|
%I M3631 N1476 #264 Apr 11 2022 21:56:51
%S 1,1,4,29,355,6942,209527,9535241,642779354,63260289423,8977053873043,
%T 1816846038736192,519355571065774021,207881393656668953041,
%U 115617051977054267807460,88736269118586244492485121,93411113411710039565210494095,134137950093337880672321868725846,261492535743634374805066126901117203
%N Number of different quasi-orders (or topologies, or transitive digraphs) with n labeled elements.
%C From _Altug Alkan_, Dec 18 2015 and Feb 28 2017: (Start)
%C 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).
%C 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.
%C In conclusion, a(19) is a number of the form 323*n - 17. (End)
%C The BII-numbers of finite topologies without their empty set are given by A326876. - _Gus Wiseman_, Aug 01 2019
%C From _Tian Vlasic_, Feb 23 2022: (Start)
%C 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.
%C 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).
%C Lower and upper bounds are known: 2^n <= a(n) <= 2^(n*(n-1)), n > 1.
%C This follows from the fact that there are 2^(n*(n-1)) reflexive relations on a set with n elements.
%C Furthermore: a(n+1) <= a(n)*(3a(n)+1). (End)
%D 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.
%D S. D. Chatterji, The number of topologies on n points, Manuscript, 1966.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 229.
%D E. D. Cooper, Representation and generation of finite partially ordered sets, Manuscript, no date.
%D E. N. Gilbert, A catalog of partially ordered systems, unpublished memorandum, Aug 08, 1961.
%D F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 243.
%D 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)
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D For further references concerning the enumeration of topologies and posets see under A001035.
%D 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.
%H V. I. Arnautov and A. V. Kochina, <a href="http://www.math.md/publications/basm/issues/y2010-n3/10375/">Method for constructing one-point expansions of a topology on a finite set and its applications</a>, Bul. Acad. Stiinte Republ. Moldav. Matem. 3 (64) (2010) 67-76.
%H Moussa Benoumhani, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Benoumhani/benoumhani11.html">The Number of Topologies on a Finite Set</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.6.
%H Moussa Benoumhani and Ali Jaballah, <a href="https://doi.org/10.1016/j.jcta.2018.07.007">Chains in lattices of mappings and finite fuzzy topological spaces</a>, Journal of Combinatorial Theory, Series A (2019) Vol. 161, 99-111.
%H M. Benoumhani and M. Kolli, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Benoumhani/benoumhani6.html">Finite topologies and partitions</a>, JIS 13 (2010) # 10.3.5.
%H Juliana Bowles and Marco B. Caminati, <a href="https://arxiv.org/abs/1705.07228">A Verified Algorithm Enumerating Event Structures</a>, arXiv:1705.07228 [cs.LO], 2017.
%H Gunnar Brinkmann and Brendan D. McKay, <a href="http://users.cecs.anu.edu.au/~bdm/papers/posets.pdf">Posets on up to 16 points</a>.
%H G. Brinkmann and B. D. McKay, <a href="http://dx.doi.org/10.1023/A:1016543307592">Posets on up to 16 Points</a>, Order 19 (2) (2002) 147-179 (Table IV).
%H J. I. Brown and S. Watson, <a href="http://dx.doi.org/10.1016/0012-365X(95)00004-G">The number of complements of a topology on n points is at least 2^n (except for some special cases)</a>, Discr. Math., 154 (1996), 27-39.
%H K. K.-H. Butler and G. Markowsky, <a href="http://www.laptop.maine.edu/Enumeration.pdf">Enumeration of finite topologies</a>, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169-184.
%H K. K.-H. Butler and G. Markowsky, <a href="/A000798/a000798_1.pdf">Enumeration of finite topologies</a>, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169-184. [Annotated scan of pages 180 and 183 only]
%H S. D. Chatterji, <a href="/A000798/a000798_10.pdf">The number of topologies on n points</a>, Manuscript, 1966 [Annotated scanned copy]
%H Tyler Clark and Tom Richmond, <a href="http://dx.doi.org/10.2140/involve.2015.8.25">The Number of Convex Topologies on a Finite Totally Ordered Set</a>, 2013, Involve, Vol. 8 (2015), No. 1, 25-32.
%H E. D. Cooper, <a href="/A000798/a000798.pdf">Representation and generation of finite partially ordered sets</a>, Manuscript, no date [Annotated scanned copy]
%H M. Erné, <a href="http://dx.doi.org/10.1007/BF01173716">Struktur- und Anzahlformeln für Topologien auf Endlichen Mengen</a>, Manuscripta Math., 11 (1974), 221-259.
%H M. Erné, <a href="/A006056/a006056.pdf">Struktur- und Anzahlformeln für Topologien auf Endlichen Mengen</a>, Manuscripta Math., 11 (1974), 221-259. (Annotated scanned copy)
%H M. Erné and K. Stege, <a href="/A006870/a006870.pdf">The number of partially ordered (labeled) sets</a>, Preprint, 1989. (Annotated scanned copy)
%H M. Erné and K. Stege, <a href="http://dx.doi.org/10.1007/BF00383446">Counting Finite Posets and Topologies</a>, Order, 8 (1991), 247-265.
%H J. W. Evans, F. Harary and M. S. Lynn, <a href="/A000798/a000798_8.pdf"> On the computer enumeration of finite topologies</a>, Commun. ACM, 10 (1967), 295-297, 313. [Annotated scanned copy]
%H J. W. Evans, F. Harary and M. S. Lynn, <a href="http://dx.doi.org/10.1145/363282.363311">On the computer enumeration of finite topologies</a>, Commun. ACM, 10 (1967), 295-297, 313.
%H S. R. Finch, <a href="/A000798/a000798_12.pdf">Transitive relations, topologies and partial orders</a>, June 5, 2003. [Cached copy, with permission of the author]
%H L. Foissy, C. Malvenuto, and F. Patras, <a href="http://arxiv.org/abs/1403.7488">B_infinity-algebras, their enveloping algebras, and finite spaces</a>, arXiv preprint arXiv:1403.7488 [math.AT], 2014.
%H Loic Foissy, Claudia Malvenuto, and Frederic Patras, <a href="http://dx.doi.org/10.1016/j.jpaa.2015.11.014">Infinitesimal and B_infinity-algebras, finite spaces, and quasi-symmetric functions</a>, Journal of Pure and Applied Algebra, Elsevier, 2016, 220 (6), pp. 2434-2458. <hal-00967351v2>.
%H L. Foissy and C. Malvenuto, <a href="http://arxiv.org/abs/1407.0476">The Hopf algebra of finite topologies and T-partitions</a>, arXiv preprint arXiv:1407.0476 [math.RA], 2014.
%H Joël Gay and Vincent Pilaud, <a href="https://arxiv.org/abs/1804.06572">The weak order on Weyl posets</a>, arXiv:1804.06572 [math.CO], 2018.
%H E. N. Gilbert, <a href="/A000798/a000798_9.pdf">A catalog of partially ordered systems</a>, unpublished memorandum, Aug 08, 1961. [Annotated scanned copy]
%H S. Giraudo, J.-G. Luque, L. Mignot and F. Nicart, <a href="http://arxiv.org/abs/1401.2010">Operads, quasiorders and regular languages</a>, arXiv preprint arXiv:1401.2010 [cs.FL], 2014.
%H D. J. Greenhoe, <a href="https://www.researchgate.net/profile/Daniel_Greenhoe/publication/281831459_Properties_of_distance_spaces_with_power_triangle_inequalities">Properties of distance spaces with power triangle inequalities</a>, ResearchGate, 2015.
%H J. Heitzig and J. Reinhold, <a href="http://dx.doi.org/10.1023/A:1006431609027">The number of unlabeled orders on fourteen elements</a>, Order 17 (2000) no. 4, 333-341.
%H Institut f. Mathematik, Univ. Hanover, <a href="http://www-ifm.math.uni-hannover.de/html/preprints.phtml">Erne/Heitzig/Reinhold papers</a>
%H G. A. Kamel, <a href="http://www.aascit.org/journal/archive2?journalId=928&paperId=2310">Partial Chain Topologies on Finite Sets</a>, Computational and Applied Mathematics Journal. Vol. 1, No. 4, 2015, pp. 174-179.
%H Dongseok Kim, Young Soo Kwon and Jaeun Lee, <a href="http://arxiv.org/abs/1206.0550">Enumerations of finite topologies associated with a finite graph</a>, arXiv preprint arXiv:1206.0550[math.CO], 2012.
%H M. Y. Kizmaz, <a href="http://arxiv.org/abs/1503.08359">On The Number Of Topologies On A Finite Set</a>, arXiv preprint arXiv:1503.08359 [math.NT], 2015-2019.
%H D. A. Klarner, <a href="/A000798/a000798_11.pdf">The number of graded partially ordered sets</a>, J. Combin. Theory, 6 (1969), 12-19. [Annotated scanned copy]
%H D. J. Kleitman and B. L. Rothschild, <a href="http://dx.doi.org/10.1090/S0002-9939-1970-0253944-9">The number of finite topologies</a>, Proc. Amer. Math. Soc., 25 (1970), 276-282.
%H Messaoud Kolli, <a href="http://www.emis.de/journals/JIS/VOL10/Kolli/messaoud30.html">Direct and Elementary Approach to Enumerate Topologies on a Finite Set</a>, J. Integer Sequences, Volume 10, 2007, Article 07.3.1.
%H Messaoud Kolli, <a href="http://dx.doi.org/10.1155/2014/798074">On the cardinality of the T_0-topologies on a finite set</a>, International Journal of Combinatorics, Volume 2014 (2014), Article ID 798074, 7 pages.
%H Sami Lazaar, Houssem Sabri, and Randa Tahri, <a href="https://doi.org/10.1007/s41980-021-00599-3">Structural and Numerical Studies of Some Topological Properties for Alexandroff Spaces</a>, Bull. Iran. Math. Soc. (2021).
%H G. Pfeiffer, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Pfeiffer/pfeiffer6.html">Counting Transitive Relations</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
%H M. Rayburn, <a href="/A000110/a000110_1.pdf">On the Borel fields of a finite set</a>, Proc. Amer. Math.. Soc., 19 (1968), 885-889. [Annotated scanned copy]
%H M. Rayburn and N. J. A. Sloane, <a href="/A000110/a000110.pdf">Correspondence, 1974</a>
%H D. Rusin, <a href="http://www.math.niu.edu/~rusin/known-math/97/finite.top">More info and references</a> [Broken link]
%H D. Rusin, <a href="/A000798/a000798.txt">More info and references</a> [Cached copy]
%H A. Shafaat, <a href="/A000798/a000798_7.pdf">On the number of topologies definable for a finite set</a>, J. Austral. Math. Soc., 8 (1968), 194-198. [Annotated scanned copy]
%H A. Shafaat, <a href="http://dx.doi.org/10.1017/S1446788700005231">On the number of topologies definable for a finite set</a>, J. Austral. Math. Soc., 8 (1968), 194-198.
%H N. J. A. Sloane, <a href="/A000112/a000112_2.pdf">List of sequences related to partial orders, circa 1972</a>
%H N. J. A. Sloane, <a href="/classic.html#POSETS">Classic Sequences</a>
%H Peter Steinbach, <a href="/A000664/a000664_8.pdf">Field Guide to Simple Graphs, Volume 4</a>, Part 8 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
%H Eric Swartz and Nicholas J. Werner, <a href="https://arxiv.org/abs/1709.05390">Zero pattern matrix rings, reachable pairs in digraphs, and Sharp's topological invariant tau</a>, arXiv:1709.05390 [math.CO], 2017.
%H Wietske Visser, Koen V. Hindriks and Catholijn M. Jonker, <a href="http://mmi.tudelft.nl/sites/default/files/visser_hindriks_jonker_2012b.pdf">Goal-based Qualitative Preference Systems</a>, 2012.
%H N. L. White, <a href="/A000798/a000798_6.pdf">Two letters to N. J. A. Sloane, 1970, with hand-drawn enclosure</a>
%H J. A. Wright, <a href="/A000798/a000798_2.pdf">Letter to N. J. A. Sloane, Nov 21 1970, with four enclosures</a>
%H J. A. Wright, <a href="/A000798/a000798_3.pdf">There are 718 6-point topologies, quasiorderings and transgraphs</a>, Preprint, 1970 [Annotated scanned copy]
%H J. A. Wright, <a href="/A000798/a000798_5.pdf">Two related abstracts, 1970 and 1972</a> [Annotated scanned copies]
%H J. A. Wright, <a href="/A000798/a000798_4.pdf">Letter to N. J. A. Sloane, Apr 06 1972, listing 18 sequences</a>
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%F a(n) = Sum_{k=0..n} Stirling2(n, k)*A001035(k).
%F E.g.f.: A(exp(x) - 1) where A(x) is the e.g.f. for A001035. - _Geoffrey Critzer_, Jul 28 2014
%F It is known that log_2(a(n)) ~ n^2/4. - _Tian Vlasic_, Feb 23 2022
%e From _Gus Wiseman_, Aug 01 2019: (Start)
%e The a(3) = 29 topologies are the following (empty sets not shown):
%e {123} {1}{123} {1}{12}{123} {1}{2}{12}{123} {1}{2}{12}{13}{123}
%e {2}{123} {1}{13}{123} {1}{3}{13}{123} {1}{2}{12}{23}{123}
%e {3}{123} {1}{23}{123} {2}{3}{23}{123} {1}{3}{12}{13}{123}
%e {12}{123} {2}{12}{123} {1}{12}{13}{123} {1}{3}{13}{23}{123}
%e {13}{123} {2}{13}{123} {2}{12}{23}{123} {2}{3}{12}{23}{123}
%e {23}{123} {2}{23}{123} {3}{13}{23}{123} {2}{3}{13}{23}{123}
%e {3}{12}{123}
%e {3}{13}{123} {1}{2}{3}{12}{13}{23}{123}
%e {3}{23}{123}
%e (End)
%t 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 *)
%Y Row sums of A326882.
%Y Cf. A001035 (labeled posets), A001930 (unlabeled topologies), A000112 (unlabeled posets), A006057.
%Y Sequences in the Erné (1974) paper: A000798, A001035, A006056, A006057, A001929, A001927, A006058, A006059, A000110.
%Y Cf. A102894, A102895, A102897, A306445, A326866, A326876, A326878, A326881.
%K nonn,nice,core,hard
%O 0,3
%A _N. J. A. Sloane_
%E Two more terms from Jobst Heitzig (heitzig(AT)math.uni-hannover.de), Jul 03 2000
%E a(17)-a(18) are from Brinkmann's and McKay's paper. - _Vladeta Jovovic_, Jun 10 2007
|
