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

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 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 general position in the affine plane.

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).

Links

Table of n, a(n) for n=0..8.

Jean-Luc Baril, Céline Moreira Dos Santos, Pizza-cutter's problem and Hamiltonian path, Mathematics Magazine (2019) Vol. 88, No. 1, 1-9. [This paper appears to say that a(9) = 3111341. I believe this is an error, and 3111341 is instead the ninth term of A090339. - N. J. A. Sloane, Feb 15 2021]

Finschi, Lukas, A graph theoretical approach for reconstruction and generation of oriented matroids, (2001). Diss., Mathematische Wissenschaften ETH Zürich, Nr. 14335, 2001. See table on page 165.

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

more,nonn,hard

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

Status

approved