Generating functions and the enumeration of lattice paths

Date
2013-08-07
Authors
Mutengwe, Phumudzo Hector
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Abstract
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.
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A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in ful lment of the requirements for the degree of Master of Science. Johannesburg, 2013.
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