Intervals of permutation class growth rates

We prove that the set of growth rates of permutation classes includes an infinite sequence of intervals whose infimum is θB ≈ 2.35526, and that it also contains every value at least λB ≈ 2.35698. These results improve on a theorem of Vatter, who determined that there are permutation classes of every growth rate at least λA ≈ 2.48187. Thus, we also refute his conjecture that the set of growth rates below λA is nowhere dense. Our proof is based upon an analysis of expansions of real numbers in non-integer bases, the study of which was initiated by Rényi in the 1950s. In particular, we prove two generalisations of a result of Pedicini concerning expansions in which the digits are drawn from sets of allowed values

Intervals of permutation class growth rates

We prove that the set of growth rates of permutation classes includes an infinite sequence of intervals whose infimum is $\theta_B\approx2.35526$, and that it also contains every value at least $\lambda_B\approx2.35698$. These results improve on a theorem of Vatter, who determined that there are permutation classes of every growth rate at least $\lambda_A\approx2.48187$. Thus, we also refute his conjecture that the set of growth rates below $\lambda_A$ is nowhere dense. Our proof is based upon an analysis of expansions of real numbers in non-integer bases, the study of which was initiated by R\'enyi in the 1950s. In particular, we prove two generalisations of a result of Pedicini concerning expansions in which the digits are drawn from sets of allowed values.Comment: 20 pages, 10 figures, ancillary files containing computer-aided calculations include

Two examples of Wilf-collapse

Two permutation classes, the X-class and subpermutations of the increasing oscillation are shown to exhibit an exponential Wilf-collapse. This means that the number of distinct enumerations of principal subclasses of each of these classes grows much more slowly than the class itself whereas a priori, based only on symmetries of the class, there is no reason to expect this. The underlying cause of the collapse in both cases is the ability to apply some form of local symmetry which, combined with a greedy algorithm for detecting patterns in these classes, yields a Wilf-collapse.Comment: Final version as accepted by DMTCS. Formatting changes onl

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 $\xi\approx 2.30522$ such that while there are uncountably many growth rates of permutation classes arbitrarily close to $\xi$, there are only countably many less than $\xi$. Here we provide a complete characterization of the growth rates less than $\xi$. In particular, this classification establishes that $\xi$ is the least accumulation point from above of growth rates and that all growth rates less than or equal to $\xi$ are achieved by finitely based classes. A significant part of this classification is achieved via a reconstruction result for sum indecomposable permutations. We conclude by refuting a suggestion of Klazar, showing that $\xi$ is an accumulation point from above of growth rates of finitely based permutation classes.Comment: To appear in Israel J. Mat

Combinatorial specifications for juxtapositions of permutation classes

We show that, given a suitable combinatorial specification for a permutation class C, one can obtain a specification for the juxtaposition (on either side) of C with Av(21) or Av(12), and that if the enumeration for C is given by a rational or algebraic generating function, so is the enumeration for the juxtaposition. Furthermore this process can be iterated, thereby providing an effective method to enumerate any "skinny" k×1 grid class in which at most one cell is non-monotone, with a guarantee on the nature of the enumeration given the nature of the enumeration of the non-monotone cell

Uncountably many enumerations of well-quasi-ordered permutation classes

We construct an uncountable family of well-quasi-ordered permutation classes, each with a distinct enumeration sequence. As there are only countably many algebraic generating functions that enumerate combinatorial objects, this shows that there are well-quasi-ordered permutation classes without algebraic generating functions, disproving a widely-held conjecture. Indeed, this shows that there are well-quasi-ordered permutation classes without even D-finite generating functions. Our construction relies on an uncountably large collection of factor-closed binary languages, and this collection also enables us to exhibit an uncountably large collection of infinite binary sequences, each with distinct linear complexity functions

Permutation classes

This is a survey on permutation classes for the upcoming book Handbook of Enumerative Combinatorics

On The Growth Of Permutation Classes

We study aspects of the enumeration of permutation classes, sets of permutations closed downwards under the subpermutation order. First, we consider monotone grid classes of permutations. We present procedures for calculating the generating function of any class whose matrix has dimensions m × 1 for some m, and of acyclic and unicyclic classes of gridded permutations. We show that almost all large permutations in a grid class have the same shape, and determine this limit shape. We prove that the growth rate of a grid class is given by the square of the spectral radius of an associated graph and deduce some facts relating to the set of grid class growth rates. In the process, we establish a new result concerning tours on graphs. We also prove a similar result relating the growth rate of a geometric grid class to the matching polynomial of a graph, and determine the effect of edge subdivision on the matching polynomial. We characterise the growth rates of geometric grid classes in terms of the spectral radii of trees. We then investigate the set of growth rates of permutation classes and establish a new upper bound on the value above which every real number is the growth rate of some permutation class. In the process, we prove new results concerning expansions of real numbers in non-integer bases in which the digits are drawn from sets of allowed values. Finally, we introduce a new enumeration technique, based on associating a graph with each permutation, and determine the generating functions for some previously unenumerated classes. We conclude by using this approach to provide an improved lower bound on the growth rate of the class of permutations avoiding the pattern 1324. In the process, we prove that, asymptotically, patterns in Łukasiewicz paths exhibit a concentrated Gaussian distribution