Analytic Combinatorics: A Multidimensional Approach is written in a reader-friendly fashion to better facilitate the understanding of the subject. Naturally, it is a firm introduction to the concept of analytic combinatorics and is a valuable tool to help readers better understand the structure and large-scale behavior of discrete objects. Primarily, the textbook is a gateway to the interactions between complex analysis and combinatorics. The study will lead readers through connections to number theory, algebraic geometry, probability and formal language theory.
The textbook starts by discussing objects that can be enumerated using generating functions, such as tree classes and lattice walks. It also introduces multivariate generating functions including the topics of the kernel method, and diagonal constructions. The second part explains methods of counting these objects, which involves deep mathematics coming from outside combinatorics, such as complex analysis and geometry.
Features
- Written with combinatorics-centric exposition to illustrate advanced analytic techniques
- Each chapter includes problems, exercises, and reviews of the material discussed in them
- Includes a comprehensive glossary, as well as lists of figures and symbols
Author(s): Marni Mishna
Series: Discrete Mathematics and its Applications
Edition: 1
Publisher: CRC Press
Year: 2019
Language: English
Pages: 252
Tags: Analytic Combinatorics
Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Preface
Symbols
Welcome to Analytic Combinatorics
Part I: Enumerative Combinatorics
1. A Primer on Combinatorial Calculus
1.1 Combinatorial Classes
1.2 Words and Walks
1.2.1 Words and Languages
1.2.2 Lattice Walks
1.2.3 What Is a Good Combinatorial Formula?
1.2.4 Bijections
1.2.5 Combinatorial Operations
1.3 Formal Power Series
1.3.1 Ordinary Generating Functions
1.3.2 Coefficient Extraction Techniques
1.4 Basic Building Blocks
1.4.1 Epsilon Class
1.4.2 Atomic Class
1.4.3 Admissible Operators and Generating Functions
1.5 Combinatorial Specifications
1.6 S-regular Classes and Regular Languages
1.6.1 Finite Automata
1.7 Tree Classes
1.7.1 Lagrange Inversion
1.8 Algebraic Classes
1.9 Discussion
1.10 Problems
2. Combinatorial Parameters
2.1 Combinatorial Parameters
2.1.1 Bivariate Generating Functions
2.2 What Can We Do with a Bivariate Generating Function?
2.2.1 Higher Moments
2.2.2 Moment Inequalities and Concentration
2.3 Deriving Multivariate Generating Functions
2.3.1 Multidimensional Parameters
2.3.2 Inherited Parameters
2.3.3 Marking Substructures
2.4 On the Number of Components
2.5 Linear Functions of Parameters
2.6 Pathlength
2.7 Catalytic Parameters and the Kernel Method
2.8 Discussion
2.9 Problems
3. Derived and Transcendental Classes
3.1 The Diagonal of a Multivariable Series
3.1.1 The Ring of Formal Laurent Series
3.1.2 Basic Manipulations
3.1.3 Algebraic Functions Are Diagonals
3.1.4 Excursion Generating Functions
3.2 The Reflection Principle
3.2.1 A One-dimensional Reflection
3.2.2 A Two-dimensional Reflection
3.3 General Finite Reflection Groups
3.3.1 A Root Systems Primer
3.3.2 Enumerating Reflectable Walks
3.3.3 A Non-simple Example: Walks in A2
3.4 Discussion
3.5 Problems
Part II: Methods for Asymptotic Analysis
4. Generating Functions as Analytic Objects
4.1 Series Expansions
4.1.1 Convergence
4.1.2 Singularities
4.2 Poles and Laurent Expansions
4.2.1 Puiseux Expansions
4.3 The Exponential Growth of Coefficients
4.4 Finding Singularities
4.5 Complex Analysis
4.5.1 Primer on Contour Integrals
4.5.2 The Residue of a Function at a Point
4.6 Asymptotic Estimates for Meromorphic Functions
4.7 The Transfer Lemma
4.8 A General Process for Coefficient Analysis
4.9 Multiple Dominant Singularities
4.10 Saddle Point Estimation
4.11 Discussion
4.12 Problems
5. Parallel Taxonomies
5.1 Rational Functions
5.2 Algebraic Functions
5.3 D-finite Functions
5.3.1 Closure Properties
5.3.2 Is It or Isn’t It?
5.3.3 G-functions
5.3.4 Combinatorial Classes with D-finite Generating Functions
5.4 Differentiably Algebraic Functions
5.5 Classification Dichotomies
5.6 The Classification of Small Step Lattice Path Models
5.6.1 A Simple Recursion
5.6.2 Models with D-finite Generating Functions
5.6.3 Models with Non-D-finite Generating Functions
5.7 Groups and the Co-growth Problem
5.7.1 Excursions on Cayley Graphs
5.7.2 Amenability vs. D-finiteness
5.8 Discussion
5.9 Problems
6. Singularities of Multivariable Rational Functions
6.1 Visualizing Domains of Convergence
6.1.1 The Univariate Case
6.1.2 The Multivariable Case
6.2 The Exponential Growth
6.3 The Height Function
6.4 Visualizing Critical Points
6.5 Examples
6.5.1 Delannoy Numbers
6.5.2 Balanced Words
6.6 Discussion
6.7 Problems
7. Integration and Multivariable Coefficient Asymptotics
7.1 A Typical Problem
7.2 Warm-up: Stirling’s Approximation
7.3 Fourier-Laplace Integrals
7.4 Easy Inventory Problems
7.5 Generalizing the Strategy to Higher Dimensions
7.5.1 Multivariate Cauchy Integral Formula
7.5.2 A Formula for Fourier-Laplace Integrals
7.5.3 How Not to Transform This Integral
7.6 Example: Simple Walks
7.6.1 Exponential Growth
7.6.2 Estimating Cauchy Integrals
7.7 A More General Strategy
7.8 Discussion
7.9 Problems
8. Multiple Points
8.1 Algebraic Geometry Basics
8.2 Critical Points
8.3 Examples
8.3.1 Tandem Walks
8.3.2 Weighted Simple Walks
8.4 A Direct Formula for Powers
8.5 The Contribution of a Transverse Multiple Point
8.6 Discussion
8.7 Problems
9. Partitions
9.1 Integer Partitions
9.2 Vector Partitions
9.2.1 Integer Points in Polytopes
9.3 Asymptotic Analysis
9.3.1 The Singular Variety and Hyperplane Arrangements
9.3.2 Reducing to the Case of Transversal Intersection
9.3.3 Algebraic Independence
9.3.4 Decomposition Dictionary
9.3.5 Decomposition into Circuit-free Denominators
9.3.6 The Complete Reduction Algorithm
9.3.7 How to Compute a Reduction Rule
9.4 Asymptotics Theorem
9.5 An Example
9.5.1 An Exact Solution
9.5.2 An Asymptotic Solution
9.5.3 The Bases with No Broken Circuits
9.5.4 The Reduction Algorithm
9.5.5 Asymptotic Formula
9.6 Discussion
9.7 Problems
Bibliography
Glossary
Index
