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E-raamat: Analytic Combinatorics: A Multidimensional Approach

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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

About the author

Marni Mishna is a professor of mathematics at Simon Fraser University in British Columbia. Her research investigates interactions between discrete structures and many diverse areas such as representation theory, functional equation theory, and algebraic geometry. Her specialty is the development of analytic tools to study the large-scale behavior of discrete objects.
Preface xiii
Symbols xvii
Welcome to Analytic Combinatorics xix
I Enumerative Combinatorics 1(78)
1 A Primer on Combinatorial Calculus
3(32)
1.1 Combinatorial Classes
4(1)
1.2 Words and Walks
4(7)
1.2.1 Words and Languages
5(1)
1.2.2 Lattice Walks
6(1)
1.2.3 What Is a Good Combinatorial Formula?
7(1)
1.2.4 Bijections
7(1)
1.2.5 Combinatorial Operations
8(3)
1.3 Formal Power Series
11(3)
1.3.1 Ordinary Generating Functions
12(1)
1.3.2 Coefficient Extraction Techniques
13(1)
1.4 Basic Building Blocks
14(3)
1.4.1 Epsilon Class
14(1)
1.4.2 Atomic Class
14(1)
1.4.3 Admissible Operators and Generating Functions
15(2)
1.5 Combinatorial Specifications
17(1)
1.6 S-regular Classes and Regular Languages
18(3)
1.6.1 Finite Automata
19(2)
1.7 Tree Classes
21(3)
1.7.1 Lagrange Inversion
23(1)
1.8 Algebraic Classes
24(4)
1.9 Discussion
28(1)
1.10 Problems
29(6)
2 Combinatorial Parameters
35(20)
2.1 Combinatorial Parameters
36(1)
2.1.1 Bivariate Generating Functions
36(1)
2.2 What Can We Do with a Bivariate Generating Function?
37(4)
2.2.1 Higher Moments
38(2)
2.2.2 Moment Inequalities and Concentration
40(1)
2.3 Deriving Multivariate Generating Functions
41(5)
2.3.1 Multidimensional Parameters
41(1)
2.3.2 Inherited Parameters
42(1)
2.3.3 Marking Substructures
43(3)
2.4 On the Number of Components
46(1)
2.5 Linear Functions of Parameters
46(1)
2.6 Pathlength
47(2)
2.7 Catalytic Parameters and the Kernel Method
49(2)
2.8 Discussion
51(1)
2.9 Problems
51(4)
3 Derived and Transcendental Classes
55(24)
3.1 The Diagonal of a Multivariable Series
56(8)
3.1.1 The Ring of Formal Laurent Series
58(1)
3.1.2 Basic Manipulations
59(1)
3.1.3 Algebraic Functions Are Diagonals
60(1)
3.1.4 Excursion Generating Functions
61(3)
3.2 The Reflection Principle
64(5)
3.2.1 A One-dimensional Reflection
65(2)
3.2.2 A Two-dimensional Reflection
67(2)
3.3 General Finite Reflection Groups
69(5)
3.3.1 A Root Systems Primer
69(2)
3.3.2 Enumerating Reflectable Walks
71(1)
3.3.3 A Non-simple Example: Walks in A2
72(2)
3.4 Discussion
74(1)
3.5 Problems
75(4)
II Methods for Asymptotic Analysis 79(134)
4 Generating Functions as Analytic Objects
81(30)
4.1 Series Expansions
82(2)
4.1.1 Convergence
82(1)
4.1.2 Singularities
83(1)
4.2 Poles and Laurent Expansions
84(3)
4.2.1 Puiseux Expansions
86(1)
4.3 The Exponential Growth of Coefficients
87(4)
4.4 Finding Singularities
91(1)
4.5 Complex Analysis
92(3)
4.5.1 Primer on Contour Integrals
92(1)
4.5.2 The Residue of a Function at a Point
93(2)
4.6 Asymptotic Estimates for Meromorphic Functions
95(3)
4.7 The Transfer Lemma
98(1)
4.8 A General Process for Coefficient Analysis
99(3)
4.9 Multiple Dominant Singularities
102(4)
4.10 Saddle Point Estimation
106(2)
4.11 Discussion
108(1)
4.12 Problems
108(3)
5 Parallel Taxonomies
111(28)
5.1 Rational Functions
112(1)
5.2 Algebraic Functions
113(2)
5.3 D-finite Functions
115(6)
5.3.1 Closure Properties
116(1)
5.3.2 Is It or Isn't It?
117(3)
5.3.3 G-functions
120(1)
5.3.4 Combinatorial Classes with D-finite Generating Functions
120(1)
5.4 Differentiably Algebraic Functions
121(2)
5.5 Classification Dichotomies
123(1)
5.6 The Classification of Small Step Lattice Path Models
124(6)
5.6.1 A Simple Recursion
125(2)
5.6.2 Models with D-finite Generating Functions
127(2)
5.6.3 Models with Non-D-finite Generating Functions
129(1)
5.7 Groups and the Co-growth Problem
130(3)
5.7.1 Excursions on Cayley Graphs
131(1)
5.7.2 Amenability vs. D-finiteness
132(1)
5.8 Discussion
133(3)
5.9 Problems
136(3)
6 Singularities of Multivariable Rational Functions
139(16)
6.1 Visualizing Domains of Convergence
140(4)
6.1.1 The Univariate Case
140(1)
6.1.2 The Multivariable Case
141(3)
6.2 The Exponential Growth
144(2)
6.3 The Height Function
146(2)
6.4 Visualizing Critical Points
148(1)
6.5 Examples
149(3)
6.5.1 Delannoy Numbers
149(1)
6.5.2 Balanced Words
150(2)
6.6 Discussion
152(1)
6.7 Problems
152(3)
7 Integration and Multivariable Coefficient Asymptotics
155(22)
7.1 A Typical Problem
156(1)
7.2 Warm-up: Stirling's Approximation
157(2)
7.3 Fourier-Laplace Integrals
159(1)
7.4 Easy Inventory Problems
160(2)
7.5 Generalizing the Strategy to Higher Dimensions
162(2)
7.5.1 Multivariate Cauchy Integral Formula
162(1)
7.5.2 A Formula for Fourier-Laplace Integrals
162(1)
7.5.3 How Not to Transform This Integral
163(1)
7.6 Example: Simple Walks
164(3)
7.6.1 Exponential Growth
164(1)
7.6.2 Estimating Cauchy Integrals
165(2)
7.7 A More General Strategy
167(5)
7.8 Discussion
172(2)
7.9 Problems
174(3)
8 Multiple Points
177(14)
8.1 Algebraic Geometry Basics
178(2)
8.2 Critical Points
180(2)
8.3 Examples
182(3)
8.3.1 Tandem Walks
182(2)
8.3.2 Weighted Simple Walks
184(1)
8.4 A Direct Formula for Powers
185(1)
8.5 The Contribution of a Transverse Multiple Point
186(2)
8.6 Discussion
188(1)
8.7 Problems
189(2)
9 Partitions
191(22)
9.1 Integer Partitions
192(2)
9.2 Vector Partitions
194(3)
9.2.1 Integer Points in Polytopes
196(1)
9.3 Asymptotic Analysis
197(7)
9.3.1 The Singular Variety and Hyperplane Arrangements
198(1)
9.3.2 Reducing to the Case of Transversal Intersection
199(2)
9.3.3 Algebraic Independence
201(1)
9.3.4 Decomposition Dictionary
202(1)
9.3.5 Decomposition into Circuit-free Denominators
202(1)
9.3.6 The Complete Reduction Algorithm
203(1)
9.3.7 How to Compute a Reduction Rule
204(1)
9.4 Asymptotics Theorem
204(1)
9.5 An Example
205(5)
9.5.1 An Exact Solution
206(1)
9.5.2 An Asymptotic Solution
207(1)
9.5.3 The Bases with No Broken Circuits
207(1)
9.5.4 The Reduction Algorithm
207(2)
9.5.5 Asymptotic Formula
209(1)
9.6 Discussion
210(1)
9.7 Problems
211(2)
Bibliography 213(12)
Glossary 225(2)
Index 227
Marni Mishna is a professor of mathematics at Simon Fraser University, BC, Canada
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