Motivated by work of Ramanujan, Freeman Dyson defined the rank of an integer partition to be its largest part minus its number of parts. If N.m; n/ denotes the number of partitions of n with rank m, then it turns out thatWe show that if ¤ 1 is a root of unity, then R. I q/ is essentially the holomorphic part of a weight 1=2 weak Maass form on a subgroup of SL 2 ./ޚ. For integers 0 Ä r < t, we use this result to determine the modularity of the generating function for N.r; tI n/, the number of partitions of n whose rank is congruent to r .mod t /. We extend the modularity above to construct an infinite family of vector valued weight 1=2 forms for the full modular group SL 2 ,/ޚ. a result which is of independent interest.
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 [12,13] 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 [11]. 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 [1] 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.
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