On the Number of Arrangements of Pseudolines

Given a simple arrangementof n pseudolines in the Euclidean plane, associate with line i the list i of the lines crossing i in the order of the crossings on line i. i = ( i 1; i 2;::; i n?1) is a permutation of f1;::;ng? fig. The vector ( 1; 2;:::; n) is an encoding for the arrangement. De ne i j = 1 if i j > i and i j = 0, otherwise. Let i = ( i 1; i 2;::; i n?1), we show that the vector ( 1; 2;:::; n) is already an encoding. We use this encoding to improve the upper bound on the number of arrangements of n pseudolines to 20:6974 n 2 . Moreover, we have enumerated arrangements with 10 pseudolines. As a by-product we determine their exact number and we can show that the maximal number of halving lines of 10 point in the plane is 13.

• Type: Conference
• Year: 1996
• Where: COMPGEOM
• Authors: Stefan Felsner
• Keywords: COMPGEOM 1996, De Ne, Discrete Geometry, Euclidean Plane, Simple Arrangementof