Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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



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
Published electronically: February 20, 2009
MathSciNet review: 2497467
Full-text PDF

Abstract | References | Similar Articles | 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.

References [Enhancements On Off] (What's this?)

  • 1. G.E. Andrews, The theory of partitions, Cambridge University Press, Cambridge, 1998. MR 1634067 (99c:11126)
  • 2. G. E. Andrews, The number of smallest parts in the partitions of $ n$, to appear in J. Reine Angew. Math.
  • 3. G. E. Andrews, Partitions, Durfee symbols, and the Atkin-Garvan moments of ranks, Invent. Math. 169 (2007), 37-73. MR 2308850 (2008d:05013)
  • 4. G. E. Andrews and F. Garvan, Dyson's crank of a partition, Bull. Amer. Math. Soc. 18 (1988), 167-171. MR 929094 (89b:11079)
  • 5. G. E. Andrews, R. Askey, and R. Roy, Special functions, Encyclopedia of Mathematics and its Applications, 71. Cambridge University Press, Cambridge, 1999. MR 1688958 (2000g:33001)
  • 6. A. O. L. Atkin and F. G. Garvan, Relations between the ranks and the cranks of partitions, Ramanujan Journal 7 (2003), 343-366. MR 2035811 (2005e:11131)
  • 7. A. O. L. Atkin and H. P. F. Swinnerton-Dyer, Some properties of partitions, Proc. London Math. Soc. 4 (1954), 84-106. MR 0060535 (15:685d)
  • 8. K. Bringmann, Aymptotics for rank partition functions, Transactions of the AMS, accepted for publication.
  • 9. K. Bringmann, On the explicit construction of higher deformations of partition statistics, Duke Math. J. 144 (2008), 195-233. MR 2437679
  • 10. K. Bringmann, F. Garvan and K. Mahlburg, Partition statistics and quasiweak Maass forms, Int. Math. Res. Not., 2009, no. 1, 63-97.
  • 11. K. Bringmann and K. Ono, Coefficients of harmonic weak Maass forms, preprint.
  • 12. K. Bringmann and K. Ono, Dyson's ranks and Maass forms, to appear in Ann. of Math.
  • 13. K. Bringmann and S. Zwegers, Rank-crank type PDE's and non-holomorphic Jacobi forms, to appear in Math. Res. Lett.
  • 14. F. Dyson, Some guesses in the theory of partitions, Eureka (Cambridge) 8 (1944), 10-15.
  • 15. F. Dyson, Mappings and symmetries of partitions, J. Combin. Theory Ser. A 51 (1989), 169-180. MR 1001259 (90f:05009)
  • 16. K. Mahlburg, Partition congruences and the Andrews-Garvan-Dyson crank, Proc. Natl. Acad. Sci. 102 (2005), no. 43, 15373-15376. MR 2188922 (2006k:11200)
  • 17. S. Ramanujan, Some properties of $ p(n)$; the number of partitions of $ n$, Proc. Camb. Phil. Soc. 19 (1919), 207-210.
  • 18. S. Ramanujan, Congruence properties of partitions, Math. Zeitschrift 9 (1921), 147-153. MR 1544457

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11P81, 05A17

Retrieve articles in all journals with MSC (2000): 11P81, 05A17

Additional Information

Kathrin Bringmann
Affiliation: Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany

Karl Mahlburg
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307

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
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society