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%I M0726 N0271 #45 Aug 29 2022 10:28:24
%S 1,1,1,2,3,5,10,24,69,384,5250,232929,28872973
%N Number of linear geometries on n (unlabeled) points.
%C For the labeled case see A056642.
%C 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.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 303, #42.
%D CRC Handbook of Combinatorial Designs, 1996, pp. 216, 697.
%D J. Doyen, Sur le nombre d'espaces linéaires non isomorphes de n points. Bull. Soc. Math. Belg. 19 1967 421-437.
%D P. Robillard, On the weighted finite linear spaces. Bull. Soc. Math. Belg. 22 (1970), 227-241.
%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 Mohamed Barakat, Reimer Behrends, Christopher Jefferson, Lukas Kühne, Martin Leuner, <a href="https://arxiv.org/abs/1907.01073">On the generation of rank 3 simple matroids with an application to Terao's freeness conjecture</a>, arXiv:1907.01073 [math.CO], 2019.
%H A. Betten and D. Betten, <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.86.1899">Linear spaces with at most 12 points</a>, J. Combinatorial Designs, Volume 7, 1999, pp. 119 - 145.
%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 J. Doyen, <a href="/A001200/a001200.pdf">Sur le nombre d'espaces linéaires non isomorphes de n points</a> [Annotated and scanned copy]
%H D. G. Glynn, <a href="https://doi.org/10.1016/0097-3165(88)90027-1">Rings of geometries II</a>, J. Combin. Theory, A 49 (1988), 26-66.
%H D. G. Glynn, <a href="https://www.researchgate.net/publication/230899821">A geometrical isomorphism algorithm</a>, Bull. ICA 7 (1993), 36-38.
%H Robert Haas, <a href="https://arxiv.org/abs/1905.12627">Cographs</a>, arXiv:1905.12627 [math.GM], 2019.
%H G. Heathcote, <a href="https://doi.org/10.1002/jcd.3180010505">Linear spaces on 16 points</a>, J. Combin. Designs, Vol. 1, No. 5 (1993), 359-378.
%H Kaplan, Nathan; Kimport, Susie; Lawrence, Rachel; Peilen, Luke; Weinreich, Max <a href="https://doi.org/10.1007/s00022-017-0391-1">Counting arcs in projective planes via Glynn’s algorithm.</a> J. Geom. 108, No. 3, 1013-1029 (2017).
%H Ch. Pietsch, <a href="https://doi.org/10.1002/jcd.3180030305">On the classification of linear spaces of order 11</a>, J. Comb. Designs, Vol. 3, No. 3 (1995), 185-193.
%Y Cf. A000041, A056642, A058731, A001548.
%K nonn,hard,more,nice
%O 0,4
%A _N. J. A. Sloane_, D.Glynn(AT)math.canterbury.ac.nz
%E Dead link to Betten paper replaced with CiteSeer entry (which has full text as cached PDF) _Robert Munafo_, Dec 13 2009
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