Partition Statistics for Cubic Partition Pairs

Abstract

In this brief note, we give two partition statistics which explain the following partition congruences: \begin{align*} b(5n+4) &\equiv 0 \pmod{5}, \\ b(7n+a) &\equiv 0 \pmod{7}, \text{if $a=2$, $3$, $4$, or $6$}. \end{align*} Here, $b(n)$ is the number of $4$-color partitions of $n$ with colors $r$, $y$, $o$, and $b$ subject to the restriction that the colors $o$ and $b$ appear only in even parts.