A326939 -id:A326939 - cp's OEIS frontend

A326945 Number of T_0 sets of subsets of {1..n} that are closed under intersection.

2, 4, 12, 96, 4404, 2725942, 151906396568, 28175293281055562650

Offset: 0

Gus Wiseman, Aug 08 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. 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) = 2 through a(2) = 12 sets of subsets:
  {}    {}        {}
  {{}}  {{}}      {{}}
        {{1}}     {{1}}
        {{},{1}}  {{2}}
                  {{},{1}}
                  {{},{2}}
                  {{1},{1,2}}
                  {{2},{1,2}}
                  {{},{1},{2}}
                  {{},{1},{1,2}}
                  {{},{2},{1,2}}
                  {{},{1},{2},{1,2}}

The non-T_0 version is A102897.
The version not closed under intersection is A326941.
The covering case is A326943.
The case without empty edges is A326959.
Cf. A003180, A182507, A316978, A319564, A326906, A326939, A326940, A326944, A326947.

Table[Length[Select[Subsets[Subsets[Range[n]]],UnsameQ@@dual[#]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]

Binomial transform of A326943.

a(5)-a(7) from Andrew Howroyd, Aug 14 2019

A326949 Number of unlabeled T_0 sets of subsets of {1..n}.

Original entry on oeis.org

2, 4, 10, 68, 3838, 37320356

Offset: 0

Gus Wiseman, Aug 08 2019

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

The non-T_0 version is A003180.
The labeled version is A326941.
The covering case is A326942 (first differences).
The case without empty edges is A326946.
Cf. A000371, A000612, A003181, A059052, A245567, A316978, A319559, A319564, A319637, A326939, A326940.

a(n) = 2 * A326946(n).

a(5) from Max Alekseyev, Oct 11 2023

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

Original entry on oeis.org

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).

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 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.

A327016 BII-numbers of finite T_0 topologies without their empty set.

Original entry on oeis.org

0, 1, 2, 5, 6, 7, 8, 17, 24, 25, 34, 40, 42, 69, 70, 71, 81, 85, 87, 88, 89, 93, 98, 102, 103, 104, 106, 110, 120, 121, 122, 127, 128, 257, 384, 385, 514, 640, 642, 1029, 1030, 1031, 1281, 1285, 1287, 1408, 1409, 1413, 1538, 1542, 1543, 1664, 1666, 1670, 1920

Offset: 1

Gus Wiseman, Aug 14 2019

nonn

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).

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.

			The sequence of all finite T_0 topologies without their empty set together with their BII-numbers begins:
   0: {}
   1: {{1}}
   2: {{2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   7: {{1},{2},{1,2}}
   8: {{3}}
  17: {{1},{1,3}}
  24: {{3},{1,3}}
  25: {{1},{3},{1,3}}
  34: {{2},{2,3}}
  40: {{3},{2,3}}
  42: {{2},{3},{2,3}}
  69: {{1},{1,2},{1,2,3}}
  70: {{2},{1,2},{1,2,3}}
  71: {{1},{2},{1,2},{1,2,3}}
  81: {{1},{1,3},{1,2,3}}
  85: {{1},{1,2},{1,3},{1,2,3}}
  87: {{1},{2},{1,2},{1,3},{1,2,3}}
  88: {{3},{1,3},{1,2,3}}

Gus Wiseman, The first 100 finite T_0 topologies, represented as posets.

T_0 topologies are A001035, with unlabeled version A000112.
BII-numbers of topologies without their empty set are A326876.
BII-numbers of T_0 set-systems are A326947.
Cf. A001930, A048793, A306445, A316978, A319564, A326031, A326872, A326875, A326939, A326941, A326959.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,1000],UnsameQ@@dual[bpe/@bpe[#]]&&SubsetQ[bpe/@bpe[#],Union[Union@@@Tuples[bpe/@bpe[#],2],DeleteCases[Intersection@@@Tuples[bpe/@bpe[#],2],{}]]]&]

cp's OEIS Frontend

A326945 Number of T_0 sets of subsets of {1..n} that are closed under intersection.

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A326949 Number of unlabeled T_0 sets of subsets of {1..n}.

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

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

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A327016 BII-numbers of finite T_0 topologies without their empty set.

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