A note on partitions in the image of pre${}_2$

Devnani and Eyyunni recently studied the maps pre${}_k$ on integer partitions, which arise from applying elementary symmetric polynomials to the parts of a partition. They asked whether there exists $n\ge 1$ such that exactly one partition of $n$ lies in the image of pre${}_2$. We show that this occurs only for $n$ in {1, 2, 4}, and that for all $n\ge 5$, at least two partitions of n are in the image of pre${}_2$.