On the Andrews-stanley Refinement of Ramanujan’s Partition Congruence modulo 5 and Generalizations

In a recent study of sign-balanced, labelled posets, Stanley introduced a new integral partition statistic srank(π) = O(π)−O(π′), where O(π) denotes the number of odd parts of the partition π and π′ is the conjugate of π. In a forthcoming paper, 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-quotientrank 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.