Available materials: Notes (PDF)
The Catalan numbers form a ubiquitous sequence in combinatorics, counting a rich variety of objects. In this talk, we will explore the fundamental ideas that explain this ubiquity. We first present a classical probability problem from the 'Green Book' where the Catalan numbers arise naturally and then show the first of a few important bijections between the various objects that the Catalan numbers count. We then introduce the basic recurrence and derive the closed-form formula via generating functions. Finally, we present a more elegant solution using the reflection principle for random walks. Time permitting, we will discuss additional distinct 'Catalan' objects and their algebraic and combinatorial significance.
This was a blackboard talk; written notes are available above.