A Conjectured Formula for the Rational \(\varvec{q},\varvec{t}\)-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 \(d^*\), giving a combinatorial proof of our conjecture on the infinite set of functions \(\{ {\mathcal {C}}_{r/s}^d: r\equiv 1 \mod s, \,\,\, d \le d^*\}\) is equivalent to a finite counting problem.