Injectivity of symmetric polynomial maps on partitions

Abstract

Introduced by Ballantine, Beck, and Merca, the elementary symmetric partition function ${\mathrm{pre}}_k$, defined on the set of partitions with at least $k$ parts, has been a topic of recent interest. We prove that ${\mathrm{pre}}_k$ is injective on the set of $m$-ary partitions for positive integers $m\ge k$, generalizing the binary $k=2$ result of Ballantine, Beck, and Merca, and complementing a result of Hadelyn, Niergarth, Li and Li showing that, for each $k\ge 3$, ${\mathrm{pre}}_k$ is not injective on partitions of $n$ with length $2k$ for infinitely many $n$. We introduce the skew Schur partition function ${\mathrm{prs}}_{{\lambda }^{\prime }/{\mu }^{\prime }}$, prove injectivity results for particular choices of ${\lambda }^{\prime },{\mu }^{\prime }$, and describe an application to representation theory.

19 pages