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In this paper, we simplify and generalize formulas for the expansion of rank2 cluster variables. In particular, we prove an equivalent, but simpler,description of the colored Dyck subpaths framework introduced by Lee andSchiffler. We then prove the conjectured bijectivity of a map constructed byFeiyang Lin between collections of colored Dyck subpaths and compatible pairs,objects introduced by Lee, Li, and Zelevinsky to study the greedy basis. We usethis bijection along with Rupel's expansion formula for quantum greedy basiselements, which sums over compatible pairs, to provide a quantum generalizationof Lee and Schiffler's colored Dyck subpaths formula.

On Dyck Path Expansion Formulas for Rank 2 Cluster Variables