Abstract.
Given a simple arrangement of n pseudolines in the Euclidean plane, associate with line i the list σ i of the lines crossing i in the order of the crossings on line i. \(\sigma_i=(\sigma^i_1,\sigma^i_2,\ldots,\sigma^i_{n-1})\) is a permutation of \(\{1,\ldots,n\} - \{i\}\) . The vector (σ 1 ,σ 2 , ...,σ_n) is an encoding for the arrangement. Define \(\tau^i_j = 1\) if \(\sigma^i_j > i\) and \(\tau^i_j = 0\) , otherwise. Let \(\tau_i=(\tau^i_1,\tau^i_2,\ldots,\tau^i_{n-1})\) , we show that the vector (τ 1 , τ 2 , ... , τ_n) is already an encoding.
We use this encoding to improve the upper bound on the number of arrangements of n pseudolines to \(2^{0.6974\cdot n^2}\) . Moreover, we have enumerated arrangements with 10 pseudolines. As a byproduct we determine their exact number and we can show that the maximal number of halving lines of 10 point in the plane is 13.
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Received December 20, 1995, and in revised form March 8, 1996.
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Felsner, S. On the Number of Arrangements of Pseudolines. Discrete Comput Geom 18, 257–267 (1997). https://doi.org/10.1007/PL00009318
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DOI: https://doi.org/10.1007/PL00009318