3AP-free permutations have no exponential growth rate

TL;DR

This paper investigates the growth rate of permutations avoiding 3-term arithmetic progressions, proving that the limit of their nth root does not exist, indicating irregular growth behavior.

Contribution

It establishes that the exponential growth rate limit for 3AP-free permutations does not exist, a novel result in permutation pattern avoidance.

Findings

The limit of the nth root of the number of 3AP-free permutations does not exist.

The growth behavior of 3AP-free permutations is irregular and non-convergent.

Provides new insights into the asymptotic properties of permutation classes avoiding arithmetic progressions.

Abstract

Let $\theta(n)$ be the number of permutations of $\{1,\dots,n\}$ with no $3$-term arithmetic progressions. We prove that $\lim_{n\to\infty}\theta(n)^{1/n}$ does not exist.