The enumeration of subclasses of the 321-avoiding permutations

Abstract

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