A241600 Number of ways of arranging n lines in the (affine) plane.

1, 1, 2, 4, 9, 47, 791, 37830

Offset: 0

Author

Max Alekseyev and N. J. A. Sloane, May 15 2014

Keywords

• nonn
• more

Comments

This is in the affine plane, rather than the projective plane, so lines are either parallel or meet in one point.

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.

a(n) might be called the size of the moduli space of n lines in the affine plane.

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.

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

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

Examples

			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.
a(1) = 1: L.
a(2) = 2: P_2, S_2.
a(3) = 4: P_3, P_2 L, S_3, G_3.
See link for illustrations of first 5 terms.
		

References

• B. Grünbaum, Arrangements and Spreads. American Mathematical Society, Providence, RI, 1972, p. 4.

Links

• Lukas Finschi, Homepage of Oriented Matroids
• L. Finschi and K. Fukuda, Complete combinatorial generation of small point set configurations and hyperplane arrangements, pp. 97-100 in Abstracts 13th Canadian Conference on Computational Geometry (CCCG '01), Waterloo, Aug. 13-15, 2001.
• Stefan Forcey, Planes and axioms, Univ. Akron (2024). See p. 3.
• Stefan Forcey, Counting plane arrangements via oriented matroids, arXiv:2504.11461 [math.HO], 2025. See pp. 5, 18.
• Komei Fukuda, Hiroyuki Miyata, and Sonoko Moriyama Complete Enumeration of Small Realizable Oriented Matroids, arXiv:1204.0645 [math.CO], 2012; Discrete Comput. Geom. 49 (2013), no. 2, 359--381. MR3017917. (Further background information.)
• Mark Haiman, with an Appendix by Ezra Miller, Commutative algebra of n points in the plane. Trends Commut. Algebra, MSRI Publ 51 (2004): 153-180. (Background)
• Sergey Kalmykov, Isolated visible infinite straight lines and their combinations, 1920-1922, private collection, on display at Tretyakov gallery. [illustrates a(1)-a(4), and part of a(5)]
• J. L. Martin, The slopes determined by n points in the plane. (Background)
• Jeremy L. Martin, The slopes determined by n points in the plane, arXiv:math/0302106 [math.AG], 2003-2006; Duke Math. J. 131 (2006), no. 1, 119-165. (Background)
• N. J. A. Sloane, Illustration of a(1)-a(5)
• N. J. A. Sloane, Exciting Number Sequences (video of talk), Mar 05 2021.
• N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences (2015 talk slides)

Crossrefs

Cf. A090338 (lines in general position), A090339 (curved lines in general position), A250001 (circles).

Cf. also A063859, A003036, A048872, A048873, A132346.

Formula

a(n) >= A000041(n). - Pablo Hueso Merino, May 10 2021

Extensions

a(6) and a(7) from Lukas Finschi, Sep 19 2014