Title: Rank Two Non-Commutative Laurent Phenomenon and Pseudo-Positivity,
Abstract: We study polynomial generalizations of the Kontsevich automorphisms acting on
the skew-field of formal rational expressions in two non-commuting variables.
Our main result is the Laurentness and pseudo-positivity of iterations of these
automorphisms. The resulting expressions are described combinatorially using a
generalization of the combinatorics of compatible pairs in a maximal Dyck path
developed by Lee, Li, and Zelevinsky. By specializing to quasi-commuting
variables we obtain pseudo-positive expressions for rank 2 quantum generalized
cluster variables. In the case that all internal exchange coefficients are
zero, this quantum specialization provides a combinatorial construction of
counting polynomials for Grassmannians of submodules in exceptional
representations of valued quivers with two vertices.