On the Injectivity of Elementary Symmetric Partitions and the Multiset Recovery Problem

The elementary symmetric partition map ${\text{pre}}_s$, introduced by Ballantine, Beck, and Merca, sends an integer partition to the summands in the evaluation of the $s$-th elementary symmetric polynomial at its parts. By encoding partition parts as prime-exponent valuation vectors, we connect ${\text{pre}}_s$ to Leo Moser’s additive Multiset Recovery Problem (1957) and prove that ${\text{pre}}_s$ is unconditionally injective on partitions of length $n$ whenever $n$ lies outside the Moser root set ${\mathcal{Z}}_s$, with no size restrictions. Furthermore, under the equal-size constraint $\mathrm{\mid }\lambda \mathrm{\mid }=\mathrm{\mid }\mu \mathrm{\mid }=N$, we prove that ${\text{pre}}_4$ is injective at the isolated singular length $n=12$, and that every fiber of ${\text{pre}}_3$ on ${\text{Part}}_6(N)$ has cardinality at most $2$, completely excluding both triplets and quartets.