Combinatorics

Informationen zum Autor 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. Klappentext mathematical gem--freshly cleaned and polishedThis 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 style* Timely and appropriate examples* Numerous well-chosen exercises* Flexible modular format* Optional sections and appendicesHighlights 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 Zusammenfassung Updating and expanding upon the successful First Edition of Combinatorics, this new edition provides the foundation for mastering combinatorics. Blending an engaging style with mathematical rigor, Professor Merris provides a uniquely flexible tool for a wide variety of approaches to combinatorics. Inhaltsverzeichnis Preface ix Chapter 1 The Mathematics of Choice 1 1.1. The Fundamental Counting Principle 2 1.2. Pascal's Triangle 10 * 1.3. Elementary Probability 21 * 1.4. Error-Correcting Codes 33 1.5. Combinatorial Identities 43 1.6. Four Ways to Choose 56 1.7. The Binomial and Multinomial Theorems 66 1.8. Partitions 76 1.9. Elementary Symmetric Functions 87 * 1.10. Combinatorial Algorithms 100 Chapter 2 The Combinatorics of Finite Functions 117 2.1. Stirling Numbers of the Second Kind 117 2.2. Bells, Balls, and Urns 128 2.3. The Principle of Inclusion and Exclusion 140 2.4. Disjoint Cycles 152 2.5. Stirling Numbers of the First Kind 161 Chapter 3 Pólya's Theory of Enumeration 175 3.1. Function Composition 175 3.2. Permutation Groups 184 3.3. Burnside's Lemma 194 3.4. Symmetry Groups 206 3.5. Color Patterns 218 3.6. Pólya's Theorem 228 3.7. The Cycle Index Polynomial 241 Chapter 4 Generating Functions 253 4.1. Difference Sequences 253 4.2. Ordinary Generating Functions 268 4.3. Applications of Generating Functions 284 4.4. Exponential Generating Functions 301 4.5. Recursive Techniques 320 Chapter 5 Enumeration in Graphs 337 5.1. The Pigeonhole Principle 338 * 5.2. Edge Colorings and Ramsey Theory 347 5.3. Chromatic Polynomials 357 * 5.4. Planar Graphs 372 5.5. Matching Polynomials 383 5.6. Oriented Graphs 394 5.7. Graphic Partitions 408 Chapter 6 Codes and Designs 421 6.1. Linear Codes 422 6.2. Decoding Algorithms 432 6.3. Latin Squares 447 6.4. Balanced Incomplete Block Designs 461 Appendix A1 Symmetric Polynomials 477 Appendix A2 Sorting Algorithms 485 Appendix A3 Matrix Theory 495 Bibliography 501 Hints and Answers to Selected Odd-Numbered Exercises 503 Index of Notation 541 Index 547 ...

Sommario

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.

Relazione

"...broad and interesting..." ( Zentralblatt Math , Vol.1035, No.10, 2004)
"...engagingly written...a robust learning tool..." ( American Mathematical Monthly , March 2004)

Dettagli sul prodotto

• Autori: Merris, R Merris, Russell Merris, Russell (California State University Merris, MERRIS RUSSELL
• Editore: Wiley, John and Sons Ltd
• Lingue: Inglese
• Formato: Copertina rigida
• Pubblicazione: 28.08.2003
• EAN: 9780471262961
• ISBN: 978-0-471-26296-1
• Pagine: 576
• Serie: Wiley Series in Discrete Mathe; Wiley Series in Discrete Mathematics and Optimization; Wiley-Interscience Series in Discrete Mathematics and Optimization
• Categorie: Scienze naturali, medicina, informatica, tecnica > Matematica > Aritmetica, algebra; Mathematik, Kombinatorik, Mathematics, Combinatorics