This text had its origins in the 1970s, when I first started teaching enumerative combinatorics and became aware of the ubiquity of Catalan numbers. Originally I just made a handwritten list for my own benefit. One of the earliest such lists has survived the ravages of time and appears in Appendix A. Over the years, the list became larger and more sophisticated. When I wrote the second volume of Enumerative Combinatorics (published in 1999), I included sixty-six combinatorial interpretations of Catalan numbers (Exercise 6.19) as well as numerous other exercises related to Catalan numbers. Since then I have continued to collect information on Catalan numbers, posting most of it on my “Catalan addendum” web page. Now the time has come to wrap up this 40 + years of compiling Catalan material, hence the present monograph. Much of it should be accessible to mathematically talented undergraduates or even high school students, while some parts will be of interest primarily to research mathematicians.

This monograph centers on 214 combinatorial interpretations of Catalan numbers (Chapters 2 and 3). Naturally some subjectivity is involved in deciding what should count as a new interpretation. It would be easy to expand the list by several hundred more entries by a little tweaking of the current items or by “transferring bijections.” For instance, there is a simple bijection φ between plane trees and ballot sequences. Thus, whenever we have a description of a Catalan object in terms of plane trees, we can apply φ and obtain a description in terms of ballot sequences. I have used my own personal tastes in deciding which such descriptions are worthwhile to include. If the reader feels that 214 is too low a number, then he or she can take solace in the solution to item 65, which discusses infinitely many combinatorial interpretations.

Also central to this monograph are the sixty-eight additional problems related to Catalan numbers in Chapters 4 and 5.