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# Generatingfunctionology

## 1st Edition

Author:
eBook ISBN: 9781483276632
Published Date: 28th January 1990
Page Count: 192
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## Description

Generatingfunctionology provides information pertinent to generating functions and some of their uses in discrete mathematics. This book presents the power of the method by giving a number of examples of problems that can be profitably thought about from the point of view of generating functions. Organized into five chapters, this book begins with an overview of the basic concepts of a generating function. This text then discusses the different kinds of series that are widely used as generating functions. Other chapters explain how to make much more precise estimates of the sizes of the coefficients of power series based on the analyticity of the function that is represented by the series. This book discusses as well the applications of the theory of generating functions to counting problems. The final chapter deals with the formal aspects of the theory of generating functions. This book is a valuable resource for mathematicians and students.

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

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

Solutions

References

Index

No. of pages:
192
Language:
English