初等对称划分的单射性与多重集恢复问题
On the Injectivity of Elementary Symmetric Partitions and the Multiset Recovery Problem
- College of Mathematics and Statistics, Chongqing University（重庆大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该论文通过素数指数编码将初等对称划分映射与多重集恢复问题关联，证明在Moser根集外映射单射，并解决等大小约束下的奇异长度单射性及纤维基数上界。
AI中文摘要：
由Ballantine、Beck和Merca引入的初等对称划分映射$\pre_s$将整数划分发送到其各部分上第$s$个初等对称多项式求值中的加数。通过将划分部分编码为素数指数估值向量，我们将$\pre_s$与Leo Moser的加法多重集恢复问题（1957年）联系起来，并证明当$n$位于Moser根集$\mathcal{Z}_s$之外时，$\pre_s$在长度为$n$的划分上是无条件单射的，且无大小限制。此外，在等大小约束$|\lambda| = |\mu| = N$下，我们证明$\pre_4$在孤立奇异长度$n = 12$处是单射的，并且$\pre_3$在$\Part_6(N)$上的每个纤维的基数至多为$2$，完全排除了三元组和四元组。
英文摘要：
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.
