Autumn 2026 Projects

Catalan Combinatorics

Faculty Mentor: Dr. Elena Hafner

Project Description: The Catalan numbers are a sequence that is ubiquitous in combinatorics. In this project, we will explore the Catalan numbers and generalizations such as the Fuss-Catalan numbers, rational Catalan numbers, and others We will study problems in enumerative combinatorics which may relate to these sequences.

For example, we will investigate certain pattern avoidance conditions in k-regular words (i.e. permutations of the multi-set containing k copies of each number in {1,…,n}). This is inspired by recent work of Laudone (arXiv:2608.21351) who studied the special case of canon permutations, and showed that the number of canon permutations avoiding any of the patterns 112, 122, 211, or 221 is counted by the Fuss-Catalan numbers.

Project Level: Intermediate: students who have taken Math 300
Additional Course Requirements: Math 461 (combinatorics) preferred
Programming Requirements: