# Counting Lattice Paths Using Fourier Methods

• Shaun Ault
• Charles Kicey
Book

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)

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

### 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