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On the Number of Arrangements of Pseudolines

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Stefan Felsner

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Institut für Informatik

Freie Universität Berlin

email: felsner@inf.fu-berlin.de

Report B 95-20

December 1995

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Server: fubinf.inf.fu-berlin.de
File: pub/reports/tr-b-95-20.ps.gz

On the Number of Arrangements of Pseudolines
Given a simple arrangement of *n* pseudolines in the
Euclidean plane, associate with line *i* the list *\sigma_i* of the
lines crossing *i* in the order of the crossings on line *i*.
*\sigma_i=(\sigma^i_1,\sigma^i_2,..,\sigma^i_{n-1})* is a permutation
of *\{1,..,n\} - \{i\}*. The vector
*(\sigma_1,\sigma_2,\ldots,\sigma_n)* is an encoding for the
arrangement. Define *\tau^i_j = 1* if *\sigma^i_j > i* and *\tau^i_j
= 0*, otherwise. Let *\tau_i=(\tau^i_1,\tau^i_2,..,\tau^i_{n-1})*,
we show that the vector
*(\tau_1,\tau_2,\ldots,\tau_n)* is already an encoding.
We use this encoding to improve the upper bound on the number of
arrangements of *n* pseudolines to *2^{0.6988\cdot 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.