  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.
