Course description
This course offers a deep dive into enumerative combinatorics through the lens of one of mathematics’ most ubiquitous sequences: the Catalan numbers. The sequence 1, 2, 5, 14, 42, … appears in arguably more distinct contexts than any other integer sequence in mathematics. The central theme is structural connectivity — moving beyond counting formulas to explore why seemingly unrelated structures share the same enumeration.
Students explore the “Catalan Garden”: the hundreds of combinatorial objects counted by the sequence. Why does the number of ways to stack coins match the number of ways to parenthesise an expression? The material is relevant not only to pure mathematicians but to computer scientists interested in data structures, and to physicists studying statistical mechanics.
The experience is highly interactive. Lectures introduce theoretical frameworks and algebraic manipulations; collaborative workshops challenge students to construct their own bijections — essentially translating hard problems into easier ones. By the end, students should be able to look past the surface of a problem to the skeletal structure connecting it to everything else.
Grading
- Quizzes 40% — proof-based, weekly, 10% each — individual
- Project 40% — group project, decided by the end of week 2
- Homework 20% — proof-based, every 2–3 days — individual