A conjectured formula for the rational $q,t$-Catalan polynomial

Abstract

We conjecture a formula for the rational $q,t$-Catalan polynomial ${\mathcal{C}}_{r/s}$ that is symmetric in $q$ and $t$ by definition. The conjecture posits that ${\mathcal{C}}_{r/s}$ can be written in terms of symmetric monomial strings indexed by maximal Dyck paths. We show that for any finite $d^{\ast }$, giving a combinatorial proof of our conjecture on the infinite set of functions $\{{\mathcal{C}}_{r/s}^d:r\equiv 1\mathrm{mod}s,d\le d^{\ast }\}$ is equivalent to a finite counting problem.