Growth rates of permutation classes: from countable to uncountable

TL;DR

This paper proves the existence of a specific algebraic number as a threshold for the growth rates of permutation classes, showing a transition from countable to uncountable sets around this point.

Contribution

It introduces the concept of concentration and structural notions of grid classes to analyze permutation class growth rates, establishing a key threshold at an algebraic number.

Findings

Existence of an algebraic number pprox. 2.30522etected as a growth rate boundary.

Uncountably many growth rates are close to this boundary, but only countably many are below it.

The paper lays groundwork for classifying growth rates up to this algebraic number.

Abstract

We establish that there is an algebraic number $\xi\approx 2.30522$ such that while there are uncountably many growth rates of permutation classes arbitrarily close to $\xi,$ there are only countably many less than $\xi$. Central to the proof are various structural notions regarding generalized grid classes and a new property of permutation classes called concentration. The classification of growth rates up to $\xi$ is completed in a subsequent paper.