Generatingfunctionology

Chapter 1: Introductory Ideas and Examples 1.1 An Easy Two Term Recurrence 1.2 A Slightly Harder Two Term Recurrence 1.3 A Three Term Recurrence 1.4 A Three Term Boundary Value Problem 1.5 Two Independent Variables 1.6 Another 2-Variable Case ExercisesChapter 2: Series 2.1 Formal Power Series 2.2 The Calculus of Formal Ordinary Power Series Generating Functions 2.3 The Calculus of Formal Exponential Generating Functions 2.4 Power Series, Analytic Theory 2.5 Some Useful Power Series 2.6 Dirichlet Series, Formal Theory ExercisesChapter 3: Cards, Decks, and Hands: The Exponential Formula 3.1 Introduction 3.2 Definitions and a Question 3.3 Examples of Exponential Families 3.4 The Main Counting Theorems 3.5 Permutations and their Cycles 3.6 Set Partitions 3.7 A Subclass of Permutations 3.8 Involutions, Etc. 3.9 2-Regular Graphs 3.10 Counting Connected Graphs 3.11 Counting Labeled Bipartite Graphs 3.12 Counting Labeled Trees 3.13 Exponential Families and Polynomials of 'Binomial Type.' 3.14 Unlabeled Cards and Hands 3.15 The Money Changing Problem 3.16 Partitions of Integers 3.17 Rooted Trees and Forests 3.18 Historical Notes ExercisesChapter 4: Applications of Generating Functions 4.1 Generating Functions Find Averages, Etc. 4.2 A Generatingfunctionological View of The Sieve Method 4.3 The 'Snake Oil' Method For Easier Combinatorial Identities 4.4 Wz Pairs Prove Harder Identities 4.5 Generating Functions and Unimodality, Convexity, Etc. 4.6 Generating Functions Prove Congruences ExercisesChapter 5: Analytic and Asymptotic Methods 5.1 The Lagrange Inversion Formula 5.2 Analyticity and Asymptotics (I): Poles 5.3 Analyticity and Asymptotics (II): Algebraic Singularities 5.4 Analyticity and Asymptotics (III): Hayman's Method ExercisesSolutionsReferencesIndex