Inequalities between ranks and cranks

Bringmann, Kathrin; Mahlburg, Karl

doi:10.1090/S0002-9939-09-09806-2

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

George E. Andrews, The theory of partitions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1998. Reprint of the 1976 original. MR 1634067
G. E. Andrews, The number of smallest parts in the partitions of $n$, to appear in J. Reine Angew. Math.
George E. Andrews, Partitions, Durfee symbols, and the Atkin-Garvan moments of ranks, Invent. Math. 169 (2007), no. 1, 37–73. MR 2308850, DOI 10.1007/s00222-007-0043-4
George E. Andrews and F. G. Garvan, Dyson’s crank of a partition, Bull. Amer. Math. Soc. (N.S.) 18 (1988), no. 2, 167–171. MR 929094, DOI 10.1090/S0273-0979-1988-15637-6
George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958, DOI 10.1017/CBO9781107325937
A. O. L. Atkin and F. G. Garvan, Relations between the ranks and cranks of partitions, Ramanujan J. 7 (2003), no. 1-3, 343–366. Rankin memorial issues. MR 2035811, DOI 10.1023/A:1026219901284
A. O. L. Atkin and P. Swinnerton-Dyer, Some properties of partitions, Proc. London Math. Soc. (3) 4 (1954), 84–106. MR 60535, DOI 10.1112/plms/s3-4.1.84
K. Bringmann, Aymptotics for rank partition functions, Transactions of the AMS, accepted for publication.
Kathrin Bringmann, On the explicit construction of higher deformations of partition statistics, Duke Math. J. 144 (2008), no. 2, 195–233. MR 2437679, DOI 10.1215/00127094-2008-035
K. Bringmann, F. Garvan and K. Mahlburg, Partition statistics and quasiweak Maass forms, Int. Math. Res. Not., 2009, no. 1, 63–97.
K. Bringmann and K. Ono, Coefficients of harmonic weak Maass forms, preprint.
K. Bringmann and K. Ono, Dyson’s ranks and Maass forms, to appear in Ann. of Math.
K. Bringmann and S. Zwegers, Rank-crank type PDE’s and non-holomorphic Jacobi forms, to appear in Math. Res. Lett.
F. Dyson, Some guesses in the theory of partitions, Eureka (Cambridge) 8 (1944), 10–15.
Freeman J. Dyson, Mappings and symmetries of partitions, J. Combin. Theory Ser. A 51 (1989), no. 2, 169–180. MR 1001259, DOI 10.1016/0097-3165(89)90043-5
Karl Mahlburg, Partition congruences and the Andrews-Garvan-Dyson crank, Proc. Natl. Acad. Sci. USA 102 (2005), no. 43, 15373–15376. MR 2188922, DOI 10.1073/pnas.0506702102
S. Ramanujan, Some properties of $p(n)$; the number of partitions of $n$, Proc. Camb. Phil. Soc. 19 (1919), 207–210.
S. Ramanujan, Congruence properties of partitions, Math. Z. 9 (1921), no. 1-2, 147–153. MR 1544457, DOI 10.1007/BF01378341