Counting lattice triangulations: Fredholm equations in combinatorics

Abstract

Let $f(m,n)$ be the number of primitive lattice triangulations of an $m\times n$ rectangle. We compute the limits $\lim_n f(m,n)^{1/n}$ for $m=2,3$. For $m=2$ we obtain the exact value of the limit, which is $(611+\sqrt{73})/36$. For $m=3$ we express the limit in terms of a certain Fredholm integral equation for generating functions. This provides a polynomial-time algorithm (with respect to the number of computed digits) for the computation of the limit with any prescribed precision. Bibliography: 13 titles.