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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:
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.
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
Overview
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 ...