Student Combinatorics: A Geometric Perspective on the q,t-Catalan Numbers

Dyck paths are combinatorial objects enumerated by the Catalan numbers. We can write a generating function for Dyck paths, using q to keep track of the area statistic, and t to keep track of the bounce statistic. Amazingly, while this function is symmetric in q and t, a bijection on Dyck paths that swaps area and bounce continues to elude even the most tenacious combinatistas. Recent work of Guoce Xin and Yingrui Zhang suggests promising results from studying a generalization of Dyck paths. Drawing inspiration from this work, we will discuss how to view Dyck paths as integer points of cones, hoping that the geometric perspective will shed new light on this problem. This is joint work with Matthias Beck, Mitsuki Hanada, Max Hlavacek, John Lentfer, and Andrés Vindas Meléndez.