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

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 $5$:

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.