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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Inequalities between ranks and cranks
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by Kathrin Bringmann and Karl Mahlburg PDF
Proc. Amer. Math. Soc. 137 (2009), 2567-2574 Request permission

Abstract:

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.
References
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Additional Information
  • Kathrin Bringmann
  • Affiliation: Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany
  • MR Author ID: 774752
  • Email: kbringma@math.uni-koeln.de
  • Karl Mahlburg
  • Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307
  • MR Author ID: 664593
  • Email: mahlburg@math.mit.edu
  • Received by editor(s): October 9, 2008
  • Received by editor(s) in revised form: October 20, 2008
  • Published electronically: February 20, 2009
  • Additional Notes: The first author was partially supported by NSF grant DMS-0757907.
    The second author was partially supported by NSA Grant 6917958.
  • Communicated by: Ken Ono
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 137 (2009), 2567-2574
  • MSC (2000): Primary 11P81; Secondary 05A17
  • DOI: https://doi.org/10.1090/S0002-9939-09-09806-2
  • MathSciNet review: 2497467