Counting lattice triangulations: Fredholm equations in combinatorics

TL;DR

This paper investigates the asymptotic behavior of the number of primitive lattice triangulations of rectangles, deriving exact and approximate limits using Fredholm equations, and provides an efficient algorithm for their computation.

Contribution

It introduces a novel approach using Fredholm integral equations to analyze lattice triangulations and offers a polynomial-time algorithm for calculating the limits with arbitrary precision.

Findings

Exact limit for 2xN triangulations: (611+√73)/36

Limit for 3xN triangulations expressed via Fredholm integral equations

Polynomial-time algorithm for computing the limits with any desired accuracy

Abstract

Let $f(m,n)$ be the number of primitive lattice triangulations of $m\times n$ rectangle. We compute the limits $\lim_n f(m,n)^{1/n}$ for $m=2$ and $3$. For $m=2$ we obtain the exact value of the limit which is equal to $(611+\sqrt{73})/36$. For $m=3$, we express the limit in terms of certain Fredholm's integral equation on generating functions. This provides a polynomial time algorithm for computation of the limit with any given precision (polynomial with respect the the number of computed digits).