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Inequalities between ranks and cranks
Authors:
Kathrin Bringmann and Karl Mahlburg
Journal:
Proc. Amer. Math. Soc. 137 (2009), 2567-2574
MSC (2000):
Primary 11P81; Secondary 05A17
Posted:
February 20, 2009
MathSciNet review:
2497467
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Additional Information
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.
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- G. E. Andrews, The number of smallest parts in the partitions of , to appear in J. Reine Angew. Math.
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- G. E. Andrews, Partitions, Durfee symbols, and the Atkin-Garvan moments of ranks, Invent. Math. 169 (2007), 37-73. MR 2308850 (2008d:05013)
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- G. E. Andrews and F. Garvan, Dyson's crank of a partition, Bull. Amer. Math. Soc. 18 (1988), 167-171. MR 929094 (89b:11079)
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- K. Bringmann, F. Garvan and K. Mahlburg, Partition statistics and quasiweak Maass forms, Int. Math. Res. Not., 2009, no. 1, 63-97.
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- K. Bringmann and K. Ono, Coefficients of harmonic weak Maass forms, preprint.
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- K. Bringmann and K. Ono, Dyson's ranks and Maass forms, to appear in Ann. of Math.
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- K. Bringmann and S. Zwegers, Rank-crank type PDE's and non-holomorphic Jacobi forms, to appear in Math. Res. Lett.
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- S. Ramanujan, Congruence properties of partitions, Math. Zeitschrift 9 (1921), 147-153. MR 1544457
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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, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307
Email:
mahlburg@math.mit.edu
DOI:
http://dx.doi.org/10.1090/S0002-9939-09-09806-2
PII:
S 0002-9939(09)09806-2
Received by editor(s):
October 9, 2008
Received by editor(s) in revised form:
October 20, 2008
Posted:
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
Article copyright:
© Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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