Partition statistics and quasiweak Maass forms

TL;DR

This paper demonstrates that hypergeometric generating functions for k-marked Durfee symbols are quasimock theta functions, revealing their automorphic properties and leading to new congruences and modularity results related to partition statistics.

Contribution

It establishes the automorphic nature of these generating functions as quasimock theta functions and explores their implications for congruences and modularity in partition theory.

Findings

Generating functions are quasimock theta functions.

Existence of infinitely many congruences for Durfee symbols.

Modularity of k-marked Durfee symbols depends on k=2 case.

Abstract

Andrews recently introduced k-marked Durfee symbols, which are a generalization of partitions that are connected to moments of Dyson's rank statistic. He used these connections to find identities relating their generating functions as well as to prove Ramanujan-type congruences for these objects and find relations between. In this paper we show that the hypergeometric generating functions for these objects are natural examples of quasimock theta functions, which are defined as the holomorphic parts of weak Maass forms and their derivatives. In particular, these generating functions may be viewed as analogs of Ramanujan's mock theta functions with arbitrarily high weight. We use the automorphic properties to prove the existence of infinitely many congruences for the Durfee symbols. Furthermore, we show that as k varies, the modularity of the k-marked Durfee symbols is precisely dictated by the case k=2. Finally, we use this relation in order to prove the existence of general congruences for rank moments in terms of level one modular forms of bounded weight.