A326881 -id:A326881 - cp's OEIS frontend

A000798 Number of different quasi-orders (or topologies, or transitive digraphs) with n labeled elements.

1, 1, 4, 29, 355, 6942, 209527, 9535241, 642779354, 63260289423, 8977053873043, 1816846038736192, 519355571065774021, 207881393656668953041, 115617051977054267807460, 88736269118586244492485121, 93411113411710039565210494095, 134137950093337880672321868725846, 261492535743634374805066126901117203

Offset: 0

N. J. A. Sloane

From Altug Alkan, Dec 18 2015 and Feb 28 2017: (Start)
a(p^k) == k+1 (mod p) for all primes p. This is proved by Kizmaz at On The Number Of Topologies On A Finite Set link. For proof see Theorem 2.4 in page 2 and 3. So a(19) == 2 (mod 19).
a(p+n) == A265042(n) (mod p) for all primes p. This is also proved by Kizmaz at related link, see Theorem 2.7 in page 4. If n=2 and p=17, a(17+2) == A265042(2) (mod 17), that is a(19) == 51 (mod 17). So a(19) is divisible by 17.
In conclusion, a(19) is a number of the form 323*n - 17. (End)
The BII-numbers of finite topologies without their empty set are given by A326876. - Gus Wiseman, Aug 01 2019
From Tian Vlasic, Feb 23 2022: (Start)
Although no general formula is known for a(n), by considering the number of topologies with a fixed number of open sets, it is possible to explicitly represent the sequence in terms of Stirling numbers of the second kind.
For example: a(n,3) = 2*S(n,2), a(n,4) = S(n,2) + 6*S(n,3), a(n,5) = 6*S(n,3) + 24*S(n,4).
Lower and upper bounds are known: 2^n <= a(n) <= 2^(n*(n-1)), n > 1.
This follows from the fact that there are 2^(n*(n-1)) reflexive relations on a set with n elements.
Furthermore: a(n+1) <= a(n)*(3a(n)+1). (End)

			From _Gus Wiseman_, Aug 01 2019: (Start)
The a(3) = 29 topologies are the following (empty sets not shown):
  {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}
(End)

K. K.-H. Butler and G. Markowsky, Enumeration of finite topologies, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169-184.
S. D. Chatterji, The number of topologies on n points, Manuscript, 1966.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 229.
E. D. Cooper, Representation and generation of finite partially ordered sets, Manuscript, no date.
E. N. Gilbert, A catalog of partially ordered systems, unpublished memorandum, Aug 08, 1961.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 243.
Levinson, H.; Silverman, R. Topologies on finite sets. II. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 699--712, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561090 (81c:54006)
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
For further references concerning the enumeration of topologies and posets see under A001035.
G.H. Patil and M.S. Chaudhary, A recursive determination of topologies on finite sets, Indian Journal of Pure and Applied Mathematics, 26, No. 2 (1995), 143-148.

V. I. Arnautov and A. V. Kochina, Method for constructing one-point expansions of a topology on a finite set and its applications, Bul. Acad. Stiinte Republ. Moldav. Matem. 3 (64) (2010) 67-76.
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Moussa Benoumhani and Ali Jaballah, Chains in lattices of mappings and finite fuzzy topological spaces, Journal of Combinatorial Theory, Series A (2019) Vol. 161, 99-111.
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Juliana Bowles and Marco B. Caminati, A Verified Algorithm Enumerating Event Structures, arXiv:1705.07228 [cs.LO], 2017.
Gunnar Brinkmann and Brendan D. McKay, Posets on up to 16 points.
G. Brinkmann and B. D. McKay, Posets on up to 16 Points, Order 19 (2) (2002) 147-179 (Table IV).
J. I. Brown and S. Watson, The number of complements of a topology on n points is at least 2^n (except for some special cases), Discr. Math., 154 (1996), 27-39.
K. K.-H. Butler and G. Markowsky, Enumeration of finite topologies, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169-184.
K. K.-H. Butler and G. Markowsky, Enumeration of finite topologies, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169-184. [Annotated scan of pages 180 and 183 only]
S. D. Chatterji, The number of topologies on n points, Manuscript, 1966 [Annotated scanned copy]
Tyler Clark and Tom Richmond, The Number of Convex Topologies on a Finite Totally Ordered Set, 2013, Involve, Vol. 8 (2015), No. 1, 25-32.
E. D. Cooper, Representation and generation of finite partially ordered sets, Manuscript, no date [Annotated scanned copy]
M. Erné, Struktur- und Anzahlformeln für Topologien auf Endlichen Mengen, Manuscripta Math., 11 (1974), 221-259.
M. Erné, Struktur- und Anzahlformeln für Topologien auf Endlichen Mengen, Manuscripta Math., 11 (1974), 221-259. (Annotated scanned copy)
M. Erné and K. Stege, The number of partially ordered (labeled) sets, Preprint, 1989. (Annotated scanned copy)
M. Erné and K. Stege, Counting Finite Posets and Topologies, Order, 8 (1991), 247-265.
J. W. Evans, F. Harary and M. S. Lynn, On the computer enumeration of finite topologies, Commun. ACM, 10 (1967), 295-297, 313. [Annotated scanned copy]
J. W. Evans, F. Harary and M. S. Lynn, On the computer enumeration of finite topologies, Commun. ACM, 10 (1967), 295-297, 313.
S. R. Finch, Transitive relations, topologies and partial orders, June 5, 2003. [Cached copy, with permission of the author]
Loic Foissy, Claudia Malvenuto, and Frederic Patras, B_infinity-algebras, their enveloping algebras, and finite spaces, arXiv preprint arXiv:1403.7488 [math.AT], 2014.
Loic Foissy, Claudia Malvenuto, and Frederic Patras, Infinitesimal and B_infinity-algebras, finite spaces, and quasi-symmetric functions, Journal of Pure and Applied Algebra, Elsevier, 2016, 220 (6), pp. 2434-2458. .
L. Foissy and C. Malvenuto, The Hopf algebra of finite topologies and T-partitions, arXiv preprint arXiv:1407.0476 [math.RA], 2014.
Joël Gay and Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018.
E. N. Gilbert, A catalog of partially ordered systems, unpublished memorandum, Aug 08, 1961. [Annotated scanned copy]
S. Giraudo, J.-G. Luque, L. Mignot and F. Nicart, Operads, quasiorders and regular languages, arXiv preprint arXiv:1401.2010 [cs.FL], 2014.
D. J. Greenhoe, Properties of distance spaces with power triangle inequalities, ResearchGate, 2015.
J. Heitzig and J. Reinhold, The number of unlabeled orders on fourteen elements, Order 17 (2000) no. 4, 333-341.
Institut f. Mathematik, Univ. Hanover, Erne/Heitzig/Reinhold papers
G. A. Kamel, Partial Chain Topologies on Finite Sets, Computational and Applied Mathematics Journal. Vol. 1, No. 4, 2015, pp. 174-179.
Dongseok Kim, Young Soo Kwon and Jaeun Lee, Enumerations of finite topologies associated with a finite graph, arXiv preprint arXiv:1206.0550[math.CO], 2012.
M. Y. Kizmaz, On The Number Of Topologies On A Finite Set, arXiv preprint arXiv:1503.08359 [math.NT], 2015-2019.
D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19. [Annotated scanned copy]
D. J. Kleitman and B. L. Rothschild, The number of finite topologies, Proc. Amer. Math. Soc., 25 (1970), 276-282.
Messaoud Kolli, Direct and Elementary Approach to Enumerate Topologies on a Finite Set, J. Integer Sequences, Volume 10, 2007, Article 07.3.1.
Messaoud Kolli, On the cardinality of the T_0-topologies on a finite set, International Journal of Combinatorics, Volume 2014 (2014), Article ID 798074, 7 pages.
Sami Lazaar, Houssem Sabri, and Randa Tahri, Structural and Numerical Studies of Some Topological Properties for Alexandroff Spaces, Bull. Iran. Math. Soc. (2021).
Götz Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
M. Rayburn, On the Borel fields of a finite set, Proc. Amer. Math.. Soc., 19 (1968), 885-889. [Annotated scanned copy]
M. Rayburn and N. J. A. Sloane, Correspondence, 1974
D. Rusin, More info and references [Broken link]
D. Rusin, More info and references [Cached copy]
A. Shafaat, On the number of topologies definable for a finite set, J. Austral. Math. Soc., 8 (1968), 194-198. [Annotated scanned copy]
A. Shafaat, On the number of topologies definable for a finite set, J. Austral. Math. Soc., 8 (1968), 194-198.
N. J. A. Sloane, List of sequences related to partial orders, circa 1972
N. J. A. Sloane, Classic Sequences
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 8 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Eric Swartz and Nicholas J. Werner, Zero pattern matrix rings, reachable pairs in digraphs, and Sharp's topological invariant tau, arXiv:1709.05390 [math.CO], 2017.
Ivo Terek, Smooth Manifolds, Williams College (2025). See p. 6.
Wietske Visser, Koen V. Hindriks and Catholijn M. Jonker, Goal-based Qualitative Preference Systems, 2012.
N. L. White, Two letters to N. J. A. Sloane, 1970, with hand-drawn enclosure
J. A. Wright, Letter to N. J. A. Sloane, Nov 21 1970, with four enclosures
J. A. Wright, There are 718 6-point topologies, quasiorderings and transgraphs, Preprint, 1970 [Annotated scanned copy]
J. A. Wright, Two related abstracts, 1970 and 1972 [Annotated scanned copies]
J. A. Wright, Letter to N. J. A. Sloane, Apr 06 1972, listing 18 sequences
Index entries for "core" sequences

Row sums of A326882.
Cf. A001035 (labeled posets), A001930 (unlabeled topologies), A000112 (unlabeled posets), A006057.
Sequences in the Erné (1974) paper: A000798, A001035, A006056, A006057, A001929, A001927, A006058, A006059, A000110.
Cf. A102894, A102895, A102897, A306445, A326866, A326876, A326878, A326881.

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

a(n) = Sum_{k=0..n} Stirling2(n, k)*A001035(k).
E.g.f.: A(exp(x) - 1) where A(x) is the e.g.f. for A001035. - Geoffrey Critzer, Jul 28 2014
It is known that log_2(a(n)) ~ n^2/4. - Tian Vlasic, Feb 23 2022

Two more terms from Jobst Heitzig (heitzig(AT)math.uni-hannover.de), Jul 03 2000

a(17)-a(18) are from Brinkmann's and McKay's paper. - Vladeta Jovovic, Jun 10 2007

A102896 Number of ACI algebras (or semilattices) on n generators with no annihilator.

Original entry on oeis.org

1, 2, 7, 61, 2480, 1385552, 75973751474, 14087648235707352472

Offset: 0

Mitch Harris, Jan 18 2005

Or, number of Moore families on an n-set, that is, families of subsets that contain the universal set {1,...,n} and are closed under intersection.
Or, number of closure operators on a set of n elements.
An ACI algebra or semilattice is a system with a single binary, idempotent, commutative and associative operation.
Also the number of set-systems on n vertices that are closed under union. The BII-numbers of these set-systems are given by A326875. - Gus Wiseman, Jul 31 2019
From Bernhard Ganter, Jul 08 2025: (Start)
Also the number of union-free families of subsets of an n-set; i.e., families of nonempty sets on n elements such that no set is a union of some others.
Also the number of intersection-free families of subsets of an n-set; i.e., of families of proper subsets on n elements such that no set is an intersection of some others.
(Note that every union-free family on an n-set is the set of union-irreducible elements of exactly one union-closed family, and each family of union-irreducible elements is union-free. Same for intersection.) (End)

			From _Gus Wiseman_, Jul 31 2019: (Start)
The a(0) = 1 through a(2) = 7 set-systems closed under union:
  {}  {}     {}
      {{1}}  {{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). [From Pierre Colomb (pierre(AT)colomb.me), Sep 04 2010]
E. H. Moore, Introduction to a Form of General Analysis, AMS Colloquium Publication 2 (1910), pp. 53-80.

Andrew J. Blumberg, Michael A. Hill, Kyle Ormsby, Angélica M. Osorno, and Constanze Roitzheim, Homotopical Combinatorics, Notices Amer. Math. Soc. (2024) Vol. 71, No. 2, 260-266. See p. 261.
Daniel Borchmann and Bernhard Ganter, Concept Lattice Orbifolds - First Steps, Proceedings of the 7th International Conference on Formal Concept Analysis (ICFCA 2009), 22-37.
Jishnu Bose, Tien Chih, Hannah Housden, Legrand Jones II, Chloe Lewis, Kyle Ormsby, and Millie Rose, Combinatorics of factorization systems on lattices, arXiv:2503.22883 [math.CO], 2025. See p. 11.
Pierre Colomb, Alexis Irlande, Olivier Raynaud and Yoan Renaud, About the Recursive Decomposition of the lattice of co-Moore Families, ICFCA 2011.
Pierre Colomb, Alexis Irlande, Olivier Raynaud, and Yoan Renaud, Recursive decomposition tree of a Moore co-family and closure algorithm, Annals of Mathematics and Artificial Intelligence, 2013, DOI 10.1007/s10472-013-9362-x.
Nachum Dershowitz, Mitchell A. Harris, and Guan-Shieng Huang, Enumeration Problems Related to Ground Horn Theories, arXiv:cs/0610054v2 [cs.LO], 2006-2008.
Michel Habib and Lhouari Nourine, The number of Moore families on n = 6, Discrete Math., 294 (2005), 291-296.
Caoilainn Kirkpatrick, Amelie el Mahmoud, Kyle Ormsby, Angélica M. Osorno, Dale Schandelmeier-Lynch, Riley Shahar, Lixing Yi, Avery Young, and Saron Zhu, Enumerating submonoids of finite commutative monoids, arXiv:2508.20786 [math.CO], 2025. See p. 16.

For set-systems closed under union:
- The covering case is A102894.
- The unlabeled case is A193674.
- The case also closed under intersection is A306445.
- Set-systems closed under overlapping union are A326866.
- The BII-numbers of these set-systems are given by A326875.
Cf. A102895, A102897, A108798, A108800, A193675, A000798, A014466, A326878, A326880, A326881.

Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],SubsetQ[#,Union@@@Tuples[#,2]]&]],{n,0,3}] (* Gus Wiseman, Jul 31 2019 *)

a(n) = Sum_{k=0..n} C(n, k)*A102894(k), where C(n, k) is the binomial coefficient.
For asymptotics see A102897.
a(n) = A102897(n)/2. - Gus Wiseman, Jul 31 2019

N. J. A. Sloane added a(6) from the Habib et al. reference, May 26 2005
Additional comments from Don Knuth, Jul 01 2005
a(7) from Pierre Colomb (pierre(AT)colomb.me), Sep 04 2010

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

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

A108800 Number of nonisomorphic systems enumerated by A102895.

Original entry on oeis.org

1, 2, 6, 28, 330, 28960, 216562364, 5592326182940100

Offset: 0

Don Knuth, Jul 01 2005

Also the number of non-isomorphic sets of sets with {} that are closed under intersection. Also the number of non-isomorphic set-systems (without {}) covering n + 1 vertices and closed under intersection. - Gus Wiseman, Aug 05 2019

			From _Gus Wiseman_, Aug 02 2019: (Start)
Non-isomorphic representatives of the a(0) = 1 through a(3) = 28 sets of sets with {} that are closed under intersection:
  {}  {}     {}            {}
      {}{1}  {}{1}         {}{1}
             {}{12}        {}{12}
             {}{1}{2}      {}{123}
             {}{2}{12}     {}{1}{2}
             {}{1}{2}{12}  {}{1}{23}
                           {}{2}{12}
                           {}{3}{123}
                           {}{1}{2}{3}
                           {}{23}{123}
                           {}{1}{2}{12}
                           {}{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}
(End)

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

Except a(0) = 1, first differences of A193675.
The connected case (i.e., with maximum) is A108798.
The same for union instead of intersection is (also) A108798.
The labeled version is A102895.
The case also closed under union is A326898.
The covering case is A326883.
Cf. A001930, A102894, A102896, A102897, A193674, A326880, A326881.

a(n > 0) = 2 * A108798(n).

a(6) added (using A193675) by N. J. A. Sloane, Aug 02 2011
Changed a(0) from 2 to 1 by Gus Wiseman, Aug 02 2019
a(7) added (using A108798) by Andrew Howroyd, Aug 10 2019

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.

Original entry on oeis.org

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

A003181 Number of P-equivalence classes of nondegenerate Boolean functions of n variables.

Original entry on oeis.org

2, 2, 8, 68, 3904, 37329264, 25626412300941056, 67516342973185974302549277749387264, 2871827610052485009904013737758920847602293486924450772201235462734479360

Offset: 0

N. J. A. Sloane

nonn

Also the number of non-isomorphic sets of subsets of {1..n} with union {1..n}. - Gus Wiseman, Aug 05 2019

			From _Gus Wiseman_, Aug 05 2019: (Start)
Non-isomorphic representatives of the a(0) = 2 through a(2) = 8 sets of subsets:
  {}    {{1}}     {{1,2}}
  {{}}  {{},{1}}  {{1},{2}}
                  {{},{1,2}}
                  {{2},{1,2}}
                  {{},{1},{2}}
                  {{},{2},{1,2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}
(End)

S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38 and 214.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..12
S. Muroga, Threshold Logic and Its Applications, Wiley, NY, 1971. [Annotated scans of a few pages]
Index entries for sequences related to Boolean functions

Cf. A000371, A001146, A003180, A003465, A055621, A007537, A326881.

b:= proc(n, i, l) `if`(n=0, 2^(w-> add(mul(2^igcd(t, l[h]),
      h=1..nops(l)), t=1..w)/w)(ilcm(l[])), `if`(i<1, 0,
      add(b(n-i*j, i-1, [l[], i$j])/j!/i^j, j=0..n/i)))
    end:
a:= n-> `if`(n=0, 2, b(n$2, [])-b(n-1$2, [])):
seq(a(n), n=0..8);  # Alois P. Heinz, Aug 14 2019

b[n_, i_, l_] := If[n == 0, 2^Function[w, Sum[Product[2^GCD[t, l[[h]]], {h, 1, Length[l]}], {t, 1, w}]/w][If[l == {}, 1, LCM @@ l]], If[i < 1, 0, Sum[b[n - i*j, i - 1, Join[l, Table[i, {j}]]]/j!/i^j, {j, 0, n/i}]]];
a[n_] := If[n == 0, 2, b[n, n, {}] - b[n - 1, n - 1, {}]];
a /@ Range[0, 8] (* Jean-François Alcover, Apr 11 2020, after Alois P. Heinz *)

a(n) = A003180(n) - A003180(n-1), for n >= 1. - Christian Sievers, Jul 22 2016

a(n) = 2 * A055621(n). - Gus Wiseman, Aug 05 2019

More terms from Christian Sievers, Jul 22 2016

Definition clarified by Ivo Timoteo, Mar 14 2017

cp's OEIS Frontend

A000798 Number of different quasi-orders (or topologies, or transitive digraphs) with n labeled elements.

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A102896 Number of ACI algebras (or semilattices) on n generators with no annihilator.

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

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

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A108800 Number of nonisomorphic systems enumerated by A102895.

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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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