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Asymptotic inequalities for positive crank and rank moments


Authors: Kathrin Bringmann and Karl Mahlburg
Journal: Trans. Amer. Math. Soc. 366 (2014), 1073-1094
MSC (2010): Primary 11P55, 05A17
Published electronically: August 16, 2013
MathSciNet review: 3130326
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Abstract: Andrews, Chan, and Kim recently introduced a modified definition of crank and rank moments for integer partitions that allows the study of both even and odd moments. In this paper, we prove the asymptotic behavior of these moments in all cases. Our main result states that the two families of moment functions are asymptotically equal, but the crank moments are also asymptotically larger than the rank moments.

Andrews, Chan, and Kim also gave a combinatorial description for the differences of the first crank and rank moments that they named the ospt-function. Our main results therefore also give the asymptotic behavior of the ospt-function (and its analogs for higher moments), and we further determine the behavior of the ospt-function modulo $ 2$ by relating its parity to Andrews' spt-function.


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Additional Information

Kathrin Bringmann
Affiliation: Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany
Email: kbringma@math.uni-koeln.de

Karl Mahlburg
Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803
Email: mahlburg@math.lsu.edu

DOI: https://doi.org/10.1090/S0002-9947-2013-05945-4
Keywords: Integer partitions, rank, crank, Tauberian theorem
Received by editor(s): May 10, 2012
Received by editor(s) in revised form: August 14, 2012
Published electronically: August 16, 2013
Additional Notes: The research of the first author was supported by the Alfried Krupp Prize for Young University Teachers of the Krupp Foundation. The second author was supported by NSF Grant DMS-1201435.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.