| Description |
1 online resource (253 pages) |
| Series |
Discrete Mathematics and Its Applications Ser |
|
Discrete Mathematics and Its Applications Ser
|
| Contents |
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 |
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1.4.3 Admissible Operators and Generating Functions1.5 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 |
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2.3.3 Marking Substructures2.4 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 |
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3.3.2 Enumerating Reflectable Walks3.3.3 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 |
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4.8 A General Process for Coefficient Analysis4.9 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 |
| Notes |
5.6.3 Models with Non-D-finite Generating Functions |
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Print version record |
| Form |
Electronic book
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| ISBN |
9781351036818 |
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1351036815 |
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