ETD Collection

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    Combinatorics of lattice paths
    (2014-09-01) Ncambalala, Thokozani Paxwell
    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.
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    Generating functions and the enumeration of lattice paths
    (2013-08-07) Mutengwe, Phumudzo Hector
    Our main focus in this research is to compute formulae for the generating function of lattice paths. We will only concentrate on two types of lattice paths, Dyck paths and Motzkin paths. We investigate di erent ways to enumerate these paths according to various parameters. We start o by studying the relationship between the Catalan numbers Cn, Fine numbers Fn and the Narayana numbers vn;k together with their corresponding generating functions. It is here where we see how the the Lagrange Inversion Formula is applied to complex generating functions to simplify computations. We then study the enumeration of Dyck paths according to the semilength and parameters such as, number of peaks, height of rst peak, number of return steps, e.t.c. We also show how some of these Dyck paths are related. We then make use of Krattenhaler's bijection between 123-avoiding permutations of length n, denoted by Sn(123), and Dyck paths of semilength n. Using this bijective relationship over Sn(123) with k descents and Dyck paths of semilength n with sum of valleys and triple falls equal to k, we get recurrence relationships between ordinary Dyck paths of semilength n and primitive Dyck paths of the same length. From these relationships, we get the generating function for Dyck paths according to semilength, number of valleys and number of triple falls. We nd di erent forms of the generating function for Motzkin paths according to length and number of plateaus with one horizontal step, then extend the discussion to the case where we have more than one horizontal step. We also study Motzkin paths where the horizontal steps have di erent colours, called the k-coloured Motzkin paths and then the k-coloured Motzkin paths which don't have any of their horizontal steps lying on the x-axis, called the k-coloured c-Motzkin paths. We nd that these two types of paths have a special relationship which can be seen from their generating functions. We use this relationship to simplify our enumeration problems.