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%I M2817 N1133 #95 Oct 01 2019 07:17:22
%S 1,1,3,9,33,139,718,4535,35979,363083,4717687,79501654,1744252509,
%T 49872339897,1856792610995,89847422244493,5637294117525695
%N Number of topologies, or transitive digraphs with n unlabeled nodes.
%D 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. <hal-00967351v2>.
%D F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 218 (but the last entry is wrong).
%D M. Kolli, On the cardinality of the T_0-topologies on a finite set, Preprint, 2014.
%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 J. A. Wright, There are 718 6-point topologies, quasi-orderings and transgraphs, Notices Amer. Math. Soc., 17 (1970), p. 646, Abstract #70T-A106.
%D J. A. Wright, personal communication.
%D For further references concerning the enumeration of topologies and posets see under A000112 and A001035.
%H C. M. Bender et al., <a href="http://arxiv.org/abs/quant-ph/0604164">Combinatorics and field theory</a>, arXiv:quant-ph/0604164, 2006.
%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 M. Benoumhani, 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 Gunnar Brinkmann and Brendan D. McKay, <a href="http://users.cecs.anu.edu.au/~bdm/papers/topologies.pdf">Counting unlabeled topologies and transitive relations</a>.
%H G. Brinkmann and B. D. McKay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/McKay/mckay170.html">Counting unlabeled topologies and transitive relations</a>, J. Integer Sequences, Volume 8, 2005.
%H Gunnar Brinkmann and Brendan D. McKay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/McKay/mckay170.html">Counting Unlabelled Topologies and Transitive Relations</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.1.
%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 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H S. R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/">Transitive relations, topologies and partial orders</a>
%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, 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, 2014
%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, 2012. - From _N. J. A. Sloane_, Nov 09 2012
%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 G. Pfeiffer, <a href="http://schmidt.nuigalway.ie/~goetz/pub/posetseq.html">Counting Transitive Relations</a>, preprint, 2004.
%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 D. Rusin, <a href="http://www.math.niu.edu/~rusin/known-math/97/finite.top">Further information and references</a> [Broken link]
%H D. Rusin, <a href="/A000112/a000112.top.txt">Further information and references</a> [Cached copy]
%H Henry Sharp, Jr., <a href="/A001930/a001930_1.pdf">Quasi-orderings and topologies on finite sets</a>, Proceedings of the American Mathematical Society 17.6 (1966): 1344-1349. [Annotated scanned copy]
%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#LOSS">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, 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 J. M. Tangen and N. J. A. Sloane, <a href="/A000666/a000666.pdf">Correspondence, 1976-1976</a>
%H R. H. Warren, <a href="/A001930/a001930.pdf">The number of topologies</a>, Houston J. Math., 8 (No. 2, 1982), 297-301. Mentions a(4)=33. [Annotated scanned copy]
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DigraphTopology.html">Digraph Topology.</a>
%H R. H. Warren, <a href="/A001930/a001930.pdf">The number of topologies</a>, Houston J. Math., 8 (No. 2, 1982), 297-301. Mentions a(4)=33. [Annotated scanned copy]
%H Wikipedia <a href="https://en.wikipedia.org/wiki/Topological_space">Topological space</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_4.pdf">Letter to N. J. A. Sloane, Apr 06 1972, listing 18 sequences</a>
%e From _Gus Wiseman_, Aug 02 2019: (Start)
%e Non-isomorphic representatives of the a(0) = 1 through a(3) = 9 topologies:
%e {} {}{1} {}{12} {}{123}
%e {}{2}{12} {}{3}{123}
%e {}{1}{2}{12} {}{23}{123}
%e {}{1}{23}{123}
%e {}{3}{23}{123}
%e {}{2}{3}{23}{123}
%e {}{3}{13}{23}{123}
%e {}{2}{3}{13}{23}{123}
%e {}{1}{2}{3}{12}{13}{23}{123}
%e (End)
%Y Cf. A000798 (labeled topologies), A001035 (labeled posets), A001930 (unlabeled topologies), A000112 (unlabeled posets), A006057, A001928, A001929.
%Y The case with unions only is A108798.
%Y The case with intersections only is (also) A108798.
%Y Partial sums are A326898 (the non-covering case).
%Y Cf. A000612, A003180, A108800, A193674, A306445, A326876, A326878, A326882.
%K nonn,hard,more,nice
%O 0,3
%A _N. J. A. Sloane_
%E a(8)-a(12) from Goetz Pfeiffer (goetz.pfeiffer(AT)nuigalway.ie), Jan 21 2004
%E a(13)-a(16) from Brinkmann's and McKay's paper, sent by _Vladeta Jovovic_, Jan 04 2006
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