On a Quadratic Relation Between Stanley-Wilf Limits and Füredi-Hajnal Limits

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 Füredi-Hajnal limit, which is the limit $\underset{n\to \infty }{\lim }\text{ex}(n,P)/n$ 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 $s_P\le 2.88c_P^2$. 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 $s_P\le F(c_P)c_P^2$. In particular, $F(c)=\log 15+o(1)\approx 2.70805\dots +o(1)$ as $c\to \infty$, and the constant improves $2.88$ once $c_P\ge 17$.

5 pages