Elementary symmetric partitions

TL;DR

This paper investigates properties of elementary symmetric partitions, deriving identities, generating functions, and formulas, and explores their generalizations and connections with color partitions.

Contribution

It introduces new identities and formulas for elementary symmetric partitions, generalizes results to d-ary partitions, and explores their connections with color partitions.

Findings

Derived identities and generating functions for pre_2 partitions

Generalized results to d-ary partitions

Explored connections with color partitions

Abstract

Let e_k(x_1,...,x_l) be an elementary symmetric polynomial and let mu = (mu_1,...,mu_l) be an integer partition. Define pre_k(mu) to be the partition whose parts are the summands in the evaluation e_k(mu_1,...,mu_l). The study of such partitions was initiated by Ballantine, Beck, and Merca who showed (among other things) that pre_2 is injective as a map on binary partitions of n. In the present work we derive a host of identities involving the sequences which count the number of parts of a given value in the image of pre_2. These include generating functions, explicit expressions, and formulas for forward differences. We generalize some of these to d-ary partitions and explore connections with color partitions. Our techniques include the use of generating functions and bijections on rooted partitions. We end with a list of conjectures and a direction for future research.