generatingfunctionology: Third Edition,9781568812793

generatingfunctionology: Third Edition

by ;
Edition: 3rd
ISBN13: 9781568812793
ISBN10: 1568812795
Format: Hardcover
Pub. Date: 2005-12-20
Publisher(s): A. K. Peters
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Summary

Generating functions, one of the most important tools in enumerative combinatorics, are a bridge between discrete mathematics and continuous analysis. Generating functions have numerous applications in mathematics, especially in Combinatorics, Probability Theory, Statistics, Theory of Markov Chains, Number Theory. One of the most important and relevant recent applications of combinatorics lies in the development of Internet search engines whose incredible capabilities dazzle even the mathematically trained user.

Table of Contents

Preface ix
Introductory Ideas and Examples
1(30)
An Easy Two Term Recurrence
4(1)
A Slightly Harder Two Term Recurrence
5(4)
A Three Term Recurrence
9(2)
A Three Term Boundary Value Problem
11(4)
Two Independent Variables
15(3)
Another 2-Variable Case
18(8)
Exercises
26(5)
Series
31(46)
Formal Power Series
31(4)
The Calculus of Formal Ordinary Power Series Generating Functions
35(6)
The Calculus of Formal Exponential Generating Functions
41(7)
Power Series, Analytic Theory
48(7)
Some Useful Power Series
55(4)
Dirichlet Series, Formal theory
59(9)
Exercises
68(9)
Cards, Decks, and Hands: The Exponential Formula
77(38)
Introduction
77(2)
Definitions and a Question
79(1)
Examples of Exponential Families
80(3)
The Main Counting Theorems
83(4)
Permutations and Their Cycles
87(1)
Set Partitions
88(1)
A Subclass of Permutations
89(1)
Involutions, etc.
90(1)
2-Regular Graphs
91(1)
Counting Connected Graphs
92(1)
Counting Labeled Bipartite Graphs
93(2)
Counting Labeled Trees
95(2)
Exponential Families and Polynomials of `Binomial Type'
97(1)
Unlabeled Cards and Hands
98(4)
The Money Changing Problem
102(5)
Partitions of Integers
107(2)
Rooted Trees and Forests
109(1)
Historical Notes
110(1)
Exercises
110(5)
Applications of Generating Functions
115(66)
Generating Functions Find Averages, etc.
115(2)
A Generatingfunctionological View of the Sieve Method
117(9)
The `Snake Oil' Method for Easier Combinatorial Identities
126(12)
WZ Pairs Prove Harder Identities
138(7)
Generating Functions and Unimodality, Convexity, etc.
145(3)
Generating Functions Prove Congruences
148(2)
The Cycle Index of the Symmetric Group
150(5)
How Many Permutations Have Square Roots?
155(4)
Counting Polyominoes
159(4)
Exact Covering Sequences
163(3)
Waiting for a String
166(1)
Blocks of 1
167(3)
Exercises
170(11)
Analytic and Asymptotic Methods
181(26)
The Lagrange Inversion Formula
181(4)
Analyticity and Asymptotics (I): Poles
185(7)
Analyticity and Asymptotics (II): Algebraic Singularities
192(4)
Analyticity and Asymptotics (III): Hayman's Method
196(7)
Exercises
203(4)
A. Using Maple and Mathematica
207(6)
A.1 Series Manipulation
208(1)
A.2 The RSolve.m Routine
209(2)
A.3 Asymptotics in Maple
211(2)
Solutions 213(26)
References 239(4)
Index 243

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