[Abstract] Alexander Berkovich and Frank G. Garvan On the Andrews-Stanley refinement of Ramanujan's partition congruence modulo 5 and generalizations Trans. Amer. Math. Soc. 358 (2006), no. 2, 703--726.


Abstract: In a recent study of sign-balanced, labelled posets Stanley, introduced a new integral partition statistic

srank(pi) = O(pi) - O(pi'),
where O(pi) denotes the number of odd parts of the partition pi and pi' is the conjugate of pi. Andrews proved the following refinement of Ramanujan's partition congruence mod 5:
p0(5n +4) = p2(5n + 4) = 0 (mod 5),
p(n) = p0(n) + p2(n),
where pi(n) (i = 0; 2) denotes the number of partitions of n with srank = i (mod 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 pi(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 crank. The second one is in terms of the 2-quotient of a partition. The third one was introduced by Garvan, Kim and Stanton. 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.

The url of this page is http://www.math.ufl.edu/~frank/abstracts/stanley.html.
Created by Francis Garvan (frank@math.ufl.edu) on Thursday, March 11, 2004.
Last update made Mon Dec 19 19:17:30 EST 2005.

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