Counting lattice triangulations: Fredholm equations in combinatorics

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).