  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.
