Combinatorics

ISBN-13: 9780471262961 / Angielski / Twarda / 2003 / 576 str.

Russell Merris

Opis

A mathematical gem-freshly cleaned and polished

This book is intended to be used as the text for a first course in combinatorics. the text has been shaped by two goals, namely, to make complex mathematics accessible to students with a wide range of abilities, interests, and motivations; and to create a pedagogical tool, useful to the broad spectrum of instructors who bring a variety of perspectives and expectations to such a course.

Features retained from the first edition: Lively and engaging writing styleTimely and appropriate examplesNumerous well-chosen exercisesFlexible modular formatOptional sections and appendices

Highlights of Second Edition enhancements: Smoothed and polished exposition, with a sharpened focus on key ideasExpanded discussion of linear codesNew optional section on algorithmsGreatly expanded hints and answers sectionMany new exercises and examples

Recenzje

“…broad and interesting…” ( Zentralblatt Math, Vol.1035, No.10, 2004)

“...engagingly written...a robust learning tool...” (American Mathematical Monthly, March 2004)

Spis treści

Preface.

Chapter 1: The Mathematics of Choice.

1.1. The Fundamental Counting Principle.

1.2. Pascal’s Triangle.

∗1.3. Elementary Probability.

∗1.4. Error–Correcting Codes.

1.5. Combinatorial Identities.

1.6. Four Ways to Choose.

1.7. The Binomial and Multinomial Theorems.

1.8. Partitions.

1.9. Elementary Symmetric Functions.

∗1.10. Combinatorial Algorithms.

Chapter 2: The Combinatorics of Finite Functions.

2.1. Stirling Numbers of the Second Kind.

2.2. Bells, Balls, and Urns.

2.3. The Principle of Inclusion and Exclusion.

2.4. Disjoint Cycles.

2.5. Stirling Numbers of the First Kind.

Chapter 3: Pólya’s Theory of Enumeration.

3.1. Function Composition.

3.2. Permutation Groups.

3.3. Burnside’s Lemma.

3.4. Symmetry Groups.

3.5. Color Patterns.

3.6. Pólya’s Theorem.

3.7. The Cycle Index Polynomial.

Chapter 4: Generating Functions.

4.1. Difference Sequences.

4.2. Ordinary Generating Functions.

4.3. Applications of Generating Functions.

4.4. Exponential Generating Functions.

4.5. Recursive Techniques.

Chapter 5: Enumeration in Graphs.

5.1. The Pigeonhole Principle.

∗5.2. Edge Colorings and Ramsey Theory.

5.3. Chromatic Polynomials.

∗5.4. Planar Graphs.

5.5. Matching Polynomials.

5.6. Oriented Graphs.

5.7. Graphic Partitions.

Chapter 6: Codes and Designs.

6.1. Linear Codes.

6.2. Decoding Algorithms.

6.3. Latin Squares.

6.4. Balanced Incomplete Block Designs.

Appendix A1: Symmetric Polynomials.

Appendix A2: Sorting Algorithms.

Appendix A3: Matrix Theory.

Bibliography.

Hints and Answers to Selected Odd–Numbered Exercises.

Index of Notation.

Index.

Note: Asterisks indicate optional sections that can be omitted without loss of continuity.

O autorze

RUSSELL MERRIS, PhD, is Professor of Mathematics and Computer Science at California State University, Hayward. Among his other books is Graph Theory, also published by Wiley.