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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the Andrews-Stanley refinement of Ramanujan's partition congruence modulo $5$ 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: In a recent study of sign-balanced, labelled posets, Stanley introduced a new integral partition statistic

\begin{displaymath}\mathrm{srank}(\pi) = {\mathcal O}(\pi) - {\mathcal O}(\pi'), \end{displaymath}

where ${\mathcal O}(\pi)$ denotes the number of odd parts of the partition $\pi$and $\pi'$ is the conjugate of $\pi$. In a forthcoming paper, Andrews proved the following refinement of Ramanujan's partition congruence mod $5$:
\begin{align*}p_0(5n+4) &\equiv p_2(5n+4) \equiv 0 \pmod{5},\\ p(n) &= p_0(n) + p_2(n), \end{align*}
where $p_i(n)$ ($i=0,2$) denotes the number of partitions of $n$ with $\mathrm{srank}\equiv i\pmod{4}$ and $p(n)$ is the number of unrestricted partitions of $n$. Andrews asked for a partition statistic that would divide the partitions enumerated by $p_i(5n+4)$ ($i=0,2$) into five equinumerous classes.

In this paper we discuss three such statistics: the ST-crank, the $2$-quotient-rank and the $5$-core-crank. The first one, while new, is intimately related to the Andrews-Garvan (1988) crank. The second one is in terms of the $2$-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 $5$. 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 $5$. Finally, we discuss some new formulas for partitions that are $5$-cores and discuss an intriguing relation between $3$-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: http://dx.doi.org/10.1090/S0002-9947-05-03751-7
PII: S 0002-9947(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