A090338 Number of ways of arranging n straight lines in general position in the (affine) plane.

1, 1, 1, 1, 1, 6, 43, 922, 38609, 3111341

Offset

0,6

Comments

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

Here we only consider arrangements of n lines in "general position", with every two lines meeting in one point and every intersection point lying on exactly two lines. See A241600 for the general case.

Two arrangements are considered the same if the lines in each arrangement can be numbered from 1 to n in such a way that, on each line, the order of crossings with the other lines is the same in the two arrangements. In particular, turning over the whole arrangement is allowed. (This does not imply that one arrangement can be continuously changed to the other (possibly after turning over) while keeping all lines straight, without changing the multiplicity of intersection points, and without a line passing through an intersection point, see the papers by Suvorov, Jaggi et al., Richter-Gebert, and Tsukamoto.)

Old name was "Number of full n-flups". The full n-flups are the topologically distinct planar configurations of n straight lines such that each line crosses each other line at exactly one intersection point and no two intersection points coincide.

Also, the number of distinct ways to divide a pancake with n straight cuts that result in the maximal number of pieces (see A000124, A000125).

References

Ringel, Gerhard. "Teilungen der Ebene durch Geraden oder topologische Geraden." Mathematische Zeitschrift 64.1 (1956): 79-102. See page 80.

Links

Table of n, a(n) for n=0..9.

Giedrius Alkauskas, Triangle unions with maximal number of sides, arXiv:2510.22584 [math.CO], 2025. See p. 2.

Tobias Christ, Database of Combinatorially Different Simple Line Arrangements

David O. H. Cutler, Jonas Karlsson, and Neil J. A. Sloane, Cutting a Pancake with an Exotic Knife, arXiv:2511.15864[math.CO], v3, April 19 2026.

Beat Jaggi, Peter Mani-Levitska, Bernd Sturmfels, and Neil White, Uniform oriented matroids without the isotopy property. Discrete Comput Geom 4, 97-100 (1989).

Jürgen Richter-Gebert, Two interesting oriented matroids, Documenta Mathematica 1 (1996), 137-148.

P. Suvorov, Isotopic but not rigidly isotopic plane systems of straight lines. In: Viro, O.Y., Vershik, A.M. (eds.) Topology and Geometry — Rohlin Seminar. Lecture Notes in Mathematics, vol 1346. Springer, Berlin, Heidelberg, pp. 545-556 (1988).

Yasuyuki Tsukamoto, New examples of oriented matroids with disconnected realization spaces, arXiv:1201.2560 [math.CO], 2012.

Jon Wild and Laurence Reeves, Illustration for a(5) = 6.

Example

See illustration of a(5), the full pentaflups. Of the six, the last shown does not have reflectional symmetry, but we do not count its mirror image as distinct. All six are drawn with lines at equally-spaced angles; it is usually (but not always) possible to achieve this (41 out of 43 of the full 6-flups, for example, have equi-angled drawings)

Crossrefs

Cf. A000124, A000125, A090339 (when the lines need not be straight), A241600, A250001.

Sequence in context: A337555 A290783 A159604 * A090339 A225159 A078810

Adjacent sequences: A090335 A090336 A090337 * A090339 A090340 A090341

Keyword

nonn,hard,more

Author

Jon Wild and Laurence Reeves, Jan 27 2004

Extensions

Edited by Max Alekseyev, May 15 2014

Further edits by N. J. A. Sloane, May 16 2014

a(9) from Christ added, and comments corrected by Günter Rote, Apr 14 2025

Status

approved