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

Mohamed Omar

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 $\displaystyle \lim_{n \to \infty} \text{ex}(n,P)/n$ 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 $c_P \geq 17$.

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

Abstract

For a permutation matrix , let denote its Stanley-Wilf limit, the exponential growth rate of the number of permutation matrices avoiding . Let denote its Füredi-Hajnal limit, which is the limit where is the maximum number of ones in an - matrix avoiding . Cibulka proved the universal quadratic bound . In this note we improve the constants in Cibulka's result through a so-called ``block contraction" argument. Defining for , this leads us to the revised inequality . In particular, as , and the constant improves once .
Paper Structure (2 sections, 4 theorems, 19 equations, 1 table)

This paper contains 2 sections, 4 theorems, 19 equations, 1 table.

Table of Contents

  1. Introduction
  2. Block contraction and lifting

Key Result

Theorem 1

For $c>0$ define Then for every permutation matrix $P$ one has Moreover $F(c)=\log 15+o(1)$ as $c\to\infty$.

Theorems & Definitions (7)

  • Theorem 1
  • Corollary 2
  • proof
  • Proposition 3
  • proof
  • Lemma 4: Klazar2000
  • proof : Proof of Theorem \ref{['thm:main']}