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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Proof of the Alder-Andrews conjecture


Authors: Claudia Alfes, Marie Jameson and Robert J. Lemke Oliver
Journal: Proc. Amer. Math. Soc. 139 (2011), 63-78
MSC (2010): Primary 11P82, 11P84
Published electronically: July 19, 2010
MathSciNet review: 2729071
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Abstract: Motivated by classical identities of Euler, Schur, and Rogers and Ramanujan, Alder investigated $ q_d(n)$ and $ Q_d(n),$ the number of partitions of $ n$ into $ d$-distinct parts and into parts which are $ \pm 1 (\operatorname{mod}d+3)$, respectively. He conjectured that $ q_d(n) \geq Q_d(n).$ Andrews and Yee proved the conjecture for $ d = 2^s-1$ and also for $ d \geq 32.$ We complete the proof of Andrews's refinement of Alder's conjecture by determining effective asymptotic estimates for these partition functions (correcting and refining earlier work of Meinardus), thereby reducing the conjecture to a finite computation.


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Additional Information

Claudia Alfes
Affiliation: Lehrstuhl A für Mathematik, RWTH Aachen, Templergraben 64, D-52062 Aachen, Germany
Address at time of publication: Fachbereich Mathematik, Technische Universität Darmstadt, Schlossegartenstrasse 7, D-64289 Darmstadt, Germany
Email: claudia.alfes@matha.rwth-aachen.de

Marie Jameson
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Address at time of publication: Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322
Email: marie.jameson@gmail.com

Robert J. Lemke Oliver
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Address at time of publication: Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322
Email: lemkeoliver@gmail.com

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10500-2
PII: S 0002-9939(2010)10500-2
Received by editor(s): March 10, 2010
Published electronically: July 19, 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.