Fibonacci and Catalan Numbers: An Introduction

Discover the properties and real–world applications of theFibonacci and the Catalan numbers

With clear explanations and easy–to–follow examples, Fibonacciand Catalan Numbers: An Introduction offers a fascinating overviewof these topics that is accessible to a broad range of readers.

Beginning with a historical development of each topic, the bookguides readers through the essential properties of the Fibonaccinumbers, offering many introductory–level examples. The authorexplains the relationship of the Fibonacci numbers to compositionsand palindromes, tilings, graph theory, and the Lucas numbers.

The book proceeds to explore the Catalan numbers, with theauthor drawing from their history to provide a solid foundation ofthe underlying properties. The relationship of the Catalan numbersto various concepts is then presented in examples dealing withpartial orders, total orders, topological sorting, graph theory,rooted–ordered binary trees, pattern avoidance, and the Narayananumbers.

The book features various aids and insights that allow readersto develop a complete understanding of the presented topics,including:

Real–world examples that demonstrate the application of theFibonacci 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 ofthe presented examples in greater depth

Illustrations that clarify and simplify the concepts

Fibonacci and Catalan Numbers is an excellent book for courseson discrete mathematics, combinatorics, and number theory,especially at the undergraduate level. Undergraduates will find thebook to be an excellent source for independent study, as well as asource of topics for research. Further, a great deal of thematerial can also be used for enrichment in high schoolcourses.

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 and : 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 

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.