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

Abstract

The elementary symmetric partition map $\pre_s$, 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 $\pre_s$ to Leo Moser's additive Multiset Recovery Problem (1957) and prove that $\pre_s$ 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 $|λ| = |μ| = N$, we prove that $\pre_4$ is injective at the isolated singular length $n = 12$, and that every fiber of $\pre_3$ on $\Part_6(N)$ has cardinality at most $2$, completely excluding both triplets and quartets.