A102894 -id:A102894 - cp's OEIS frontend

A326882 Irregular triangle read by rows where T(n,k) is the number of finite topologies with n points and k nonempty open sets, 0 <= k <= 2^n - 1.

1, 0, 1, 0, 1, 2, 1, 0, 1, 6, 9, 6, 6, 0, 1, 0, 1, 14, 43, 60, 72, 54, 54, 20, 24, 0, 12, 0, 0, 0, 1, 0, 1, 30, 165, 390, 630, 780, 955, 800, 900, 500, 660, 240, 390, 120, 190, 10, 100, 0, 60, 0, 0, 0, 20, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Gus Wiseman, Aug 01 2019

			Triangle begins:
  1
  0  1
  0  1  2  1
  0  1  6  9  6  6  0  1
  0  1 14 43 60 72 54 54 20 24  0 12  0  0  0  1
Row n = 3 counts the following topologies:
{}{123} {}{1}{123}  {}{1}{12}{123} {}{1}{2}{12}{123}  {}{1}{2}{12}{13}{123}
        {}{2}{123}  {}{1}{13}{123} {}{1}{3}{13}{123}  {}{1}{2}{12}{23}{123}
        {}{3}{123}  {}{1}{23}{123} {}{2}{3}{23}{123}  {}{1}{3}{12}{13}{123}
        {}{12}{123} {}{2}{12}{123} {}{1}{12}{13}{123} {}{1}{3}{13}{23}{123}
        {}{13}{123} {}{2}{13}{123} {}{2}{12}{23}{123} {}{2}{3}{12}{23}{123}
        {}{23}{123} {}{2}{23}{123} {}{3}{13}{23}{123} {}{2}{3}{13}{23}{123}
                    {}{3}{12}{123}
                    {}{3}{13}{123}      {}{1}{2}{3}{12}{13}{23}{123}
                    {}{3}{23}{123}

Andrew Howroyd, Table of n, a(n) for n = 0..254
Wikipedia, Topological space

Row lengths are A000079.
Row sums are A000798.
Cf. A001930, A014466, A102894, A102895, A102896, A102897, A306445, A326876, A326878, A326881.
Columns: A281774 and refs therein.

Table[Length[Select[Subsets[Subsets[Range[n]],{k}],MemberQ[#,{}]&&MemberQ[#,Range[n]]&&SubsetQ[#,Union[Union@@@Tuples[#,2],Intersection@@@Tuples[#,2]]]&]],{n,0,4},{k,2^n}]

Terms a(31) and beyond from Andrew Howroyd, Aug 10 2019

A326880 BII-numbers of set-systems that are closed under nonempty intersection.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 23, 24, 25, 26, 27, 29, 31, 32, 33, 34, 35, 38, 39, 40, 41, 42, 43, 46, 47, 56, 57, 58, 59, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 85, 87, 88

Offset: 1

Gus Wiseman, Jul 29 2019

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.

The enumeration of these set-systems by number of covered vertices is A326881.

			Most small numbers are in the sequence, but the sequence of non-terms together with the set-systems with those BII-numbers begins:
  20: {{1,2},{1,3}}
  22: {{2},{1,2},{1,3}}
  28: {{1,2},{3},{1,3}}
  30: {{2},{1,2},{3},{1,3}}
  36: {{1,2},{2,3}}
  37: {{1},{1,2},{2,3}}
  44: {{1,2},{3},{2,3}}
  45: {{1},{1,2},{3},{2,3}}
  48: {{1,3},{2,3}}
  49: {{1},{1,3},{2,3}}
  50: {{2},{1,3},{2,3}}
  51: {{1},{2},{1,3},{2,3}}
  52: {{1,2},{1,3},{2,3}}
  53: {{1},{1,2},{1,3},{2,3}}
  54: {{2},{1,2},{1,3},{2,3}}
  55: {{1},{2},{1,2},{1,3},{2,3}}
  60: {{1,2},{3},{1,3},{2,3}}
  61: {{1},{1,2},{3},{1,3},{2,3}}
  62: {{2},{1,2},{3},{1,3},{2,3}}
  84: {{1,2},{1,3},{1,2,3}}

John Tyler Rascoe, Table of n, a(n) for n = 1..10000

Cf. A006126, A048793, A102894, A102895, A102896, A102897, A306445, A326031, A326872, A326874, A326875, A326876, A326881.

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

from itertools import count, islice, combinations
def bin_i(n): #binary indices
    return([(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1'])
def a_gen():
    for n in count(0):
        E,f = [bin_i(k) for k in bin_i(n)],0
        for i in combinations(E,2):
            x = list(set(i[0])&set(i[1]))
            if x not in E and len(x) > 0:
                f += 1
                break
        if f < 1:
            yield n
A326880_list = list(islice(a_gen(), 100)) # John Tyler Rascoe, Mar 07 2025

A326881 Number of set-systems with {} that are closed under intersection and cover n vertices.

Original entry on oeis.org

1, 1, 5, 71, 4223, 2725521, 151914530499, 28175294344381108057

Offset: 0

Gus Wiseman, Jul 30 2019

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

The case also closed under union is A000798.
The connected case (i.e., with maximum) is A102894.
The same for union instead of intersection is (also) A102894.
The non-covering case is A102895.
The BII-numbers of these set-systems (without the empty set) are A326880.
The unlabeled case is A326883.
Cf. A003465, A014466, A102896, A102897, A193674, A193675, A306445, A307249, A326878.

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

Inverse binomial transform of A102895. - Andrew Howroyd, Aug 10 2019

a(5)-a(7) from Andrew Howroyd, Aug 10 2019

A326906 Number of sets of subsets of {1..n} that are closed under union and cover all n vertices.

Original entry on oeis.org

2, 2, 8, 90, 4542, 2747402, 151930948472, 28175295407840207894

Offset: 0

Gus Wiseman, Aug 03 2019

Differs from A102895 in having a(0) = 2 instead of 1.

			The a(0) = 2 through a(2) = 8 sets of subsets:
  {}    {{1}}     {{1,2}}
  {{}}  {{},{1}}  {{},{1,2}}
                  {{1},{1,2}}
                  {{2},{1,2}}
                  {{},{1},{1,2}}
                  {{},{2},{1,2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}

The case without empty sets is A102894.
The case with a single covering edge is A102895.
Binomial transform is A102897.
The case also closed under intersection is A326878 for n > 0.
The same for intersection instead of union is (also) A326906.
The unlabeled version is A326907.
Cf. A000798, A102896, A102897, A108800, A193675, A306445, A326880, A326881, A326883.

Table[Length[Select[Subsets[Subsets[Range[n]]],Union@@#==Range[n]&&SubsetQ[#,Union@@@Tuples[#,2]]&]],{n,0,3}]

a(n) = 2 * A102894(n).

A326883 Number of unlabeled set-systems with {} that are closed under intersection and cover n vertices.

Original entry on oeis.org

1, 1, 4, 22, 302, 28630, 216533404, 5592325966377736

Offset: 0

Gus Wiseman, Jul 30 2019

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 22 set-systems:
  {{}}  {{}{1}}  {{}{12}}        {{}{123}}
                 {{}{1}{2}}      {{}{1}{23}}
                 {{}{2}{12}}     {{}{3}{123}}
                 {{}{1}{2}{12}}  {{}{1}{2}{3}}
                                 {{}{23}{123}}
                                 {{}{1}{3}{23}}
                                 {{}{2}{3}{123}}
                                 {{}{3}{13}{23}}
                                 {{}{1}{23}{123}}
                                 {{}{3}{23}{123}}
                                 {{}{1}{2}{3}{23}}
                                 {{}{1}{2}{3}{123}}
                                 {{}{2}{3}{13}{23}}
                                 {{}{1}{3}{23}{123}}
                                 {{}{2}{3}{23}{123}}
                                 {{}{3}{13}{23}{123}}
                                 {{}{1}{2}{3}{13}{23}}
                                 {{}{1}{2}{3}{23}{123}}
                                 {{}{2}{3}{13}{23}{123}}
                                 {{}{1}{2}{3}{12}{13}{23}}
                                 {{}{1}{2}{3}{13}{23}{123}}
                                 {{}{1}{2}{3}{12}{13}{23}{123}}

The case also closed under union is A001930.
The connected case (i.e., with maximum) is A108798.
The same for union instead of intersection is (also) A108798.
The non-covering case is A108800.
The labeled case is A326881.
Cf. A000798, A102894, A102895, A102897, A193674, A193675, A306445, A326875, A326878, A326880.

a(n) = A108800(n) - A108800(n-1) for n > 0. - Andrew Howroyd, Aug 10 2019

a(5)-a(7) from Andrew Howroyd, Aug 10 2019

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

A326902 Number of set-systems (without {}) covering n vertices that are closed under intersection.

Original entry on oeis.org

1, 1, 3, 19, 319, 21881, 16417973, 1063459099837, 225402359008808647339

Offset: 0

Gus Wiseman, Aug 04 2019

A set-system is a finite set of finite nonempty sets, so no two edges of a set-system that is closed under intersection can be disjoint.

			The a(0) = 1 through a(3) = 19 set-systems:
  {}  {{1}}  {{1,2}}      {{1,2,3}}
             {{1},{1,2}}  {{1},{1,2,3}}
             {{2},{1,2}}  {{2},{1,2,3}}
                          {{3},{1,2,3}}
                          {{1,2},{1,2,3}}
                          {{1,3},{1,2,3}}
                          {{2,3},{1,2,3}}
                          {{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 case closed under union and intersection is A006058.
The case with union instead of intersection is A102894.
The unlabeled version is A108800(n - 1).
The non-covering case is A326901.
The connected case is A326903.
Cf. A000798, A001930, A102895, A102898, A182507, A326866, A326876, A326878, A326882, A326900, A326901, A326904.

Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]

Inverse binomial transform of A326901. - Andrew Howroyd, Aug 10 2019

a(5)-a(8) from Andrew Howroyd, Aug 10 2019

A326903 Number of set-systems (without {}) on n vertices that are closed under intersection and have an edge containing all of the vertices, or Moore families without {}.

Original entry on oeis.org

0, 1, 3, 16, 209, 11851, 8277238, 531787248525, 112701183758471199051

Offset: 0

Gus Wiseman, Aug 04 2019

A set-system is a finite set of finite nonempty sets, so no two edges of such a set-system can be disjoint.

If {} is allowed, we get Moore families (A102896, cf A102895).

			The a(1) = 1 through a(3) = 16 set-systems:
  {{1}}  {{1,2}}      {{1,2,3}}
         {{1},{1,2}}  {{1},{1,2,3}}
         {{2},{1,2}}  {{2},{1,2,3}}
                      {{3},{1,2,3}}
                      {{1,2},{1,2,3}}
                      {{1,3},{1,2,3}}
                      {{2,3},{1,2,3}}
                      {{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}}

M. Habib and L. Nourine, The number of Moore families on n = 6, Discrete Math., 294 (2005), 291-296.

The case closed under union and intersection is A006058.
The case with union instead of intersection is A102894.
The unlabeled version is A193674.
The case without requiring the maximum edge is A326901.
The covering case is A326902.
Cf. A000798, A001930, A102895, A102898, A326866, A326876, A326878, A326882, A326904.

Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],MemberQ[#,Range[n]]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]

a(n) = A326901(n) / 2 for n > 0. - Andrew Howroyd, Aug 10 2019

a(5)-a(8) from Andrew Howroyd, Aug 10 2019

A326913 BII-numbers of set-systems (without {}) closed under union and intersection.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 16, 17, 24, 32, 34, 40, 64, 65, 66, 68, 69, 70, 72, 80, 81, 85, 88, 96, 98, 102, 104, 120, 128, 256, 257, 384, 512, 514, 640, 1024, 1025, 1026, 1028, 1029, 1030, 1152, 1280, 1281, 1285, 1408, 1536, 1538, 1542, 1664, 1920, 2048, 2056, 2176

Offset: 1

Gus Wiseman, Aug 04 2019

nonn

A set-system is a finite set of finite nonempty sets, so no two edges of a set-system that is closed under intersection can be disjoint.

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 closed under union and intersection together with their BII-numbers begins:
   0: {}
   1: {{1}}
   2: {{2}}
   4: {{1,2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   8: {{3}}
  16: {{1,3}}
  17: {{1},{1,3}}
  24: {{3},{1,3}}
  32: {{2,3}}
  34: {{2},{2,3}}
  40: {{3},{2,3}}
  64: {{1,2,3}}
  65: {{1},{1,2,3}}
  66: {{2},{1,2,3}}
  68: {{1,2},{1,2,3}}
  69: {{1},{1,2},{1,2,3}}
  70: {{2},{1,2},{1,2,3}}
  72: {{3},{1,2,3}}

Cf. A048793, A102894, A102895, A102896, A102897, A326031, A326875, A326876, A326878, A326880, A326901.

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

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

cp's OEIS Frontend

A326882 Irregular triangle read by rows where T(n,k) is the number of finite topologies with n points and k nonempty open sets, 0 <= k <= 2^n - 1.

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A326880 BII-numbers of set-systems that are closed under nonempty intersection.

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A326881 Number of set-systems with {} that are closed under intersection and cover n vertices.

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A326906 Number of sets of subsets of {1..n} that are closed under union and cover all n vertices.

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A326883 Number of unlabeled set-systems with {} that are closed under intersection and cover n vertices.

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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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A326902 Number of set-systems (without {}) covering n vertices that are closed under intersection.

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A326903 Number of set-systems (without {}) on n vertices that are closed under intersection and have an edge containing all of the vertices, or Moore families without {}.

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