On the explicit construction of higher deformations of partition statistics

TL;DR

This paper advances the understanding of partition statistics by constructing a new class of functions called quasiweak Maass forms, linking them to higher deformations of partition generating functions and proving conjectures of Andrews.

Contribution

It introduces quasiweak Maass forms, a new class of functions with quasimodular components, and applies them to prove conjectures related to partition statistics.

Findings

Constructed quasiweak Maass forms with quasimodular components.

Proved two conjectures of Andrews regarding partition statistics.

Developed a new method for transformation laws over incomplete lattices.

Abstract

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.