A note on multiset reconstruction from sum and pairwise products

Ballantine, Beck, and Merca defined a map $\mathrm{pre}_2$, which sends an integer partition $\lambda = (\lambda_1, \dots, \lambda_{\ell})$ to the set $\{\lambda_i\lambda_j : 1 \leq i < j \leq \ell\}$ consisting of the pairwise products of parts of $\lambda$. The same three authors and Sagan conjectured that for each $n$, the map $\mathrm{pre}_2$ is injective on the set of integer partitions of $n$. In this note, we prove their conjecture.