A326945 -id:A326945 - 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

A326943 Number of T_0 sets of subsets of {1..n} that cover all n vertices and are closed under intersection.

Original entry on oeis.org

2, 2, 6, 70, 4078, 2704780, 151890105214, 28175292217767880450

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(3) = 6 sets of subsets:
  {}    {{1}}     {{1},{1,2}}
  {{}}  {{},{1}}  {{2},{1,2}}
                  {{},{1},{2}}
                  {{},{1},{1,2}}
                  {{},{2},{1,2}}
                  {{},{1},{2},{1,2}}

The non-T_0 version is A326906.
The case without empty edges is A309615.
The non-covering version is A326945.
The version not closed under intersection is A326939.
Cf. A003180, A003181, A003465, A059052, A059201, A245567, A316978, A319564, A319637, A326940, A326941, A326942, A326947.

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

Inverse binomial transform of A326945.

a(n) = Sum_{k=0..n} Stirling1(n,k)*A326906(k). - Andrew Howroyd, Aug 14 2019

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

A326944 Number of T_0 sets of subsets of {1..n} that cover all n vertices, contain {}, and are closed under intersection.

Original entry on oeis.org

1, 1, 4, 58, 3846, 2685550, 151873991914, 28175291154649937052

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) = 1 through a(2) = 4 sets of subsets:
  {{}}  {{},{1}}  {{},{1},{2}}
                  {{},{1},{1,2}}
                  {{},{2},{1,2}}
                  {{},{1},{2},{1,2}}

The version not closed under intersection is A059201.
The non-T_0 version is A326881.
The version where {} is not necessarily an edge is A326943.
Cf. A003181, A003465, A055621, A182507, A245567, A316978, A319564, A326906, A326939, A326941, A326945, A326947.

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

a(n) = Sum_{k=0..n} Stirling1(n,k)*A326881(k). - Andrew Howroyd, Aug 14 2019

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

A309615 Number of T_0 set-systems covering n vertices that are closed under intersection.

Original entry on oeis.org

1, 1, 2, 12, 232, 19230, 16113300, 1063117943398, 225402329237199496416

Offset: 0

Gus Wiseman, Aug 11 2019

First differs from A182507 at a(5) = 19230, A182507(5) = 12848.

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) = 12 set-systems:
  {}  {{1}}  {{1},{1,2}}  {{1},{1,2},{1,3}}
             {{2},{1,2}}  {{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 version with empty edges allowed is A326943.

Cf. A003465, A059052, A059201, A319637, A326880, A326881, A326901, A326902, A326905, A326944, A326945.

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

a(n) = A326943(n) - A326944(n).
a(n) = Sum_{k = 1..n} s(n,k) * A326901(k - 1) where s = A048994.
a(n) = Sum_{k = 1..n} s(n,k) * A326902(k) where s = A048994.

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

A334252 Number of closure operators on a set of n elements which satisfy the T_0 separation axiom.

Original entry on oeis.org

1, 2, 5, 44, 2179, 1362585, 75953166947, 14087646640499308474

Offset: 0

Joshua Moerman, Apr 20 2020

The T_0 axiom states that the closure of {x} and {y} are different for distinct x and y.

			The a(0) = 1 through a(2) = 5 set-systems of closed sets:
{{}}  {{}}      {{1,2},{1}}
      {{1},{}}  {{1,2},{2}}
                {{1,2},{1},{}}
                {{1,2},{2},{}}
                {{1,2},{1},{2},{}}

R. S. R. Myers, J. Adámek, S. Milius, and H. Urbat, Coalgebraic constructions of canonical nondeterministic automata, Theoretical Computer Science, 604 (2015), 81-101.
B. Venkateswarlu and U. M. Swamy, T_0-Closure Operators and Pre-Orders, Lobachevskii Journal of Mathematics, 39 (2018), 1446-1452.

The number of all closure operators is given in A102896.
For strict T0 closure operators, see A334253.
For T1 closure operators, see A334254.
Cf. A326943, A326944, A326945.

a(n) = Sum_{k=0..n} Stirling1(n,k) * A102896(k). - Andrew Howroyd, Apr 20 2020

a(6)-a(7) from Andrew Howroyd, Apr 20 2020

A334253 Number of strict closure operators on a set of n elements which satisfy the T_0 separation axiom.

Original entry on oeis.org

1, 1, 3, 35, 2039, 1352390, 75945052607, 14087646108883940225

Offset: 0

Joshua Moerman, Apr 20 2020

The T_0 axiom states that the closure of {x} and {y} are different for distinct x and y.

A closure operator is strict if the empty set is closed.

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

R. S. R. Myers, J. Adámek, S. Milius, and H. Urbat, Coalgebraic constructions of canonical nondeterministic automata, Theoretical Computer Science, 604 (2015), 81-101.
B. Venkateswarlu and U. M. Swamy, T_0-Closure Operators and Pre-Orders, Lobachevskii Journal of Mathematics, 39 (2018), 1446-1452.

The number of all strict closure operators is given in A102894.
For all T0 closure operators, see A334252.
For strict T1 closure operators, see A334255.
A strict closure operator which preserves unions is called topological, see A001035.
Cf. A326943, A326944, A326945.

a(n) = Sum_{k=0..n} Stirling1(n,k) * A102894(k). - Andrew Howroyd, Apr 20 2020

a(6)-a(7) from Andrew Howroyd, Apr 20 2020

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

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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A326943 Number of T_0 sets of subsets of {1..n} that cover all n vertices and are closed under intersection.

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A326944 Number of T_0 sets of subsets of {1..n} that cover all n vertices, contain {}, and are closed under intersection.

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A309615 Number of T_0 set-systems covering n vertices that are closed under intersection.

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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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A334252 Number of closure operators on a set of n elements which satisfy the T_0 separation axiom.

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A334253 Number of strict closure operators on a set of n elements which satisfy the T_0 separation axiom.

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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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