A326906 -id:A326906 - cp's OEIS frontend

A001035 Number of partially ordered sets ("posets") with n labeled elements (or labeled acyclic transitive digraphs).

1, 1, 3, 19, 219, 4231, 130023, 6129859, 431723379, 44511042511, 6611065248783, 1396281677105899, 414864951055853499, 171850728381587059351, 98484324257128207032183, 77567171020440688353049939, 83480529785490157813844256579, 122152541250295322862941281269151, 241939392597201176602897820148085023

Offset: 0

N. J. A. Sloane

From Altug Alkan, Dec 22 2015: (Start)
a(p^k) == 1 (mod p) and a(n + p) == a(n + 1) (mod p) for all primes p.
a(0+19) == a(0+1) (mod 19) or a(19^1) == 1 (mod 19), that is, a(19) mod 19 = 1.
a(2+17) == a(2+1) (mod 17). So a(19) == 19 (mod 17), that is, a(19) mod 17 = 2.
a(6+13) == a(6+1) (mod 13). So a(19) == 6129859 (mod 13), that is, a(19) mod 13 = 8.
a(8+11) == a(8+1) (mod 11). So a(19) == 44511042511 (mod 11), that is, a(19) mod 11 = 1.
a(12+7) == a(12+1) (mod 7). So a(19) == 171850728381587059351 (mod 7), that is, a(19) mod 7 = 1.
a(14+5) == a(14+1) (mod 5). So a(19) == 77567171020440688353049939 (mod 5), that is, a(19) mod 5 = 4.
a(16+3) == a(16+1) (mod 3). So a(19) == 122152541250295322862941281269151 (mod 3), that is, a(19) mod 3 = 1.
a(17+2) == a(17+1) (mod 2). So a(19) mod 2 = 1.
In conclusion, a(19) is a number of the form 2*3*5*7*11*13*17*19*n - 1615151, that is, 9699690*n - 1615151.
Additionally, for n > 0, note that the last digit of a(n) has the simple periodic pattern: 1,3,9,9,1,3,9,9,1,3,9,9,...
(End)
Number of rank n sublattices of the Boolean algebra B_n. - Kevin Long, Nov 20 2018
a(n) is the number of n X n idempotent Boolean relation matrices (A121337) that have rank n. - Geoffrey Critzer, Aug 16 2023
a(19) == 163279579 (mod 232792560). - Didier Garcia, Feb 06 2025

			R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, Chap. 3, page 98, Fig. 3-1 shows the unlabeled posets with <= 4 points.
From _Gus Wiseman_, Aug 14 2019: (Start)
Also the number of T_0 topologies with n points. For example, the a(0) = 1 through a(3) = 19 topologies are:
  {}  {}{1}  {}{1}{12}     {}{1}{12}{123}
             {}{2}{12}     {}{1}{13}{123}
             {}{1}{2}{12}  {}{2}{12}{123}
                           {}{2}{23}{123}
                           {}{3}{13}{123}
                           {}{3}{23}{123}
                           {}{1}{2}{12}{123}
                           {}{1}{3}{13}{123}
                           {}{2}{3}{23}{123}
                           {}{1}{12}{13}{123}
                           {}{2}{12}{23}{123}
                           {}{3}{13}{23}{123}
                           {}{1}{2}{12}{13}{123}
                           {}{1}{2}{12}{23}{123}
                           {}{1}{3}{12}{13}{123}
                           {}{1}{3}{13}{23}{123}
                           {}{2}{3}{12}{23}{123}
                           {}{2}{3}{13}{23}{123}
                           {}{1}{2}{3}{12}{13}{23}{123}
(End)

G. Birkhoff, Lattice Theory, Amer. Math. Soc., 1961, p. 4.
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 427.
K. K.-H. Butler, A Moore-Penrose inverse for Boolean relation matrices, pp. 18-28 of Combinatorial Mathematics (Proceedings 2nd Australian Conf.), Lect. Notes Math. 403, 1974.
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. "The number of partially ordered sets. I." Journal of Korean Mathematical Society 11.1 (1974).
S. D. Chatterji, The number of topologies on n points, Manuscript, 1966.
L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 60, 229.
M. Erné, Struktur- und Anzahlformeln für Topologien auf endlichen Mengen, PhD dissertation, Westfälische Wilhelms-Universität zu Münster, 1972.
M. Erné and K. Stege, The number of labeled orders on fifteen elements, personal communication.
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).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, Chap. 3, pages 96ff; Vol. 2, Problem 5.39, p. 88.

Christian Bean, Émile Nadeau, Jay Pantone, and Henning Ulfarsson, Permutations avoiding bipartite partially ordered patterns have a regular insertion encoding, The Electronic Journal of Combinatorics, Volume 31, Issue 3 (2024); arXiv preprint, arXiv:2312.07716 [math.CO], 2023.
Juliana Bowles and Marco B. Caminati, A Verified Algorithm Enumerating Event Structures, arXiv:1705.07228 [cs.LO], 2017.
G. Brinkmann and B. D. McKay, Posets on up to 16 Points, Order 19 (2) (2002) 147-179.
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, The number of partially ordered sets, Journal of Combinatorial Theory, Series B 13.3 (1972): 276-289.
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]
K. K.-H. Butler and G. Markowsky, The number of partially ordered sets. II., J. Korean Math. Soc 11 (1974): 7-17.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
S. D. Chatterji, The number of topologies on n points, Manuscript, 1966. [Annotated scanned copy]
Narendrakumar R. Dasre and Pritam Gujarathi, Approximating the Bounds for Number of Partially Ordered Sets with n Labeled Elements, Computing in Engineering and Technology, Advances in Intelligent Systems and Computing, Vol. 1025, Springer (Singapore 2019), 349-356.
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.
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]
S. R. Finch, Transitive relations, topologies and partial orders.
S. R. Finch, Transitive relations, topologies and partialorders, June 5, 2003. [Cached copy, with permission of the author]
Eldar Fischer, Johann A. Makowsky, and Vsevolod Rakita, MC-finiteness of restricted set partition functions, arXiv:2302.08265 [math.CO], 2023.
Didier Garcia, Proof of a(19) formula [in French].
Didier Garcia, Two conjectures concerning a(n) [in French].
Joël Gay and Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018.
G. Grekos, Letter to N. J. A. Sloane, Oct 31 1994, with attachments.
J. Heitzig and J. Reinhold, The number of unlabeled orders on fourteen elements, Order 17 (2000) no. 4, 333-341.
Richard Kenyon, Maxim Kontsevich, Oleg Ogievetsky, Cosmin Pohoata, Will Sawin, and Senya Shlosman, The miracle of integer eigenvalues, arXiv:2401.05291 [math.CO], 2024. See p. 4.
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. - From _N. J. A. Sloane_, Nov 09 2012
M. Y. Kizmaz, On The Number Of Topologies On A Finite Set, arXiv preprint arXiv:1503.08359 [math.NT], 2015.
D. J. Kleitman and B. L. Rothschild, Asymptotic enumeration of partial orders on a finite set, Trans. Amer. Math. Soc., 205 (1975) 205-220.
G. Kreweras, Dénombrement des ordres étagés, Discrete Math., 53 (1985), 147-149.
Institut f. Mathematik, Univ. Hanover, Erne/Heitzig/Reinhold papers.
Sami Lazaar, Houssem Sabri, and Randa Tahri, Structural and Numerical Studies of Some Topological Properties for Alexandroff Spaces, Bull. Iran. Math. Soc. (2021).
N. Lygeros and P. Zimmermann, Computation of P(14), the number of posets with 14 elements: 1.338.193.159.771.
G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
Bob Proctor, Chapel Hill Poset Atlas.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Ivo Rosenberg, The number of maximal closed classes in the set of functions over a finite domain, J. Combinatorial Theory Ser. A 14 (1973), 1-7.
Ivo Rosenberg and N. J. A. Sloane, Correspondence, 1971.
D. Rusin, Further information and references. [Broken link]
D. Rusin, Further information and references. [Cached 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, List of sequences related to partial orders, circa 1972.
N. J. A. Sloane, Classic Sequences.
Gus Wiseman, Hasse diagrams of the a(4) = 219 posets.
J. A. Wright, There are 718 6-point topologies, quasiorderings and transgraphs, Preprint, 1970. [Annotated scanned copy]
J. A. Wright, Letter to N. J. A. Sloane, Apr 06 1972, listing 18 sequences.
Index entries for sequences related to posets

Cf. A000798 (labeled topologies), A001930 (unlabeled topologies), A000112 (unlabeled posets), A006057.
Sequences in the Erné (1974) paper: A000798, A001035, A006056, A006057, A001929, A001927, A006058, A006059, A000110.
Cf. A316978, A319564, A326876, A326906, A326939, A326943, A326944, A326947.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Select[Subsets[Subsets[Range[n]]],MemberQ[#,{}]&&MemberQ[#,Range[n]]&&UnsameQ@@dual[#]&&SubsetQ[#,Union@@@Tuples[#,2]]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}] (* Gus Wiseman, Aug 14 2019 *)

A000798(n) = Sum_{k=0..n} Stirling2(n,k)*a(k).
Related to A000112 by Erné's formulas: a(n+1) = -s(n, 1), a(n+2) = n*a(n+1) + s(n, 2), a(n+3) = binomial(n+4, 2)*a(n+2) - s(n, 3), where s(n, k) = sum(binomial(n+k-1-m, k-1)*binomial(n+k, m)*sum((m!)/(number of automorphisms of P)*(-(number of antichains of P))^k, P an unlabeled poset with m elements), m=0..n).
From Altug Alkan, Dec 22 2015: (Start)
a(p^k) == 1 (mod p) for all primes p and for all nonnegative integers k.
a(n + p) == a(n + 1) (mod p) for all primes p and for all nonnegative integers n.
If n = 1, then a(1 + p) == a(2) (mod p), that is, a(p + 1) == 3 (mod p).
If n = p, then a(p + p) == a(p + 1) (mod p), that is, a(2*p) == a(p + 1) (mod p).
In conclusion, a(2*p) == 3 (mod p) for all primes p.
(End)
a(n) = Sum_{k=0..n} Stirling1(n,k)*A000798(k). - Tian Vlasic, Feb 25 2022

a(15)-a(16) from Jobst Heitzig (heitzig(AT)math.uni-hannover.de), Jul 03 2000

a(17)-a(18) from Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 02 2008

A102897 Number of ACI algebras (or semilattices) on n generators.

Original entry on oeis.org

2, 4, 14, 122, 4960, 2771104, 151947502948, 28175296471414704944

Offset: 0

Mitch Harris, Jan 18 2005

Also counts Horn functions on n variables, Boolean functions whose set of truth assignments are closed under 'and', or equivalently, the Boolean functions that can be written as a conjunction of Horn clauses, clauses with at most one negative literal.
Also, number of families of subsets of {1,...,n} that are closed under intersection (because we can throw in the universe, or take it out, without affecting anything else).
An ACI algebra or semilattice is a system with a single binary, idempotent, commutative and associative operation.
Also the number of finite sets of finite subsets of {1..n} that are closed under union. - Gus Wiseman, Aug 03 2019

			a(2) = 14: Let the points be labeled a, b. We want the number of collections of subsets of {a, b} which are closed under intersection. 0 subsets: 1 way ({}), 1 subset: 4 ways (0; a; b; ab), 2 subsets: 5 ways (0,a; 0,b; 0,ab; a,ab; b,ab) [not a,b because their intersection, 0, would be missing], 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 14.
From _Gus Wiseman_, Aug 03 2019: (Start)
The a(0) = 2 through a(2) = 14 sets of subsets closed under union:
  {}    {}        {}
  {{}}  {{}}      {{}}
        {{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}}
                  {{},{1},{2},{1,2}}
(End)

V. B. Alekseev, On the number of intersection semilattices [in Russian], Diskretnaya Mat. 1 (1989), 129-136.
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.
G. Burosch, J. Demetrovics, G. O. H. Katona, D. J. Kleitman and A. A. Sapozhenko, On the number of closure operations, in Combinatorics, Paul Erdős is Eighty (Volume 1), Keszthely: Bolyai Society Mathematical Studies, 1993, 91-105.
P. Colomb, A. Irlande and O. Raynaud, Counting of Moore Families for n=7, International Conference on Formal Concept Analysis (2010)
Alfred Horn, Journal of Symbolic Logic 16 (1951), 14-21. [See Lemma 7.]
D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.1, p. 79.
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.
D. E. Knuth, HORN-COUNT

For nonempty set systems of the same type, see A121921.
Regarding sets of subsets closed under union:
- The case with an edge containing all of the vertices is A102895.
- The case without empty edges is A102896.
- The case with intersection instead of union is (also) A102897.
- The unlabeled version is A193675.
- The case closed under both union and intersection is A306445.
- The BII-numbers of set-systems closed under union are A326875.
- The covering case is A326906.
Cf. A102894, A108798, A108800, A193674, A193675, A326866, A326880, A326900.

Table[Length[Select[Subsets[Subsets[Range[n]]],SubsetQ[#,Union@@@Tuples[#,2]]&]],{n,0,3}] (* Gus Wiseman, Aug 03 2019 *)

a(n) = 2*A102896(n) = Sum_{k=0..n} C(n, k)*A102895(k), where C(n, k) is the binomial coefficient

Asymptotically, log_2 a(n) ~ binomial(n, floor(n/2)) for all of A102894, A102895, A102896 and this sequence [Alekseev; Burosch et al.]

Additional comments from Don Knuth, Jul 01 2005

A006058 Number of connected labeled T_4-topologies with n points.

Original entry on oeis.org

1, 1, 3, 16, 145, 2111, 47624, 1626003, 82564031, 6146805142, 662718022355, 102336213875523, 22408881211102698, 6895949927379360277, 2958271314760111914191, 1756322140048351303019576

Offset: 0

N. J. A. Sloane

From Gus Wiseman, Aug 05 2019: (Start)
For n > 0, also the number of topologies covering {1..n} whose nonempty open sets have nonempty intersection. Also the number of topologies covering {1..n} whose nonempty open sets are pairwise intersecting. For example, the a(0) = 1 through a(3) = 16 topologies (empty sets not shown) are:
  {}  {{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}}
(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Herman Jamke, Table of n, a(n) for n = 0..19
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)

Sequences in the Erné (1974) paper: A000798, A001035, A006056, A006057, A001929, A001927, A006058, A006059, A000110.

Cf. A001930, A003465, A108798, A306445, A326878, A326906.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],Intersection[#1,#2]=={}&],Union@@#==Range[n]&&SubsetQ[#,Union[Union@@@Tuples[#,2],Intersection@@@Tuples[#,2]]]&]],{n,0,4}] (* Gus Wiseman, Aug 05 2019 *)
A000798 = Append[Cases[Import["https://oeis.org/A000798/b000798.txt", "Table"], {, }][[All, 2]], 0];
a[n_] := If[n == 0, 1, Sum[ Binomial[n, k] A000798[[k+1]], {k, 0, n-1}]];
a /@ Range[0, Length[A000798]-1] (* Jean-François Alcover, Jan 01 2020 *)

From Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 02 2008: (Start)
a(n) = Sum_{k=0..n-1} binomial(n, k)*A000798(k) if n>=1.
E.g.f.: Z4(x) = A(x)*(exp(x)-1) + 1 where A(x) denotes the e.g.f. for A000798. (End)
a(n) = A326909(n) - A000798(n). - Gus Wiseman, Aug 05 2019

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 02 2008

A326943 Number of T_0 sets of subsets of {1..n} that cover all n vertices and are closed under intersection.

Original entry on oeis.org

2, 2, 6, 70, 4078, 2704780, 151890105214, 28175292217767880450

Offset: 0

Gus Wiseman, Aug 08 2019

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks 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).

			The a(0) = 2 through a(3) = 6 sets of subsets:
  {}    {{1}}     {{1},{1,2}}
  {{}}  {{},{1}}  {{2},{1,2}}
                  {{},{1},{2}}
                  {{},{1},{1,2}}
                  {{},{2},{1,2}}
                  {{},{1},{2},{1,2}}

The non-T_0 version is A326906.
The case without empty edges is A309615.
The non-covering version is A326945.
The version not closed under intersection is A326939.
Cf. A003180, A003181, A003465, A059052, A059201, A245567, A316978, A319564, A319637, A326940, A326941, A326942, A326947.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Select[Subsets[Subsets[Range[n]]],Union@@#==Range[n]&&UnsameQ@@dual[#]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]

Inverse binomial transform of A326945.

a(n) = Sum_{k=0..n} Stirling1(n,k)*A326906(k). - Andrew Howroyd, Aug 14 2019

a(5)-a(7) from Andrew Howroyd, Aug 14 2019

A326944 Number of T_0 sets of subsets of {1..n} that cover all n vertices, contain {}, and are closed under intersection.

Original entry on oeis.org

1, 1, 4, 58, 3846, 2685550, 151873991914, 28175291154649937052

Offset: 0

Gus Wiseman, Aug 08 2019

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

The version not closed under intersection is A059201.
The non-T_0 version is A326881.
The version where {} is not necessarily an edge is A326943.
Cf. A003181, A003465, A055621, A182507, A245567, A316978, A319564, A326906, A326939, A326941, A326945, A326947.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Select[Subsets[Subsets[Range[n]]],MemberQ[#,{}]&&Union@@#==Range[n]&&UnsameQ@@dual[#]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]

a(n) = Sum_{k=0..n} Stirling1(n,k)*A326881(k). - Andrew Howroyd, Aug 14 2019

a(5)-a(7) from Andrew Howroyd, Aug 14 2019

A326945 Number of T_0 sets of subsets of {1..n} that are closed under intersection.

Original entry on oeis.org

2, 4, 12, 96, 4404, 2725942, 151906396568, 28175293281055562650

Offset: 0

Gus Wiseman, Aug 08 2019

			The a(0) = 2 through a(2) = 12 sets of subsets:
  {}    {}        {}
  {{}}  {{}}      {{}}
        {{1}}     {{1}}
        {{},{1}}  {{2}}
                  {{},{1}}
                  {{},{2}}
                  {{1},{1,2}}
                  {{2},{1,2}}
                  {{},{1},{2}}
                  {{},{1},{1,2}}
                  {{},{2},{1,2}}
                  {{},{1},{2},{1,2}}

The non-T_0 version is A102897.
The version not closed under intersection is A326941.
The covering case is A326943.
The case without empty edges is A326959.
Cf. A003180, A182507, A316978, A319564, A326906, A326939, A326940, A326944, A326947.

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

Binomial transform of A326943.

a(5)-a(7) from Andrew Howroyd, Aug 14 2019

A326904 Number of unlabeled set-systems (without {}) on n vertices that are closed under intersection.

Original entry on oeis.org

1, 2, 4, 10, 38, 368, 29328, 216591692, 5592326399531792

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.

Apart from the offset the same as A193675. - R. J. Mathar, Aug 09 2019

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 10 set-systems:
  {}  {}     {}           {}
      {{1}}  {{1}}        {{1}}
             {{1,2}}      {{1,2}}
             {{2},{1,2}}  {{1,2,3}}
                          {{2},{1,2}}
                          {{3},{1,2,3}}
                          {{2,3},{1,2,3}}
                          {{3},{1,3},{2,3}}
                          {{3},{2,3},{1,2,3}}
                          {{3},{1,3},{2,3},{1,2,3}}

The covering case is A108800(n - 1).
The case with an edge containing all of the vertices is A193674(n - 1).
The case with union instead of intersection is A193674.
The labeled version is A326901.
Cf. A000798, A001930, A006058, A102895, A102898, A326876, A326866, A326878, A326882, A326903, A326906.

a(n > 0) = 2 * A193674(n - 1).

A326907 Number of non-isomorphic sets of subsets of {1..n} that are closed under union and cover all n vertices. First differences of A193675.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Aug 03 2019

Differs from A108800 in having a(0) = 2 instead of 1.

			Non-isomorphic representatives of the a(0) = 2 through a(3) = 28 sets of sets:
  {}    {{1}}    {{12}}          {{123}}
  {{}}  {{}{1}}  {{}{12}}        {{}{123}}
                 {{2}{12}}       {{3}{123}}
                 {{}{2}{12}}     {{23}{123}}
                 {{1}{2}{12}}    {{}{3}{123}}
                 {{}{1}{2}{12}}  {{}{23}{123}}
                                 {{1}{23}{123}}
                                 {{3}{23}{123}}
                                 {{13}{23}{123}}
                                 {{}{1}{23}{123}}
                                 {{}{3}{23}{123}}
                                 {{}{13}{23}{123}}
                                 {{2}{3}{23}{123}}
                                 {{2}{13}{23}{123}}
                                 {{3}{13}{23}{123}}
                                 {{12}{13}{23}{123}}
                                 {{}{2}{3}{23}{123}}
                                 {{}{2}{13}{23}{123}}
                                 {{}{3}{13}{23}{123}}
                                 {{}{12}{13}{23}{123}}
                                 {{2}{3}{13}{23}{123}}
                                 {{3}{12}{13}{23}{123}}
                                 {{}{2}{3}{13}{23}{123}}
                                 {{}{3}{12}{13}{23}{123}}
                                 {{2}{3}{12}{13}{23}{123}}
                                 {{}{2}{3}{12}{13}{23}{123}}
                                 {{1}{2}{3}{12}{13}{23}{123}}
                                 {{}{1}{2}{3}{12}{13}{23}{123}}

The case without empty sets is A108798.
The case with a single covering edge is A108800.
First differences of A193675.
The case also closed under intersection is A326898 for n > 0.
The labeled version is A326906.
The same for union instead of intersection is (also) A326907.
Cf. A001930, A102895, A108798, A193674, A193675, A326880, A326881, A326883, A326898, A326908.

a(7) added from A108800 by Andrew Howroyd, Aug 10 2019

A326909 Number of sets of subsets of {1..n} closed under union and intersection and covering all of the vertices.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Aug 04 2019

nonn

Differs from A326878 in having a(0) = 2 instead of 1.

			The a(0) = 2 through a(2) = 7 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}}

Covering sets of subsets are A000371.
The case without empty sets is A108798.
The case with a single covering edge is A326878.
The unlabeled version is A326898 for n > 0.
The case closed only under union is A326906.
The case closed only under intersection is (also) A326906.
Cf. A000798, A001930, A003465, A006058, A306445, A326876, A326882, A326907, A326908.

Table[Length[Select[Subsets[Subsets[Range[n]]],Union@@#==Range[n]&&SubsetQ[#,Union[Union@@@Tuples[#,2],Intersection@@@Tuples[#,2]]]&]],{n,0,3}]
(* Second program: *)
A000798 = Cases[Import["https://oeis.org/A000798/b000798.txt", "Table"], {, }][[All, 2]];
A006058 = Cases[Import["https://oeis.org/A006058/b006058.txt", "Table"], {, }][[All, 2]];
a[n_] := A006058[[n + 1]] + A000798[[n + 1]];
a /@ Range[0, 18] (* Jean-François Alcover, Dec 30 2019 *)

a(n) = A000798(n) + A006058(n). - Jean-François Alcover, Dec 30 2019, after Gus Wiseman's comment in A006058.

a(18) from A000798+A006058 by Jean-François Alcover, Dec 30 2019

cp's OEIS Frontend

A001035 Number of partially ordered sets ("posets") with n labeled elements (or labeled acyclic transitive digraphs).

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A102897 Number of ACI algebras (or semilattices) on n generators.

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A006058 Number of connected labeled T_4-topologies with n points.

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A326943 Number of T_0 sets of subsets of {1..n} that cover all n vertices and are closed under intersection.

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A326944 Number of T_0 sets of subsets of {1..n} that cover all n vertices, contain {}, and are closed under intersection.

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A326945 Number of T_0 sets of subsets of {1..n} that are closed under intersection.

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A326904 Number of unlabeled set-systems (without {}) on n vertices that are closed under intersection.

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