On Dyck Path Expansion Formulas for Rank 2 Cluster Variables

TL;DR

This paper simplifies and extends formulas for rank 2 cluster variables using Dyck paths, proving a bijection and providing a quantum generalization of existing combinatorial formulas.

Contribution

It introduces a simpler description of the Dyck subpaths framework and proves the bijectivity of a key map, extending the formulas to quantum cluster algebras.

Findings

Simplified the Dyck subpaths expansion formula.

Proved bijectivity of the map between Dyck subpaths and compatible pairs.

Extended formulas to quantum cluster variables.

Abstract

In this paper, we simplify and generalize formulas for the expansion of rank 2 cluster variables. In particular, we prove an equivalent, but simpler, description of the colored Dyck subpaths framework introduced by Lee and Schiffler. We then prove the conjectured bijectivity of a map constructed by Feiyang Lin between collections of colored Dyck subpaths and compatible pairs, objects introduced by Lee, Li, and Zelevinsky to study the greedy basis. We use this bijection along with Rupel's expansion formula for quantum greedy basis elements, which sums over compatible pairs, to provide a quantum generalization of Lee and Schiffler's colored Dyck subpaths formula.