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. 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. 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. Chapter 3 deals with non-intersecting pairs of lattice paths. 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.

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. 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. the area enclosed by or the number of contacts between two paths or the number of occurrences of certain patterns in a path.

Several parameters appearing in lattice paths will be analyzed, e.g.

This thesis deals with enumerative as well as asymptotic aspects of directed lattice paths.

2020 lattice pattern