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Congruences for Andrews' spt-function modulo powers of $ 5$, $ 7$ and $ 13$


Author: F. G. Garvan
Journal: Trans. Amer. Math. Soc. 364 (2012), 4847-4873
MSC (2010): Primary 11P83, 11F33, 11F37; Secondary 11P82, 05A15, 05A17
DOI: https://doi.org/10.1090/S0002-9947-2012-05513-9
Published electronically: April 11, 2012
MathSciNet review: 2922612
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Abstract: Congruences are found modulo powers of $ 5$, $ 7$ and $ 13$ for Andrews' smallest parts partition function $ \mbox {spt}(n)$. These congruences are reminiscent of Ramanujan's partition congruences modulo powers of $ 5$, $ 7$ and $ 11$. Recently, Ono proved explicit Ramanujan-type congruences for $ \mbox {spt}(n)$ modulo $ \ell $ for all primes $ \ell \ge 5$ which were conjectured earlier by the author. We extend Ono's method to handle the powers of $ 5$, $ 7$ and $ 13$ congruences. We need the theory of weak Maass forms as well as certain classical modular equations for the Dedekind eta-function.


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Additional Information

F. G. Garvan
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611-8105
Email: fgarvan@ufl.edu

DOI: https://doi.org/10.1090/S0002-9947-2012-05513-9
Keywords: Andrews’ spt-function, weak Maass forms, congruences, partitions, modular forms
Received by editor(s): November 20, 2010
Published electronically: April 11, 2012
Additional Notes: The author was supported in part by NSA Grant H98230-09-1-0051.
Dedicated: Dedicated to my friend and mentor Michael D. Hirschhorn on the occasion of his 63rd birthday
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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