The Catalan numbers are one of the most important sequences in combinatorics, which studies the mathematics of counting and arranging finite discrete structures. Combinatorics is an essential branch of mathematics as it provides efficient techniques for enumerating quantities that would otherwise be difficult to quantify by conventional methods. The Catalan numbers have several equivalent definitions, including the recursive and closed forms. While the closed form is more straightforward, the recursive definition is more helpful in solving problems in combinatorics. Through the recurrence and bijective proofs, the Catalan numbers allow deriving various combinatorial objects. We will explore properties and patterns exhibited by Catalan objects like lattice paths, Dyck paths, balanced parentheses, rooted trees, full binary trees, multiplication schemes, and polygon triangulations. The recursive formula reveals the relationships between these structures, while bijective mappings prove their cardinalities are the same.

# Catalan Numbers: The Beauty of Bijections

by Cassie Kloss | Mar 21, 2024 | Math Seminar, Spring 2024 | 6 comments

This sounds like a great topic to talk about with many different areas of mathematics being intertwined. I have had a hard time learning about recursion in the past in computer science classes. However, I am always interested in expanding my knowledge when it comes to recursion in math and computer science. I do not know much about these topics, but I am very excited to hear about them!

This topic seems very interesting. I can’t wait to learn about the recursive and closed forms and to see the different properties and patterns that come from Catalan numbers. Looking forward to your talk!

The topic of your seminar sounds very complex and interesting. I have no prior knowledge of catalan numbers so I am intrigued to learn about them. I can not wait to hear you dive into what catalan numbers are during your talk!

I have never heard much about Catalan numbers before which is why I am very excited about this talk. Both your teaser show how your talk is interesting!! Good luck Cassie

Before your talk, I had heard of the Catalan numbers, but was relatively unfamiliar with their significance mathematically. Your talk provided some insight into that, which was great! Watching the calculations behind determining the lattice paths using the Catalan numbers was very interesting and I can see how this sequence and recursion can be applied to fields like computer science and combinatorics.

I don’t think I had ever heard of the Catalan numbers before your talk but you explained it well and definitely cleared up some confusion I had surrounding the concept. It was very interesting when you were showing how all the different paths could be drawn!