A002773 Number of nonisomorphic simple matroids (or geometries) with n points.

1, 1, 1, 2, 4, 9, 26, 101, 950, 376467

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• more

References

• Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 138.
• Knuth, Donald E. "The asymptotic number of geometries." Journal of Combinatorial Theory, Series A 16.3 (1974): 398-400.
• 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).

Links

• N. Bansal, R. Pendavingh, and J. G. van der Pol, On the number of matroids, arXiv:1206.6270v1 [math.CO], 2012.
• Nikhil Bansal, Rudi A. Pendavingh, and Jorn G. van der Pol, On the number of matroids, Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms. SIAM, 2013; full version in Combinatorica, 35:3 (2015), 253-277.
• J. E. Blackburn, H. H. Crapo, and D. A. Higgs, A catalogue of combinatorial geometries, Math. Comp 27 (1973), 155-166.
• Henry H. Crapo and Gian-Carlo Rota, On the foundations of combinatorial theory. II. Combinatorial geometries, Studies in Appl. Math. 49 (1970), 109-133.
• Henry H. Crapo and Gian-Carlo Rota, On the foundations of combinatorial theory. II. Combinatorial geometries, Studies in Appl. Math. 49 (1970), 109-133. [Annotated scanned copy of pages 126 and 127 only]
• W. M. B. Dukes, Tables of matroids.
• W. M. B. Dukes, Counting and Probability in Matroid Theory, Ph.D. Thesis, Trinity College, Dublin, 2000.
• W. M. B. Dukes, The number of matroids on a finite set, arXiv:math/0411557 [math.CO], 2004.
• W. M. B. Dukes, On the number of matroids on a finite set, Séminaire Lotharingien de Combinatoire 51 (2004), Article B51g.
• Dillon Mayhew and Gordon F. Royle, Matroids with nine elements, arXiv:math/0702316 [math.CO], 2007.
• Dillon Mayhew and Gordon F. Royle, Matroids with nine elements, J. Combin. Theory Ser. B 98(2) (2008), 415-431.
• M. J. Piff, An upper bound for the number of matroids, J. Combinatorial Theory Ser. B, vol 14 (1973), pp. 241-245.
• Gordon Royle and Dillon Mayhew, 9-element matroids.
• N. J. A. Sloane, Initial terms (* denotes 5 points in general position in 4-space).
• Eric Weisstein's World of Mathematics, Matroid.
• Index entries for sequences related to matroids

Crossrefs

Cf. A055545, A056642. Row sums of A058730.

Formula

Limit_{ n -> oo } (log_2 log_2 a(n))/n = 1. [Knuth]

2^n/n^(3/2) << log a(n) << 2^n/n, proved by Knuth and Piff respectively. - Charles R Greathouse IV, Mar 20 2021

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

Extensions

a(9) from Gordon Royle, Dec 23 2006