Combinatorics of lattice paths

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Master of Science. Johannesburg, 2014.This dissertation consists of ve chapters which deal with lattice paths such as Dyck paths, skew Dyck paths and generalized Motzkin paths. They never go below the horizontal axis. We derive the generating functions to enumerate lattice paths according to di erent parameters. These parameters include strings of length 2, 3, 4 and r for all r 2 f2; 3; 4; g, area and semi-base, area and semi-length, and semi-base and semi-perimeter. The coe cients in the series expansion of these generating functions give us the number of combinatorial objects we are interested to count. In particular 1. Chapter 1 is an introduction, here we derive some tools that we are going to use in the subsequent Chapters. We rst state the Lagrange inversion formula which is a remarkable tool widely use to extract coe cients in generating functions, then we derive some generating functions for Dyck paths, skew Dyck paths and Motzkin paths. 2. In Chapter 2 we use generating functions to count the number of occurrences of strings in a Dyck path. We rst derive generating functions for strings of length 2, 3, 4 and r for all r 2 f2; 3; 4; g, we then extract the coe cients in the generating functions to get the number of occurrences of strings in the Dyck paths of semi-length n. 3. In Chapter 3, Sections 3.1 and 3.2 we derive generating functions for the relationship between strings of lengths 2 and 3 and the relationship between strings of lengths 3 and 4 respectively. In Section 3.3 we derive generating functions for the low occurrences of the strings of lengths 2, 3 and 4 and lastly Section 3.4 deals with derivations of generating functions for the high occurrences of some strings . 4. Chapter 4, Subsection 4.1.1 deals with the derivation of the generating functions for skew paths according to semi-base and area, we then derive the generating functions according to area. In Subsection 4.1.2, we consider the same as in Section 4.1.1, but here instead of semi-base we use semi-length. The last section 4.2, we use skew paths to enumerate the number of super-diagonal bar graphs according to perimeter. 5. Chapter 5 deals with the derivation of recurrences for the moments of generalized Motzkin paths, in particular we consider those Motzkin paths that never touch the x-axis except at (0,0) and at the end of the path