Inequalities between ranks and cranks

By Kathrin Bringmann, Karl Mahlburg. 2009

51 citations recorded by the source corpus.

Higher moments of the partition rank and crank statistics have been studied for their connections to combinatorial objects such as Durfee symbols, as well as for their connections to harmonic Maass forms. This paper proves the first several cases of (and strengthens) a conjecture due to Garvan, which states that the moments of the crank function are always larger than the moments of the rank function. Furthermore, asymptotic estimates for these differences are also proven.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/b6732fdc9f6acdfa9d371e839467d8bed0038c16

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

Citations in the corpus

Asymptotic inequalities for positive crank and rank moments (2012, result): “Atkin and Garvan described algebraic and arithmetic relationships between crank and rank moments, but the nature of their exact numerical values was first explored by Garvan, who conjectured a strong inequality between the two families of moments in [24]; a related set of conjectures was later given”
Asymptotics for rank and crank moments (2009, methodology): “We next follow the arguments in [13] and use the Hardy-Ramanujan circle method, following the original framework chronicled in [1, 21, 24].”
Higher Order SPT-Functions (2010, result): “Analytic and arithmetic properties of higher order rank moments were studied by Bringmann, Lovejoy and Osburn [9] and by Bringmann, the author and Mahlburg [8].”
Taylor coefficients of mock-Jacobi forms and moments of partition statistics (2014, background): “The cases k = 2 and 3 of Garvan’s Conjecture were previously proven by the first two authors [10] for sufficiently large n.”