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

TL;DR

The paper addresses when the Stanley–Wilf limit $L(\pi)$ grows polynomially with the pattern size, focusing on blockable permutations formed by inflations $\sigma[P_1,\dots,P_c]$. It introduces a general $c$-blockable framework and develops a recursive, density-based method to bound the associated extremal function, improving on prior bounds for this family. The main result shows that if each component $P_i$ has polynomially-bounded $L(P_i)$, then $L(B_{\{\mathcal P_1,\dots,\mathcal P_c\}}) = O(k^{4a+16c^2+64ac^2\ln c})$ with $a = \max a_i$, and hence polynomial growth follows via $L(\pi)=O(c(\pi)^2)$. The method extends beyond splittable cases and broadens the class of permutation families known to exhibit subquadratic or polynomial growth, offering tools relevant to permutation-avoidance growth questions and property testing contexts.

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 $π$.

Figures (2)

• Figure 1: A $4$-blockable permutation
• Figure 2: Density Argument

Theorems & Definitions (18)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Theorem 1.5: Marcus-Tardos Theorem marcus_tardos_2004
• Definition 1.6
• Lemma 1.7
• Definition 1.8
• Definition 1.9
• Theorem 1.10