A319637 -id:A319637 - cp's OEIS frontend

A326942 Number of unlabeled T_0 sets of subsets of {1..n} that cover all n vertices.

2, 2, 6, 58, 3770

Offset: 0

Gus Wiseman, Aug 07 2019

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},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges).

			Non-isomorphic representatives of the a(0) = 2 through a(2) = 6 sets of subsets:
  {}    {{1}}     {{1},{2}}
  {{}}  {{},{1}}  {{2},{1,2}}
                  {{},{1},{2}}
                  {{},{2},{1,2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}

The non-T_0 version is A003181.
The case without empty edges is A319637.
The labeled version is A326939.
The non-covering version is A326949 (partial sums).
Cf. A000371, A003180, A055621, A059201, A316978, A319559, A319564, A326907, A326941, A326943, A326946.

a(n) = 2 * A319637(n).

A326959 Number of T_0 set-systems covering a subset of {1..n} that are closed under intersection.

Original entry on oeis.org

1, 2, 5, 22, 297, 20536, 16232437, 1063231148918, 225402337742595309857

Offset: 0

Gus Wiseman, Aug 13 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}}. The T_0 condition means that the dual is strict (no repeated edges).

			The a(0) = 1 through a(3) = 22 set-systems:
  {}  {}     {}           {}
      {{1}}  {{1}}        {{1}}
             {{2}}        {{2}}
             {{1},{1,2}}  {{3}}
             {{2},{1,2}}  {{1},{1,2}}
                          {{1},{1,3}}
                          {{2},{1,2}}
                          {{2},{2,3}}
                          {{3},{1,3}}
                          {{3},{2,3}}
                          {{1},{1,2},{1,3}}
                          {{2},{1,2},{2,3}}
                          {{3},{1,3},{2,3}}
                          {{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},{1,2},{1,3},{1,2,3}}
                          {{2},{1,2},{2,3},{1,2,3}}
                          {{3},{1,3},{2,3},{1,2,3}}

The covering case is A309615.
T_0 set-systems are A326940.
The version with empty edges allowed is A326945.
Cf. A051185, A058891, A059201, A316978, A319559, A309615, A319637, A326943, A326944, A326946, A326947, A326959.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],UnsameQ@@dual[#]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]

Binomial transform of A309615.

a(5)-a(8) from Andrew Howroyd, Aug 14 2019

A330294 Number of non-isomorphic fully chiral set-systems on n vertices.

Original entry on oeis.org

1, 2, 3, 10, 899

Offset: 0

Gus Wiseman, Dec 10 2019

A set-system is a finite set of finite nonempty sets. It is fully chiral if every permutation of the covered vertices gives a different representative.

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 10 set-systems:
  0  0    0        0
     {1}  {1}      {1}
          {2}{12}  {2}{12}
                   {1}{3}{23}
                   {2}{13}{23}
                   {3}{23}{123}
                   {2}{3}{13}{23}
                   {1}{3}{23}{123}
                   {2}{13}{23}{123}
                   {2}{3}{13}{23}{123}

The labeled version is A330282.
Partial sums of A330295 (the covering case).
Unlabeled costrict (or T_0) set-systems are A326946.
BII-numbers of fully chiral set-systems are A330226.
Non-isomorphic fully chiral multiset partitions are A330227.
Fully chiral partitions are A330228.
Fully chiral factorizations are A330235.
MM-numbers of fully chiral multisets of multisets are A330236.
Cf. A000612, A016031, A055621, A083323, A283877, A319637, A330098, A330231, A330232, A330234.

A330295 Number of non-isomorphic fully chiral set-systems covering n vertices.

Original entry on oeis.org

1, 1, 1, 7, 889

Offset: 0

Gus Wiseman, Dec 10 2019

A set-system is a finite set of finite nonempty sets. It is fully chiral if every permutation of the covered vertices gives a different representative.

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 7 set-systems:
  0  {1}  {1}{12}  {1}{2}{13}
                   {1}{12}{23}
                   {1}{12}{123}
                   {1}{2}{12}{13}
                   {1}{2}{13}{123}
                   {1}{12}{23}{123}
                   {1}{2}{12}{13}{123}

The labeled version is A330229.
First differences of A330294 (the non-covering case).
Unlabeled costrict (or T_0) set-systems are A326946.
BII-numbers of fully chiral set-systems are A330226.
Non-isomorphic fully chiral multiset partitions are A330227.
Fully chiral partitions are A330228.
Fully chiral factorizations are A330235.
MM-numbers of fully chiral multisets of multisets are A330236.
Cf. A000612, A016031, A055621, A083323, A283877, A319637, A330098, A330231, A330232, A330234, A330282.

A368731 Number of non-isomorphic n-element sets of nonempty subsets of {1..n}.

Original entry on oeis.org

1, 1, 2, 10, 97, 2160, 126862, 21485262, 11105374322, 18109358131513, 95465831661532570, 1660400673336788987026, 96929369602251313489896310, 19268528295096123543660356281600, 13203875101002459910158494602665950757, 31517691852305548841992346407978317698725021

Offset: 0

Gus Wiseman, Jan 07 2024

nonn

			Non-isomorphic representatives of the a(3) = 10 set-systems:
  {{1},{2},{3}}
  {{1},{2},{1,2}}
  {{1},{2},{1,3}}
  {{1},{2},{1,2,3}}
  {{1},{1,2},{1,3}}
  {{1},{1,2},{2,3}}
  {{1},{1,2},{1,2,3}}
  {{1},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,3},{1,2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

The case of graphs is A001434, labeled A116508.
Labeled version is A136556, covering A054780, binomial transform of A367916.
The case of labeled covering graphs is A367863, binomial transform A367862.
These include the set-systems ranked by A367917.
The covering case is A368186, for graphs A006649, connected A057500.
Requiring all edges to be singletons or pairs gives A368598.
A003465 counts covers with any number of edges, unlabeled A055621.
A046165 counts minimal covers, ranks A309326.
A058891 counts set-systems, unlabeled A000612, without singletons A016031.
A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.
Cf. A000088, A007716, A283877, A367901, A367902, A367903, A368599.

brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]},{i,Length[p]}])], {p,Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute /@ Subsets[Subsets[Range[n],{1,n}],{n}]]],{n,0,4}]

a(n) = polcoef(G(n, n), n) \\ G defined in A368186. - Andrew Howroyd, Jan 11 2024

Terms a(6) and beyond from Andrew Howroyd, Jan 11 2024

A326948 Number of connected T_0 set-systems on n vertices.

Original entry on oeis.org

1, 1, 3, 86, 31302, 2146841520, 9223371978880250448, 170141183460469231408869283342774399392, 57896044618658097711785492504343953919148780260559635830120038252613826101856

Offset: 0

Gus Wiseman, Aug 08 2019

nonn

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}}. The T_0 condition means that the dual is strict (no repeated edges).

			The a(3) = 86 set-systems:
  {12}{13}         {1}{2}{13}{123}     {1}{2}{3}{13}{23}
  {12}{23}         {1}{2}{23}{123}     {1}{2}{3}{13}{123}
  {13}{23}         {1}{3}{12}{13}      {1}{2}{3}{23}{123}
  {1}{2}{123}      {1}{3}{12}{23}      {1}{2}{12}{13}{23}
  {1}{3}{123}      {1}{3}{12}{123}     {1}{2}{12}{13}{123}
  {1}{12}{13}      {1}{3}{13}{23}      {1}{2}{12}{23}{123}
  {1}{12}{23}      {1}{3}{13}{123}     {1}{2}{13}{23}{123}
  {1}{12}{123}     {1}{3}{23}{123}     {1}{3}{12}{13}{23}
  {1}{13}{23}      {1}{12}{13}{23}     {1}{3}{12}{13}{123}
  {1}{13}{123}     {1}{12}{13}{123}    {1}{3}{12}{23}{123}
  {2}{3}{123}      {1}{12}{23}{123}    {1}{3}{13}{23}{123}
  {2}{12}{13}      {1}{13}{23}{123}    {1}{12}{13}{23}{123}
  {2}{12}{23}      {2}{3}{12}{13}      {2}{3}{12}{13}{23}
  {2}{12}{123}     {2}{3}{12}{23}      {2}{3}{12}{13}{123}
  {2}{13}{23}      {2}{3}{12}{123}     {2}{3}{12}{23}{123}
  {2}{23}{123}     {2}{3}{13}{23}      {2}{3}{13}{23}{123}
  {3}{12}{13}      {2}{3}{13}{123}     {2}{12}{13}{23}{123}
  {3}{12}{23}      {2}{3}{23}{123}     {3}{12}{13}{23}{123}
  {3}{13}{23}      {2}{12}{13}{23}     {1}{2}{3}{12}{13}{23}
  {3}{13}{123}     {2}{12}{13}{123}    {1}{2}{3}{12}{13}{123}
  {3}{23}{123}     {2}{12}{23}{123}    {1}{2}{3}{12}{23}{123}
  {12}{13}{23}     {2}{13}{23}{123}    {1}{2}{3}{13}{23}{123}
  {12}{13}{123}    {3}{12}{13}{23}     {1}{2}{12}{13}{23}{123}
  {12}{23}{123}    {3}{12}{13}{123}    {1}{3}{12}{13}{23}{123}
  {13}{23}{123}    {3}{12}{23}{123}    {2}{3}{12}{13}{23}{123}
  {1}{2}{3}{123}   {3}{13}{23}{123}    {1}{2}{3}{12}{13}{23}{123}
  {1}{2}{12}{13}   {12}{13}{23}{123}
  {1}{2}{12}{23}   {1}{2}{3}{12}{13}
  {1}{2}{12}{123}  {1}{2}{3}{12}{23}
  {1}{2}{13}{23}   {1}{2}{3}{12}{123}

Andrew Howroyd, Table of n, a(n) for n = 0..11

The same with covering instead of connected is A059201, with unlabeled version A319637.
The non-T_0 version is A323818 (covering) or A326951 (not-covering).
The non-connected version is A326940, with unlabeled version A326946.
Cf. A000371, A003465, A245567, A316978, A319559, A319564, A326939, A326941, A326947.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&Length[csm[#]]<=1&&UnsameQ@@dual[#]&]],{n,0,3}]

Logarithmic transform of A059201.

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).

A327013 Number of non-isomorphic T_0 set-systems covering a subset of {1..n} that are closed under intersection.

Original entry on oeis.org

1, 2, 3, 6, 23, 282, 28033

Offset: 0

Gus Wiseman, Dec 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}}. The T_0 condition means that the dual is strict (no repeated edges).

			Non-isomorphic representatives of the a(1) = 2 through a(4) = 23 set-systems:
    0    0        0                 0
    {1}  {1}      {1}               {1}
         {1}{12}  {1}{12}           {1}{12}
                  {1}{12}{13}       {1}{12}{13}
                  {1}{12}{123}      {1}{12}{123}
                  {1}{12}{13}{123}  {1}{12}{13}{14}
                                    {1}{12}{13}{123}
                                    {1}{12}{13}{124}
                                    {1}{12}{123}{124}
                                    {1}{12}{13}{1234}
                                    {1}{12}{123}{1234}
                                    {1}{12}{13}{14}{123}
                                    {1}{12}{13}{123}{124}
                                    {1}{12}{13}{14}{1234}
                                    {1}{12}{13}{123}{1234}
                                    {1}{12}{13}{124}{1234}
                                    {1}{12}{123}{124}{1234}
                                    {1}{12}{13}{14}{123}{124}
                                    {1}{12}{13}{14}{123}{1234}
                                    {1}{12}{13}{123}{124}{1234}
                                    {1}{12}{13}{14}{123}{124}{134}
                                    {1}{12}{13}{14}{123}{124}{1234}
                                    {1}{12}{13}{14}{123}{124}{134}{1234}

The labeled version is A326959.
T_0 set-systems are A326940.
Cf. A051185, A059201, A319559, A309615, A319637, A326944, A326945, A326946.

a(5)-a(6) from Andrew Howroyd, Dec 21 2019

cp's OEIS Frontend

A326942 Number of unlabeled T_0 sets of subsets of {1..n} that cover all n vertices.

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A326959 Number of T_0 set-systems covering a subset of {1..n} that are closed under intersection.

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A330294 Number of non-isomorphic fully chiral set-systems on n vertices.

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A330295 Number of non-isomorphic fully chiral set-systems covering n vertices.

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A368731 Number of non-isomorphic n-element sets of nonempty subsets of {1..n}.

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A326948 Number of connected T_0 set-systems on n vertices.

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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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A327013 Number of non-isomorphic T_0 set-systems covering a subset of {1..n} that are closed under intersection.

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