Elementary symmetric polynomials and a potentially injective family of maps on partitions

TL;DR

This paper disproves a recent conjecture about the injectivity of maps on partitions derived from elementary symmetric polynomials, introduces a modified conjecture, and explores relationships between these maps.

Contribution

It provides an infinite family of counterexamples to the conjecture, offers alternative proofs for some subcases, and analyzes the image size of specific maps.

Findings

Disproved the conjecture for certain cases using counterexamples.

Provided alternative proofs for the case k=2.

Discussed lower bounds for the image of pre_2.

Abstract

In this article, we provide an infinite family of examples to disprove a recent conjecture due to Ballantine and her collaborators on the injectivity of a class of maps, namely pre_k, defined on integer partitions. These maps arise from applying the sequence of elementary symmetric polynomials to integer partitions, where pre_k is associated with the kth polynomial. Subsequently, we state a modified version of their conjecture. Throwing fresh light on these class of maps, we study the inter-relationships between them, deviating from the approaches so far, which study these maps one at a time. Though one case of the conjecture (k=2) has now been settled independently by the work of Ballantine and collaborators, and Li, we provide alternate proofs of three subcases corresponding to this settled case. We also discuss lower bounds for the number of partitions of n which are in the image of the map pre_2.