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Explicit bounds for the number of $ p$-core partitions

Authors: Byungchan Kim and Jeremy Rouse
Journal: Trans. Amer. Math. Soc. 366 (2014), 875-902
MSC (2010): Primary 11P82; Secondary 11P55, 11F66
Published electronically: August 19, 2013
MathSciNet review: 3130320
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Abstract: In this article, we derive explicit bounds on $ c_{t} (n)$, the number of $ t$-core partitions of $ n$. In the case when $ t = p$ is prime, we express the generating function $ f(z)$ as the sum

$\displaystyle f(z) = e_{p} E(z) + \sum _{i} r_{i} g_{i}(z) $

of an Eisenstein series and a sum of normalized Hecke eigenforms. We combine the Hardy-Littlewood circle method with properties of the adjoint square lifting from automorphic forms on $ \rm {GL}(2)$ to $ \rm {GL}(3)$ to bound $ R(p) := \sum _{i} \vert r_{i}\vert$, solving a problem raised by Granville and Ono.

In the case of general $ t$, we use a combination of techniques to bound $ c_{t}(n)$ and as an application prove that for all $ n \geq 0$, $ n \ne t+1$,

$\displaystyle c_{t+1}(n) \geq c_{t}(n) $

provided $ 4 \leq t \leq 198$, as conjectured by Stanton.

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Additional Information

Byungchan Kim
Affiliation: School of Liberal Arts, Seoul National University of Science and Technology, 172 Gongreung 2 dong, Nowongu, Seoul,139-743, Korea

Jeremy Rouse
Affiliation: Department of Mathematics, Wake Forest University, Winston-Salem, North Carolina 27109

Received by editor(s): November 30, 2009
Received by editor(s) in revised form: May 22, 2012
Published electronically: August 19, 2013
Additional Notes: The second author was supported by NSF grant DMS-0901090
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.