Small permutation classes

2007/12/31 by Vincent Vatter · 5 citations
Computer Science · Engineering · Mathematics · #Advanced Combinatorial Mathematics #Coding theory and cryptography #Combinatorics #Mathematics #Permutation (music) #graph theory and CDMA systems #math.CO

paper · pdf · doi:10.1112/plms/pdr017

published as Proc. London Math. Soc. 103 (2011), 879--921

openalex publication_date 2011/07/08 · arxiv created 2016/04/05 · arxiv updated 2016/04/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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, the substitution decomposition, and atomicity (a.k.a. the joint embedding property).

Citations

• A Combinatorial Problem in Geometry
• Overview of some general results in combinatorial enumeration
• Simple permutations and pattern restricted permutations

Cited by

• Intervals of Permutation Class Growth Rates
• Growth rates of permutation classes: Categorization up to the uncountability threshold
• Growth rates of permutation classes: from countable to uncountable
• Juxtaposing Catalan permutation classes with monotone ones
• Stanley–Wilf conjecture [wikipedia]