Abstract: Compatible subsets of edges in a maximal Dyck path were introduced by Lee, Li, and Zelevinsky as a tool for constructing nice bases for rank two cluster algebras. In this talk, I will present a generalization of this combinatorics and give two applications. The first application is a combinatorial construction of non-commutative rank two generalized cluster variables which proves a conjecture of Kontsevich. The second application gives a combinatorial description of the cells in an affine paving of rank two quiver Grassmannians, this part is joint work with Thorsten Weist.