On a Quadratic Relation Between Stanley-Wilf Limits and F\"{u}redi-Hajnal Limits

TL;DR

This paper refines the known quadratic bound relating Stanley-Wilf and F"{u}redi-Hajnal limits for permutation matrices by improving the constants through a new block contraction method, especially for larger limits.

Contribution

It introduces a block contraction technique to improve the universal quadratic bound between Stanley-Wilf and F"{u}redi-Hajnal limits, refining the constant factor in the inequality.

Findings

Improved the constant from 2.88 to approximately 2.708 for large limits.

Established a new bound involving a function F(c) that asymptotically approaches 2.708.

Demonstrated the bound's effectiveness for c_P ≥ 17.

Abstract

For a permutation matrix $P$, let $s_P$ denote its Stanley-Wilf limit, the exponential growth rate of the number of $n\times n$ permutation matrices avoiding $P$. Let $c_P$ denote its F\"{u}redi-Hajnal limit, which is the limit $\displaystyle \lim_{n \to \infty} \text{ex}(n,P)/n$ where $\text{ex}(n,P)$ is the maximum number of ones in an $n\times n$ $0$-$1$ matrix avoiding $P$. Cibulka proved the universal quadratic bound $s_P\leq 2.88\,c_P^2$. In this note we improve the constants in Cibulka's result through a so-called ``block contraction" argument. Defining \[ F(c)=\inf_{t\in\mathbb{N}} \frac{(t!)^{1/t}\,15^{\,c/t}}{c}, \] for $c>0$, this leads us to the revised inequality $s_P\leq F(c_P)\,c_P^2$. In particular, $F(c)=\log 15+o(1) \approx 2.70805\ldots +o(1)$ as $c\to\infty$, and the constant improves $2.88$ once $c_P \geq 17$.