Number theoretic properties of generating functions related to Dyson's rank for partitions into distinct parts
Author:
Maria Monks
Journal:
Proc. Amer. Math. Soc. 138 (2010), 481494
MSC (2000):
Primary 11P82, 11P83
Published electronically:
September 10, 2009
MathSciNet review:
2557166
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Abstract: Let denote the number of partitions of into distinct parts. We show that Dyson's rank provides a combinatorial interpretation of the wellknown fact that is almost always divisible by . This interpretation gives rise to a new false theta function identity that reveals surprising analytic properties of one of Ramanujan's mock theta functions, which in turn gives generating functions for values of certain Dirichlet functions at nonpositive integers.
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 K. Alladi, Partition identities involving gaps and weights, Transactions of the Amer. Math. Soc. 349, No. 12 (1997), 50015019. MR 1401759 (98c:05012)
 2.
 G. E. Andrews, Simplicity and surprise in Ramanujan's ``lost'' notebook, Amer. Math. Monthly 104, No. 10 (1997), 918925. MR 1490910 (99c:11016)
 3.
 G. E. Andrews, F. J. Dyson, and D. Hickerson, Partitions and indefinite quadratic forms, Invent. Math. 91 (1988), 391407. MR 928489 (89f:11071)
 4.
 G. E. Andrews, K. Ono, and J. JiménezUrroz, series identities and values of certain functions, Duke Math. J. 108, No. 3 (2001), 395419. MR 1838657 (2002e:11055)
 5.
 G. E. Andrews and F. G. Garvan, Dyson's crank of a partition, Bull. Amer. Math. Soc. 18, No. 2 (1988) 167171. MR 929094 (89b:11079)
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 T. M. Apostol, Introduction to Analytic Number Theory, SpringerVerlag, New York, 1976. MR 0434929 (55:7892)
 7.
 A. O. L. Atkin and P. SwinnertonDyer, Some properties of partitions, Proc. London Math. Soc. (3) 4 (1954), 84106. MR 0060535 (15:685d)
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 K. Bringmann and K. Ono, Dyson's ranks and Maass forms, Annals of Mathematics, to appear.
 9.
 H. Cohen, identities for Maass waveforms, Invent. Math. 91, No. 3 (1988), 409422. MR 928490 (89f:11072)
 10.
 F. J. Dyson, Some guesses in the theory of partitions, Eureka, vol. 8, Cambridge, 1944, 1015.
 11.
 F. G. Garvan, New combinatorial interpretations of Ramanujan's partition congruences mod , , and , Transactions of the Amer. Math. Soc. 305, No. 1 (1988), 4777. MR 920146 (89b:11081)
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 B. Gordon and K. Ono, Divisibility properties of certain partition functions by powers of primes, The Ramanujan Journal 1 (1997), 2535. MR 1607526 (99c:11128)
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 J. Lovejoy, Divisibility and distribution of partitions into distinct parts, Adv. Math. 158 (2001), 253263. MR 1822684 (2002k:11185)
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 J. Lovejoy, The number of partitions into distinct parts modulo powers of 5, Bull. London Math. Soc. 35, No. 1 (2003), 4146. MR 1934430 (2003g:11119)
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 K. Ono, Distribution of the partition function modulo , Annals of Mathematics (2) 151 (2000), 293307. MR 1745012 (2000k:11115)
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 K. Ono, The Web of Modularity: Arithmetic of Coefficients of Modular Forms and series, CBMS Reg. Conf. Ser. Math., 102, American Mathematical Society, Providence, RI, 2004. MR 2020489 (2005c:11053)
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 Ø. Rødseth, Dissections of the generating functions of and , Arbok Univ. Bergen Mat.Natur. 1969, No. 12 (1970), 312. MR 0434959 (55:7922)
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 J. J. Sylvester and F. Franklin. A constructive theory of partitions, arranged in three acts, an interact and an exodion. Amer. J. Math. 5 (1882), 251330. MR 1505328
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Additional Information
Maria Monks
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email:
monks@mit.edu
DOI:
http://dx.doi.org/10.1090/S000299390910076X
Received by editor(s):
April 20, 2009
Received by editor(s) in revised form:
June 7, 2009
Published electronically:
September 10, 2009
Communicated by:
Ken Ono
Article copyright:
© Copyright 2009
Maria Monks
