An Improved Lower Bound on the Number of Pseudoline Arrangements

TL;DR

This paper addresses the longstanding problem of bounding the growth rate of the number $B_n$ of non-isomorphic simple pseudoline arrangements by establishing a new lower bound with $c^- > 0.2721$. It introduces a patch-based construction using $k$ parallel bundles of lines and local perturbations (reroutings) within patches, combining two counting techniques: dynamic programming and the Lindström-Gessel-Viennot lemma. The authors show that, for $k=4,6,12$, the corresponding counts $F_k(n)$ satisfy $F_k(n) \ge 2^{c n^2 - O(n)}$ with increasing constants, and then apply a step that resolves parallel bundles to derive $B_n \ge F_k(n) (B_m)^k$, yielding $c^- \ge \frac{k}{k-1} c$; this leads to $c^- > 0.2721$. The result significantly improves prior lower bounds and demonstrates a scalable computational–combinatorial approach that could extend to larger $k$ or related geometric arrangements, potentially influencing bounds in related combinatorial geometry problems.

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.

Figures (12)

• Figure 1: Left: An arrangement of $k=3$ bundles of parallel lines and a collection of interior-disjoint patches (highlighted red) such that each multicrossing point is covered by a patch. Right: A partial pseudoline arrangement with the same parallel bundles obtained by rerouting within the patches.
• Figure 2: Construction with 4 bundles as in DumitrescuMandal2020.
• Figure 3: An illustration of the tiling for region $R_4$ in the 4-slope construction. The patch $P_4$ is a square of side length $8\frac{\sqrt{2}}{2}$ and area $32$.
• Figure 4: Construction with 6 bundles as in DumitrescuMandal2020.
• Figure 5: Three of the four types of patches for our construction on $k=6$ bundles: \ref{['fig:types_of_tiles_6_slopes_6']} For $R_6$ we use a hexagonal tiling such that each patch $P_6$ contains exactly 7 crossings of order 6 and 14 crossings of order 3. \ref{['fig:types_of_tiles_6_slopes_5']} For $R_5$ we use a hexagonal tiling such that each patch $P_5$ contains exactly 12 crossings of order 5. \ref{['fig:types_of_tiles_6_slopes_4']} For $R_4$ we use a rectangular tiling such that each patch $P_4$ contains exactly $8 \cdot 4 = 32$ crossings of order 4.

Theorems & Definitions (6)

• Theorem 1
• Proposition 2
• Proposition 3
• Proposition 4
• Lemma 5
• Lemma 6: Lindström Linstrom1973, Gessel & Viennot Gessel1985