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

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.