On the Andrews-Stanley refinement of Ramanujan’s partition congruence modulo $5$ and generalizations

Abstract:

In a recent study of sign-balanced, labelled posets, Stanley introduced a new integral partition statistic \begin{equation*} \mathrm {srank}(\pi ) = {\mathcal O}(\pi ) - {\mathcal O}(\pi ’), \end{equation*} 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.