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

Hongshu Lin, Wenston J. T. Zang

Let $b_{t,i}(n)$ denote the total number of $i$-hooks in $t$-regular partitions of $n$. Singh and Barman conjectured that $b_{t+1,2}(n)\ge b_{t,2}(n)$ holds for all $t\ge 3$ and…