A326960 -id:A326960 - cp's OEIS frontend

A059052 Number of n-element unlabeled ordered T_0-antichains; T_1-hypergraphs (with empty hyperedge but without multiple hyperedges) on n labeled nodes.

2, 4, 4, 72, 38040, 4020463392, 18438434825136728352, 340282363593610211921722192165556850240, 115792089237316195072053288318104625954343609704705784618785209431974668731584

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Dec 19 2000

nonn

An antichain on a set is a T_0-antichain if for every two distinct points of the set there exists a member of the antichain containing one but not the other point. T_1-hypergraph is a hypergraph which for every ordered pair (u,v) of distinct nodes has a hyperedge containing u but not v.

			a(3) = 2 + 13 + 26 + 22 + 8 + 1. a(6) = 2^64 - 30*2^48 + 120*2^40 + 60*2^36 + 60*2^34 - 12*2^33 - 345*2^32 - 720*2^30 + 810*2^28 + 120*2^27 + 480*2^26 + 360*2^25 - 480*2^24 - 720*2^23 - 240*2^22 - 540*2^21 + 1380*2^20 + 750*2^19 + 60*2^18 - 210*2^17 - 1535*2^16 - 1820*2^15 + 2250*2^14 + 1800*2^13 - 2820*2^12 + 300*2^11 + 2040*2^10 + 340*2^9 - 1815*2^8 + 510*2^7 - 1350*2^6 + 1350*2^5 + 274*2^4 - 548*2^3 + 120*2^2.

V. Jovovic, G. Kilibarda, On enumeration of the class of all monotone Boolean functions, in preparation.

V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, in Russian, Diskretnaya Matematika, 11 (1999), no. 4, 127-138.
V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, English translation, in Discrete Mathematics and Applications, 9, (1999), no. 6.

Cf. A059048, A059049, A059050, A059051, A326960.

a(n) = Sum_{m=0..2^n} A(n, m), where A(n, m) is number of n-element ordered T_0-antichains on an unlabeled m-set. Cf. A059048.

a(n) = row sums of A059048.

A326961 Number of set-systems covering n vertices where every vertex is the unique common element of some subset of the edges, also called covering T_1 set-systems.

Original entry on oeis.org

1, 1, 2, 36, 19020, 2010231696, 9219217412568364176, 170141181796805105960861096082778425120, 57896044618658097536026644159052312977171804852352892309392604715987334365792

Offset: 0

Gus Wiseman, Aug 12 2019

nonn

Same as A059523 except with a(1) = 1 instead of 2.

Alternatively, these are set-systems covering n vertices whose dual is a (strict) antichain. 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. An antichain is a set of sets, none of which is a subset of any other.

			The a(3) = 36 set-systems:
  {{1}{2}{3}}        {{12}{13}{23}{123}}     {{2}{3}{12}{13}{23}}
  {{12}{13}{23}}     {{1}{2}{3}{12}{13}}     {{2}{3}{12}{13}{123}}
  {{1}{2}{3}{12}}    {{1}{2}{3}{12}{23}}     {{2}{12}{13}{23}{123}}
  {{1}{2}{3}{13}}    {{1}{2}{3}{13}{23}}     {{3}{12}{13}{23}{123}}
  {{1}{2}{3}{23}}    {{1}{2}{12}{13}{23}}    {{1}{2}{3}{12}{13}{23}}
  {{1}{2}{13}{23}}   {{1}{2}{3}{12}{123}}    {{1}{2}{3}{12}{13}{123}}
  {{1}{2}{3}{123}}   {{1}{2}{3}{13}{123}}    {{1}{2}{3}{12}{23}{123}}
  {{1}{3}{12}{23}}   {{1}{2}{3}{23}{123}}    {{1}{2}{3}{13}{23}{123}}
  {{2}{3}{12}{13}}   {{1}{3}{12}{13}{23}}    {{1}{2}{12}{13}{23}{123}}
  {{1}{12}{13}{23}}  {{1}{2}{13}{23}{123}}   {{1}{3}{12}{13}{23}{123}}
  {{2}{12}{13}{23}}  {{1}{3}{12}{23}{123}}   {{2}{3}{12}{13}{23}{123}}
  {{3}{12}{13}{23}}  {{1}{12}{13}{23}{123}}  {{1}{2}{3}{12}{13}{23}{123}}

Covering set-systems are A003465.
Covering T_0 set-systems are A059201.
The version with empty edges allowed is A326960.
The non-covering version is A326965.
Covering set-systems whose dual is a weak antichain are A326970.
The unlabeled version is A326974.
The BII-numbers of T_1 set-systems are A326979.
Cf. A058891, A059052, A059523, A323818, A326972, A326973, A326976, A326977.

tmQ[eds_]:=Union@@Select[Intersection@@@Rest[Subsets[eds]],Length[#]==1&]==Union@@eds;
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&tmQ[#]&]],{n,0,3}]

Inverse binomial transform of A326965.

A326970 Number of set-systems covering n vertices whose dual is a weak antichain.

Original entry on oeis.org

1, 1, 3, 43, 19251

Offset: 0

Gus Wiseman, Aug 10 2019

A set-system is a finite set of finite nonempty sets. The dual of a set-system has, for each vertex, one edges 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(3) = 43 set-systems:
  {123}  {1}{23}  {1}{2}{3}     {1}{2}{3}{12}
         {2}{13}  {12}{13}{23}  {1}{2}{3}{13}
         {3}{12}  {1}{23}{123}  {1}{2}{3}{23}
                  {2}{13}{123}  {1}{2}{13}{23}
                  {3}{12}{123}  {1}{2}{3}{123}
                                {1}{3}{12}{23}
                                {2}{3}{12}{13}
                                {1}{12}{13}{23}
                                {2}{12}{13}{23}
                                {3}{12}{13}{23}
                                {12}{13}{23}{123}
.
  {1}{2}{3}{12}{13}     {1}{2}{3}{12}{13}{23}    {1}{2}{3}{12}{13}{23}{123}
  {1}{2}{3}{12}{23}     {1}{2}{3}{12}{13}{123}
  {1}{2}{3}{13}{23}     {1}{2}{3}{12}{23}{123}
  {1}{2}{12}{13}{23}    {1}{2}{3}{13}{23}{123}
  {1}{2}{3}{12}{123}    {1}{2}{12}{13}{23}{123}
  {1}{2}{3}{13}{123}    {1}{3}{12}{13}{23}{123}
  {1}{2}{3}{23}{123}    {2}{3}{12}{13}{23}{123}
  {1}{3}{12}{13}{23}
  {2}{3}{12}{13}{23}
  {1}{2}{13}{23}{123}
  {1}{3}{12}{23}{123}
  {2}{3}{12}{13}{123}
  {1}{12}{13}{23}{123}
  {2}{12}{13}{23}{123}
  {3}{12}{13}{23}{123}

Covering set-systems are A003465.
Covering set-systems whose dual is strict are A059201.
The T_1 case is A326961.
The BII-numbers of these set-systems are A326966.
The non-covering case is A326968.
The unlabeled version is A326973.
Cf. A006126, A059052, A059523, A326950, A326960, A326965, A326969, A326971, A326974, A326975, A326978.

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],{1,n}]],Union@@#==Range[n]&&stableQ[dual[#],SubsetQ]&]],{n,0,3}]

Inverse binomial transform of A326968.

A326951 Number of unlabeled sets of subsets of {1..n} where every covered vertex is the unique common element of some subset of the edges.

Original entry on oeis.org

2, 4, 8, 40, 2464

Offset: 0

Gus Wiseman, Aug 13 2019

Alternatively, these are unlabeled 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. An antichain is a set of subsets where no edge is a subset of any other.

			Non-isomorphic representatives of the a(0) = 2 through a(2) = 8 sets of subsets:
  {}    {}        {}
  {{}}  {{}}      {{}}
        {{1}}     {{1}}
        {{},{1}}  {{},{1}}
                  {{1},{2}}
                  {{},{1},{2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}

Unlabeled sets of subsets are A003180.
Unlabeled T_0 sets of subsets are A326949.
The labeled version is A326967.
The case without empty edges is A326972.
The covering case is A327011 (first differences).
Cf. A003181, A059052, A326960, A326965, A326969, A326974, A326976.

a(n) = 2 * A326972(n).

a(n) = Sum_{k = 0..n} A327011(k).

A334255 Number of strict closure operators on a set of n elements which satisfy the T_1 separation axiom.

Original entry on oeis.org

1, 1, 1, 8, 545, 702525, 66960965307

Offset: 0

Joshua Moerman, Apr 24 2020

The T_1 axiom states that all singleton sets {x} are closed.

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

			The a(3) = 8 set-systems of closed sets:
  {{1,2,3},{1},{2},{3},{}}
  {{1,2,3},{1,2},{1},{2},{3},{}}
  {{1,2,3},{1,3},{1},{2},{3},{}}
  {{1,2,3},{2,3},{1},{2},{3},{}}
  {{1,2,3},{1,2},{1,3},{1},{2},{3},{}}
  {{1,2,3},{1,2},{2,3},{1},{2},{3},{}}
  {{1,2,3},{1,3},{2,3},{1},{2},{3},{}}
  {{1,2,3},{1,2},{1,3},{2,3},{1},{2},{3},{}}

Dmitry I. Ignatov, Supporting iPython code for counting closure systems w.r.t. the T_1 separation axiom, Github repository
Dmitry I. Ignatov, Supporting iPython notebook
Eric Weisstein's World of Mathematics, Separation Axioms
Wikipedia, Separation Axiom

The number of all strict closure operators is given in A102894.
For all strict T_0 closure operators, see A334253.
For T_1 closure operators, see A334254.
Cf. A326960, A326961, A326979.

Table[Length[
  Select[Subsets[Subsets[Range[n]]],
   And[MemberQ[#, {}], MemberQ[#, Range[n]],
     SubsetQ[#, Intersection @@@ Tuples[#, 2]],
     SubsetQ[#, Map[{#} &, Range[n]]]] &]], {n, 0, 4}] (* Tian Vlasic, Jul 29 2022 *)

a(6) from Dmitry I. Ignatov, Jul 03 2022

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

Original entry on oeis.org

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.

A334254 Number of closure operators on a set of n elements which satisfy the T_1 separation axiom.

Original entry on oeis.org

1, 2, 1, 8, 545, 702525, 66960965307

Offset: 0

Joshua Moerman, Apr 20 2020

The T_1 axiom states that all singleton sets {x} are closed.

For n>1, this property implies strictness (meaning that the empty set is closed).

			The a(3) = 8 set-systems of closed sets:
  {{1,2,3},{1},{2},{3},{}}
  {{1,2,3},{1,2},{1},{2},{3},{}}
  {{1,2,3},{1,3},{1},{2},{3},{}}
  {{1,2,3},{2,3},{1},{2},{3},{}}
  {{1,2,3},{1,2},{1,3},{1},{2},{3},{}}
  {{1,2,3},{1,2},{2,3},{1},{2},{3},{}}
  {{1,2,3},{1,3},{2,3},{1},{2},{3},{}}
  {{1,2,3},{1,2},{1,3},{2,3},{1},{2},{3},{}}

Dmitry I. Ignatov, On the Cryptomorphism between Davis' Subset Lattices, Atomic Lattices, and Closure Systems under T1 Separation Axiom, arXiv:2209.12256 [cs.DM], 2022.
Dmitry I. Ignatov, Supporting iPython code for counting closure systems w.r.t. the T_1 separation axiom, Github repository
Dmitry I. Ignatov, PDF of the supporting iPython notebook
S. Mapes, Finite atomic lattices and resolutions of monomial ideals, J. Algebra, 379 (2013), 259-276.
Eric Weisstein's World of Mathematics, Separation Axioms
Wikipedia, Separation Axiom

The number of all closure operators is given in A102896.
For T_0 closure operators, see A334252.
For strict T_1 closure operators, see A334255, the only difference is a(1).
Cf. A326960, A326961, A326979.

a(6) from Dmitry I. Ignatov, Jul 03 2022

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).

A355517 Number of nonisomorphic systems enumerated by A334254; that is, the number of inequivalent closure operators on a set of n elements where all singletons are closed.

Original entry on oeis.org

1, 2, 1, 4, 50, 7443, 95239971

Offset: 0

Dmitry I. Ignatov, Jul 05 2022

The T_1 axiom states that all singleton sets {x} are closed.

For n>1, this property implies strictness (meaning that the empty set is closed).

			a(0) = 1 counts the empty set, while a(1) = 2 counts {{1}} and {{},{1}}.
For a(2) = 1 the closure system is as follows:  {{1,2},{1},{2},{}}.
The a(3) = 4 inequivalent set-systems of closed sets are:
  {{1,2,3},{1},{2},{3},{}}
  {{1,2,3},{1,2},{1},{2},{3},{}}
  {{1,2,3},{1,2},{1,3},{1},{2},{3},{}}
  {{1,2,3},{1,2},{1,3},{2,3},{1},{2},{3},{}}.

Dmitry I. Ignatov, Supporting iPython code for counting nonequivalent closure systems w.r.t. the T_1 separation axiom, Github repository
Eric Weisstein's World of Mathematics, Separation Axioms
Wikipedia, Separation Axiom

The number of all closure operators is given in A102896, while A193674 contains the number of all nonisomorphic ones.
For T_1 closure operators and their strict counterparts, see A334254 and A334255, respectively; the only difference is a(1).
Cf. A326960, A326961, A326979.

cp's OEIS Frontend

A059052 Number of n-element unlabeled ordered T_0-antichains; T_1-hypergraphs (with empty hyperedge but without multiple hyperedges) on n labeled nodes.

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A326961 Number of set-systems covering n vertices where every vertex is the unique common element of some subset of the edges, also called covering T_1 set-systems.

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A326970 Number of set-systems covering n vertices whose dual is a weak antichain.

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A326951 Number of unlabeled 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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A334255 Number of strict closure operators on a set of n elements which satisfy the T_1 separation axiom.

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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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A334254 Number of closure operators on a set of n elements which satisfy the T_1 separation axiom.

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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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A355517 Number of nonisomorphic systems enumerated by A334254; that is, the number of inequivalent closure operators on a set of n elements where all singletons are closed.

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