An Introduction to Catalan Numbers

1. Front Matter

   Pages i-xii

   PDF

2. Introduction

   • Steven Roman

   Pages 1-5

3. Dyck Words

   • Steven Roman

   Pages 7-11

4. The Catalan Numbers

   • Steven Roman

   Pages 13-20

5. Catalan Numbers and Paths

   • Steven Roman

   Pages 21-22

6. Catalan Numbers and Trees

   • Steven Roman

   Pages 23-28

7. Catalan Numbers and Geometric Widgits

   • Steven Roman

   Pages 29-38

8. Catalan Numbers and Algebraic Widgits

   • Steven Roman

   Pages 39-43

9. Catalan Numbers and Interval Structures

   • Steven Roman

   Pages 45-50

10. Catalan Numbers and Partitions

    • Steven Roman

    Pages 51-60

11. Catalan Numbers and Permutations

    • Steven Roman

    Pages 61-71

12. Catalan Numbers and Semiorders

    • Steven Roman

    Pages 73-86

13. Back Matter

    Pages 87-121

    PDF

This textbook provides an introduction to the Catalan numbers and their remarkable properties, along with their various applications in combinatorics. Intended to be accessible to students new to the subject, the book begins with more elementary topics before progressing to more mathematically sophisticated topics. Each chapter focuses on a specific combinatorial object counted by these numbers, including paths, trees, tilings of a staircase, null sums in Z_n+1, interval structures, partitions, permutations, semiorders, and more. Exercises are included at the end of book, along with hints and solutions, to help students obtain a better grasp of the material. The text is ideal for undergraduate students studying combinatorics, but will also appeal to anyone with a mathematical background who has an interest in learning about the Catalan numbers.

“Roman does an admirable job of providing an introduction to Catalan numbers of a different nature from the previous ones. He has made an excellent choice of topics in order to convey the flavor of Catalan combinatorics. [Readers] will acquire a good feeling for why so many mathematicians are enthralled by the remarkable ubiquity and elegance of Catalan numbers.”

 - From the foreword by Richard Stanley

• Algebraic Widgits
• Catalan numbers
• Combinatorics
• Dyck Words
• Geometric Widgits
• Interval Structures
• Permutations
• Semiorders

“The pace of this book is of an introductory nature, the coverage of Catalan numbers is rigorous and will provide the reader with a firm grasp of many of the properties of these numbers. … a scholarly work and one that number theorists will find well worth reading.” (James Van Speybroeck, Computing Reviews, April, 2016)

“The book is supplemented by a set of exercises for those who want to go further. The reader is helped by 70 carefully designed figures throughout the book. The 24 enumeration problems are selected carefully to show Catalan numbers from very different viewpoints. Several of these problems likely have their most readable write-up in this book. A must for anyone, who wants to understand the significance of Catalan numbers!” (László Székely, zbMATH 1342.05002, 2016)

• California State University, Fullerton , IRVINE, USA

  Steven Roman

Steven Roman is Professor Emeritus of Mathematics at California State University Fullerton.  He is the author of numerous other mathematics textbooks, including Field Theory (2006), Advanced Linear Algebra(2008), Fundamentals of Group Theory (2012), Introduction to the Mathematics of Finance (2012), and An Introduction to Catalan Numbers (2015).                        

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