# An Introduction to Catalan Numbers

• Steven Roman
Textbook

Part of the Compact Textbooks in Mathematics book series (CTM)

1. Front Matter
Pages i-xii
2. Steven Roman
Pages 1-5
3. Steven Roman
Pages 7-11
4. Steven Roman
Pages 13-20
5. Steven Roman
Pages 21-22
6. Steven Roman
Pages 23-28
7. Steven Roman
Pages 29-38
8. Steven Roman
Pages 39-43
9. Steven Roman
Pages 45-50
10. Steven Roman
Pages 51-60
11. Steven Roman
Pages 61-71
12. Steven Roman
Pages 73-86
13. Back Matter
Pages 87-121

### Introduction

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 Zn+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

### Keywords

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

#### Authors and affiliations

• Steven Roman
• 1
1. 1.California State University, Fullerton IRVINEUSA