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The Andrews-Stanley partition function and $ p(n)$: congruences


Author: Holly Swisher
Journal: Proc. Amer. Math. Soc. 139 (2011), 1175-1185
MSC (2010): Primary 11P82, 11P83
DOI: https://doi.org/10.1090/S0002-9939-2010-10551-8
Published electronically: August 24, 2010
MathSciNet review: 2748412
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Abstract: R. Stanley formulated a partition function $ t(n)$ which counts the number of partitions $ \pi$ for which the number of odd parts of $ \pi$ is congruent to the number of odd parts in the conjugate partition $ \pi'$ $ \pmod{4}$. In light of G. E. Andrews' work on this subject, it is natural to ask for relationships between $ t(n)$ and the usual partition function $ p(n)$. In particular, Andrews showed that the $ \pmod{5}$ Ramanujan congruence for $ p(n)$ also holds for $ t(n)$. In this paper we extend his observation by showing that there are infinitely many arithmetic progressions $ An + B$ such that for all $ n\geq 0$,

$\displaystyle t(An+B) \equiv p(An+B) \equiv 0 \pmod{l^j} $

whenever $ l\geq 5$ is prime and $ j\geq 1$.


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Additional Information

Holly Swisher
Affiliation: Department of Mathematics, Oregon State University, Corvallis, Oregon 97301
Email: swisherh@math.oregonstate.edu

DOI: https://doi.org/10.1090/S0002-9939-2010-10551-8
Received by editor(s): April 16, 2010
Published electronically: August 24, 2010
Communicated by: Kathrin Bringmann
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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