Combinatorics of two second-order mock theta functions

We introduce combinatorial interpretations of the coefficients of two second-order mock theta functions. Then, we provide a bijection that relates the two combinatorial interpretations for each function. By studying other special cases of the multivariate identity proved by the bijection, we obtain new combinatorial interpretations for the coefficients of Watson's third-order mock theta function ω(q) and Ramanujan's third-order mock theta function f(q).