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arXiv · 2409.11268

Elementary symmetric partitions

Cristina Ballantine·George Beck·Mircea Merca·Bruce Sagan
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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.

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Cristina Ballantine, George Beck, Mircea Merca, Bruce Sagan. 2024-09-17. Elementary symmetric partitions. https://arxiv.org/abs/2409.11268

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