# Fibonacci and Catalan Numbers: An Introduction

John Wiley & Sons, 21 Şub 2012 - 448 sayfa
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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### İçindekiler

 Compositions and Palindromes 2 Tilings Divisibility Properties of the Fibonacci Numbers 9 Optics Botany and the Fibonacci Numbers 18 More on α and β Applications in Trigonometry Physics 50 Examples from Graph Theory An Introduction to the Lucas Numbers 60 The Lucas Numbers Further Properties and Examples 77 Matrices The Inverse Tangent Function and an Infinite Sum113 8 The gcd Property for the Fibonacci Numbers 14
 Young Tableaux Compositions and Vertices and Arcs 74 Triangulating the Interior of a Convex Polygon 82 Some Examples from Graph Theory 85 Partial Orders Total Orders and Topological Sorting 6 Sequences and a Generating Tree 11 Maximal Cliques a Computer Science Example and the Tennis Ball Problem 17 The Catalan Numbers at Sporting Events 22 A Recurrence Relation for the Catalan Numbers 26

 Alternate Fibonacci Numbers 17 One Final Example? 29 References 30 The Catalan Numbers 43 Historical Background 44 A First Example A Formula for the Catalan Numbers 46 Some Further Initial Examples 54 Dyck Paths Peaks and Valleys 62
 Rooted Ordered Binary Trees Pattern Avoidance and Data 31 Staircases Arrangements of Coins The Handshaking Problem 32 The Narayana Numbers 33 Related Number Sequences The Motzkin Numbers The Fine 34 Generalized Catalan Numbers 35 Solutions for the OddNumbered Exercises 39 Index 38 Telif Hakkı

### Yazar hakkında (2012)

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.