

Offset: 0

Gus Wiseman, Oct 29 2018

A strict antichain is a finite set of finite nonempty sets, none of which is a subset of any other.

			The a(2) = 1 through a(9) = 11 partitions:
  (11)  (111)  (211)   (2111)   (222)     (2221)     (2222)      (3222)
               (1111)  (11111)  (2211)    (22111)    (3221)      (22221)
                                (3111)    (31111)    (22211)     (32211)
                                (21111)   (211111)   (32111)     (33111)
                                (111111)  (1111111)  (41111)     (222111)
                                                     (221111)    (321111)
                                                     (311111)    (411111)
                                                     (2111111)   (2211111)
                                                     (11111111)  (3111111)
                                                                 (21111111)
                                                                 (111111111)
The a(8) = 9 integer partitions together with a realizing strict antichain for each (the parts of the partition count the appearances of each vertex in the antichain):
     (41111): {{1,2},{1,3},{1,4},{1,5}}
      (3221): {{1,2},{1,3},{1,4},{2,3}}
     (32111): {{1,3},{1,2,4},{1,2,5}}
    (311111): {{1,2},{1,3},{1,4,5,6}}
      (2222): {{1,2},{1,3,4},{2,3,4}}
     (22211): {{1,2,3,4},{1,2,3,5}}
    (221111): {{1,2,3},{1,2,4,5,6}}
   (2111111): {{1,2},{1,3,4,5,6,7}}
  (11111111): {{1,2,3,4,5,6,7,8}}

Cf. A000070, A000569, A006126, A096827, A209816, A283877, A293606, A293993, A306005, A318361, A319721, A321176, A321184.

submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
stableQ[u_]:=Apply[And,Outer[#1==#2||!submultisetQ[#1,#2]&&!submultisetQ[#2,#1]&,u,u,1],{0,1}];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
anti[m_]:=Select[mps[m],And[And@@UnsameQ@@@#,UnsameQ@@#,Min@@Length/@#>1,stableQ[#]]&];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[strnorm[n],anti[#]!={}&]],{n,8}]

cp's OEIS Frontend

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

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A327425 Number of unlabeled antichains of nonempty sets covering n vertices where every two vertices appear together in some edge (cointersecting).

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A319643 Number of non-isomorphic weight-n strict multiset partitions whose dual is an antichain of (not necessarily distinct) multisets.

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A321678 Number of non-isomorphic weight-n strict antichains of sets with no singletons.

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

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A321185 Number of integer partitions of n that are the vertex-degrees of some strict antichain of sets with no singletons.

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Mathematica