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A001930
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Number of topologies, or transitive digraphs with n unlabeled nodes.
(Formerly M2817 N1133)
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33
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1, 1, 3, 9, 33, 139, 718, 4535, 35979, 363083, 4717687, 79501654, 1744252509, 49872339897, 1856792610995, 89847422244493, 5637294117525695
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,3
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REFERENCES
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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>.
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.
For further references concerning the enumeration of topologies and posets see under A000112 and A001035.
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LINKS
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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. [Annotated scan of pages 180 and 183 only]
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.)
R. H. Warren, The number of topologies, Houston J. Math., 8 (No. 2, 1982), 297-301. Mentions a(4)=33. [Annotated scanned copy]
R. H. Warren, The number of topologies, Houston J. Math., 8 (No. 2, 1982), 297-301. Mentions a(4)=33. [Annotated scanned copy]
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EXAMPLE
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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)
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CROSSREFS
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The case with unions only is A108798.
The case with intersections only is (also) A108798.
Partial sums are A326898 (the non-covering case).
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KEYWORD
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nonn,hard,more,nice
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AUTHOR
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EXTENSIONS
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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
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STATUS
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approved
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