An Improved Lower Bound on the Number of Pseudoline Arrangements
<|reference_start|>An Improved Lower Bound on the Number of Pseudoline Arrangements: 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 $2^{\Theta(n^2)}$ 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 Lindstr\"om-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.<|reference_end|>
@article{kühnast2024an,
    title={An Improved Lower Bound on the Number of Pseudoline Arrangements},
    author={Fernando Cort'es K"uhnast, Justin Dallant, Stefan Felsner, Manfred
  Scheucher},
    journal={arXiv preprint arXiv:2402.13107},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.13107},
    primaryClass={math.CO cs.CG cs.DM}
}