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arXiv · 2606.00420

Counterexamples regarding elementary symmetric partitions

Vixail Hadelyn·Harper Niergarth·Weiyou Li·Wenhui Li
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Abstract

Ballantine, Beck, and Merca defined the elementary symmetric partition map pre$_j$ that sends a partition $\lambda$ to a larger partition whose parts are the summands appearing in the evaluation of the $j$-th elementary symmetric polynomial on $\lambda$. They conjectured that pre$_j$ is injective on the set of partitions of $n$ with length $\ell \geq j$. The $\ell = j$ case was disproved by Devnani and Eyyunni; they instead conjectured the statement to be true for $\ell > j$. In this article, we answer this refined conjecture in the negative by proving that pre$_j$ is not injective on partitions of $n$ with length $2j$ for $j \geq 3$. We also prove that the analogous map prh$_j$ defined via the complete homogenous symmetric polynomial is injective on the set of all partitions.

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Vixail Hadelyn, Harper Niergarth, Weiyou Li, Wenhui Li. 2026-05-29. Counterexamples regarding elementary symmetric partitions. https://arxiv.org/abs/2606.00420

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