## Duke Mathematical Journal

### On the explicit construction of higher deformations of partition statistics

Kathrin Bringmann

#### 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

#### Article information

Source
Duke Math. J. Volume 144, Number 2 (2008), 195-233.

Dates
First available in Project Euclid: 14 August 2008

http://projecteuclid.org/euclid.dmj/1218716298

Digital Object Identifier
doi:10.1215/00127094-2008-035

Mathematical Reviews number (MathSciNet)
MR2437679

Zentralblatt MATH identifier
1154.11034

Subjects
Primary: 11P82: Analytic theory of partitions

#### Citation

Bringmann, Kathrin. On the explicit construction of higher deformations of partition statistics. Duke Math. J. 144 (2008), no. 2, 195--233. doi:10.1215/00127094-2008-035. http://projecteuclid.org/euclid.dmj/1218716298.

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