Small permutation classes
Proceedings of the London Mathematical Society, 103 (2011), 879–921.
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:
- 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.
- This work is briefly described in a talk I gave at the Rutgers Experimental Mathematics Seminar, and in a colloquium I gave at Dartmouth College.