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Counting Lattice Paths Using Fourier Methods

  • Shaun Ault
  • Charles Kicey

Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Also part of the Lecture Notes in Applied and Numerical Harmonic Analysis book sub series (LN-ANHA)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Shaun Ault, Charles Kicey
    Pages 1-22
  3. Shaun Ault, Charles Kicey
    Pages 23-44
  4. Shaun Ault, Charles Kicey
    Pages 45-67
  5. Shaun Ault, Charles Kicey
    Pages 69-87
  6. Back Matter
    Pages 89-136

About this book

Introduction

This monograph introduces a novel and effective approach to counting lattice paths by using the discrete Fourier transform (DFT) as a type of periodic generating function. Utilizing a previously unexplored connection between combinatorics and Fourier analysis, this method will allow readers to move to higher-dimensional lattice path problems with ease. The technique is carefully developed in the first three chapters using the algebraic properties of the DFT, moving from one-dimensional problems to higher dimensions. In the following chapter, the discussion turns to geometric properties of the DFT in order to study the corridor state space. Each chapter poses open-ended questions and exercises to prompt further practice and future research. Two appendices are also provided, which cover complex variables and non-rectangular lattices, thus ensuring the text will be self-contained and serve as a valued reference.

Counting Lattice Paths Using Fourier Methods is ideal for upper-undergraduates and graduate students studying combinatorics or other areas of mathematics, as well as computer science or physics. Instructors will also find this a valuable resource for use in their seminars. Readers should have a firm understanding of calculus, including integration, sequences, and series, as well as a familiarity with proofs and elementary linear algebra.

Keywords

Lattice Path Discrete Fourier Transform Corridor Numbers Complex Variables Combinatorics

Authors and affiliations

  • Shaun Ault
    • 1
  • Charles Kicey
    • 2
  1. 1.Department of MathematicsValdosta State UniversityValdostaUSA
  2. 2.Department of MathematicsValdosta State UniversityValdostaUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-030-26696-7
  • Copyright Information Springer Nature Switzerland AG 2019
  • Publisher Name Birkhäuser, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-030-26695-0
  • Online ISBN 978-3-030-26696-7
  • Series Print ISSN 2296-5009
  • Series Online ISSN 2296-5017
  • Buy this book on publisher's site