A326961 -id:A326961 - cp's OEIS frontend

A059201 Number of T_0-covers of a labeled n-set.

1, 1, 4, 96, 31692, 2147001636, 9223371991763269704, 170141183460469231473432887375376674952, 57896044618658097711785492504343953920509909728243389682424010192567186540224

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Jan 16 2001

A cover of a set is a T_0-cover if for every two distinct points of the set there exists a member (block) of the cover containing one but not the other point.
From Gus Wiseman, Aug 13 2019: (Start)
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). For example, the a(2) = 4 covers are:
  {{1},{2}}
  {{1},{1,2}}
  {{2},{1,2}}
  {{1},{2},{1,2}}
(End)

G. C. Greubel, Table of n, a(n) for n = 0..11
Vladeta Jovovic, T_0-covers of a labeled 3-set

Row sums of A059202.
Cf. A059203, A059084, A059085, A059086, A059087, A059088, A059089.
Covering set-systems are A003465.
The unlabeled version is A319637.
The version with empty edges allowed is A326939.
The non-covering version is A326940.
BII-numbers of T_0 set-systems are A326947.
The same with connected instead of covering is A326948.
The T_1 version is A326961.
Cf. A245567, A316978, A319559, A319564, A323818, A326941, A326946, A326970.

Table[Sum[StirlingS1[n + 1, k]*2^(2^(k - 1) - 1), {k, 0, n + 1}], {n,0,5}] (* G. C. Greubel, Dec 28 2016 *)
dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&UnsameQ@@dual[#]&]],{n,0,3}] (* Gus Wiseman, Aug 13 2019 *)

a(n) = Sum_{i=0..n+1} stirling1(n+1, i)*2^(2^(i-1)-1).
a(n) = Sum_{m=0..2^n-1} A059202(n,m).
Inverse binomial transform of A326940 and exponential transform of A326948. - Gus Wiseman, Aug 13 2019

A326965 Number of set-systems on n vertices where every covered vertex is the unique common element of some subset of the edges.

Original entry on oeis.org

1, 2, 5, 46, 19181, 2010327182, 9219217424630040409, 170141181796805106025395618012972506978, 57896044618658097536026644159052312978532934306727333157337631572314050272137

Offset: 0

Gus Wiseman, Aug 10 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}}. An antichain is a set-system where no edge is a subset of any other. This sequence counts set-systems whose dual is a (strict) antichain, also called T_1 set-systems.

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

Set-systems are A058891.
T_0 set-systems are A326940.
The covering case is A326961.
The version with empty edges allowed is A326967.
Set-systems whose dual is a weak antichain are A326968.
The unlabeled version is A326972.
The BII_numbers of these set-systems are A326979.
Cf. A059052, A326951, A326966, A326970, A326971, A326976, A326977.

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

Binomial transform of A326961.

a(n) = A326967(n)/2.

A326974 Number of unlabeled set-systems covering n vertices where every vertex is the unique common element of some subset of the edges, also called unlabeled covering T_1 set-systems.

Original entry on oeis.org

1, 1, 2, 16, 1212

Offset: 0

Gus Wiseman, Aug 11 2019

Alternatively, these are unlabeled 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. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. An antichain is a set-system where no edge is a subset of any other.

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 16 set-systems:
  {}  {{1}}  {{1},{2}}        {{1},{2},{3}}
             {{1},{2},{1,2}}  {{1,2},{1,3},{2,3}}
                              {{1},{2},{3},{2,3}}
                              {{1},{2},{1,3},{2,3}}
                              {{1},{2},{3},{1,2,3}}
                              {{3},{1,2},{1,3},{2,3}}
                              {{1},{2},{3},{1,3},{2,3}}
                              {{1,2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{2,3},{1,2,3}}
                              {{2},{3},{1,2},{1,3},{2,3}}
                              {{1},{2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{1,2},{1,3},{2,3}}
                              {{3},{1,2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{1,3},{2,3},{1,2,3}}
                              {{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Unlabeled covers are A055621.
The same with T_0 instead of T_1 is A319637.
The labeled version is A326961.
The non-covering version is A326972 (partial sums).
Unlabeled covering set-systems whose dual is a weak antichain are A326973.
Cf. A000612, A059523, A319559, A326946, A326951, A326965, A326970, A326971, A326976, A326977, A326979.

a(n > 0) = A326972(n) - A326972(n - 1).

A326979 BII-numbers of T_1 set-systems.

Original entry on oeis.org

0, 1, 2, 3, 7, 8, 9, 10, 11, 15, 25, 27, 30, 31, 42, 43, 45, 47, 51, 52, 53, 54, 55, 59, 60, 61, 62, 63, 75, 79, 91, 94, 95, 107, 109, 111, 115, 116, 117, 118, 119, 123, 124, 125, 126, 127, 128, 129, 130, 131, 135, 136, 137, 138, 139, 143, 153, 155, 158, 159

Offset: 1

Gus Wiseman, Aug 13 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_1 condition means that the dual is a (strict) antichain, meaning that none of its edges is a subset of any other.

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 T_1 set-systems together with their BII-numbers begins:
   0: {}
   1: {{1}}
   2: {{2}}
   3: {{1},{2}}
   7: {{1},{2},{1,2}}
   8: {{3}}
   9: {{1},{3}}
  10: {{2},{3}}
  11: {{1},{2},{3}}
  15: {{1},{2},{1,2},{3}}
  25: {{1},{3},{1,3}}
  27: {{1},{2},{3},{1,3}}
  30: {{2},{1,2},{3},{1,3}}
  31: {{1},{2},{1,2},{3},{1,3}}
  42: {{2},{3},{2,3}}
  43: {{1},{2},{3},{2,3}}
  45: {{1},{1,2},{3},{2,3}}
  47: {{1},{2},{1,2},{3},{2,3}}
  51: {{1},{2},{1,3},{2,3}}
  52: {{1,2},{1,3},{2,3}}

BII-numbers of T_0 set-systems are A326947.
T_1 set-systems are counted by A326965, A326961 (covering), A326972 (unlabeled), and A326974 (unlabeled covering).
BII-numbers of set-systems whose dual is a weak antichain are A326966.
Cf. A059201, A059523, A319639, A326970, A326973, A326976, A326977.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
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}];
Select[Range[0,100],UnsameQ@@dual[bpe/@bpe[#]]&&stableQ[dual[bpe/@bpe[#]],SubsetQ]&]

A059523 Number of n-element unlabeled ordered T_0-antichains without isolated vertices; number of T_1-hypergraphs (without empty edge and without multiple edges) on n labeled vertices.

Original entry on oeis.org

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

Offset: 0

Vladeta Jovovic and Goran Kilibarda, Jan 20 2001; revised Jun 03 2004

nonn

			Number of k-element T_1-hipergraphs (without empty edge and without multiple edges) on 3 labeled vertices is
C(7,k)-6*C(5,k)+6*C(4,k)+3*C(3,k)-6*C(2,k)+2*C(1,k),k=0..7; so a(3)=2+11+15+7+1=36=2^7-6*2^5+6*2^4+3*2^3-6*2^2+2*2.

V. Jovovic, T_1-hyper graphs on a labeled 3-set
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.
Goran Kilibarda and Vladeta Jovovic, Enumeration of some classes of T_0-hypergraphs, arXiv:1411.4187 [math.CO], 2014.

Cf. A003465, A059201, A059052, A326961.

a(n) = A059052(n)/2.

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.

A326977 Number of integer partitions of n such that the dual of the multiset partition obtained by factoring each part into prime numbers is a (strict) antichain, also called T_1 integer partitions.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 36, 49, 64, 85, 109, 141, 181, 234, 294, 375, 470, 589, 733, 917, 1131, 1401, 1720, 2113, 2581, 3153, 3833, 4655, 5631, 6801, 8192, 9849, 11816, 14148, 16899, 20153, 23990, 28503, 33815, 40038, 47330, 55858, 65841, 77475

Offset: 0

Gus Wiseman, Aug 13 2019

nonn

The dual of a multiset partition has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. An antichain is a set of multisets, none of which is a submultiset of any other.

			The a(0) = 1 through a(7) = 14 partitions:
  ()  (1)  (2)   (3)    (4)     (5)      (33)      (7)
           (11)  (21)   (22)    (32)     (42)      (43)
                 (111)  (31)    (41)     (51)      (52)
                        (211)   (221)    (222)     (322)
                        (1111)  (311)    (321)     (331)
                                (2111)   (411)     (421)
                                (11111)  (2211)    (511)
                                         (3111)    (2221)
                                         (21111)   (3211)
                                         (111111)  (4111)
                                                   (22111)
                                                   (31111)
                                                   (211111)
                                                   (1111111)

T_0 integer partitions are A319564.
Set-systems whose dual is a (strict) antichain are A326965.
The version where the dual is a weak antichain is A326978.
T_1 factorizations (whose dual is a strict antichain) are A327012.
Cf. A000041, A319728, A326961, A326974, A326976, A326979.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]
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}];
submultQ[cap_,fat_]:=And@@Function[i,Count[fat,i]>=Count[cap,i]]/@Union[List@@cap];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@dual[primeMS/@#]&&stableQ[dual[primeMS/@#],submultQ]&]],{n,0,30}]

A326966 BII-numbers of set-systems whose dual is a weak antichain.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 8, 9, 10, 11, 12, 15, 16, 18, 25, 27, 30, 31, 32, 33, 42, 43, 45, 47, 51, 52, 53, 54, 55, 59, 60, 61, 62, 63, 64, 75, 76, 79, 82, 91, 94, 95, 97, 107, 109, 111, 115, 116, 117, 118, 119, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 135

Offset: 1

Gus Wiseman, Aug 13 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}}. A weak antichain is a multiset of sets, none of which is a proper subset of any other.

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 set-systems whose dual is a weak antichain together with their BII-numbers begins:
   0: {}
   1: {{1}}
   2: {{2}}
   3: {{1},{2}}
   4: {{1,2}}
   7: {{1},{2},{1,2}}
   8: {{3}}
   9: {{1},{3}}
  10: {{2},{3}}
  11: {{1},{2},{3}}
  12: {{1,2},{3}}
  15: {{1},{2},{1,2},{3}}
  16: {{1,3}}
  18: {{2},{1,3}}
  25: {{1},{3},{1,3}}
  27: {{1},{2},{3},{1,3}}
  30: {{2},{1,2},{3},{1,3}}
  31: {{1},{2},{1,2},{3},{1,3}}
  32: {{2,3}}
  33: {{1},{2,3}}

Set-systems whose dual is a weak antichain are counted by A326968, with covering case A326970, unlabeled version A326971, and unlabeled covering version A326973.
BII-numbers of set-systems whose dual is strict (T_0) are A326947.
BII-numbers of set-systems whose dual is a (strict) antichain (T_1) are A326979.
Cf. A059523, A319639, A326961, A326965, A326969, A326974, A326975, A326978.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
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}];
Select[Range[0,100],stableQ[dual[bpe/@bpe[#]],SubsetQ]&]

A326960 Number of sets of subsets of {1..n} covering all n vertices whose dual is a (strict) antichain, also called covering T_1 sets of subsets.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Aug 13 2019

nonn

Same as A059052 except with a(1) = 2 instead of 4.
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.
Alternatively, these are sets of subsets of {1..n} covering all n vertices where every vertex is the unique common element of some subset of the edges.

			The a(0) = 2 through a(2) = 4 sets of subsets:
  {}    {{1}}     {{1},{2}}
  {{}}  {{},{1}}  {{},{1},{2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}

Covering sets of subsets are A000371.
Covering T_0 sets of subsets are A326939.
The case without empty edges is A326961.
The non-covering version is A326967.
Cf. A003181, A059052, A059523, A319639, A326951, A326965, A326974, A326976, A326977, A326979.

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

Binomial transform of A326967.

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

cp's OEIS Frontend

A059201 Number of T_0-covers of a labeled n-set.

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A326965 Number of set-systems on n vertices where every covered vertex is the unique common element of some subset of the edges.

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A326974 Number of unlabeled set-systems covering n vertices where every vertex is the unique common element of some subset of the edges, also called unlabeled covering T_1 set-systems.

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A326979 BII-numbers of T_1 set-systems.

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A059523 Number of n-element unlabeled ordered T_0-antichains without isolated vertices; number of T_1-hypergraphs (without empty edge and without multiple edges) on n labeled vertices.

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

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A326977 Number of integer partitions of n such that the dual of the multiset partition obtained by factoring each part into prime numbers is a (strict) antichain, also called T_1 integer partitions.

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A326966 BII-numbers of set-systems whose dual is a weak antichain.

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A326960 Number of sets of subsets of {1..n} covering all n vertices whose dual is a (strict) antichain, also called covering T_1 sets of subsets.

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