Elementary Symmetric Polynomials and a Potentially Injective Family of Maps on Partitions

Ballantine et al. [‘Partitions and elementary symmetric polynomials: an experimental approach’, Ramanujan J. 66 (2) (2025), Article no. 34] proposed two conjectures on the injectivity of a class of maps p r e k preₖ p r e Subscript k defined on integer partitions. These maps arise from applying the sequence of elementary symmetric polynomials to integer partitions. We provide an infinite family of examples to disprove the conjecture for k ≥ 3 k≥ 3 k greater than or equals 3 and state a modified version of it. Throwing fresh light on this class of maps, we study the inter-relationships between them, deviating from the approaches so far, which study these maps one at a time. While the conjecture for k = 2 $k=2$ k equals 2 has now been settled, we provide alternate proofs of three subcases. We also discuss lower bounds for the number of partitions of n that are in the image of the map p r e 2 pre₂ p r e 2 .