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%I M1197 N0462 #83 Jan 13 2022 01:31:07
%S 1,1,1,2,4,9,26,101,950,376467
%N Number of nonisomorphic simple matroids (or geometries) with n points.
%D Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 138.
%D Knuth, Donald E. "The asymptotic number of geometries." Journal of Combinatorial Theory, Series A 16.3 (1974): 398-400.
%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).
%H N. Bansal, R. Pendavingh, and J. G. van der Pol, <a href="http://arxiv.org/abs/1206.6270">On the number of matroids</a>, arXiv:1206.6270v1 [math.CO], 2012.
%H Nikhil Bansal, Rudi A. Pendavingh, and Jorn G. van der Pol, <a href="https://doi.org/10.1007/s00493-014-3029-z">On the number of matroids</a>, Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms. SIAM, 2013; full version in Combinatorica, 35:3 (2015), 253-277.
%H J. E. Blackburn, H. H. Crapo, and D. A. Higgs, <a href="https://doi.org/10.1090/S0025-5718-1973-0419270-0">A catalogue of combinatorial geometries</a>, Math. Comp 27 (1973), 155-166.
%H Henry H. Crapo and Gian-Carlo Rota, <a href="https://doi.org/10.1002/sapm1970492109">On the foundations of combinatorial theory. II. Combinatorial geometries</a>, Studies in Appl. Math. 49 (1970), 109-133.
%H Henry H. Crapo and Gian-Carlo Rota, <a href="/A002773/a002773.pdf">On the foundations of combinatorial theory. II. Combinatorial geometries</a>, Studies in Appl. Math. 49 (1970), 109-133. [Annotated scanned copy of pages 126 and 127 only]
%H W. M. B. Dukes, <a href="http://www.stp.dias.ie/~dukes/matroid.html">Tables of matroids</a>.
%H W. M. B. Dukes, <a href="https://web.archive.org/web/20030208144026/http://www.stp.dias.ie/~dukes/phd.html">Counting and Probability in Matroid Theory</a>, Ph.D. Thesis, Trinity College, Dublin, 2000.
%H W. M. B. Dukes, <a href="https://arxiv.org/abs/math/0411557">The number of matroids on a finite set</a>, arXiv:math/0411557 [math.CO], 2004.
%H W. M. B. Dukes, <a href="http://emis.impa.br/EMIS/journals/SLC/wpapers/s51dukes.html">On the number of matroids on a finite set</a>, Séminaire Lotharingien de Combinatoire 51 (2004), Article B51g.
%H Dillon Mayhew and Gordon F. Royle, <a href="https://arxiv.org/abs/math/0702316">Matroids with nine elements</a>, arXiv:math/0702316 [math.CO], 2007.
%H Dillon Mayhew and Gordon F. Royle, <a href="https://doi.org/10.1016/j.jctb.2007.07.005">Matroids with nine elements</a>, J. Combin. Theory Ser. B 98(2) (2008), 415-431.
%H M. J. Piff, <a href="https://doi.org/10.1016/0095-8956(73)90006-3">An upper bound for the number of matroids</a>, J. Combinatorial Theory Ser. B, vol 14 (1973), pp. 241-245.
%H Gordon Royle and Dillon Mayhew, <a href="https://web.archive.org/web/20080828102733/http://people.csse.uwa.edu.au/gordon/matroid-integer-sequences.html">9-element matroids</a>.
%H N. J. A. Sloane, <a href="/A002773/a002773.gif">Initial terms (* denotes 5 points in general position in 4-space)</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Matroid.html">Matroid</a>.
%H <a href="/index/Mat#matroid">Index entries for sequences related to matroids</a>
%F Limit_{ n -> oo } (log_2 log_2 a(n))/n = 1. [Knuth]
%F 2^n/n^(3/2) << log a(n) << 2^n/n, proved by Knuth and Piff respectively. - _Charles R Greathouse IV_, Mar 20 2021
%F Bansal, Pendavingh, & van der Pol prove an upper bound almost matching the lower bound above: log a(n) <= 2*sqrt(2/Pi)*2^n/n^(3/2)*(1 + o(1)). - _Charles R Greathouse IV_, Mar 20 2021
%Y Cf. A055545, A056642. Row sums of A058730.
%K nonn,nice,more
%O 0,4
%A _N. J. A. Sloane_
%E a(9) from _Gordon Royle_, Dec 23 2006
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