|  | Grimaldi, Ralph
Fibonacci and Catalan Numbers
An Introduction
 1. Edition March 2012
87.90 Euro
2012. 380 Pages, Hardcover
ISBN 978-0-470-63157-7 - John Wiley & Sons
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| Short description
In this one-of-a-kind book, Ralph Grimaldi uses his extensive experience from the classroom and as a leader of mini-courses to present an accessible, single resource on the topics of Fibonacci Numbers and Catalan Numbers. The book builds a complete foundation for each topic through historical background, introductory-level examples, and in-depth discussion of related topics, as well as real-world examples and detailed exercise sets. It is ideal as a text or supplement for courses on discrete mathematics, combinatorics, and number theory at the upper-undergraduate and graduate levels, as well as a resource for undergraduate research courses, independent study projects, and senior/graduate theses.
From the 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
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