Fibonacci and Catalan Numbers

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 Numbers33
8. Chess Pieces on Chessboards 40
9. Optics, Botany, and the Fibonacci Numbers 46
10. Solving Linear Recurrence Relations: The Binet Form forFn 51
11. More on alpha and beta: Applications in Trigonometry,Physics, Continued Fractions, Probability, the Associative Law, andComputer Science 65
12. Examples from Graph Theory: An Introduction to the LucasNumbers 79
13. The Lucas Numbers: Further Properties and Examples 100
14. Matrices, The Inverse Tangent Function, and an Infinite Sum113
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 Sorting205
26. Sequences and a Generating Tree 211
27. Maximal Cliques, a Computer Science Example, and the TennisBall 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 theSecond Time 236
31. Rooted Ordered Binary Trees, Pattern Avoidance, and DataStructures 238
32. Staircases, Arrangements of Coins, Handshaking Problem, andNoncrossing Partitions 250
33. The Narayana Numbers 268
34. Related Number Sequences: The Motzkin Numbers, The FineNumbers, and The Schröder Numbers 282
35. Generalized Catalan Numbers 290
36. One Final Example? 296
Solutions for the Odd-Numbered Exercises 301
Index 355