  We define two-parameter generalizations of two combinatorial constructions of
Andrews: the kth symmetrized rank moment and the k-marked Durfee symbol. We
prove that three specializations of the associated generating functions are
so-called quasimock theta functions, while a fourth specialization gives
quasimodular forms. We then define a two-parameter generalization of Andrews'
smallest parts function and note that this leads to quasimock theta functions
as well. The automorphic properties are deduced using q-series identities
relating the relevant generating functions to known mock theta functions.
