We show that for large enough $n$, the number of non-isomorphic pseudoline arrangements of order $n$ is greater than $2^{c\cdot n^2}$ for some constant $c > 0.2604$, improving the previous best bound of $c>0.2083$ by Dumitrescu and Mandal (2020). Arrangements of pseudolines (and in particular arrangements of lines) are important objects appearing in many forms in discrete and computational geometry. They have strong ties for example with oriented matroids, sorting networks and point configurations. Let $B_n$ be the number of non-isomorphic pseudoline arrangements of order $n$ and let $b_n := \log_2(B_n)$. The problem of estimating $b_n$ dates back to Knuth, who conjectured that $b_n \leq 0.5n^2 + o(n^2)$ and derived the first bounds $n^2/6-O(n) \leq b_n \leq 0.7924(n^2+n)$. Both the upper and the lower bound have been improved a couple of times since. For the upper bound, it was first improved to $b_n < 0.6988n^2$ (Felsner, 1997), then $b_n < 0.6571 n^2$ by Felsner and Valtr (2011), for large enough $n$. In the same paper, Felsner and Valtr improved the constant in the lower bound to $c> 0.1887$, which was subsequently improved by Dumitrescu and Mandal to $c>0.2083$. Our new bound is based on a construction which starts with one of the constructions of Dumitrescu and Mandal and breaks it into constant sized pieces. We then use software to compute the contribution of each piece to the overall number of pseudoline arrangements. This method adds a lot of flexibility to the construction and thus offers many avenues for future tweaks and improvements which could lead to further tightening of the lower bound.

翻译：我们证明，对于足够大的 $n$，存在常数 $c > 0.2604$，使得 $n$ 阶非同构伪线排列的数量大于 $2^{c\cdot n^2}$，改进了 Dumitrescu 和 Mandal（2020）之前的最佳下界 $c>0.2083$。伪线排列（特别是线的排列）是离散与计算几何中多种形式出现的重要研究对象，例如与定向拟阵、排序网络和点配置有紧密联系。令 $B_n$ 表示 $n$ 阶非同构伪线排列的数量，$b_n := \log_2(B_n)$。估计 $b_n$ 的问题可追溯到 Knuth，他猜想 $b_n \leq 0.5n^2 + o(n^2)$，并首次推导出下界 $n^2/6 - O(n) \leq b_n \leq 0.7924(n^2+n)$。此后，上下界均被多次改进。对于上界，首先由 Felsner（1997）改进为 $b_n < 0.6988n^2$，随后 Felsner 和 Valtr（2011）对足够大的 $n$ 改进为 $b_n < 0.6571 n^2$。在同一论文中，Felsner 和 Valtr 将下界常数改进为 $c>0.1887$，之后 Dumitrescu 和 Mandal 进一步将其提升至 $c>0.2083$。我们的新下界基于一种构造：该构造以 Dumitrescu 和 Mandal 的构造之一为起点，并将其分解为恒定大小的片段。随后，我们利用软件计算每个片段对伪线排列总数的贡献。该方法为构造增加了极大的灵活性，从而为未来的调整与改进提供了多种途径，有望进一步收紧下界。