A note on multiset reconstruction from sum and pairwise products

Ballantine, Beck, and Merca defined a map ${\mathrm{p}\mathrm{r}\mathrm{e}}_2$, which sends an integer partition $\lambda =({\lambda }_1,\dots ,{\lambda }_{\mathrm{\ell }})$ to the set $\{{\lambda }_i{\lambda }_j:1\le i<j\le \mathrm{\ell }\}$ consisting of the pairwise products of parts of $\lambda$. The same three authors and Sagan conjectured that for each $n$, the map ${\mathrm{p}\mathrm{r}\mathrm{e}}_2$ is injective on the set of integer partitions of $n$. In this note, we prove their conjecture.