Fibonacci and Catalan Numbers - An Introduction

Informationen zum Autor RALPH P. GRIMALDI, PHD, is Professor of Mathematics at Rose-Hulman Institute of Technology. With more than forty years of experience in academia, Dr. Grimaldi has published numerous articles in discrete mathematics, combinatorics, and graph theory. Over the past twenty years, he has developed and led mini-courses and workshops examining the Fibonacci and the Catalan numbers. Klappentext Discover the properties and real-world applications of the Fibonacci and the Catalan numbersWith clear explanations and easy-to-follow examples, Fibonacci and Catalan Numbers: An Introduction offers a fascinating overview of these topics that is accessible to a broad range of readers.Beginning with a historical development of each topic, the book guides readers through the essential properties of the Fibonacci numbers, offering many introductory-level examples. The author explains the relationship of the Fibonacci numbers to compositions and palindromes, tilings, graph theory, and the Lucas numbers.The book proceeds to explore the Catalan numbers, with the author drawing from their history to provide a solid foundation of the underlying properties. The relationship of the Catalan numbers to various concepts is then presented in examples dealing with partial orders, total orders, topological sorting, graph theory, rooted-ordered binary trees, pattern avoidance, and the Narayana numbers.The book features various aids and insights that allow readers to develop a complete understanding of the presented topics, including:* Real-world examples that demonstrate the application of the Fibonacci and the Catalan numbers to such fields as sports, botany, chemistry, physics, and computer science* More than 300 exercises that enable readers to explore many of the presented examples in greater depth* Illustrations that clarify and simplify the conceptsFibonacci and Catalan Numbers is an excellent book for courses on discrete mathematics, combinatorics, and number theory, especially at the undergraduate level. Undergraduates will find the book to be an excellent source for independent study, as well as a source of topics for research. Further, a great deal of the material can also be used for enrichment in high school courses. Zusammenfassung The material has been extensively class-tested for over ten years at both the author's own university and other institutions. The book is uniquely organized into two main sections, one on Fibonacci Numbers and one on Catalan Numbers, each containing subsections that explore related topics in intricate detail. Inhaltsverzeichnis Preface xi Part One The Fibonacci Numbers 1. Historical Background 3 2. The Problem of the Rabbits 5 3. The Recursive Definition 7 4. Properties of the Fibonacci Numbers 8 5. Some Introductory Examples 13 6. Compositions and Palindromes 23 7. Tilings: Divisibility Properties of the Fibonacci Numbers 33 8. Chess Pieces on Chessboards 40 9. Optics, Botany, and the Fibonacci Numbers 46 10. Solving Linear Recurrence Relations: The Binet Form for Fn 51 11. More on ¿ and ß : Applications in Trigonometry, Physics, Continued Fractions, Probability, the Associative Law, and Computer Science 65 12. Examples from Graph Theory: An Introduction to the Lucas Numbers 79 13. The Lucas Numbers: Further Properties and Examples 100 14. Matrices, The Inverse Tangent Function, and an Infinite Sum 113 15. The gcd Property for the Fibonacci Numbers 121 16. Alternate Fibonacci Numbers 126 17. One Final Example? 140 Part Two The Catalan Numbers 18. Historical Background 147 19. A First Example: A Formula for the Catalan Numbers 150 20. Some Further Initial Examples 159 21. Dyck Paths, Peaks, and Valleys 169 22. Young Tableaux, Comp...

List of contents

Preface xi
 
Part One. The Fibonacci Numbers
 
1. Historical Background 3
 
2. The Problem of the Rabbits 5
 
3. The Recursive Definition 7
 
4. Properties of the Fibonacci Numbers 8
 
5. Some Introductory Examples 13
 
6. Composition and Palindromes 23
 
7. Tilings: Divisibility Properties of the Fibonacci Numbers 33
 
8. Chess Pieces on Chessboards 40
 
9. Optics, Botany, and the Fibonacci Numbers 46
 
10. Solving Linear Recurrence Relations: The Binet Form for Fn 51
 
11. More on ± and ²: Applications in Trigonometry, Physics, Continued Fractions, Probability, the Associative Law, and Computer Science 65
 
12. Examples from Graph Theory: An Introduction to the Lucas Numbers 79
 
13. The Lucas Numbers: Further Properties and Examples 100
 
14. Matrices, The Inverse Tangent Function, and an Infinite Sum 113
 
15. The ged Property for the Fibonacci Numbers 121
 
16. Alternate Fibonacci Numbers 126
 
17. One Final Example? 140
 
Part Two. The Catalan Numbers
 
18. Historical Background 147
 
19. A First Example: A Formula for the Catalan Numbers 150
 
20. Some Further Initial Examples 159
 
21. Dyck Paths, Peaks, and Valleys 169
 
22. Young Tableaux, Compositions, and Vertices and Ares 183
 
23. Triangulating the Interior of a Convex Polygon 192
 
24. Some Examples from Graph Theory 195
 
25. Partial Orders, Total Orders, and Topological Sorting 205
 
26. Sequences and a Generating Tree 211
 
27. Maximal Cliques, a Computer Science Example, and the Tennis Ball Problem 219
 
28. The Catalan Numbers at Sporting Events 226
 
29. A Recurrence Relation for the Catalan Numbers 231
 
30. Triangulating the Interior of a Convex Polygon for the Second Time 236
 
31. Rooted Ordered Binary Trees, Pattern Avoidance, and Data Structures 238
 
32. Staircases, Arrangements of Coins, Handshaking Problem, and Noncrossing Partitions 250
 
33. The Narayana Numbers 268
 
34. Related Number Sequences: The Motzkin Numbers, The Fine Numbers, and The Schröder Numbers 282
 
35. Generalized Catalan Numbers 290
 
36. One Final Example? 296
 
Solutions for the Odd-Numbered Exercises 301
 
Index 355

Product details

• Authors: Grimaldi, R Grimaldi, Ralph Grimaldi, Ralph P. Grimaldi
• Publisher: Wiley, John and Sons Ltd
• Languages: English
• Product format: Hardback
• Released: 23.03.2012
• EAN: 9780470631577
• ISBN: 978-0-470-63157-7
• No. of pages: 380
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Probability theory, stochastic theory, mathematical statistics; Mathematik, Zahlentheorie, Kombinatorik, Diskrete Mathematik, Mathematics, Combinatorics, Discrete Mathematics, Number Theory