# Book:Herbert S. Wilf/generatingfunctionology

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## Herbert S. Wilf:

## Herbert S. Wilf: *generatingfunctionology*

Published $\text {1990}$, **Academic Press, Inc.**

### Subject Matter

### Contents

- Preface

- Preface to the Second Edition

**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
- Exercises

**Chapter 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
- Exercises

**Chapter 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
- Exercises

**Chapter 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
- 4.7 The cycle index of the symmetric group
- 4.8 How many permutations have square roots?
- 4.9 Counting polyominoes
- 4.10 Exact covering sequences
- Exercises

**Chapter 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
- Exercises

- Appendix: Using
*Maple*^{TM}and*Mathematica*^{TM}

- Appendix: Using

**Solutions**

**References**