CINXE.COM
A000798 - OEIS
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 3.2 Final//EN"> <html> <head> <link rel="stylesheet" href="/styles.css"> <meta name="format-detection" content="telephone=no"> <meta http-equiv="content-type" content="text/html; charset=utf-8"> <meta name=viewport content="width=device-width, initial-scale=1"> <meta name="keywords" content="OEIS,integer sequences,Sloane" /> <title>A000798 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A000798" function redir() { var host = document.location.hostname; if(host != "oeis.org" && host != "127.0.0.1" && !/^([0-9.]+)$/.test(host) && host != "localhost" && host != "localhost.localdomain") { document.location = "https"+":"+"//"+"oeis"+".org/" + myURL; } } function sf() { if(document.location.pathname == "/" && document.f) document.f.q.focus(); } </script> </head> <body bgcolor=#ffffff onload="redir();sf()"> <div class=loginbar> <div class=login> <a href="/login?redirect=%2fA000798">login</a> </div> </div> <div class=center><div class=top> <center> <div class=donors> The OEIS is supported by <a href="http://oeisf.org/#DONATE">the many generous donors to the OEIS Foundation</a>. </div> <div class=banner> <a href="/"><img class=banner border="0" width="600" src="/banner2021.jpg" alt="A000798 - OEIS"></a> </div> </center> </div></div> <div class=center><div class=pagebody> <div class=searchbarcenter> <form name=f action="/search" method="GET"> <div class=searchbargreet> <div class=searchbar> <div class=searchq> <input class=searchbox maxLength=1024 name=q value="" title="Search Query"> </div> <div class=searchsubmit> <input type=submit value="Search" name=go> </div> <div class=hints> <span class=hints><a href="/hints.html">Hints</a></span> </div> </div> <div class=searchgreet> (Greetings from <a href="/welcome">The On-Line Encyclopedia of Integer Sequences</a>!) </div> </div> </form> </div> <div class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> A000798 </div> <div class=seqname> Number of different quasi-orders (or topologies, or transitive digraphs) with n labeled elements. <br><font size=-1>(Formerly M3631 N1476)</font> </div> </div> <div class=scorerefs> 84 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>1, 1, 4, 29, 355, 6942, 209527, 9535241, 642779354, 63260289423, 8977053873043, 1816846038736192, 519355571065774021, 207881393656668953041, 115617051977054267807460, 88736269118586244492485121, 93411113411710039565210494095, 134137950093337880672321868725846, 261492535743634374805066126901117203</div> <div class=seqdatalinks> (<a href="/A000798/list">list</a>; <a href="/A000798/graph">graph</a>; <a href="/search?q=A000798+-id:A000798">refs</a>; <a href="/A000798/listen">listen</a>; <a href="/history?seq=A000798">history</a>; <a href="/search?q=id:A000798&fmt=text">text</a>; <a href="/A000798/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>0,3</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>From <a href="/wiki/User:Altug_Alkan">Altug Alkan</a>, Dec 18 2015 and Feb 28 2017: (Start)</div> <div class=sectline>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).</div> <div class=sectline>a(p+n) == <a href="/A265042" title="a(n) = the unique number k such that T(p + n) == k mod p for all primes p, where T(n) = A000798(n) = number of topologies on...">A265042</a>(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) == <a href="/A265042" title="a(n) = the unique number k such that T(p + n) == k mod p for all primes p, where T(n) = A000798(n) = number of topologies on...">A265042</a>(2) (mod 17), that is a(19) == 51 (mod 17). So a(19) is divisible by 17.</div> <div class=sectline>In conclusion, a(19) is a number of the form 323*n - 17. (End)</div> <div class=sectline>The BII-numbers of finite topologies without their empty set are given by <a href="/A326876" title="BII-numbers of finite topologies without their empty set.">A326876</a>. - <a href="/wiki/User:Gus_Wiseman">Gus Wiseman</a>, Aug 01 2019</div> <div class=sectline>From <a href="/wiki/User:Tian_Vlasic">Tian Vlasic</a>, Feb 23 2022: (Start)</div> <div class=sectline>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.</div> <div class=sectline>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).</div> <div class=sectline>Lower and upper bounds are known: 2^n <= a(n) <= 2^(n*(n-1)), n > 1.</div> <div class=sectline>This follows from the fact that there are 2^(n*(n-1)) reflexive relations on a set with n elements.</div> <div class=sectline>Furthermore: a(n+1) <= a(n)*(3a(n)+1). (End)</div> </div> </div> <div class=section> <div class=sectname>REFERENCES</div> <div class=sectbody> <div class=sectline>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.</div> <div class=sectline>S. D. Chatterji, The number of topologies on n points, Manuscript, 1966.</div> <div class=sectline>L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 229.</div> <div class=sectline>E. D. Cooper, Representation and generation of finite partially ordered sets, Manuscript, no date.</div> <div class=sectline>E. N. Gilbert, A catalog of partially ordered systems, unpublished memorandum, Aug 08, 1961.</div> <div class=sectline>F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 243.</div> <div class=sectline>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)</div> <div class=sectline>N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).</div> <div class=sectline>N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).</div> <div class=sectline>For further references concerning the enumeration of topologies and posets see under <a href="/A001035" title="Number of partially ordered sets ("posets") with n labeled elements (or labeled acyclic transitive digraphs).">A001035</a>.</div> <div class=sectline>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.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline><a href="/A000798/b000798.txt">Table of n, a(n) for n=0..18.</a></div> <div class=sectline>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.</div> <div class=sectline>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.</div> <div class=sectline>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.</div> <div class=sectline>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.</div> <div class=sectline>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.</div> <div class=sectline>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>.</div> <div class=sectline>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).</div> <div class=sectline>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.</div> <div class=sectline>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.</div> <div class=sectline>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]</div> <div class=sectline>S. D. Chatterji, <a href="/A000798/a000798_10.pdf">The number of topologies on n points</a>, Manuscript, 1966 [Annotated scanned copy]</div> <div class=sectline>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.</div> <div class=sectline>E. D. Cooper, <a href="/A000798/a000798.pdf">Representation and generation of finite partially ordered sets</a>, Manuscript, no date [Annotated scanned copy]</div> <div class=sectline>M. Ern茅, <a href="http://dx.doi.org/10.1007/BF01173716">Struktur- und Anzahlformeln f眉r Topologien auf Endlichen Mengen</a>, Manuscripta Math., 11 (1974), 221-259.</div> <div class=sectline>M. Ern茅, <a href="/A006056/a006056.pdf">Struktur- und Anzahlformeln f眉r Topologien auf Endlichen Mengen</a>, Manuscripta Math., 11 (1974), 221-259. (Annotated scanned copy)</div> <div class=sectline>M. Ern茅 and K. Stege, <a href="/A006870/a006870.pdf">The number of partially ordered (labeled) sets</a>, Preprint, 1989. (Annotated scanned copy)</div> <div class=sectline>M. Ern茅 and K. Stege, <a href="http://dx.doi.org/10.1007/BF00383446">Counting Finite Posets and Topologies</a>, Order, 8 (1991), 247-265.</div> <div class=sectline>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]</div> <div class=sectline>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.</div> <div class=sectline>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]</div> <div class=sectline>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.</div> <div class=sectline>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>.</div> <div class=sectline>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.</div> <div class=sectline>Jo毛l 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.</div> <div class=sectline>E. N. Gilbert, <a href="/A000798/a000798_9.pdf">A catalog of partially ordered systems</a>, unpublished memorandum, Aug 08, 1961. [Annotated scanned copy]</div> <div class=sectline>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.</div> <div class=sectline>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.</div> <div class=sectline>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.</div> <div class=sectline>Institut f. Mathematik, Univ. Hanover, <a href="http://www-ifm.math.uni-hannover.de/html/preprints.phtml">Erne/Heitzig/Reinhold papers</a></div> <div class=sectline>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.</div> <div class=sectline>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.</div> <div class=sectline>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.</div> <div class=sectline>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]</div> <div class=sectline>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.</div> <div class=sectline>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.</div> <div class=sectline>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.</div> <div class=sectline>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).</div> <div class=sectline>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.</div> <div class=sectline>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]</div> <div class=sectline>M. Rayburn and N. J. A. Sloane, <a href="/A000110/a000110.pdf">Correspondence, 1974</a></div> <div class=sectline>D. Rusin, <a href="http://www.math.niu.edu/~rusin/known-math/97/finite.top">More info and references</a> [Broken link]</div> <div class=sectline>D. Rusin, <a href="/A000798/a000798.txt">More info and references</a> [Cached copy]</div> <div class=sectline>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]</div> <div class=sectline>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.</div> <div class=sectline>N. J. A. Sloane, <a href="/A000112/a000112_2.pdf">List of sequences related to partial orders, circa 1972</a></div> <div class=sectline>N. J. A. Sloane, <a href="/classic.html#POSETS">Classic Sequences</a></div> <div class=sectline>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 <a href="/A000088" title="Number of simple graphs on n unlabeled nodes.">A000088</a>, <a href="/A008406" title="Triangle T(n,k) read by rows, giving number of graphs with n nodes (n >= 1) and k edges (0 <= k <= n(n-1)/2).">A008406</a>, <a href="/A000055" title="Number of trees with n unlabeled nodes.">A000055</a>, <a href="/A000664" title="Number of graphs with n edges.">A000664</a>, respectively.)</div> <div class=sectline>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.</div> <div class=sectline>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.</div> <div class=sectline>N. L. White, <a href="/A000798/a000798_6.pdf">Two letters to N. J. A. Sloane, 1970, with hand-drawn enclosure</a></div> <div class=sectline>J. A. Wright, <a href="/A000798/a000798_2.pdf">Letter to N. J. A. Sloane, Nov 21 1970, with four enclosures</a></div> <div class=sectline>J. A. Wright, <a href="/A000798/a000798_3.pdf">There are 718 6-point topologies, quasiorderings and transgraphs</a>, Preprint, 1970 [Annotated scanned copy]</div> <div class=sectline>J. A. Wright, <a href="/A000798/a000798_5.pdf">Two related abstracts, 1970 and 1972</a> [Annotated scanned copies]</div> <div class=sectline>J. A. Wright, <a href="/A000798/a000798_4.pdf">Letter to N. J. A. Sloane, Apr 06 1972, listing 18 sequences</a></div> <div class=sectline><a href="/index/Cor#core">Index entries for "core" sequences</a></div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>a(n) = Sum_{k=0..n} Stirling2(n, k)*<a href="/A001035" title="Number of partially ordered sets ("posets") with n labeled elements (or labeled acyclic transitive digraphs).">A001035</a>(k).</div> <div class=sectline>E.g.f.: A(exp(x) - 1) where A(x) is the e.g.f. for <a href="/A001035" title="Number of partially ordered sets ("posets") with n labeled elements (or labeled acyclic transitive digraphs).">A001035</a>. - <a href="/wiki/User:Geoffrey_Critzer">Geoffrey Critzer</a>, Jul 28 2014</div> <div class=sectline>It is known that log_2(a(n)) ~ n^2/4. - <a href="/wiki/User:Tian_Vlasic">Tian Vlasic</a>, Feb 23 2022</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>From <a href="/wiki/User:Gus_Wiseman">Gus Wiseman</a>, Aug 01 2019: (Start)</div> <div class=sectline>The a(3) = 29 topologies are the following (empty sets not shown):</div> <div class=sectline> {123} {1}{123} {1}{12}{123} {1}{2}{12}{123} {1}{2}{12}{13}{123}</div> <div class=sectline> {2}{123} {1}{13}{123} {1}{3}{13}{123} {1}{2}{12}{23}{123}</div> <div class=sectline> {3}{123} {1}{23}{123} {2}{3}{23}{123} {1}{3}{12}{13}{123}</div> <div class=sectline> {12}{123} {2}{12}{123} {1}{12}{13}{123} {1}{3}{13}{23}{123}</div> <div class=sectline> {13}{123} {2}{13}{123} {2}{12}{23}{123} {2}{3}{12}{23}{123}</div> <div class=sectline> {23}{123} {2}{23}{123} {3}{13}{23}{123} {2}{3}{13}{23}{123}</div> <div class=sectline> {3}{12}{123}</div> <div class=sectline> {3}{13}{123} {1}{2}{3}{12}{13}{23}{123}</div> <div class=sectline> {3}{23}{123}</div> <div class=sectline>(End)</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>Table[Length[Select[Subsets[Subsets[Range[n], {1, n}]], Union@@#==Range[n]&&SubsetQ[#, Union[Union@@@Tuples[#, 2], DeleteCases[Intersection@@@Tuples[#, 2], {}]]]&]], {n, 0, 3}] (* <a href="/wiki/User:Gus_Wiseman">Gus Wiseman</a>, Aug 01 2019 *)</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Row sums of <a href="/A326882" title="Irregular triangle read by rows where T(n,k) is the number of finite topologies with n points and k nonempty open sets, 0 <=...">A326882</a>.</div> <div class=sectline>Cf. <a href="/A001035" title="Number of partially ordered sets ("posets") with n labeled elements (or labeled acyclic transitive digraphs).">A001035</a> (labeled posets), <a href="/A001930" title="Number of topologies, or transitive digraphs with n unlabeled nodes.">A001930</a> (unlabeled topologies), <a href="/A000112" title="Number of partially ordered sets ("posets") with n unlabeled elements.">A000112</a> (unlabeled posets), <a href="/A006057" title="Number of topologies on n labeled points satisfying axioms T_0-T_4.">A006057</a>.</div> <div class=sectline>Sequences in the Ern茅 (1974) paper: <a href="/A000798" title="Number of different quasi-orders (or topologies, or transitive digraphs) with n labeled elements.">A000798</a>, <a href="/A001035" title="Number of partially ordered sets ("posets") with n labeled elements (or labeled acyclic transitive digraphs).">A001035</a>, <a href="/A006056" title="Number of topologies on n labeled points satisfying the T_4 axiom.">A006056</a>, <a href="/A006057" title="Number of topologies on n labeled points satisfying axioms T_0-T_4.">A006057</a>, <a href="/A001929" title="Number of connected topologies on n labeled points.">A001929</a>, <a href="/A001927" title="Number of connected partially ordered sets with n labeled points.">A001927</a>, <a href="/A006058" title="Number of connected labeled T_4-topologies with n points.">A006058</a>, <a href="/A006059" title="Number of connected labeled T_0-T_4-topologies with n points.">A006059</a>, <a href="/A000110" title="Bell or exponential numbers: number of ways to partition a set of n labeled elements.">A000110</a>.</div> <div class=sectline>Cf. <a href="/A102894" title="Number of ACI algebras or semilattices on n generators, with no identity or annihilator.">A102894</a>, <a href="/A102895" title="Number of ACI algebras or semilattices on n generators with no identity element.">A102895</a>, <a href="/A102897" title="Number of ACI algebras (or semilattices) on n generators.">A102897</a>, <a href="/A306445" title="Number of collections of subsets of {1, 2, ..., n} that are closed under union and intersection.">A306445</a>, <a href="/A326866" title="Number of connectedness systems on n vertices.">A326866</a>, <a href="/A326876" title="BII-numbers of finite topologies without their empty set.">A326876</a>, <a href="/A326878" title="Number of topologies whose points are a subset of {1..n}.">A326878</a>, <a href="/A326881" title="Number of set-systems with {} that are closed under intersection and cover n vertices.">A326881</a>.</div> <div class=sectline>Sequence in context: <a href="/A231498" title="Dimensions of algebraic generators of Hopf algebra ASM.">A231498</a> <a href="/A168602" title="G.f. satisfies: A(x) = 1 + x*A(x)^2*A(2x).">A168602</a> <a href="/A368452" title="Expansion of e.g.f. exp(-x) / (1 + log(1 - 3*x)/3).">A368452</a> * <a href="/A135485" title="a(n) = Sum_{i=1..n} prime(i)^(i-1), where prime(i) denotes i-th prime number.">A135485</a> <a href="/A210526" title="E.g.f.: Sum_{n>=0} tan(n*x)^n / n!.">A210526</a> <a href="/A221079" title="E.g.f.: Sum_{n>=0} arctanh(n*x)^n/n!.">A221079</a></div> <div class=sectline>Adjacent sequences: <a href="/A000795" title="Sali茅 numbers: expansion of cosh x / cos x = Sum_{n >= 0} a(n)*x^(2n)/(2n)!.">A000795</a> <a href="/A000796" title="Decimal expansion of Pi (or digits of Pi).">A000796</a> <a href="/A000797" title="Numbers that are not the sum of 4 tetrahedral numbers.">A000797</a> * <a href="/A000799" title="a(n) = floor(2^n / n).">A000799</a> <a href="/A000800" title="Sum of upward diagonals of Eulerian triangle.">A000800</a> <a href="/A000801" title="a(n) = Sum_{k = 1..n} floor(2^k / k).">A000801</a></div> </div> </div> <div class=section> <div class=sectname>KEYWORD</div> <div class=sectbody> <div class=sectline><span title="a sequence of nonnegative numbers">nonn</span>,<span title="an exceptionally nice sequence">nice</span>,<span title="an important sequence">core</span>,<span title="next term not known, may be hard to find. please extend this sequence">hard</span></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:N._J._A._Sloane">N. J. A. Sloane</a></div> </div> </div> <div class=section> <div class=sectname>EXTENSIONS</div> <div class=sectbody> <div class=sectline>Two more terms from Jobst Heitzig (heitzig(AT)math.uni-hannover.de), Jul 03 2000</div> <div class=sectline>a(17)-a(18) are from Brinkmann's and McKay's paper. - <a href="/wiki/User:Vladeta_Jovovic">Vladeta Jovovic</a>, Jun 10 2007</div> </div> </div> <div class=section> <div class=sectname>STATUS</div> <div class=sectbody> <div class=sectline>approved</div> </div> </div> </div> <div class=space10></div> </div> </div></div> <p> <div class=footerpad></div> <div class=footer> <center> <div class=bottom> <div class=linksbar> <a href="/">Lookup</a> <a href="/wiki/Welcome"><font color="red">Welcome</font></a> <a href="/wiki/Main_Page"><font color="red">Wiki</font></a> <a href="/wiki/Special:RequestAccount">Register</a> <a href="/play.html">Music</a> <a href="/plot2.html">Plot 2</a> <a href="/demo1.html">Demos</a> <a href="/wiki/Index_to_OEIS">Index</a> <a href="/webcam">WebCam</a> <a href="/Submit.html">Contribute</a> <a href="/eishelp2.html">Format</a> <a href="/wiki/Style_Sheet">Style Sheet</a> <a href="/transforms.html">Transforms</a> <a href="/ol.html">Superseeker</a> <a href="/recent">Recents</a> </div> <div class=linksbar> <a href="/community.html">The OEIS Community</a> </div> <div class=linksbar> Maintained by <a href="http://oeisf.org">The OEIS Foundation Inc.</a> </div> <div class=dbinfo>Last modified February 17 19:52 EST 2025. Contains 380975 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>