On the AndrewsStanley refinement of Ramanujan's partition congruence modulo and generalizations
Authors:
Alexander Berkovich and Frank G. Garvan
Journal:
Trans. Amer. Math. Soc. 358 (2006), 703726
MSC (2000):
Primary 11P81, 11P83; Secondary 05A17, 05A19
Published electronically:
March 10, 2005
MathSciNet review:
2177037
Fulltext PDF Free Access
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Abstract: In a recent study of signbalanced, labelled posets, Stanley introduced a new integral partition statistic
where denotes the number of odd parts of the partition and is the conjugate of . In a forthcoming paper, Andrews proved the following refinement of Ramanujan's partition congruence mod : where () denotes the number of partitions of with and is the number of unrestricted partitions of . Andrews asked for a partition statistic that would divide the partitions enumerated by () into five equinumerous classes. In this paper we discuss three such statistics: the STcrank, the quotientrank and the corecrank. The first one, while new, is intimately related to the AndrewsGarvan (1988) crank. The second one is in terms of the quotient of a partition. The third one was introduced by Garvan, Kim and Stanton in 1990. We use it in our combinatorial proof of the Andrews refinement. Remarkably, the Andrews result is a simple consequence of a stronger refinement of Ramanujan's congruence mod . This more general refinement uses a new partition statistic which we term the BGrank. We employ the BGrank to prove new partition congruences modulo . Finally, we discuss some new formulas for partitions that are cores and discuss an intriguing relation between cores and the AndrewsGarvan crank.
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Additional Information
Alexander Berkovich
Affiliation:
Department of Mathematics, University of Florida, Gainesville, Florida 326118105
Email:
alexb@math.ufl.edu
Frank G. Garvan
Affiliation:
Department of Mathematics, University of Florida, Gainesville, Florida 326118105
Email:
frank@math.ufl.edu
DOI:
http://dx.doi.org/10.1090/S0002994705037517
PII:
S 00029947(05)037517
Keywords:
Partitions,
$t$cores,
ranks,
cranks,
Stanley's statistic,
Ramanujan's congruences
Received by editor(s):
January 12, 2004
Received by editor(s) in revised form:
February 24, 2004
Published electronically:
March 10, 2005
Article copyright:
© Copyright 2005 American Mathematical Society
