

A000798


Number of different quasiorders (or topologies, or transitive digraphs) with n labeled elements.
(Formerly M3631 N1476)


49



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

0,3


COMMENTS

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


REFERENCES

Moussa Benoumhani, The Number of Topologies on a Finite Set, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.6.
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), 2739.
K. K.H. Butler and G. Markowsky, Enumeration of finite topologies, Proc. 4th SE Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169184.
S. D. Chatterji, The number of topologies on n points, Manuscript, 1966.
Tyler Clark and Tom Richmond, The Number of Convex Topologies on a Finite Totally Ordered Set, 2013, to appear in Involve; http://people.wku.edu/tom.richmond/Papers/CountConvexTopsFTOsets.pdf
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 229.
E. D. Cooper, Representation and generation of finite partially ordered sets, Manuscript, no date.
J. W. Evans, F. Harary and M. S. Lynn, On the computer enumeration of finite topologies, Commun. ACM, 10 (1967), 295297, 313.
E. N. Gilbert, A catalog of partially ordered systems, unpublished memorandum, Aug 08, 1961.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.
J. Heitzig and J. Reinhold, The number of unlabeled orders on fourteen elements, Order 17 (2000) no. 4, 333341.
D. J. Kleitman and B. L. Rothschild, The number of finite topologies, Proc. Amer. Math. Soc., 25 (1970), 276282.
Messaoud Kolli, "Direct and Elementary Approach to Enumerate Topologies on a Finite Set", J. Integer Sequences, Volume 10, 2007, Article 07.3.1.
M. Kolli, On the cardinality of the T_0topologies on a finite set, Preprint, 2014.
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. 699712, Congress. Numer., XXIIIXXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561090 (81c:54006)  From N. J. A. Sloane, Jun 05 2012
A. Shafaat, On the number of topologies definable for a finite set, J. Austral. Math. Soc., 8 (1968), 194198.
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.


LINKS

Table of n, a(n) for n=0..18.
Gunnar Brinkmann and Brendan D. McKay, Posets on up to 16 points.
G. Brinkmann, B. D. McKay, Posets on up to 16 Points, Order 19 (2) (2002) 147179 (Table IV).
K. K.H. Butler and G. Markowsky, Enumeration of finite topologies, Proc. 4th SE Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169184
K. K.H. Butler and G. Markowsky, Enumeration of finite topologies, Proc. 4th SE Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169184. [Annotated scan of pages 180 and 183 only]
S. D. Chatterji, The number of topologies on n points, Manuscript, 1966 [Annotated scanned copy]
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), 221259.
M. Erné and K. Stege, Counting Finite Posets and Topologies, Order, 8 (1991), 247265.
J. W. Evans, F. Harary and M. S. Lynn, On the computer enumeration of finite topologies, Commun. ACM, 10 (1967), 295297, 313. [Annotated scanned copy]
S. R. Finch, Transitive relations, topologies and partial orders
L. Foissy, C. Malvenuto, F. Patras, B_infinityalgebras, their enveloping algebras, and finite spaces, arXiv preprint arXiv:1403.7488 [math.AT], 2014.
L. Foissy and C. Malvenuto, The Hopf algebra of finite topologies and Tpartitions, arXiv preprint arXiv:1407.0476 [math.RA], 2014.
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.
Institut f. Mathematik, Univ. Hanover, Erne/Heitzig/Reinhold papers
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.
D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 1219. [Annotated scanned copy]
G. 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), 885889. [Annotated scanned copy]
M. Rayburn and N. J. A. Sloane, Correspondence, 1974
D. Rusin, More info and references
A. Shafaat, On the number of topologies definable for a finite set, J. Austral. Math. Soc., 8 (1968), 194198. [Annotated scanned copy]
N. J. A. Sloane, List of sequences related to partial orders, circa 1972
N. J. A. Sloane, Classic Sequences
Wietske Visser, Koen V. Hindriks and Catholijn M. Jonker, Goalbased Qualitative Preference Systems, 2012.
N. L. White, Two letters to N. J. A. Sloane, 1970, with handdrawn enclosure
J. A. Wright, Letter to N. J. A. Sloane, Nov 21 1970, with four enclosures
J. A. Wright, There are 718 6point 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


FORMULA

Related to A001035 by A000798(n) = Sum 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


CROSSREFS

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.
Sequence in context: A137646 A231498 A168602 * A135485 A210526 A221079
Adjacent sequences: A000795 A000796 A000797 * A000799 A000800 A000801


KEYWORD

nonn,nice,core,hard,changed


AUTHOR

N. J. A. Sloane


EXTENSIONS

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


STATUS

approved



