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Counting Lattice Paths Using Fourier Methods

  • Book
  • © 2019

Overview

Authors:
  1. Shaun Ault

    1. Department of Mathematics, Valdosta State University, Valdosta, USA

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  2. Charles Kicey

    1. Department of Mathematics, Valdosta State University, Valdosta, USA

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    You can also search for this author in PubMed   Google Scholar

  • Introduces a unique technique to count lattice paths by using the discrete Fourier transform
  • Explores the interconnection between combinatorics and Fourier methods
  • Motivates students to move from one-dimensional problems to higher dimensions
  • Presents numerous exercises with selected solutions appearing at the end

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

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

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Table of contents (4 chapters)

  1. Front Matter

    Pages i-xii
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  2. Lattice Paths and Corridors

    • Shaun Ault, Charles Kicey
    Pages 1-22

  3. One-Dimensional Lattice Walks

    • Shaun Ault, Charles Kicey
    Pages 23-44

  4. Lattice Walks in Higher Dimensions

    • Shaun Ault, Charles Kicey
    Pages 45-67

  5. Corridor State Space

    • Shaun Ault, Charles Kicey
    Pages 69-87

  6. Back Matter

    Pages 89-136
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Keywords

About this book

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.

Authors and Affiliations

  • Department of Mathematics, Valdosta State University, Valdosta, USA

    Shaun Ault, Charles Kicey

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