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

TL;DR

This paper proposes a conjectured symmetric formula for the rational $q,t$-Catalan polynomial using maximal Dyck paths, linking it to a finite combinatorial counting problem.

Contribution

It introduces a conjecture for the rational $q,t$-Catalan polynomial's symmetric formula and relates its proof to a finite combinatorial counting problem.

Findings

Conjectured a symmetric formula for $ ext{Catalan}_ {r/s}$.

Linked the proof to a finite counting problem.

Established equivalence for functions with fixed $d^*$.

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 $\{ \mathcal{C}_{r/s}^d: r\equiv 1 \mod s, \,\,\, d \leq d^*\}$ is equivalent to a finite counting problem.