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# Fibonacci and Catalan Numbers: An Introduction

Ralph P. Grimaldi

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 α and β: 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