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No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections or of the United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Rosewood Drive, Danvers, MA , , fax ISBNX acid-free paper 1.

Combinatorial analysis I. M47 This book is dedicated to my wife, Karen Diehl Merris. The Fundamental Counting Principle 2 1. Error-Correcting Codes 33 1. Combinatorial Identities 43 1. Four Ways to Choose 56 1. The Binomial and Multinomial Theorems 66 1. Partitions 76 1. Stirling Numbers of the Second Kind 2. Bells, Balls, and Urns 2. The Principle of Inclusion and Exclusion 2. Disjoint Cycles 2. Function Composition 3.

Permutation Groups 3. Symmetry Groups 3. Color Patterns 3. The Cycle Index Polynomial Note: Asterisks indicate optional sections that can be omitted without loss of continuity.

Chapter 4 Generating Functions 4. Difference Sequences 4. Ordinary Generating Functions 4. Applications of Generating Functions 4. Exponential Generating Functions 4. Recursive Techniques Chapter 5 Enumeration in Graphs 5. Edge Colorings and Ramsey Theory 5. Planar Graphs 5. Matching Polynomials 5. Oriented Graphs 5. Graphic Partitions Chapter 6 Codes and Designs 6.

Linear Codes 6. Decoding Algorithms 6. Latin Squares 6. Preface This book is intended to be used as the text for a course in combinatorics at the level of beginning upper division students. It has been shaped by two goals: to make some fairly deep mathematics accessible to students with a wide range of abilities, interests, and motivations and to create a pedagogical tool useful to the broad spec- trum of instructors who bring a variety of perspectives and expectations to such a course.

Following a basic foundation in Chapters 1 and 2, each instructor is free to pick and choose the most appropriate topics from the remaining four chapters. As sum- marized in the chart below, Chapters 3—6 are completely independent of each other. All of these topics are reviewed in Appendix A3. Strategies that promote student engagement are a lively writing style, timely and appropriate examples, interesting historical anecdotes, a variety of exercises tem- pered and enlivened by suitable hints and answers , and judicious use of footnotes and appendices to touch on topics better suited to more advanced students.

These are things about which there is general agreement, at least in principle. There is less agreement about how to focus student energies on attainable objec- tives, in part because focusing on some things inevitably means neglecting others. If the course is approached as a last chance to expose students to this marvelous sub- ject, it probably will be. These topics appear and reappear throughout the text.

While other celebrated examples, e. For the sake of argument, let us stipulate that these roles could just as well have been reversed. The issue is that beginning upper division students cannot be expected to absorb, much less appreci- ate, all of these special arrays and sequences in a single semester. Material new to the second edition includes an optional section on algo- rithms, several new examples, and many new exercises, some designed to guide students to discover and prove nontrivial results for themselves.

Finally, the section of hints and answers has been expanded by an order of magnitude. If it seems desirable to cover some but not all of Chapter 3, there are many natural places to exit in favor of something else, e. Optional Sections 1. With the same caveat, Section 1. The material in Section 6. The book contains much more material then can be covered in a single semester.

Among the possible syllabi for a one semester course are the following: Chapters 1, 2, and 4 and Sections 3. I am especially grateful for the tireless assistance of Cynthia Johnson and Ken Rebman.

Thus, the idea of an irrational is deeper than the idea of an integer. In the version presented here, one is faced with a sequence of decisions, each of which involves some number of choices. It is from situations like this that the chapter derives its name. Special cases of these numbers are addressed from a combinatorial perspective in Section 1.

Section 1. Sections 1.



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This book is intended to be used as the text for a first course in combinatorics. Russell Merris. A mathematical gem—freshly cleaned and polished This book is intended to be used as the text for a first course in combinatorics. Features retained from the first edition: Lively and engaging writing style Timely and appropriate examples Numerous well-chosen exercises Flexible modular format Optional sections and appendices Highlights of Second Edition enhancements: Smoothed and polished exposition, with a sharpened focus on key ideas Expanded discussion of linear codes New optional section on algorithms Greatly expanded hints and answers section Many new exercises and examples. Chapter 2 The Combinatorics of Finite Functions. Chapter 4 Generating Functions.

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