Asymptotics of the number of lattice triangulations of rectangles of width 4 and 5

TL;DR

This paper investigates the asymptotic growth of the number of primitive lattice triangulations of rectangles with widths 4 and 5, using Fredholm integral equations to derive and numerically solve for their limits.

Contribution

It extends previous work by deriving and solving integral equations for the asymptotics of lattice triangulations for larger rectangle widths.

Findings

Derived systems of Fredholm integral equations for m=4,5

Numerically computed high-precision asymptotic limits

Extended understanding of lattice triangulation enumeration

Abstract

Let $f(m,n)$ be the number of primitive lattice triangulations of an $m \times n$ rectangle. We express the limits $\lim_n f(m,n)^{1/n}$ for $m = 4$ and $m=5$ in terms of certain systems of Fredholm integral equations on generating functions (the case $m\le3$ was treated in a previous paper). Solving these equations numerically, we compute approximate values of these limits with a rather high precision.