Duke Mathematical Journal

On the explicit construction of higher deformations of partition statistics

Kathrin Bringmann

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The modularity of the partition-generating function has many important consequences: for example, asymptotics and congruences for p(n). In a pair of articles, Bringmann and Ono [11], [12] connected the rank, a partition statistic introduced by Dyson [18], to weak Maass forms, a new class of functions that are related to modular forms and that were first considered in [14]. Here, we take a further step toward understanding how weak Maass forms arise from interesting partition statistics by placing certain 2-marked Durfee symbols introduced by Andrews [1] into the framework of weak Maass forms. To do this, we construct a new class of functions that we call quasi-weak Maass forms because they have quasi-modular forms as components. As an application, we prove two conjectures of Andrews [1, Conjectures 11, 13]. It seems that this new class of functions will play an important role in better understanding weak Maass forms of higher weight themselves and also their derivatives. As a side product, we introduce a new method that enables us to prove transformation laws for generating functions over incomplete lattices

Article information

Duke Math. J. Volume 144, Number 2 (2008), 195-233.

First available in Project Euclid: 14 August 2008

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11P82: Analytic theory of partitions
Secondary: 05A17: Partitions of integers [See also 11P81, 11P82, 11P83]


Bringmann, Kathrin. On the explicit construction of higher deformations of partition statistics. Duke Math. J. 144 (2008), no. 2, 195--233. doi:10.1215/00127094-2008-035. https://projecteuclid.org/euclid.dmj/1218716298

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  • G. E. Andrews, On the theorems of Watson and Dragonette for Ramanujan's mock theta functions, Amer. J. Math. 88 (1966), 454--490.
  • —, ``Mock theta functions'' in Theta Functions --.-Bowdoin 1987, Part 2 (Brunswick, Me., 1987), Proc. Sympos. Pure Math. 49, Part 2, Amer. Math. Soc., Providence, 1989, 283--297.
  • —, Partitions, Durfee symbols, and the Atkin-Garvan moments of ranks, Invent. Math. 169 (2007), 37--73.
  • —, The number of smallest parts in the partitions of n, preprint, 2007.
  • G. E. Andrews, F. Dyson, and D. Hickerson, Partitions and indefinite quadratic forms, Invent. Math. 91 (1988), 391--407.
  • A. O. L. Atkin and F. G. Garvan, Relations between the ranks and cranks of partitions, Ramanujan J. 7 (2003), 343--366.
  • A. O. L. Atkin and P. Swinnerton-Dyer, Some properties of partitions, Proc. London Math. Soc. (3) 4 (1954), 84--106.
  • K. Bringmann, Asymptotics for rank partition functions, to appear in Trans. Amer. Math. Soc.
  • —, On certain congruences for Dyson's ranks, to appear in Int. J. Number Theory.
  • K. Bringmann, F. Garvan, and K. Mahlburg, Higher rank moments, in preparation.
  • K. Bringmann and K. Ono, The $f(q)$ mock theta function conjecture and partition ranks, Invent. Math. 165 (2006), 243--266.
  • —, Dyson's ranks and Maass forms, to appear in Ann. of Math. (2).
  • K. Bringmann, K. Ono, and R. C. Rhoades, Eulerian series as modular forms, to appear in J. Amer. Math. Soc.
  • J. H. Bruinier and J. Funke, On two geometric theta lifts, Duke Math. J. 125 (2004), 45--90.
  • Y.-S. Choi, Tenth order mock theta functions in Ramanujan's lost notebook, Invent. Math. 136 (1999), 497--569.
  • H. Cohen, $q$-identities for Maass waveforms, Invent. Math. 91 (1988), 409--422.
  • L. A. Dragonette, Some asymptotic formulae for the mock theta series of Ramanujan, Trans. Amer. Math. Soc. 72 (1952), 474--500.
  • F. Dyson, Some guesses in the theory of partitions, Eureka 8 (1944), 10--15.
  • D. Hickerson, On the seventh order mock theta functions, Invent. Math. 94 (1988), 661--677.
  • J. Lehner, A partition function connected with the modulus five, Duke Math. J. 8 (1941), 631--655.
  • K. Ono, Distribution of the partition function modulo $m$, Ann. of Math. (2) 151 (2000), 293--307.
  • S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Springer, Berlin, 1988.
  • G. Shimura, On modular forms of half integral weight, Ann. of Math. (2) 97 (1973), 440--481.
  • S. Treneer, Congruences for the coefficients of weakly holomorphic modular forms, Proc. London Math. Soc. (3) 93 (2006), 304--324.
  • G. N. Watson, ``The final problem: An account of the mock theta functions'' in Ramanujan: Essays and Surveys, Hist. Math. 22, Amer. Math. Soc., Providence, 2001, 325--347.
  • S. P. Zwegers, ``Mock $\theta$-functions and real analytic modular forms'' in $q$-Series with Applications to Combinatorics, Number Theory, and Physics (Urbana, Ill., 2000), Contemp. Math. 291, Amer. Math. Soc., Providence, 2001, 269--277.