A000112 -id:A000112 - 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.
Moussa Benoumhani, The Number of Topologies on a Finite Set, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.6.
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.
M. Benoumhani and M. Kolli, Finite topologies and partitions, JIS 13 (2010) # 10.3.5.
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

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

Original entry on oeis.org

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

A001930 Number of topologies, or transitive digraphs with n unlabeled nodes.

Original entry on oeis.org

1, 1, 3, 9, 33, 139, 718, 4535, 35979, 363083, 4717687, 79501654, 1744252509, 49872339897, 1856792610995, 89847422244493, 5637294117525695

Offset: 0

N. J. A. Sloane

			From _Gus Wiseman_, Aug 02 2019: (Start)
Non-isomorphic representatives of the a(0) = 1 through a(3) = 9 topologies:
  {}  {}{1}  {}{12}        {}{123}
             {}{2}{12}     {}{3}{123}
             {}{1}{2}{12}  {}{23}{123}
                           {}{1}{23}{123}
                           {}{3}{23}{123}
                           {}{2}{3}{23}{123}
                           {}{3}{13}{23}{123}
                           {}{2}{3}{13}{23}{123}
                           {}{1}{2}{3}{12}{13}{23}{123}
(End)

Loic Foissy, Claudia Malvenuto, 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. .
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 218 (but the last entry is wrong).
M. Kolli, On the cardinality of the T_0-topologies on a finite set, Preprint, 2014.
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).
J. A. Wright, There are 718 6-point topologies, quasi-orderings and transgraphs, Notices Amer. Math. Soc., 17 (1970), p. 646, Abstract #70T-A106.
J. A. Wright, personal communication.
For further references concerning the enumeration of topologies and posets see under A000112 and A001035.

C. M. Bender et al., Combinatorics and field theory, arXiv:quant-ph/0604164, 2006.
Moussa Benoumhani, The Number of Topologies on a Finite Set, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.6.
M. Benoumhani and M. Kolli, Finite topologies and partitions, JIS 13 (2010) # 10.3.5
Gunnar Brinkmann and Brendan D. McKay, Counting unlabeled topologies and transitive relations.
G. Brinkmann and B. D. McKay, Counting unlabeled topologies and transitive relations, J. Integer Sequences, Volume 8, 2005.
Gunnar Brinkmann and Brendan D. McKay, Counting Unlabelled Topologies and Transitive Relations, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.1.
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]
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
S. R. Finch, Transitive relations, topologies and partial orders
S. R. Finch, Transitive relations, topologies and partial orders, June 5, 2003. [Cached copy, with permission of the author]
L. Foissy, C. Malvenuto, and F. Patras, B_infinity-algebras, their enveloping algebras, and finite spaces, arXiv preprint arXiv:1403.7488 [math.AT], 2014-2015.
Misha Gavrilovich and Misha Rabinovich, The Quillen negation monoid of a category, and Schreier graphs of its action on classes of morphisms, 2024. See p. 11.
Dongseok Kim, Young Soo Kwon and Jaeun Lee, Enumerations of finite topologies associated with a finite graph, arXiv preprint arXiv:1206.0550, 2012. - From _N. J. A. Sloane_, Nov 09 2012
Messaoud Kolli, Direct and Elementary Approach to Enumerate Topologies on a Finite Set, J. Integer Sequences, Volume 10, 2007, Article 07.3.1.
G. Pfeiffer, Counting Transitive Relations, preprint, 2004.
G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
D. Rusin, Further information and references [Broken link]
D. Rusin, Further information and references [Cached copy]
Henry Sharp, Jr., Quasi-orderings and topologies on finite sets, Proceedings of the American Mathematical Society 17.6 (1966): 1344-1349. [Annotated scanned copy]
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.
J. M. Tangen and N. J. A. Sloane, Correspondence, 1976-1976
R. H. Warren, The number of topologies, Houston J. Math., 8 (No. 2, 1982), 297-301. Mentions a(4)=33. [Annotated scanned copy]
Eric Weisstein's World of Mathematics, Digraph Topology.
R. H. Warren, The number of topologies, Houston J. Math., 8 (No. 2, 1982), 297-301. Mentions a(4)=33. [Annotated scanned copy]
Wikipedia Topological space
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

Cf. A000798 (labeled topologies), A001035 (labeled posets), A001930 (unlabeled topologies), A000112 (unlabeled posets), A006057, A001928, A001929.
The case with unions only is A108798.
The case with intersections only is (also) A108798.
Partial sums are A326898 (the non-covering case).
Cf. A000612, A003180, A108800, A193674, A306445, A326876, A326878, A326882.

a(8)-a(12) from Goetz Pfeiffer (goetz.pfeiffer(AT)nuigalway.ie), Jan 21 2004

a(13)-a(16) from Brinkmann's and McKay's paper, sent by Vladeta Jovovic, Jan 04 2006

A001927 Number of connected partially ordered sets with n labeled points.

Original entry on oeis.org

1, 1, 2, 12, 146, 3060, 101642, 5106612, 377403266, 40299722580, 6138497261882, 1320327172853172, 397571105288091506, 166330355795371103700, 96036130723851671469482, 76070282980382554147600692, 82226869197428315925408327266, 120722306604121583767045993825620, 239727397782668638856762574296226842

Offset: 0

N. J. A. Sloane

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.
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).

C. M. Bender et al., Combinatorics and field theory, arXiv:quant-ph/0604164, 2006.
G. Brinkmann, B. D. McKay, Posets on up to 16 Points, Order 19 (2) (2002) 147-179 (Table II, up to 18 points)
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]
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, Counting Finite Posets and Topologies, Order, 8 (1991), 247-265.
N. J. A. Sloane, List of sequences related to partial orders, circa 1972
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. A000112, A001035, A000608, A066303, A342501 (refined by rank).

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

A001035 = {1, 1, 3, 19, 219, 4231, 130023, 6129859, 431723379, 44511042511, 6611065248783, 1396281677105899, 414864951055853499, 171850728381587059351, 98484324257128207032183, 77567171020440688353049939, 83480529785490157813844256579, 122152541250295322862941281269151, 241939392597201176602897820148085023};
max = Length[A001035]-1;
B[x_] = Sum[A001035[[k+1]]*x^k/k!, {k, 0, max}];
A[x_] = 1 + Log[B[x]];
CoefficientList[A[x] + O[x]^(max-1), x]*Range[0, max-2]! (* Jean-François Alcover, Apr 17 2014, updated Aug 30 2018 *)

E.g.f. A(x)=log(B(x)) where B(x) is e.g.f. of A001035.

More terms from Christian G. Bower, Dec 12 2001

a(17)-a(18) using data from A001035 from Alois P. Heinz, Aug 30 2018

A001929 Number of connected topologies on n labeled points.

Original entry on oeis.org

1, 1, 3, 19, 233, 4851, 158175, 7724333, 550898367, 56536880923, 8267519506789, 1709320029453719, 496139872875425839, 200807248677750187825, 112602879608997769049739, 86955243134629606109442219, 91962123875462441868790125305, 132524871920295877733718959290203, 259048612476248175744581063815546423

Offset: 0

N. J. A. Sloane

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.
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).

C. M. Bender et al., Combinatorics and Field theory, arXiv:quant-ph/0604164, 2006.
G. Brinkmann and B. D. McKay, Posets on up to 16 Points, Order 19 (2) (2002) 147-179, Table IV up to 18 points
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]
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, Counting Finite Posets and Topologies, Order, 8 (1991), 247-265.
N. J. A. Sloane, List of sequences related to partial orders, circa 1972
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

Cf. A001928, A001930.

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

A001035 = {1, 1, 3, 19, 219, 4231, 130023, 6129859, 431723379, 44511042511, 6611065248783, 1396281677105899, 414864951055853499, 171850728381587059351, 98484324257128207032183, 77567171020440688353049939, 83480529785490157813844256579, 122152541250295322862941281269151, 241939392597201176602897820148085023};
max = Length[A001035]-1;
B[x_] = Sum[A001035[[k+1]]*x^k/k!, {k, 0, max}];
A[x_] = 1 + Log[B[x]];
A001927 = CoefficientList[ A[x] + O[x]^(max-1), x]*Range[0, max-2]!;
a[n_] := Sum[StirlingS2[n, k] *A001927[[k+1]], {k, 0, n}];
Table[a[n], {n, 0, max -2}] (* Jean-François Alcover, Aug 30 2018, after Vladeta Jovovic *)

a(n) = Sum_{k=0..n} Stirling2(n,k)*A001927(k). - Vladeta Jovovic, Apr 10 2006

More terms from Vladeta Jovovic, Apr 10 2006

a(17)-a(18) using data from A001035 from Alois P. Heinz, Aug 30 2018

A121337 Number of idempotent relations on n labeled elements.

Original entry on oeis.org

1, 2, 11, 123, 2360, 73023, 3465357

Offset: 0

Florian Kammüller (flokam(AT)cs.tu-berlin.de), Aug 28 2006

A relation r is idempotent if r ; r = r, where ; denotes sequential composition.
From Geoffrey Critzer, Oct 18 2023 : (Start)
a(n) is also the number of maximal subgroups in the semigroup of binary relations on [n].  See Butler and Markowski link.
A binary relation is idempotent iff it is both dense (A355730) and transitive (A006905).
A binary relation is idempotent iff it is both limit dominating (A366194) and limit dominated (A366722).  See Gregory, Kirkland, and Pullman link.
A binary relation R on [n] is idempotent iff the following biconditional statement holds for all x,y in [n]:  There is a cyclic traverse from x to y in G(R) iff (x,y) is in R.  Here, G(R) is the directed graph with self loops allowed (A002416) corresponding to R.  See Rosenblatt link.
Let Q be a quasi-order (A000798) on [n].  Let D(X) be the relation {(x,x):x is in X}.  Let S be a subset of [n] such that: (i) For all x in S, the class in the equivalence relation Q intersect Q^(-1) containing (x,x) is a singleton and (ii) for all x,y in S, the component containing x is not covered by the component containing y in the condensation of G(Q) .  Here, the condensation of G(Q) is the acyclic digraph (A003024) obtained from G(Q) by replacing every strongly connected component (SCC) by a single vertex and all directed edges from one SCC to another with a single directed edge.  Then a relation is idempotent iff it is of the form Q-D(S).  See Schein link.  (End)

			a(2) = 11 because given a set {a,b} of two elements, of the 2^(2*2) = 16 relations on the set, only 5 are not idempotent. - _Michael Somos_, Jul 28 2013
These 5 relations that are not idempotent are: {(a,b)}, {(b,a)}, {(a,b),(b,a)}, {(a,b),(b,a),(b,b)}, {(a,a),(a,b),(b,a)}. - _Geoffrey Critzer_, Aug 07 2016

F. Kammüller, Interactive Theorem Proving in Software Engineering, Habilitationsschrift, Technische Universitaet Berlin (2006).
Ki Hang Kim, Boolean Matrix Theory and Applications, Marcel Decker, 1982.

G. Brinkmann and B. D. McKay, Counting unlabeled topologies and transitive relations, J. Integer Sequences, Volume 8, 2005.
K. K.-H. Butler and G. Markowsky, The Number of Maximal Subgroups of the Semigroup of Binary Relations, Kyungpook Math. J. Vol 12, June 1972.
D. A. Gregory, S. Kirkland, and N. J. Pullman, Power convergent Boolean matrices, Linear Algebra and its Applications, Volume 179, 15 January 1993, Pages 105-117.
F. Kammüller, Counting idempotent relations.
F. Kammüller and J. W. Sanders, Idempotent relations in Isabelle/HOL, International Colloquium on Theoretical Aspects of Computing, ICTAC'04. Volume 3407 of Lecture Notes in Computer Science, Springer-Verlag (2005).
G. Pfeiffer, Counting transitive relations, J. Integer Sequences, Volume 7, 2004, #3.
D. Rosenblatt, On the graphs of finite Boolean relation matrices, Journal of Research of the National Bureau of Standards, 67B No. 4, 1963.
B. M. Schein, A construction for idempotent binary relations, Proc. Japan Acad., Vol. 46, No. 3 (1970), pp. 246-247.

Cf. A000798 (labeled quasi-orders (or topologies)), A001930 (unlabeled quasi-orders), A001035 (labeled partial orders), A000112 (unlabeled partial orders), A002416, A003024, A366722, A366194, A355730, A006905.

Row sums of A360984.

Prepend[Table[Length[Select[Tuples[Tuples[{0, 1}, n], n], (MatrixPower[#, 2] /. x_ /; x > 0 -> 1) == # &]], {n, 1, 4}], 1] (* Geoffrey Critzer, Aug 07 2016 *)

Offset corrected by James Mitchell, Jul 28 2013

a(1) corrected by Philippe Beaudoin, Aug 11 2015

A000608 Number of connected partially ordered sets with n unlabeled elements.

Original entry on oeis.org

1, 1, 1, 3, 10, 44, 238, 1650, 14512, 163341, 2360719, 43944974, 1055019099, 32664984238, 1303143553205, 66900392672168, 4413439778321689

Offset: 0

N. J. A. Sloane, Simon Plouffe, J. A. Wright

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.
E. N. Gilbert, A catalog of partially ordered systems, unpublished memorandum, Aug 08, 1961.
G. Melançon, 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).

Gunnar Brinkmann and Brendan D. McKay, Counting unlabeled topologies and transitive relations.
G. Brinkmann and B. D. McKay, Counting unlabeled topologies and transitive relations, J. Integer Sequences, Volume 8, 2005.
G. Brinkmann and B. D. McKay, Posets on up to 16 Points [On Brendan McKay's home page]
G. Brinkmann and B. D. McKay, Posets on up to 16 Points, Order 19 (2) (2002) 147-179.
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]
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
E. N. Gilbert, A catalog of partially ordered systems, unpublished memorandum, Aug 08, 1961. [Annotated scanned copy]
Henry Sharp, Jr., Quasi-orderings and topologies on finite sets, Proceedings of the American Mathematical Society 17.6 (1966): 1344-1349. [Annotated scanned copy]
N. J. A. Sloane, List of sequences related to partial orders, circa 1972
N. J. A. Sloane, Transforms
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 10 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
N. L. White, Two letters to N. J. A. Sloane, 1970, with hand-drawn enclosure
Wikipedia, Illustration n=4, Illustration n=5, Illustration n=6" (2021)
J. A. Wright, There are 718 6-point topologies, quasiorderings and transgraphs, Preprint, 1970. [Annotated scanned copy]
J. A. Wright, There are 718 6-point topologies, quasi-orderings and transgraphs, Notices Amer. Math. Soc., 17 (1970), p. 646, Abstract #70T-A106.
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 sequences related to posets

Inverse Euler transform of A000112.

Cf. A263864 (multiset transform), A342500 (refined by rank).

A000112 = Cases[Import["https://oeis.org/A000112/b000112.txt", "Table"], {, }][[All, 2]];
(* EulerInvTransform is defined in A022562 *)
{1} ~Join~ EulerInvTransform[Rest[A000112]] (* Jean-François Alcover, Dec 04 2019, updated Mar 17 2020 *)

More terms from Christian G. Bower, who pointed out connection with A000112, Jan 21 1998 and Dec 12 2001

More terms from Vladeta Jovovic, Jan 04 2006; corrected Jan 15 2006

A245567 Number of antichain covers of a labeled n-set such that for every two distinct elements in the n-set, there is a set in the antichain cover containing one of the elements but not the other.

Original entry on oeis.org

2, 1, 1, 5, 76, 5993, 7689745, 2414465044600, 56130437141763247212112, 286386577668298408602599478477358234902247

Offset: 0

Patrick De Causmaecker, Jul 25 2014

This is the number of antichain covers such that the induced partition contains only singletons. The induced partition of {{1,2},{2,3},{1,3},{3,4}} is {{1},{2},{3},{4}}, while the induced partition of {{1,2,3},{2,3,4}} is {{1},{2,3},{4}}.
This sequence is related to A006126. See 1st formula.
The sequence is also related to Dedekind numbers through Stirling numbers of the second kind. See 2nd formula.
Sets of subsets of the described type are said to be T_0. - Gus Wiseman, Aug 14 2019

			For n = 0, a(0) = 2 by the antisets {}, {{}}.
For n = 1, a(1) = 1 by the antiset {{1}}.
For n = 2, a(2) = 1 by the antiset {{1},{2}}.
For n = 3, a(3) = 5 by the antisets {{1},{2},{3}}, {{1,2},{1,3}}, {{1,2},{2,3}}, {{1,3},{2,3}}, {{1,2},{1,3},{2,3}}.

Patrick De Causmaecker and Stefan De Wannemacker, On the number of antichains of sets in a finite universe, arXiv:1407.4288 [math.CO], 2014.

Cf. A000372 (Dedekind numbers), A006126 (Number of antichain covers of a labeled n-set).
Sequences counting and ranking T_0 structures:
  A000112 (unlabeled topologies),
  A001035 (topologies),
  A059201 (covering set-systems),
  A245567 (antichain covers),
  A309615 (covering set-systems closed under intersection),
  A316978 (factorizations),
  A319559 (unlabeled set-systems by weight),
  A319564 (integer partitions),
  A319637 (unlabeled covering set-systems),
  A326939 (covering sets of subsets),
  A326940 (set-systems),
  A326941 (sets of subsets),
  A326943 (covering sets of subsets closed under intersection),
  A326944 (covering sets of subsets with {} and closed under intersection),
  A326945 (sets of subsets closed under intersection),
  A326946 (unlabeled set-systems),
  A326947 (BII-numbers of set-systems),
  A326948 (connected set-systems),
  A326949 (unlabeled sets of subsets),
  A326950 (antichains),
  A326959 (set-systems closed under intersection),
  A327013 (unlabeled covering set-systems closed under intersection),
  A327016 (BII-numbers of topologies).

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[Subsets[Subsets[Range[n]]],Union@@#==Range[n]&&stableQ[#,SubsetQ]&&UnsameQ@@dual[#]&]],{n,0,3}] (* Gus Wiseman, Aug 14 2019 *)

A000372(n) = Sum_{k=0..n} S(n+1,k+1)*a(k).
a(n) = A006126(n) - Sum_{k=1..n-1} S(n,k)*a(k).
Were n > 0 and S(n,k) is the number of ways to partition a set of n elements into k nonempty subsets.
Inverse binomial transform of A326950, if we assume a(0) = 1. - Gus Wiseman, Aug 14 2019

Definition corrected by Patrick De Causmaecker, Oct 10 2014

a(9), based on A000372, from Patrick De Causmaecker, Jun 01 2023

A006455 Number of partial orders on {1,2,...,n} that are contained in the usual linear order (i.e., xRy => x

Original entry on oeis.org

1, 1, 2, 7, 40, 357, 4824, 96428, 2800472, 116473461, 6855780268, 565505147444, 64824245807684

Offset: 0

N. J. A. Sloane

Also known as naturally labeled posets. - David Bevan, Nov 16 2023
Also the number of n X n upper triangular Boolean matrices B with all diagonal entries 1 such that B = B^2.
The asymptotic values derived by Brightwell et al. are relevant only for extremely large values of n. The average number of linear extensions (topological sorts) of a random partial order on {1,...,n} is n! S_n / N_n, where S_n is this sequence and N_n is sequence A001035

			a(3) = 7: {}, {1R2}, {1R3}, {2R3}, {1R2, 1R3}, {1R3, 2R3}, {1R2, 1R3, 2R3}.

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

S. P. Avann, The lattice of natural partial orders, Aequationes Mathematicae 8 (1972), 95-102.
David Bevan, Gi-Sang Cheon and Sergey Kitaev, On naturally labelled posets and permutations avoiding 12-34, arXiv:2311.08023 [math.CO], 2023.
Graham Brightwell, Hans Jürgen Prömel and Angelika Steger, The average number of linear extensions of a partial order, Journal of Combinatorial Theory A73 (1996), 193-206.
S. R. Finch, Transitive relations, topologies and partial orders
S. R. Finch, Transitive relations, topologies and partial orders, June 5, 2003. [Cached copy, with permission of the author]
Joël Gay and Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018.
L. H. Harper, The Range of a Steiner Operation, arXiv preprint arXiv:1608.07747 [math.CO], 2016.
N. Hindman and N. J. A. Sloane, Correspondence, 1981-1991
Florent Hivert and Nicolas M. Thiéry, Controlling the C3 Super Class Linearization Algorithm for Large Hierarchies of Classes, Order (2022).
Adam King, A. Laubmeier, K. Orans, and A. Godbole, Universal and Overlap Cycles for Posets, Words, and Juggling Patterns, arXiv preprint arXiv:1405.5938 [math.CO], 2014.
D. E. Knuth, POSETS, program for n = 10, 11, 12.
J.-G. Luque, L. Mignot and F. Nicart, Some Combinatorial Operators in Language Theory, arXiv preprint arXiv:1205.3371 [cs.FL], 2012. - _N. J. A. Sloane_, Oct 22 2012
Index entries for sequences related to posets

Cf. A000112, A001035, A323502.

E.g.f.: exp(S(x)-1) where S(x)is the e.g.f. for A323502. - Ludovic Schwob, Mar 29 2024

Additional comments and more terms from Don Knuth, Dec 03 2001

A048172 Number of labeled series-parallel graphs with n edges.

Original entry on oeis.org

1, 3, 19, 195, 2791, 51303, 1152019, 30564075, 935494831, 32447734143, 1257770533339, 53884306900515, 2528224238464471, 128934398091500823, 7101273378743303779, 420078397130637237915, 26563302733186339752511, 1788055775343964413724143, 127652707703771090396080939

Offset: 1

N. J. A. Sloane

Labeled N-free posets. - Detlef Pauly (dettodet(AT)yahoo.de), Dec 27 2002

Ronald C. Read, Graphical enumeration by cycle-index sums: first steps toward a unified treatment, Research Report CORR 91-19, University of Waterloo, Sept 1991.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.39.

Vincenzo Librandi, Table of n, a(n) for n = 1..100
F. Chapoton, F. Hivert, J.-C. Novelli, A set-operad of formal fractions and dendriform-like sub-operads, arXiv preprint arXiv:1307.0092 [math.CO], 2013.
Frédéric Fauvet, L. Foissy, D. Manchon, Operads of finite posets, arXiv preprint arXiv:1604.08149 [math.CO], 2016.
S. R. Finch, Series-parallel networks, July 7, 2003. [Cached copy, with permission of the author]
Vladimir Kruchinin, The method for obtaining expressions for coefficients of reverse generating functions, arXiv:1211.3244 [math.CO], 2012.
R. P. Stanley, Enumeration of posets generated by disjoint unions and ordinal sums, Proc. Amer. Math. Soc. 45 (1974), 295-299
Index entries for reversions of series
Index entries for sequences related to posets

Cf. A048174, A058349, A058350.

Cf. A000112 (unlabeled posets), A001035 (labeled posets), A003430 (unlabeled analog).

with(gfun):
f := series((ln(1+x)-x^2/(1+x)), x, 21):
egf := seriestoseries(f, 'revogf'):
seriestolist(egf, 'Laplace');

lim = 19; Join[{1}, Drop[ CoefficientList[ InverseSeries[ Series[x + 2*(1 - Cosh[x]) , {x, 0, lim}], y] + InverseSeries[ Series[-Log[1 - x] - x^2/(1 - x),{x, 0, lim}], y], y], 2]*Range[2, lim]!] (* Jean-François Alcover, Sep 21 2011, after g.f. *)
m = 17; Rest[CoefficientList[InverseSeries[Series[Log[1+x]-x^2/(1+x), {x, 0, m}], x], x]*Table[k!,{k, 0, m}]](* Jean-François Alcover, Apr 18 2011 *)

h(n,k):=if n=k then 0 else (-1)^(n-k)*binomial(n-k-1,k-1); a(n):=if n=1 then 1 else -sum((k!/n!*stirling1(n,k)+sum(binomial(k,j)*sum((j)!/(i)!*stirling1(i,j)*h(n-i,k-j),i,j,n-k+j),j,1,k-1)+h(n,k))*a(k),k,1,n-1); /* Vladimir Kruchinin, Sep 08 2010 */

x='x+O('x^55);
s=-log(1-x)-x^2/(1-x);
A048174=Vec(serlaplace(serreverse(s)));
t=x+2*(1-cosh(x));
A058349=Vec(serlaplace(serreverse(t)));
A048172=A048174+A058349;  A048172[1]-=1;
A048172 /* Joerg Arndt, Feb 04 2011 */

a(n) = A058349(n) + A048174(n).
a(n) = A058349(n) + A058350(n) (n>=2).
Reference (by Ronald C. Read) gives generating functions.
E.g.f. is reversion of log(1+x)-x^2/(1+x).
a(n)=if n=1 then 1 else -sum((k!/n!*stirling1(n,k)+sum(binomial(k,j)*sum((j)!/(i)!*stirling1(i,j)*h(n-i,k-j),i,j,n-k+j),j,1,k-1)+h(n,k))*a(k),k,1,n-1), h(n,k)=if n=k then 0 else (-1)^(n-k)*binomial(n-k-1,k-1), n>0. - Vladimir Kruchinin, Sep 08 2010
a(n) ~ sqrt((5+3*sqrt(5))/10) * n^(n-1) / (exp(n) * (2 - sqrt(5) + log((1+sqrt(5))/2))^(n-1/2)). - Vaclav Kotesovec, Feb 25 2014

More terms from Joerg Arndt, Feb 04 2011

cp's OEIS Frontend

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A001927 Number of connected partially ordered sets with n labeled points.

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A000608 Number of connected partially ordered sets with n unlabeled elements.

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