Mock modular forms and singular combinatorial series
A celebrated result of Bringmann and Ono shows that the combinatorial rank generating function exhibits automorphic properties after being completed by the addition of a non-holomorphic integral. Since then, automorphic properties of various related combinatorial families have been studied. Here, extending work of Andrews and Bringmann, we study general infinite families of combinatorial $q$-series pertaining to $k$-marked Durfee symbols, in which we allow additional singularities. We show that these singular combinatorial families are essentially mixed mock and quasimock modular forms, and provide their explicit non-holomorphic completions. As a special case of our work, we consider $k=3$, and provide an asymptotic expansion for the associated partition rank statistic, solving a special case of an open problem of Andrews.