Small permutation classes

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).

Download the paper:

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• source: zip

Related links:

• Steven Finch's short summary of growth rates.
• Michael Albert's FPSAC 2008 talk about growth rates.
• The open problem about slicing axes-parallel rectangles with lines is now listed at The Open Problems Project.

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