On partitions associated with elementary symmetric polynomials
Cristina Ballantine, Shaheen Nazir, Bridget Eileen Tenner, Karlee Westrem, Chenchen Zhao
TL;DR
This work studies the elementary symmetric partition map $\mathop{\mathrm{pre}}_k$ on partitions, proving injectivity on a natural large subset $\mathcal{S}_k$ by recovering multiplicity data $m_{\lambda}(i)$ from the image $\nu=\mathop{\mathrm{pre}}_k(\lambda)$. It derives consequences for counting parts of fixed size in images, via rooted and colored partitions, and obtains generating functions and forward-difference relations that generalize prior Beck-type conjectures. The paper also connects injectivity to plethysm, showing that full injectivity would allow reconstruction of the entire family $\{e_r\}$ from $\{e_r[ e_k ]\}$ and discusses a Vieta-type interpretation. Open questions remain about the full injectivity for general $k$ and the precise description of the image sets, inviting further exploration of plethysm and partition behavior.
Abstract
The elementary symmetric partition function is a map on the set of partitions. It sends a partition lambda to the partition whose parts are the summands in the evaluation of the elementary symmetric function on the parts of lambda. These elementary symmetric partition functions have been studied before, and are related to plethysm. In this note, we study properties of the elementary symmetric partition functions, particularly related to injectivity and the number of parts appearing in their image partitions.