A059523 -id:A059523 - cp's OEIS frontend

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

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.

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.

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

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

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

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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.

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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.

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

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