Skip to content # Catalan Numbers

## Activity Guide

# This guide looks a set of related problems and the methodology behind Catalan Numbers.

Davis, Tom. (2020). Catalan Numbers. In S. Kohli (Ed.), Math Circle Activity Database. American Institute of Mathematics. https://mathcircles.org/activity/catalan-numbers/

Contributor(s): Tom Davis

Math Circles: MTC Network

- Activity Description
- Topics Covered
- Session Styles
- Mathematical Practices

Suppose you have n pairs of parentheses and you would like to form valid groupings of them, how many groupings are there for each value of n? How many “mountain ranges” can you form with n upstrokes and n downstrokes that all stay above the original line? What about counting the number of ways to triangulate a regular polygon with n + 2 sides? We begin with a set of problems that will be shown to be completely equivalent. The solution to each problem is the same sequence of numbers called the Catalan numbers.