Elementary symmetric partitions

2024/09/17 by Cristina Ballantine, Ballantine, Cristina, George R. Beck +5 · 1 citation
Mathematics · #05A17 #05A19 (Secondary) #05E05 (Primary) 05A15 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2409.11268

openalex publication_date 2024/09/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let e_k(x₁,...,x_l) be an elementary symmetric polynomial and let mu = (mu₁,...,mu_l) be an integer partition. Define pre_k(mu) to be the partition whose parts are the summands in the evaluation e_k(mu₁,...,mu_l). The study of such partitions was initiated by Ballantine, Beck, and Merca who showed (among other things) that pre₂ 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 pre₂. 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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• On partitions associated with elementary symmetric polynomials

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