Cataland: A Romance of Many Bijections
Cataland: A Romance of Many Bijections
This is a course taught at the Lodha Genius - Ashoka University Programme 2026 for grades 11 and 12.
This course offers a deep dive into Enumerative Combinatorics through the lens of one of mathematics’ most ubiquitous and fascinating sequences: the Catalan numbers. The Catalan sequence 1, 2, 5, 14, 42, … appears in arguably more distinct contexts than any other integer sequence in all of mathematics. The central theme of this course is structural connectivity, around the Catalan sequence and the objects that this sequence counts. We will move beyond simple counting formulas to explore why seemingly unrelated structures share the same underlying enumeration.
Students will explore the "Catalan Garden" – the hundreds of combinatorial objects counted by the Catalan sequence. For instance, we will ask and answer questions like: Why does the number of ways to stack coins match the number of ways to parenthesise an equation? It is relevant not only to pure mathematicians but also to computer scientists interested in data structures (sorting, trees) and to physicists studying statistical mechanics.
The learning experience is highly interactive: Lectures will introduce theoretical frameworks and algebraic manipulations, while collaborative Workshops will challenge students to construct their own bijections, essentially "translating" hard problems into easier ones. By the end of the course, students will develop a good combinatorial intuition. They will learn to look past the surface of a mathematical problem to see the hidden skeletal structure that connects it to the broader mathematical universe.
Catalan Numbers, Richard P. Stanley, Cambridge University Press, 2015.
An Introduction to Catalan Numbers, Steven Roman, Brikhäuser, 2015.
Catalan Numbers with Applications, Thomas Koshy, Oxford University Press, 2009.
Fibonacci and Catalan Numbers: An Introduction, Ralph P. Grimaldi, John Wiley and Sons, 2012.
Catalan Numbers Page, Igor Pak, University of California, Los Angeles.
Lecture Time: TBA
Lecture Venue: TBA
Attendance Policy: 100% attendance is required. This is a fast-paced course with daily lectures, and missing even one will impact learning.
Quizzes: A regular proof-based exam, once every week, contributing 40% of the total score (10% per week). This is an individual component.
Project: This will be a group project, to be decided by the end of the 2nd week. This will contribute 40% of the total score.
Homework: Proof-based mathematical questions will be assigned every 2-3 days; this is an individual component, and will contribute 20% of the total score.