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Fibonacci and Catalan Numbers: An Introduction

ISBN: 978-1-118-30438-9
May 2012
Fibonacci and Catalan Numbers: An Introduction (1118304381) cover image
Discover the properties and real-world applications of the Fibonacci and the Catalan numbers

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

Beginning with a historical development of each topic, the book guides readers through the essential properties of the Fibonacci numbers, offering many introductory-level examples. The author explains the relationship of the Fibonacci numbers to compositions and palindromes, tilings, graph theory, and the Lucas numbers.

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

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

  • Real-world examples that demonstrate the application of the Fibonacci 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 of the presented examples in greater depth

  • Illustrations that clarify and simplify the concepts

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

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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 

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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.

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