Inequalities and asymptotics for hook lengths in ℓ-regular partitions and ℓ-distinct partitions

2025/01/19 by Eunmi Kim, Kim, Eunmi · 1 citation
Mathematics · #05A17 #11P82 #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Inequalities and Applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2501.10916

openalex publication_date 2025/01/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article, we study hook lengths in ℓ-regular partitions and ℓ-distinct partitions. More precisely, we establish hook length inequalities between ℓ-regular partitions and ℓ-distinct partitions for hook lengths 2 and 3, by deriving asymptotic formulas for the total number of hooks of length t in both partition classes, for t = 1, 2, 3. From these asymptotics, we show that the ratio of the total number of hooks of length t in ℓ-regular partitions to those in ℓ-distinct partitions tends to a constant that depends on ℓ and t. We also provide hook length inequalities within ℓ-regular partitions and within ℓ-distinct partitions.

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• Proof of Singh and Barman's conjecture on hook length biases

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