On the Andrews-Stanley Refinement of Ramanujan's Partition Congruence Modulo 5

Abstract

In a recent study of sign-balanced, labelled posets Stanley [13], 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' the conjugate of pi. In [1] Andrews proved the following refinement of Ramanujan's partition congruence mod 5: p[0](5n +4) = p[2](5n + 4) = 0 (mod 5), p(n) = p[0](n) + p[2](n), where p[i](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 p[i](5n + 4) (i = 0, 2) into five equinumerous classes. In this paper we discuss two such statistics. The first one, while new, is intimately related to the Andrews-Garvan [2] crank. The second one is in terms of the 5-core crank, introduced by Garvan, Kim and Stanton [9]. Finally, we discuss some new formulas for partitions that are 5-cores.