Partitions with Parts Separated by Parity and Maass Forms

Recently, Andrews initiated the study of partitions with parts separated by parity in connection with Ramanujan's mock theta 
functions. Various families of partitions built from Andrews' ideas have very similar hypergeometric constructions but wildly differing modular structures. We study eight such examples and give their relations to various types of modular objects, using this relationship to compute the asymptotic growth of these partition families. We give special emphasis to an example connecting to the Ramanujan sigma function, whose modular structure is tied to Maass forms by famous work of Andrews-Dyson-Hickerson and Cohen. In this special setting, we use a new approach towards the error to modularity of false indefinite 
theta functions to compute a full Rademacher-type expansion for this special family. The techniques critically involve the relationship between false indefinite theta functions and mock Maass forms as defined by Zwegers.