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Counting : The Art of Enumerative Combinatorics :  The Art of Enumerative Combinatorics - George E. Martin

Counting : The Art of Enumerative Combinatorics

The Art of Enumerative Combinatorics


Published: June 2001
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Counting: The Art of Enumerative Combinatorics provides an introduction to discrete mathematics that addresses questions that begin, How many ways are there to...For example, How many ways are there to order a collection of 12 ice cream cones if 8 flavors are available? At the end of the book the reader should be able to answer such nontrivial counting questions as, How many ways are there to color the faces of a cube if k colors are available with each face having exactly one color? or How many ways are there to stack n poker chips, each of which can be red, white, blue, or green, such that each red chip is adjacent to at least 1 green chip? Since there are no prerequisites, this book can be used for college courses in combinatorics at the sophomore level for either computer science or mathematics students. The first five chapters have served as the basis for a graduate course for in-service teachers. Chapter 8 introduces graph theory.

From the reviews:

"Much of Martin's charming and accessible text could be used with bright school students. ... The book is rounded off by a section called `The back of the book' which includes solutions and discussion of many exercises. George E. Martin is a remarkable writer who brings combinatorics alive. He has written a splendid introduction that requires very few prerequisites, yet soon delivers the reader into some highly effective methods of counting. The book is highly recommended." (S. C. Russen, The Mathematical Gazette, Vol. 88 (551), 2004)

"This truly is an undergraduate mathematics text; parts of it could be the text for a high school combinatorics course. The author has made a successful effort to illuminate and teach the elementary parts of combinatorics. He uses examples and problems to teach; there are 245 problems in Chapter 1! ... If I were not retired and had been asked to teach an undergraduate course in combinatorics, I would have liked to use this book." (W. Moser, Mathematical Reviews, Issue 2002 g)

"This book is a nice textbook on enumerative combinatorics to undergraduates. It introduces the most important ideas ... . A lot of `easy' applications are given and homework is listed (with hints). The book also touches some elementary graph enumeration problems. The text is clear and easy to follow. It is even suitable to learn it alone, which is also aided by nice exam problems." (Peter L. Erdoes, Zentralblatt MATH, Vol. 968, 2001)

"The teaching of topics in discrete mathematics is becoming increasingly popular and this text contains chapters on a number of pertinent areas for exposure at an elementary level. ... The author uses non-worked discovery-type examples to lead into observations about the material. ... There are many interesting exercises for the student to attempt. These are spread throughout the various chapters and are effective in developing interest in the topics. The book contains a `Back of the Book' section rather than an Answers section." (M. J. Williams, The Australian Mathematical Society Gazette, Vol. 29 (1), 2002)

Elementary Enumeration
Counting Is Hardp. 1
Conventionsp. 2
Permutationsp. 3
A Discussion Questionp. 4
The Pigeonhole Principlep. 4
n Chose r by Way of MISSISSIPPIp. 5
The Round Tablep. 7
The Birthday Problemp. 10
n Chose r with Repetitionp. 11
Ice Cream Cones - The Double Dipp. 17
Block Walkingp. 18
Quickies and Knightsp. 19
The Binomial Theoremp. 22
Homework for a Weekp. 23
Three Hour Examsp. 24
The Principle of Inclusion and Exclusion
Introduction to PIEp. 27
Proof of PIEp. 30
Derangementsp. 32
Partitionsp. 34
Balls into Boxesp. 35
A Plethora of Problemsp. 38
Eating Outp. 39
Generating Functions
What Is x?p. 43
An Algebraic Excursion to [Riemann integral][[x]]p. 44
Introducing Generating Functionsp. 46
Clotheslinesp. 47
Examples and Homeworkp. 52
Computationsp. 57
Exponential Generating Functionsp. 59
Comprehensive Examsp. 64
Symmetry Groupsp. 67
Legendre's Theoremp. 71
Permutation Groupsp. 74
Generatorsp. 77
Cyclic Groupsp. 80
Equivalence and Isomorphismp. 82
The Definitionp. 85
Burnside's Lemmap. 88
Applications of Burnside's Lemmap. 94
Polya's Pattern Inventoryp. 100
Necklacesp. 108
Recurrence Relations
Examples of Recurrence Relationsp. 113
The Fibonacci Numbersp. 117
A Dozen Recurrence Problemsp. 121
Solving Recurrence Relationsp. 122
The Catalan Numbersp. 125
Nonhomogeneous Recurrence Relationsp. 133
Mathematical Induction
The Principle of Mathematical Inductionp. 137
The Strong Form of Mathematical Inductionp. 142
Hall's Marriage Theoremp. 146
The Vocabulary of Graph Theoryp. 153
Walks, Trails, Circuits, Paths, and Cyclesp. 160
Treesp. 168
Degree Sequencesp. 175
Euler's Formulap. 177
The Back of the Bookp. 183
Indexp. 247
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780387952253
ISBN-10: 038795225X
Series: Springer Undergraduate Texts in Mathematics and Technology
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 252
Published: June 2001
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 24.6 x 16.2  x 2.4
Weight (kg): 0.56