On the Andrews-Stanley refinement of Ramanujan's partition congruence modulo and generalizations

Authors:
Alexander Berkovich and Frank G. Garvan

Journal:
Trans. Amer. Math. Soc. **358** (2006), 703-726

MSC (2000):
Primary 11P81, 11P83; Secondary 05A17, 05A19

Published electronically:
March 10, 2005

MathSciNet review:
2177037

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Abstract | References | Similar Articles | Additional Information

Abstract: In a recent study of sign-balanced, 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 ST-crank, the -quotient-rank and the -core-crank. The first one, while new, is intimately related to the Andrews-Garvan (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 BG-rank. We employ the BG-rank 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 Andrews-Garvan crank.

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Additional Information

**Alexander Berkovich**

Affiliation:
Department of Mathematics, University of Florida, Gainesville, Florida 32611-8105

Email:
alexb@math.ufl.edu

**Frank G. Garvan**

Affiliation:
Department of Mathematics, University of Florida, Gainesville, Florida 32611-8105

Email:
frank@math.ufl.edu

DOI:
https://doi.org/10.1090/S0002-9947-05-03751-7

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