Improved Lower Bound on the Number of Pseudoline Arrangements

TL;DR

The paper tackles the problem of bounding the number $B_n$ of non-isomorphic simple pseudoline arrangements by proving a new asymptotic lower bound $b_n=\log_2 B_n \ge c\,n^2$ with $c>0.2604$ for large $n$, improving the previous $c>0.2083$. It builds on the Dumitrescu–Mandal construction with a modular, constant-sized subembedding counting approach, implemented in software to aggregate many local contributions. A region-based counting framework, together with the Lindström–Gessel–Viennot determinant method for counting vertex-disjoint paths, yields large local counts that, when combined recursively, produce the improved global bound. The methodology provides a flexible framework for further tightening by exploring alternative embeddings, larger substructures, or more powerful counting techniques, and has implications for related combinatorial objects such as oriented matroids and wiring diagrams.

Abstract

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.

Figures (47)

• Figure 1: A pseudoline arrangement of order $4$ with its natural labeling.
• Figure 2: Matoušek's lower bound construction.
• Figure 3: Illustrations $(A)$ and $(B)$ represent two different embeddings of the same pseudochord arrangement, while $(C)$ is a different arrangement with the same matching. Illustration $(D)$ is an embedding of a matching distinct from the three others.
• Figure 4: Embedding of a $(3,2,4)$-matching.
• Figure 5: Example of an embedded matching with independent chords. The blue chord $c_1$ is independent from both $c_2$ and $c_3$.

Theorems & Definitions (11)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Proposition 6
• Proposition 7
• Definition 8
• Lemma 9: Lindström Linstrom1973, Gessel & Viennot Gessel1985
• Theorem 10