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

2025/10/22 by Lin, Hongshu, Wenston J. T. Zang, Zang, Wenston J. T.
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2510.19185

openalex publication_date 2025/10/22 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28

Abstract

Let b_t,i(n) denote the total number of i-hooks in t-regular partitions of n. Singh and Barman conjectured that b_t+1,2(n) ≥ b_t,2(n) holds for all t≥ 3 and n≥ 0. This conjecture was known to hold for t=3 due to work of Barman Mahanta and Singh. In this paper, we prove this conjecture.

Citations

• Inequalities and asymptotics for hook lengths in ℓ-regular partitions and ℓ-distinct partitions
• Hook length inequalities for t-regular partitions in the t-aspect
• Hook length biases in ordinary and t-regular partitions
• Inequalities and asymptotics for hook numbers in restricted partitions
• Hook length biases and general linear partition inequalities
• The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications
• Seiberg-Witten Theory and Random Partitions

Related

• On the hook length biases of the 2- and 3-regular partitions
• On hook length biases in t-regular partitions
• Hook length biases in ordinary and t-regular partitions
• The inequality on the number of 1-hooks, 2-hooks and 3-hooks in t-regular partitions
• Finite analogs of partition bias related to hook length two and a variant of Sylvester's map