A326959 -id:A326959 - cp's OEIS frontend

A245567 Number of antichain covers of a labeled n-set such that for every two distinct elements in the n-set, there is a set in the antichain cover containing one of the elements but not the other.

2, 1, 1, 5, 76, 5993, 7689745, 2414465044600, 56130437141763247212112, 286386577668298408602599478477358234902247

Offset: 0

Patrick De Causmaecker, Jul 25 2014

This is the number of antichain covers such that the induced partition contains only singletons. The induced partition of {{1,2},{2,3},{1,3},{3,4}} is {{1},{2},{3},{4}}, while the induced partition of {{1,2,3},{2,3,4}} is {{1},{2,3},{4}}.
This sequence is related to A006126. See 1st formula.
The sequence is also related to Dedekind numbers through Stirling numbers of the second kind. See 2nd formula.
Sets of subsets of the described type are said to be T_0. - Gus Wiseman, Aug 14 2019

			For n = 0, a(0) = 2 by the antisets {}, {{}}.
For n = 1, a(1) = 1 by the antiset {{1}}.
For n = 2, a(2) = 1 by the antiset {{1},{2}}.
For n = 3, a(3) = 5 by the antisets {{1},{2},{3}}, {{1,2},{1,3}}, {{1,2},{2,3}}, {{1,3},{2,3}}, {{1,2},{1,3},{2,3}}.

Patrick De Causmaecker and Stefan De Wannemacker, On the number of antichains of sets in a finite universe, arXiv:1407.4288 [math.CO], 2014.

Cf. A000372 (Dedekind numbers), A006126 (Number of antichain covers of a labeled n-set).
Sequences counting and ranking T_0 structures:
  A000112 (unlabeled topologies),
  A001035 (topologies),
  A059201 (covering set-systems),
  A245567 (antichain covers),
  A309615 (covering set-systems closed under intersection),
  A316978 (factorizations),
  A319559 (unlabeled set-systems by weight),
  A319564 (integer partitions),
  A319637 (unlabeled covering set-systems),
  A326939 (covering sets of subsets),
  A326940 (set-systems),
  A326941 (sets of subsets),
  A326943 (covering sets of subsets closed under intersection),
  A326944 (covering sets of subsets with {} and closed under intersection),
  A326945 (sets of subsets closed under intersection),
  A326946 (unlabeled set-systems),
  A326947 (BII-numbers of set-systems),
  A326948 (connected set-systems),
  A326949 (unlabeled sets of subsets),
  A326950 (antichains),
  A326959 (set-systems closed under intersection),
  A327013 (unlabeled covering set-systems closed under intersection),
  A327016 (BII-numbers of topologies).

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[Subsets[Subsets[Range[n]]],Union@@#==Range[n]&&stableQ[#,SubsetQ]&&UnsameQ@@dual[#]&]],{n,0,3}] (* Gus Wiseman, Aug 14 2019 *)

A000372(n) = Sum_{k=0..n} S(n+1,k+1)*a(k).
a(n) = A006126(n) - Sum_{k=1..n-1} S(n,k)*a(k).
Were n > 0 and S(n,k) is the number of ways to partition a set of n elements into k nonempty subsets.
Inverse binomial transform of A326950, if we assume a(0) = 1. - Gus Wiseman, Aug 14 2019

Definition corrected by Patrick De Causmaecker, Oct 10 2014

a(9), based on A000372, from Patrick De Causmaecker, Jun 01 2023

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

Original entry on oeis.org

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

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],{}]]]&]

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

			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

A245567 Number of antichain covers of a labeled n-set such that for every two distinct elements in the n-set, there is a set in the antichain cover containing one of the elements but not the other.

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

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

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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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