Hook length biases for self-conjugate partitions and partitions with distinct odd parts

2024/06/26 by Catherine Cossaboom, Cossaboom, Catherine
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2406.18480

openalex publication_date 2024/06/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We establish a hook length bias between self-conjugate partitions and partitions of distinct odd parts, demonstrating that there are more hooks of fixed length t ≥ 2 among self-conjugate partitions of n than among partitions of distinct odd parts of n for sufficiently large n. More precisely, we derive asymptotic formulas for the total number of hooks of fixed length t in both classes. This resolves a conjecture of Ballantine, Burson, Craig, Folsom, and Wen.

Related

• Hook lengths in self-conjugate partitions
• Hook length biases and general linear partition inequalities
• Hook length biases in ordinary and t-regular partitions
• Inequalities and asymptotics for hook lengths in ℓ-regular partitions and ℓ-distinct partitions
• Finite analogs of partition bias related to hook length two and a variant of Sylvester's map