Studies on several parameters in lattice paths

Abstract

This thesis deals with enumerative as well as asymptotic aspects of directed lattice paths. Several parameters appearing in lattice paths will be analyzed, e.g. the area enclosed by or the number of contacts between two paths or the number of occurrences of certain patterns in a path. The first chapter gives an overview over the history of lattice path theory as well as basic definitions and an overview of the applications of lattice paths in mathematical models arising in natural sciences or computer science. In the second chapter the methods used in enumerative and asymptotic combinatorics will be introduced: combinatorial classes and their generating functions for exact enumeration as well as singularity analysis for asymptotic results. There are two lattice path configurations on which this thesis mostly focuses: non-intersecting pairs (or tuples) of paths and paths which avoid patterns, i.e., fixed sequences of consecutivesteps. Chapter 3 deals with non-intersecting pairs of lattice paths. We will derive results abouttheir average number of contacts as well as the average area between them.Chapter 4 deals with pattern avoidance in lattice paths. First, the vectorial kernel methoddeveloped by Andrei Asinowski, Axel Bacher, Cyril Banderier, and Bernhard Gittenberger will be introduced, since it is a very powerful tool for enumerating lattice paths avoiding a fixed pattern as well as enumerating the occurrences of a fixed pattern in a lattice path. Then it will be generalized in two directions: for enumerating lattice paths with longer steps and for enumerating lattice paths which avoid several patterns at once. The tools developed in this section have also been used to prove a conjecture by David Callan about the asymptotic behavior of the expected number of ascents in Schröder paths.In Chapter 5 we will combine the methods from Chapter 3 and 4 for studying patternavoidance as well as the lower height in pairs of non-intersecting lattice paths. Some of the results in this thesis have already been published in scientific papers. For acomplete list of the publications this thesis is based on, see page IV.