AIM Research Talk: Polyhedral geometry of refined q,t-Catalan numbers

Many problems in algebraic combinatorics have geometric objects hidden in the background, and bringing this geometry to the foreground can reveal structure that is difficult to see from formulas alone. We study a refinement of the q,t-Catalan numbers introduced by Xin and Zhang in 2022 and 2023 using tools from polyhedral geometry. These refined polynomials depend on a vector of parameters k and recover the classical q,t-Catalan numbers when every entry of k is equal to one.

We begin with an introduction to polyhedral cones and their connection to multivariable generating functions. We then construct cones arising from the constraints that define k-Dyck paths and show how their area and bounce statistics can be encoded geometrically. This viewpoint provides a new interpretation of Xin and Zhang’s generating functions and recovers their q,t-symmetry results for parameter vectors of length three and for the four-parameter family (k, k, k, k). It also leads to extensions, including the family (k, k+m, k+m, k+m), and offers insight into related generalizations and open problems concerning refined q,t-Catalan numbers. No prior experience with polyhedral geometry will be assumed. The talk is based on joint work with Matthias Beck, Mitsuki Hanada, Max Hlavacek, John Lentfer, and Katie Waddle.