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

TL;DR

This paper proves Singh and Barman's conjecture that the number of 2-hooks in (t+1)-regular partitions of n is at least as many as in t-regular partitions, for all t ≥ 3 and n ≥ 0.

Contribution

The paper provides a proof confirming the conjecture about hook length biases in t-regular partitions, extending previous results known for t=3.

Findings

Confirmed the inequality for all t ≥ 3 and n ≥ 0

Extended the known case t=3 to all t≥3

Established a new understanding of hook length distributions in regular partitions

Abstract

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