For each positive integer n we use the concept of ‘admissible arrays on n symbols’ to define a set of positive integers $Q(n)$ which is determined solely by number theoretical and combinatorial constraints and whose computation reduces to a finite problem. In earlier joint work with M. Scheutzow, it was shown that the set $Q(n)$ is intimately connected to the set of periods of periodic points of classes of non-expansive nonlinear maps defined on the positive cone in $\mathbb{R}^n$. In this paper we continue the characterization of $Q(n)$ and present precise asymptotic estimates for the largest element of $Q(n)$. For example, if $\gamma(n)$ denotes the largest element of $Q(n)$, then we show that $\lim_{n \to \infty} (n\log n)^{-1/2}\log \gamma(n) = 1$. We also discuss why understanding further details about the fine structure of $Q(n)$ involves some delicate number theoretical issues.