ON THE EXPLICIT CONSTRUCTION OF HIGHER DEFORMATIONS OF PARTITION STATISTICS

By Kathrin Bringmann. 2007

139 citations recorded by the source corpus.

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 rst 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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/f8e73ca0fda1775a77f71ee10c8ef3915c997a75

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In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

Citations in the corpus

Unearthing the visions of a master: harmonic Maass forms and number theory (2008, background): “In this direction there are papers by Bringmann, Garvan, Lovejoy, Mahlburg, and Osburn [48, 50, 52], in which theorems concerning the automorphic properties of partition generating functions, asymptotics, and congruences are obtained for various partition statistics such as those arising in the work”
Zagier-type dualities and lifting maps for harmonic Maass–Jacobi forms (2010, background): “Bringmann [3] and Bringmann, Garvan, and Mahlburg [4] examined quasiharmonic Maass forms, i.”
Rank and crank moments for overpartitions (2008, background): “(1.2) Since their introduction by Atkin and Garvan [4], the rank and crank moments and their linear combinations have been the subject of a number of works [2, 3, 5, 6, 16, 17].”
Congruences for Andrews' Smallest Parts Partition Function and New Congruences for Dyson's Rank (2007, unknown): “h prime ℓ>3 there is a positive proportion of the primes Qsuch that (7.4) spt(ℓQ4n+µ) ≡ 0 (mod ℓ), for all integers k, n≥ 0 and µis an integer constant at depends on ℓand Q. We note that Bringmann [13] proved this result by a diﬀerent method. In a recent paper, Bringmann, Ono and Rhoades [16] hav”