On the growth rate of the Stanley-Wilf limit of blockable permutations

Abstract

Given a permutation $\pi$, let $\text{Av}_n(\pi)$ be the number of permutations of length $n$ that avoid $\pi$ as a subpermutation. The celebrated resolution of the Stanley-Wilf conjecture by Marcus and Tardos confirmed that the limit $L(\pi) = \lim_{n \to \infty} |\text{Av}_n(\pi)|^{1/n}$ exists. A central and challenging question concerns the behavior of $L(\pi)$ as a function of the pattern length $|\pi|$. While Fox proved that $L(\pi)$ is exponential in $|\pi|$ for almost all permutations, it is known that $L(\pi)$ grows polynomially for specific structural classes. For instance, $L(\pi)$ is known to be quadratic in $|\pi|$ when $\pi$ is a monotone or a layered permutation. In this paper, we address this question for {\it blockable} permutations $\pi$.