Growth rates of permutation classes: from countable to uncountable

摘要

We establish that there is an algebraic number xi approximate to 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.