Fibonacci and Catalan Numbers

An Introduction

Inbunden, Engelska, 2012

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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.

Produktinformation

Utgivningsdatum2012-03-23
Mått164 x 241 x 26 mm
Vikt676 g
FormatInbunden
SpråkEngelska
Antal sidor380
FörlagJohn Wiley & Sons Inc
ISBN9780470631577

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Preface xiPart One The Fibonacci Numbers1. Historical Background 32. The Problem of the Rabbits 53. The Recursive Definition 74. Properties of the Fibonacci Numbers 85. Some Introductory Examples 136. Compositions and Palindromes 237. Tilings: Divisibility Properties of the Fibonacci Numbers 338. Chess Pieces on Chessboards 409. Optics, Botany, and the Fibonacci Numbers 4610. Solving Linear Recurrence Relations: The Binet Form for Fn5111. More on α and β: Applications in Trigonometry, Physics, Continued Fractions, Probability, the Associative Law, and Computer Science 6512. Examples from Graph Theory: An Introduction to the Lucas Numbers 7913. The Lucas Numbers: Further Properties and Examples 10014. Matrices, The Inverse Tangent Function, and an Infinite Sum 11315. The gcd Property for the Fibonacci Numbers 12116. Alternate Fibonacci Numbers 12617. One Final Example? 140Part Two The Catalan Numbers18. Historical Background 14719. A First Example: A Formula for the Catalan Numbers 15020. Some Further Initial Examples 15921. Dyck Paths, Peaks, and Valleys 16922. Young Tableaux, Compositions, and Vertices and Arcs 18323. Triangulating the Interior of a Convex Polygon 19224. Some Examples from Graph Theory 19525. Partial Orders, Total Orders, and Topological Sorting 20526. Sequences and a Generating Tree 21127. Maximal Cliques, a Computer Science Example, and the Tennis Ball Problem 21928. The Catalan Numbers at Sporting Events 22629. A Recurrence Relation for the Catalan Numbers 23130. Triangulating the Interior of a Convex Polygon for the Second Time 23631. Rooted Ordered Binary Trees, Pattern Avoidance, and Data Structures 23832. Staircases, Arrangements of Coins, The Handshaking Problem, and Noncrossing Partitions 25033. The Narayana Numbers 26834. Related Number Sequences: The Motzkin Numbers, The Fine Numbers, and The Schröder Numbers 28235. Generalized Catalan Numbers 29036. One Final Example? 296Solutions for the Odd-Numbered Exercises 301Index 355