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