Small permutation classes
release_ohni3j7emngnzautbn3wlrjumi
by
Vincent Vatter
2016
Abstract
We establish a phase transition for permutation classes (downsets of
permutations under the permutation containment order): there is an algebraic
number κ, approximately 2.20557, for which there are only countably many
permutation classes of growth rate (Stanley-Wilf limit) less than κ but
uncountably many permutation classes of growth rate κ, answering a
question of Klazar. We go on to completely characterize the possible
sub-κ 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).
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