On the explicit construction of higher deformations of partition statistics

Abstract

The modularity of the partition-generating function has many important consequences: for example, asymptotics and congruences for $p(n)$. In a pair of articles, Bringmann and Ono [11], [12] connected the rank, a partition statistic introduced by Dyson [18], to weak Maass forms, a new class of functions that are related to modular forms and that were first considered in [14]. Here, we take a further step toward understanding how weak Maass forms arise from interesting partition statistics by placing certain $2$-marked Durfee symbols introduced by Andrews [1] into the framework of weak Maass forms. To do this, we construct a new class of functions that we call quasi-weak Maass forms because they have quasi-modular forms as components. As an application, we prove two conjectures of Andrews [1, Conjectures 11, 13]. It seems that this new class of functions will play an important role in better understanding weak Maass forms of higher weight themselves and also their derivatives. As a side product, we introduce a new method that enables us to prove transformation laws for generating functions over incomplete lattices