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

 

Resolution of a conjecture of Andrews and Lewis involving cranks of partitions


Author: Daniel M. Kane
Journal: Proc. Amer. Math. Soc. 132 (2004), 2247-2256
MSC (2000): Primary 11P82, 11P83
Published electronically: March 3, 2004
MathSciNet review: 2052400
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Abstract: Andrews and Lewis have conjectured that the sign of the number of partitions of $n$ with crank congruent to 0 mod 3, minus the number of partitions of $n$ with crank congruent to 1 mod 3, is determined by the congruence class of $n$ mod 3 apart from a finite number of specific exceptions. We prove this by using the ``circle method" to approximate the value of this difference to great enough accuracy to determine its sign for all sufficiently large $n$.


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Additional Information

Daniel M. Kane
Affiliation: 2814 Regent Street, Madison, Wisconsin 53705
Address at time of publication: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
Email: dankane@mit.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07353-8
PII: S 0002-9939(04)07353-8
Keywords: Circle method, partitions, cranks
Received by editor(s): February 18, 2003
Received by editor(s) in revised form: May 15, 2003
Published electronically: March 3, 2004
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2004 American Mathematical Society