On the least exponential growth admitting uncountably many closed permutation classes

不可数多个well-quasi-ordered排列类的枚举

Uncountably many enumerations of well-quasi-ordered permutation classes

本文构造了不可数多个well-quasi-ordered排列类，每个类有独特的枚举序列，推翻了所有well-quasi-ordered排列类具有代数生成函数的猜想，证明许多此类类缺乏D-有限或D-代数生成函数。

Growth rates of permutation classes: from countable to uncountable

We establish that there is an algebraic number $ξ\approx 2.30522$ such that while there are uncountably many growth rates of permutation classes arbitrarily close to $ξ,$ there are only countably many less than $ξ$. 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 $ξ$ is completed in a subsequent paper.

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 …

Growth rates of permutation classes: categorization up to the uncountability threshold

In the antecedent paper to this it was established that there is an algebraic number $ξ\approx 2.30522$ such that while there are uncountably many growth rates of permutation classes arbitrarily close to $ξ$, there are only countably many less than $ξ$. Here we provide a complete characterization of the growth rates less than $ξ$. In particular, this classification establishes that $ξ$ is the least accumulation point from above of growth rates and that all growth rates less than or equal to $ξ$ are achieved by finitely based classes. A significant part of this classification is achieved via a …

Minimal prime ages, words and permutation graphs Extended abstract

This paper is a contribution to the study of hereditary classes of finite graphs. We classify these classes according to the number of prime structures they contain. We consider such classes that are \emph{minimal prime}: classes that contain infinitely many primes but every proper hereditary subclass contains only finitely many primes. We give a complete characterization of such classes. In fact, each one of these classes is a well quasi ordered age and there are uncountably many of them. Eleven of these ages remain well quasi ordered when labels in a well quasi ordering are added. Among the …

Pin 类 II：小 Pin 类

Pin classes II: Small pin classes

研究 Pin 类的结构特性，发现生长速率相变点 μ≈3.28277，揭示 Pin 类的生长速率分布及分类。