Counterexamples regarding elementary symmetric partitions

Abstract

Ballantine, Beck, and Merca defined the elementary symmetric partition map pre${}_j$ that sends a partition $\hat{I}»$ to a larger partition whose parts are the summands appearing in the evaluation of the $j$-th elementary symmetric polynomial on $\hat{I}»$. They conjectured that pre${}_j$ is injective on the set of partitions of $n$ with length $\ell \ge j$. 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 $j\ge 3$. We also prove that the analogous map prh${}_j$ defined via the complete homogenous symmetric polynomial is injective on the set of all partitions.

18 pages. Comments welcome! v2. Small correction to Conjecture 1.3, updated a reference v3. Added Remark 1.4 and acknowledgments, noted progress made on Question 5.3