cp's OEIS Frontend

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

%I A090338 #51 May 01 2025 08:31:01
%S A090338 1,1,1,1,1,6,43,922,38609,3111341
%N A090338 Number of ways of arranging n straight lines in general position in the (affine) plane.
%C A090338 This is in the affine plane, rather than the projective plane, so two lines are either parallel or meet in one point.
%C A090338 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.
%C A090338 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.)
%C A090338 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.
%C A090338 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).
%H A090338 Tobias Christ, <a href="https://geometry.inf.ethz.ch/christt/linearr/">Database of Combinatorially Different Simple Line Arrangements</a>
%H A090338 Beat Jaggi, Peter Mani-Levitska, Bernd Sturmfels, and Neil White, <a href="https://doi.org/10.1007/BF02187717">Uniform oriented matroids without the isotopy property</a>. Discrete Comput Geom 4, 97-100 (1989).
%H A090338 Jürgen Richter-Gebert, <a href="https://doi.org/10.4171/dm/7">Two interesting oriented matroids</a>, Documenta Mathematica 1 (1996), 137-148.
%H A090338 P. Suvorov, <a href="https://doi.org/10.1007/BFb0082793">Isotopic but not rigidly isotopic plane systems of straight lines</a>. 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).
%H A090338 Yasuyuki Tsukamoto, <a href="https://arxiv.org/abs/1201.2560">New examples of oriented matroids with disconnected realization spaces</a> (2012)
%H A090338 Jon Wild and Laurence Reeves, <a href="/A090338/a090338.gif">Illustration for a(5) = 6</a>.
%e A090338 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)
%Y A090338 Cf. A000124, A000125, A090339 (when the lines need not be straight), A241600, A250001.
%K A090338 nonn,hard,more
%O A090338 0,6
%A A090338 _Jon Wild_ and _Laurence Reeves_, Jan 27 2004
%E A090338 Edited by _Max Alekseyev_, May 15 2014
%E A090338 Further edits by _N. J. A. Sloane_, May 16 2014
%E A090338 a(9) from Christ added, and comments corrected by _Günter Rote_, Apr 14 2025