  The modularity of the partition generating function has many important
consequences, for example asymptotics and congruences for $p(n)$. In a series
of papers the author and Ono \cite{BO1,BO2} connected the rank, a partition
statistic introduced by Dyson, to weak Maass forms, a new class of functions
which are related to modular forms and which were first considered in
\cite{BF}. Here we do a further step towards understanding how weak Maass forms
arise from interesting partition statistics by placing certain 2-marked Durfee
symbols introduced by Andrews \cite{An1} into the framework of weak Maass
forms. To do this we construct a new class of functions which we call quasiweak
Maass forms because they have quasimodular forms as components. As an
application we prove two conjectures of Andrews. 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 which enables us to prove transformation laws
for generating functions over incomplete lattices.
