Small permutation classes

TL;DR

This paper proves a phase transition in permutation classes at a specific algebraic growth rate, distinguishing between countably many classes below it and uncountably many at it, and characterizes growth rates below this threshold.

Contribution

It introduces the concept of generalized grid classes and fully characterizes sub-κ growth rates of permutation classes, answering longstanding open questions.

Findings

Existence of a critical algebraic number κ ≈ 2.20557 for permutation class growth rates.

Countably many classes below κ, uncountably many at κ.

Complete characterization of sub-κ growth rates.

Abstract

We establish a phase transition for permutation classes (downsets of permutations under the permutation containment order): there is an algebraic number $\kappa$, approximately 2.20557, for which there are only countably many permutation classes of growth rate (Stanley-Wilf limit) less than $\kappa$ but uncountably many permutation classes of growth rate $\kappa$, answering a question of Klazar. We go on to completely characterize the possible sub-$\kappa$ growth rates of permutation classes, answering a question of Kaiser and Klazar. Central to our proofs are the concepts of generalized grid classes (introduced herein), partial well-order, and atomicity (also known as the joint embedding property).