Small permutation classes

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).