An Improved Lower Bound on the Number of Pseudoline Arrangements

Abstract

Arrangements of pseudolines are classic objects in discrete and computational geometry. They have been studied with increasing intensity since their introduction almost 100 years ago. The study of the number $B_n$ of non-isomorphic simple arrangements of $n$ pseudolines goes back to Goodman and Pollack, Knuth, and others. It is known that $B_n$ is in the order of $2^{Θ(n^2)}$ and finding asymptotic bounds on $b_n = \frac{\log_2(B_n)}{n^2}$ remains a challenging task. In 2011, Felsner and Valtr showed that $0.1887 \leq b_n \le 0.6571$ for sufficiently large $n$. The upper bound remains untouched but in 2020 Dumitrescu and Mandal improved the lower bound constant to $0.2083$. Their approach utilizes the known values of $B_n$ for up to $n=12$. We tackle the lower bound by utilizing dynamic programming and the Lindström-Gessel-Viennot lemma. Our new bound is $b_n \geq 0.2721$ for sufficiently large $n$. The result is based on a delicate interplay of theoretical ideas and computer assistance.

This article is to appear in the proceedings of the 40th International Symposium on Computational Geometry (SoCG 2024). It is a merge of the following two independent submissions: 1) Justin Dallant -- Improved Lower Bound on the Number of Pseudoline Arrangements 2) Fernando Cortés Kühnast, Stefan Felsner, and Manfred Scheucher -- An Improved Lower Bound on the Number of Pseudoline Arrangements