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On the Injectivity of Elementary Symmetric Partitions and the Multiset Recovery Problem
Abstract
The elementary symmetric partition map pres, 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 pres to Leo Moser's additive Multiset Recovery Problem (1957) and prove that pres is unconditionally injective on partitions of length n whenever n lies outside the Moser root set 𝒵s, with no size restrictions. Furthermore, under the equal-size constraint |λ| = |μ| = N, we prove that pre4 is injective at the isolated singular length n = 12, and that every fiber of pre3 on Part6(N) has cardinality at most $2$, completely excluding both triplets and quartets.
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BibTeX
@misc{sun2026injectivity,
title = {{On the Injectivity of Elementary Symmetric Partitions and the Multiset Recovery Problem}},
author = {Ziyao Sun},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21922},
eprint = {2609.21922},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21922},
}
RIS
TY - UNPB
TI - On the Injectivity of Elementary Symmetric Partitions and the Multiset Recovery Problem
AU - Ziyao Sun
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.21922
UR - https://arxiv.org/abs/2609.21922
AB - 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 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.
ER -