A001200 Number of linear geometries on n (unlabeled) points.

1, 1, 1, 2, 3, 5, 10, 24, 69, 384, 5250, 232929, 28872973

Offset: 0

Author

N. J. A. Sloane, D.Glynn(AT)math.canterbury.ac.nz

Keywords

• nonn
• hard
• more
• nice

Comments

For the labeled case see A056642.

Also a(n) = 1 + number of non-isomorphic simple rank-3 matroids on n elements (see A058731); a(n) = number of non-isomorphic 2-partitions of a set of size n. For 1-partitions see A000041.

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 303, #42.
• CRC Handbook of Combinatorial Designs, 1996, pp. 216, 697.
• J. Doyen, Sur le nombre d'espaces linéaires non isomorphes de n points. Bull. Soc. Math. Belg. 19 1967 421-437.
• P. Robillard, On the weighted finite linear spaces. Bull. Soc. Math. Belg. 22 (1970), 227-241.
• 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

• Mohamed Barakat, Reimer Behrends, Christopher Jefferson, Lukas Kühne, Martin Leuner, On the generation of rank 3 simple matroids with an application to Terao's freeness conjecture, arXiv:1907.01073 [math.CO], 2019.
• A. Betten and D. Betten, Linear spaces with at most 12 points, J. Combinatorial Designs, Volume 7, 1999, pp. 119 - 145.
• J. E. Blackburn, H. H. Crapo, and D. A. Higgs, A catalogue of combinatorial geometries, Math. Comp 27 1973 155-166.
• J. Doyen, Sur le nombre d'espaces linéaires non isomorphes de n points [Annotated and scanned copy]
• D. G. Glynn, Rings of geometries II, J. Combin. Theory, A 49 (1988), 26-66.
• D. G. Glynn, A geometrical isomorphism algorithm, Bull. ICA 7 (1993), 36-38.
• Robert Haas, Cographs, arXiv:1905.12627 [math.GM], 2019.
• G. Heathcote, Linear spaces on 16 points, J. Combin. Designs, Vol. 1, No. 5 (1993), 359-378.
• Nathan Kaplan, Susie Kimport, Rachel Lawrence, Luke Peilen and Max Weinreich, Counting arcs in projective planes via Glynn's algorithm, J. Geom. 108, No. 3, 1013-1029 (2017).
• Ch. Pietsch, On the classification of linear spaces of order 11, J. Comb. Designs, Vol. 3, No. 3 (1995), 185-193.

Crossrefs

Cf. A000041, A056642, A058731, A001548.