OFFSET
0,3
REFERENCES
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.
LINKS
Ibtsam A. R. Alroily and Brahim Chaourar, A super-multiplicative inequality for the number of finite unlabeled arbitrary and T_0 topologies, arXiv:2511.17263 [math.CO], 2025. See p. 2.
Carl M. Bender, Dorje C. Brody, and Bernhard K. Meister, Combinatorics and field theory, arXiv:quant-ph/0604164, 2006.
Moussa Benoumhani, The Number of Topologies on a Finite Set, J. Int. Seq. 9 (2006), Art. 06.2.6.
Moussa Benoumhani and Messaoud Kolli, Finite topologies and partitions, J. Int. Seq. 13 (2010), Art. 10.3.5
Gunnar Brinkmann and Brendan D. McKay, Counting unlabeled topologies and transitive relations, J. Int. Seq. 8 (2005), Art. 05.2.1. See also Austral. Nat'l Univ.
Kim Ki-Hang Butler and George Markowsky, Enumeration of finite topologies, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. (1973) Vol. 8, 169-184. See also Annotated scan of pages 180 and 183 only.
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Int. Seq. 3 (2000), Art. 00.1.5.
Sangmin Chun, Gangyong Lee, Mauricio Medina-Barcenas, and Cosmin S. Roman, Rudimentary Structural Matrix Rings, J. Alg. Appl. (2026). See p. 13.
Steven R. Finch, Transitive relations, topologies and partial orders, 2003.
Steven R. Finch, Transitive relations, topologies and partial orders, June 5, 2003. [Cached copy, with permission of the author]
Loïc Foissy, Claudia Malvenuto, and Frédéric Patras, B_infinity-algebras, their enveloping algebras, and finite spaces, arXiv preprint arXiv:1403.7488 [math.AT], 2014-2015.
Loic Foissy, Claudia Malvenuto, and Frédéric 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. <hal-00967351v2>.
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:1206.0550 [math.CO], 2012. - From N. J. A. Sloane, Nov 09 2012
Messaoud Kolli, Direct and Elementary Approach to Enumerate Topologies on a Finite Set, J. Int. Seq. 10 (2007), Art. 07.3.1.
Götz Pfeiffer, Counting Transitive Relations, J. Int. Seq. 7 (2004), Art. 04.3.2.
D. Rusin, Further information and references.
D. Rusin, Further information and references [Cached copy]
Henry Sharp, Jr., Quasi-orderings and topologies on finite sets, Proc. Amer. Math. Soc. 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(2) (1982), 297-301. Mentions a(4)=33. [Annotated scanned copy]
Eric Weisstein's World of Mathematics, Digraph Topology.
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.
EXAMPLE
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)
CROSSREFS
KEYWORD
nonn,hard,more,nice
AUTHOR
EXTENSIONS
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
STATUS
approved