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15 August 2008 On the explicit construction of higher deformations of partition statistics
Kathrin Bringmann
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Duke Math. J. 144(2): 195-233 (15 August 2008). DOI: 10.1215/00127094-2008-035


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


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Kathrin Bringmann. "On the explicit construction of higher deformations of partition statistics." Duke Math. J. 144 (2) 195 - 233, 15 August 2008.


Published: 15 August 2008
First available in Project Euclid: 14 August 2008

zbMATH: 1154.11034
MathSciNet: MR2437679
Digital Object Identifier: 10.1215/00127094-2008-035

Primary: 11P82
Secondary: 05A17

Rights: Copyright © 2008 Duke University Press


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Vol.144 • No. 2 • 15 August 2008
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