Characterising inflations of monotone grid classes of permutations

We characterise those permutation classes whose simple permutations are monotone griddable. This characterisation is obtained by identifying a set of nine substructures, at least one of which must occur in any simple permutation containing a long sum of 21s

Uniquely-Wilf classes

Two permutations in a class are Wilf-equivalent if, for every size, $n$, the number of permutations in the class of size $n$ containing each of them is the same. Those infinite classes that have only one equivalence class in each size for this relation are characterised provided either that they avoid at least one permutation of size 3, or at least three permutations of size 4.Comment: Updated to DMTCS styl

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

The enumeration of subclasses of the 321-avoiding permutations

This thesis is dedicated to the enumeration of subclasses of 321-avoiding permutations, using a combination of theoretical and experimental investigations. The thesis is organised as follows: Chapter 1 provides the necessary definitions and preliminaries, discusses the current state of research on the subject of enumerating Av(321) and its subclasses, then gives an introduction on the basic problem of containment check for 321-avoiding permutations, the process of which is used throughout our work. The main results of this study are explained in Chapter 2 and 3. Chapter 2 focuses on the implementation aspects of enumerating 321-avoiding classes, where the main goal is to develop efficient algorithms to generate all permutations up to a certain length contained in classes of the form Av(321, π). The permutation counts are then used to guess the generating function by fitting a rational function to the computed data. In Chapter 3, we deal with the more theoretical problem of enumerating 321-avoiding polynomial classes given a structural description. In particular, we propose a method which computes the grid class of such a class given its basis. We then use this information to enumerate the class using an improved version of a known algorithm