Fibonacci and Catalan Numbers: An Introduction

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John Wiley & Sons, 21 Şub 2012 - 448 sayfa
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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.

 

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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
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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.

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