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%I #71 Dec 06 2023 14:09:15
%S 1,1,2,4,9,47,791,37830
%N Number of ways of arranging n lines in the (affine) plane.
%C This is in the affine plane, rather than the projective plane, so lines are either parallel or meet in one point.
%C Two arrangements are considered the same if one can be continuously changed to the other while keeping all lines straight, without changing the multiplicity of intersection points, and without a line passing through an intersection point. Turning over is also allowed.
%C a(n) might be called the size of the moduli space of n lines in the affine plane.
%C The subsequence giving the number of arrangements G_n of n lines in "general position" (with every two lines meeting in one point and every intersection point lying on exactly two lines) is given by A090338.
%C The moduli space of n points in the affine plane has been studied by several people (see for example Haiman and Miller, 2004; Martin, 2003). There is no direct connection with this problem, but these references are included for background information. - _N. J. A. Sloane_, Sep 13 2014
%C Lukas Finschi points out (email, Sep 19 2014) that a(n) = A063859(n)+1 for n <= 7 (but not for larger n). - _N. J. A. Sloane_, Sep 20 2014
%D B. Grünbaum, Arrangements and Spreads. American Mathematical Society, Providence, RI, 1972, p. 4.
%H Lukas Finschi, <a href="https://finschi.com/math/om/">Homepage of Oriented Matroids</a>
%H L. Finschi and K. Fukuda, <a href="http://www.cccg.ca/proceedings/2001/finschi-1053.ps.gz">Complete combinatorial generation of small point set configurations and hyperplane arrangements</a>, pp. 97-100 in Abstracts 13th Canadian Conference on Computational Geometry (CCCG '01), Waterloo, Aug. 13-15, 2001.
%H Komei Fukuda, Hiroyuki Miyata, and Sonoko Moriyama <a href="http://arxiv.org/abs/1204.0645">Complete Enumeration of Small Realizable Oriented Matroids</a>, arXiv:1204.0645 [math.CO], 2012; Discrete Comput. Geom. 49 (2013), no. 2, 359--381. MR3017917. (Further background information.)
%H Mark Haiman, with an Appendix by Ezra Miller, <a href="http://math.berkeley.edu/~mhaiman/ftp/msri-talks-2002/msri-comm-alg.pdf">Commutative algebra of n points in the plane</a>. Trends Commut. Algebra, MSRI Publ 51 (2004): 153-180. (Background)
%H J. L. Martin, <a href="http://www.math.umn.edu/math/slopes.pdf">The slopes determined by n points in the plane</a>. (Background)
%H Jeremy L. Martin, <a href="https://arxiv.org/abs/math/0302106">The slopes determined by n points in the plane</a>, arXiv:math/0302106 [math.AG], 2003-2006; Duke Math. J. 131 (2006), no. 1, 119-165. (Background)
%H N. J. A. Sloane, <a href="/A241600/a241600.pdf">Illustration of a(1)-a(5)</a>
%H N. J. A. Sloane, <a href="https://www.youtube.com/watch?v=9ogbsh8KuEM">Exciting Number Sequences</a> (video of talk), Mar 05 2021.
%H N. J. A. Sloane, <a href="/A241600/a241600_1.pdf">The On-Line Encyclopedia of Integer Sequences</a> (2015 talk slides)
%F a(n) >= A000041(n). - _Pablo Hueso Merino_, May 10 2021
%e Let P_n = n parallel lines, S_n = star of n lines through a point, G_n = n lines in general position, L = P_1 = S_1 = G_1 = a single line.
%e a(1) = 1: L.
%e a(2) = 2: P_2, S_2.
%e a(3) = 4: P_3, P_2 L, S_3, G_3.
%e See link for illustrations of first 5 terms.
%Y Cf. A090338 (lines in general position), A090339 (curved lines in general position), A250001 (circles).
%Y Cf. also A063859, A003036, A048872, A048873, A132346.
%K nonn,more
%O 0,3
%A _Max Alekseyev_ and _N. J. A. Sloane_, May 15 2014
%E a(6) and a(7) from Lukas Finschi, Sep 19 2014
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