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

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