A102896 -id:A102896 - cp's OEIS frontend

A193674 Number of nonisomorphic systems enumerated by A102896; that is, the number of inequivalent closure operators (or Moore families).

1, 2, 5, 19, 184, 14664, 108295846, 2796163199765896

Offset: 0

Don Knuth, Jul 01 2005

Also the number of unlabeled n-vertex set-systems (A003180) closed under union. - Gus Wiseman, Aug 01 2019

			From _Gus Wiseman_, Aug 01 2019: (Start)
Non-isomorphic representatives of the a(0) = 1 through a(3) = 19 set-systems closed under union:
  {}  {}     {}               {}
      {{1}}  {{1}}            {{1}}
             {{1,2}}          {{1,2}}
             {{2},{1,2}}      {{1,2,3}}
             {{1},{2},{1,2}}  {{2},{1,2}}
                              {{3},{1,2,3}}
                              {{1},{2},{1,2}}
                              {{2,3},{1,2,3}}
                              {{1},{2,3},{1,2,3}}
                              {{3},{2,3},{1,2,3}}
                              {{1,3},{2,3},{1,2,3}}
                              {{2},{3},{2,3},{1,2,3}}
                              {{2},{1,3},{2,3},{1,2,3}}
                              {{3},{1,3},{2,3},{1,2,3}}
                              {{1,2},{1,3},{2,3},{1,2,3}}
                              {{2},{3},{1,3},{2,3},{1,2,3}}
                              {{3},{1,2},{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}}
(End)

D. E. Knuth, The Art of Computer Programming, Vol. 4, Section 7.1.1

Daniel Borchmann, Bernhard Ganter, Concept Lattice Orbifolds - First Steps, Proceedings of the 7th International Conference on Formal Concept Analysis (ICFCA 2009), 22-37 (Reference points to A108799).
G. Brinkmann and R. Deklerck, Generation of Union-Closed Sets and Moore Families, arXiv:1701.03751 [math.CO], 2017.
G. Brinkmann and R. Deklerck, Generation of Union-Closed Sets and Moore Families, Journal of Integer Sequences, Vol.21 (2018), Article 18.1.7.
P. Colomb, A. Irlande and O. Raynaud, Counting of Moore Families for n=7, International Conference on Formal Concept Analysis (2010).

Cf. A102894, A102895, A102897.
The labeled case is A102896.
The covering case is A108798.
The same for intersection instead of union is A108800.
The case with empty edges allowed is A193675.
Cf. A000612, A001930, A003180, A306445, A326875, A326883.

a(n) = A193675(n)/2.

a(6) received Aug 17 2005
a(6) corrected by Pierre Colomb, Aug 02 2011
a(7) from Gunnar Brinkmann, Feb 07 2018

A367903 Number of sets of nonempty subsets of {1..n} contradicting a strict version of the axiom of choice.

Original entry on oeis.org

0, 0, 1, 67, 30997, 2147296425, 9223372036784737528, 170141183460469231731687303625772608225, 57896044618658097711785492504343953926634992332820282019728791606173188627779

Offset: 0

Gus Wiseman, Dec 05 2023

The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.

			The a(2) = 1 set-system is {{1},{2},{1,2}}.
The a(3) = 67 set-systems have following 21 non-isomorphic representatives:
  {{1},{2},{1,2}}
  {{1},{2},{3},{1,2}}
  {{1},{2},{3},{1,2,3}}
  {{1},{2},{1,2},{1,3}}
  {{1},{2},{1,2},{1,2,3}}
  {{1},{2},{1,3},{2,3}}
  {{1},{2},{1,3},{1,2,3}}
  {{1},{1,2},{1,3},{2,3}}
  {{1},{1,2},{1,3},{1,2,3}}
  {{1},{1,2},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3}}
  {{1},{2},{3},{1,2},{1,2,3}}
  {{1},{2},{1,2},{1,3},{2,3}}
  {{1},{2},{1,2},{1,3},{1,2,3}}
  {{1},{2},{1,3},{2,3},{1,2,3}}
  {{1},{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3}}
  {{1},{2},{3},{1,2},{1,3},{1,2,3}}
  {{1},{2},{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Wikipedia, Axiom of choice.

Multisets of multisets of this type are ranked by A355529.
The version without singletons is A367769.
The version for simple graphs is A367867, covering A367868.
The version allowing empty edges is A367901.
The complement is A367902, without singletons A367770, ranks A367906.
For a unique choice (instead of none) we have A367904, ranks A367908.
These set-systems have ranks A367907.
An unlabeled version is A368094, for multiset partitions A368097.
A000372 counts antichains, covering A006126, nonempty A014466.
A003465 counts covering set-systems, unlabeled A055621.
A058891 counts set-systems, unlabeled A000612.
A059201 counts covering T_0 set-systems.
A323818 counts covering connected set-systems.
A326031 gives weight of the set-system with BII-number n.
Cf. A007716, A083323, A092918, A102896, A283877, A306445, A355739, A355740, A367862, A367905, A368409, A368413.

Table[Length[Select[Subsets[Rest[Subsets[Range[n]]]], Select[Tuples[#],UnsameQ@@#&]=={}&]],{n,0,3}]

a(n) + A367904(n) + A367772(n) = A058891(n+1) = 2^(2^n-1).

a(5)-a(8) from Christian Sievers, Jul 26 2024

A367902 Number of sets of nonempty subsets of {1..n} satisfying a strict version of the axiom of choice.

Original entry on oeis.org

1, 2, 7, 61, 1771, 187223, 70038280, 90111497503, 397783376192189

Offset: 0

Gus Wiseman, Dec 05 2023

The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.

			The a(2) = 7 set-systems:
  {}
  {{1}}
  {{2}}
  {{1,2}}
  {{1},{2}}
  {{1},{1,2}}
  {{2},{1,2}}

Wikipedia, Axiom of choice.

The version for simple graphs is A133686, covering A367869.
The version without singletons is A367770.
The complement allowing empty edges is A367901.
The complement is A367903, without singletons A367769, ranks A367907.
For a unique choice we have A367904, ranks A367908.
These set-systems have ranks A367906.
A000372 counts antichains, covering A006126, nonempty A014466.
A003465 counts covering set-systems, unlabeled A055621.
A058891 counts set-systems, unlabeled A000612.
A059201 counts covering T_0 set-systems.
A323818 counts covering connected set-systems.
A326031 gives weight of the set-system with BII-number n.
Cf. A007716, A083323, A092918, A102896, A283877, A306445, A355739, A355740, A367862, A367905, A370636.

Table[Length[Select[Subsets[Subsets[Range[n]]], Select[Tuples[#],UnsameQ@@#&]!={}&]],{n,0,3}]

a(n) = A370636(2^n-1). - Alois P. Heinz, Mar 09 2024

a(6)-a(8) from Christian Sievers, Jul 25 2024

A367901 Number of sets of subsets of {1..n} contradicting a strict version of the axiom of choice.

Original entry on oeis.org

1, 2, 9, 195, 63765, 4294780073, 18446744073639513336, 340282366920938463463374607341656713953, 115792089237316195423570985008687907853269984665640564039457583610129753447747

Offset: 0

Gus Wiseman, Dec 05 2023

nonn

The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.

			The a(2) = 9 sets of sets:
  {{}}
  {{},{1}}
  {{},{2}}
  {{},{1,2}}
  {{},{1},{2}}
  {{},{1},{1,2}}
  {{},{2},{1,2}}
  {{1},{2},{1,2}}
  {{},{1},{2},{1,2}}

Wikipedia, Axiom of choice.

The version for simple graphs is A367867, covering A367868.
The complement is counted by A367902, no singletons A367770, ranks A367906.
The version without empty edges is A367903, ranks A367907.
For a unique choice (instead of none) we have A367904, ranks A367908.
A000372 counts antichains, covering A006126, nonempty A014466.
A003465 counts covering set-systems, unlabeled A055621.
A058891 counts set-systems, unlabeled A000612.
A059201 counts covering T_0 set-systems.
A323818 counts covering connected set-systems, unlabeled A323819.
A326031 gives weight of the set-system with BII-number n.
Cf. A007716, A083323, A092918, A102896, A283877, A306445, A355739, A355740, A367769, A367862, A367905.

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

a(n) = 2^2^n - A367902(n). - Christian Sievers, Aug 01 2024

a(5)-a(8) from Christian Sievers, Aug 01 2024

A367904 Number of sets of nonempty subsets of {1..n} with only one possible way to choose a sequence of different vertices of each edge.

Original entry on oeis.org

1, 2, 6, 38, 666, 32282, 3965886, 1165884638, 792920124786, 1220537093266802, 4187268805038970806, 31649452354183112810198, 522319168680465054600480906, 18683388426164284818805590810122, 1439689660962836496648920949576152046, 237746858936806624825195458794266076911118

Offset: 0

Gus Wiseman, Dec 08 2023

nonn

			The set-system Y = {{1},{1,2},{2,3}} has choices (1,1,2), (1,1,3), (1,2,2), (1,2,3), of which only (1,2,3) has all different elements, so Y is counted under a(3).
The a(0) = 1 through a(2) = 6 set-systems:
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1},{2}}
             {{1},{1,2}}
             {{2},{1,2}}

Christian Sievers, Table of n, a(n) for n = 0..77

The maximal case (n subsets) is A003024.
The version for at least one choice is A367902.
The version for no choices is A367903, no singletons A367769, ranks A367907.
These set-systems have ranks A367908, nonzero A367906.
A000372 counts antichains, covering A006126, nonempty A014466.
A003465 counts covering set-systems, unlabeled A055621.
A058891 counts set-systems, unlabeled A000612.
A059201 counts covering T_0 set-systems.
A323818 counts covering connected set-systems, unlabeled A323819.
Cf. A102896, A133686, A283877, A306445, A355741, A367770, A367862, A367869, A367901, A367905.

Table[Length[Select[Subsets[Subsets[Range[n]]], Length[Select[Tuples[#],UnsameQ@@#&]]==1&]],{n,0,3}]

a(n) = A367902(n) - A367772(n). - Christian Sievers, Jul 26 2024

Binomial transform of A003024. - Christian Sievers, Aug 12 2024

a(5)-a(8) from Christian Sievers, Jul 26 2024

More terms from Christian Sievers, Aug 12 2024

A306445 Number of collections of subsets of {1, 2, ..., n} that are closed under union and intersection.

Original entry on oeis.org

2, 4, 13, 74, 732, 12085, 319988, 13170652, 822378267, 76359798228, 10367879036456, 2029160621690295, 565446501943834078, 221972785233309046708, 121632215040070175606989, 92294021880898055590522262, 96307116899378725213365550192, 137362837456925278519331211455157, 266379254536998812281897840071155592

Offset: 0

Yuan Yao, Feb 15 2019

nonn

			For n = 0, the empty collection and the collection containing the empty set only are both valid.
For n = 1, the 2^(2^1)=4 possible collections are also all closed under union and intersection.
For n = 2, there are only 3 invalid collections, namely the collections containing both {1} and {2} but not both {1,2} and the empty set. Hence there are 2^(2^2)-3 = 13 valid collections.
From _Gus Wiseman_, Jul 31 2019: (Start)
The a(0) = 2 through a(4) = 13 sets of sets:
  {}    {}        {}
  {{}}  {{}}      {{}}
        {{1}}     {{1}}
        {{},{1}}  {{2}}
                  {{1,2}}
                  {{},{1}}
                  {{},{2}}
                  {{},{1,2}}
                  {{1},{1,2}}
                  {{2},{1,2}}
                  {{},{1},{1,2}}
                  {{},{2},{1,2}}
                  {{},{1},{2},{1,2}}
(End)

R. Stanley, Enumerative Combinatorics, volume 1, second edition, Exercise 3.46.

The covering case with {} is A000798.
The case closed under union only is A102897.
The case closed under intersection only is (also) A102897.
The BII-numbers of these set-systems are A326876.
Cf. A001930, A102895, A102896, A326866, A326878, A326882.

Table[Length[Select[Subsets[Subsets[Range[n]]],SubsetQ[#,Union[Union@@@Tuples[#,2],Intersection@@@Tuples[#,2]]]&]],{n,0,3}] (* Gus Wiseman, Jul 31 2019 *)
A000798 = Cases[Import["https://oeis.org/A000798/b000798.txt", "Table"], {, }][[All, 2]];
a[n_] := 1 + Sum[Binomial[n, i]*Binomial[i, i - d]*A000798[[d + 1]], {d, 0, n}, {i, d, n}];
a /@ Range[0, Length[A000798] - 1] (* Jean-François Alcover, Dec 30 2019 *)

import math
# Sequence A000798
topo = [1, 1, 4, 29, 355, 6942, 209527, 9535241, 642779354, 63260289423, 8977053873043, 1816846038736192, 519355571065774021, 207881393656668953041, 115617051977054267807460, 88736269118586244492485121, 93411113411710039565210494095, 134137950093337880672321868725846, 261492535743634374805066126901117203]
def nCr(n, r):
    return math.factorial(n) // (math.factorial(r) * math.factorial(n-r))
for n in range(len(topo)):
    ans = 1
    for d in range(n+1):
        for i in range(d, n+1):
            ans += nCr(n,i) * nCr(i, i-d) * topo[d]
    print(n, ans)

a(n) = 1 + Sum_{d=0..n} Sum_{i=d..n} C(n,i)*C(i,i-d)*A000798(d). (Follows by caseworking on the maximal and minimal set in the collection.)

E.g.f.: exp(x) + exp(x)^2*B(exp(x)-1) where B(x) is the e.g.f. for A001035 (after Stanley reference above). - Geoffrey Critzer, Jan 19 2024

a(16)-a(18) from A000798 by Jean-François Alcover, Dec 30 2019

A326866 Number of connectedness systems on n vertices.

Original entry on oeis.org

1, 2, 8, 96, 6720, 8130432, 1196099819520

Offset: 0

Gus Wiseman, Jul 29 2019

We define a connectedness system (investigated by Vim van Dam in 2002) to be a set of finite nonempty sets (edges) that is closed under taking the union of two overlapping edges.

			The a(0) = 1 through a(2) = 8 connectedness systems:
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1,2}}
             {{1},{2}}
             {{1},{1,2}}
             {{2},{1,2}}
             {{1},{2},{1,2}}

Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.

The case without singletons is A072446.
The unlabeled case is A326867.
The connected case is A326868.
Binomial transform of A326870 (the covering case).
The BII-numbers of these set-systems are A326872.
Cf. A072444, A072447, A102896, A306445.

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

a(n) = 2^n * A072446(n).

a(6) corrected by Christian Sievers, Oct 26 2023

A102894 Number of ACI algebras or semilattices on n generators, with no identity or annihilator.

Original entry on oeis.org

1, 1, 4, 45, 2271, 1373701, 75965474236, 14087647703920103947

Offset: 0

Mitch Harris, Jan 18 2005

Or, number of families of subsets of {1, ..., n} that are closed under intersection and contain both the universe and the empty set.
An ACI algebra or semilattice is a system with a single binary, idempotent, commutative and associative operation.
Also the number of set-systems covering n vertices that are closed under union. The BII-numbers of these set-systems are given by A326875. - Gus Wiseman, Aug 01 2019
Number of strict closure operators on a set of n elements, where the closure operator is said to be strict if the empty set is closed. - Tian Vlasic, Jul 30 2022

			From _Gus Wiseman_, Aug 01 2019: (Start)
The a(3) = 45 set-systems with {} and {1,2,3} that are closed under intersection are the following ({} and {1,2,3} not shown). The BII-numbers of these set-systems are given by A326880.
0   {1}   {1}{2}   {1}{2}{3}    {1}{2}{3}{12}   {1}{2}{3}{12}{13}
    {2}   {1}{3}   {1}{2}{12}   {1}{2}{3}{13}   {1}{2}{3}{12}{23}
    {3}   {2}{3}   {1}{2}{13}   {1}{2}{3}{23}   {1}{2}{3}{13}{23}
    {12}  {1}{12}  {1}{2}{23}   {1}{2}{12}{13}
    {13}  {1}{13}  {1}{3}{12}   {1}{2}{12}{23}
    {23}  {1}{23}  {1}{3}{13}   {1}{3}{12}{13}        {1}{2}{3}{12}{13}{23}
          {2}{12}  {1}{3}{23}   {1}{3}{13}{23}
          {2}{13}  {2}{3}{12}   {2}{3}{12}{23}
          {2}{23}  {2}{3}{13}   {2}{3}{13}{23}
          {3}{12}  {2}{3}{23}
          {3}{13}  {1}{12}{13}
          {3}{23}  {2}{12}{23}
                   {3}{13}{23}
(End)

G. Birkhoff, Lattice Theory. American Mathematical Society, Colloquium Publications, Vol. 25, 3rd ed., Providence, RI, 1967.
Maria Paola Bonacina and Nachum Dershowitz, Canonical Inference for Implicational Systems, in Automated Reasoning, Lecture Notes in Computer Science, Volume 5195/2008, Springer-Verlag.
E. H. Moore, Introduction to a Form of General Analysis, AMS Colloquium Publication 2 (1910), pp. 53-80.

Maria Paola Bonacina and Nachum Dershowitz, Canonical ground Horn theories, Lecture Notes in Computer Science 7797, 35-71 (2013).
P. Colomb, A. Irlande and O. Raynaud, Counting of Moore Families for n=7, International Conference on Formal Concept Analysis (2010).
N. Dershowitz, G. S. Huang and M. Harris, Enumeration Problems Related to Ground Horn Theories, arXiv:cs/0610054v2 [cs.LO], 2006-2008.
Christopher S. Flippen, Minimal Sets, Union-Closed Families, and Frankl's Conjecture, Master's thesis, Virginia Commonwealth Univ., 2023.
M. Habib and L. Nourine, The number of Moore families on n = 6, Discrete Math., 294 (2005), 291-296.

Regarding set-systems covering n vertices closed under union:
- The non-covering case is A102896.
- The BII-numbers of these set-systems are A326875.
- The case with intersection instead of union is A326881.
- The unlabeled case is A108798.
Cf. A003465, A072447, A102895, A102897, A108800, A193674, A193675, A306445, A326870, A326880, A326883.

Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&SubsetQ[#,Union@@@Tuples[#,2]]&]],{n,0,3}] (* Gus Wiseman, Aug 01 2019 *)

Inverse binomial transform of A102896.

For asymptotics see A102897.

Additional comments from Don Knuth, Jul 01 2005

A326878 Number of topologies whose points are a subset of {1..n}.

Original entry on oeis.org

1, 2, 7, 45, 500, 9053, 257151, 11161244, 725343385, 69407094565, 9639771895398, 1919182252611715, 541764452276876719, 214777343584048313318, 118575323291814379721651, 90492591258634595795504697, 94844885130660856889237907260, 135738086271526574073701454370969, 263921383510041055422284977248713291

Offset: 0

Gus Wiseman, Jul 30 2019

nonn

			The a(0) = 1 through a(2) = 7 topologies:
  {{}}  {{}}      {{}}
        {{},{1}}  {{},{1}}
                  {{},{2}}
                  {{},{1,2}}
                  {{},{1},{1,2}}
                  {{},{2},{1,2}}
                  {{},{1},{2},{1,2}}

Wikipedia Topological space

Binomial transform of A000798 (the covering case).

Cf. A001035, A001930, A003465, A014466, A102896, A102897, A306445, A326866, A326876.

Table[Length[Select[Subsets[Subsets[Range[n]]],MemberQ[#,{}]&&SubsetQ[#,Union[Union@@@Tuples[#,2],Intersection@@@Tuples[#,2]]]&]],{n,0,4}]
(* Second program: *)
A000798 = Cases[Import["https://oeis.org/A000798/b000798.txt", "Table"], {, }][[All, 2]];
a[n_] := Sum[Binomial[n, k]*A000798[[k+1]], {k, 0, n}];
a /@ Range[0, Length[A000798]-1] (* Jean-François Alcover, Dec 30 2019 *)

From Geoffrey Critzer, Jul 12 2022: (Start)
E.g.f.: exp(x)*A(exp(x)-1) where A(x) is the e.g.f. for A001035.
a(n) = Sum_{k=0..n} binomial(n,k)*A000798(k). (End)

A102895 Number of ACI algebras or semilattices on n generators with no identity element.

Original entry on oeis.org

1, 2, 8, 90, 4542, 2747402, 151930948472, 28175295407840207894

Offset: 0

Mitch Harris, Jan 18 2005

An ACI algebra or semilattice is a system with a single binary, idempotent, commutative and associative operation.

Or, number of families of subsets of {1, ..., n} that are closed under intersection and contain the empty set.

			a(2) = 8: Let the points be labeled a, b and let 0 denote the empty set. We want the number of collections of subsets of {a, b} which are closed under intersection and contain the empty subset. 0 subsets: 0 ways, 1 subset: 1 way (0), 2 subsets: 3 ways (0,a; 0,b; 0,ab), 3 subsets: 3 ways (0,a,b; 0,a,ab; 0,b,ab), 4 subsets: 1 way (0,a,b,ab), for a total of 8.
From _Gus Wiseman_, Aug 02 2019: (Start)
The a(0) = 1 through a(2) = 8 sets of sets with {} that are closed under intersection are:
  {{}}  {{}}      {{}}
        {{},{1}}  {{},{1}}
                  {{},{2}}
                  {{},{1,2}}
                  {{},{1},{2}}
                  {{},{1},{1,2}}
                  {{},{2},{1,2}}
                  {{},{1},{2},{1,2}}
(End)

G. Birkhoff, Lattice Theory. American Mathematical Society, Colloquium Publications, Vol. 25, 3rd ed., Providence, RI, 1967.
Maria Paola Bonacina and Nachum Dershowitz, Canonical Inference for Implicational Systems, in Automated Reasoning, Lecture Notes in Computer Science, Volume 5195/2008, Springer-Verlag.
P. Colomb, A. Irlande and O. Raynaud, Counting of Moore Families for n=7, International Conference on Formal Concept Analysis (2010)
E. H. Moore, Introduction to a Form of General Analysis, AMS Colloquium Publication 2 (1910), pp. 53-80.

N. Dershowitz, G. S. Huang and M. Harris, Enumeration Problems Related to Ground Horn Theories, arXiv:cs/0610054v2 [cs.LO], 2006-2008.
M. Habib and L. Nourine, The number of Moore families on n = 6, Discrete Math., 294 (2005), 291-296.

The connected case (i.e., with maximum) is A102894.
The same for union instead of intersection is A102896.
The unlabeled version is A108800.
The case also closed under union is A326878.
The BII-numbers of these set-systems (without the empty set) are A326880.
The covering case is A326881.
Cf. A000798, A102897, A108798, A193674, A193675, A306445, A326883.

Table[Length[Select[Subsets[Subsets[Range[n]]],MemberQ[#,{}]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}] (* Gus Wiseman, Aug 02 2019 *)

For asymptotics see A102897.

a(n > 0) = 2 * A102894(n).

Additional comments from Don Knuth, Jul 01 2005

Changed a(0) from 2 to 1 by Gus Wiseman, Aug 02 2019

cp's OEIS Frontend

A193674 Number of nonisomorphic systems enumerated by A102896; that is, the number of inequivalent closure operators (or Moore families).

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A367903 Number of sets of nonempty subsets of {1..n} contradicting a strict version of the axiom of choice.

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A367902 Number of sets of nonempty subsets of {1..n} satisfying a strict version of the axiom of choice.

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A367901 Number of sets of subsets of {1..n} contradicting a strict version of the axiom of choice.

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A367904 Number of sets of nonempty subsets of {1..n} with only one possible way to choose a sequence of different vertices of each edge.

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A306445 Number of collections of subsets of {1, 2, ..., n} that are closed under union and intersection.

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A326866 Number of connectedness systems on n vertices.

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