Lattice Paths

This thesis is a survey of some of the well known results in lattice path theory. Chapter 1 looks into the history of lattice paths. That is, when it began and how it was popularized. Chapter 3 focuses on general lattices and lattice paths. It later looks into different types of properties of some lattice paths. This is divided into two types: quarter-plane and self-avoiding walks. Chapter 4 and 5 explore some of the properties of quarter-plane walks and self-avoiding walks, respectively

Some permutations on Dyck words

We examine three permutations on Dyck words. The first one, α, is related to the Baker and Norine theorem on graphs, the second one, β, is the symmetry, and the third one is the composition of these two. The first two permutations are involutions and it is not difficult to compute the number of their fixed points, while the third one has cycles of different lengths. We show that the lengths of these cycles are odd numbers. This result allows us to give some information about the interplay between α and β, and a characterization of the fixed points of α∘β