The inequality on the number of $1$-hooks, $2$-hooks and $3$-hooks in $t$-regular partitions

Let $b_{n,k}$ denote the number of hooks of length $k$ in all the $t$-regular partitions of $n$. Singh and Barman raised the question of finding the relation between $b_{t,2}(n)$ and $b_{t,1}(n)$. Kim showed that there exists $N$ such that $b_{t,2}(n)\ge b_{t,1}(n)$ and $b_{t,2}(n)\ge b_{t,3}(n)$ for $n>N$. In this paper, we find an explicit bound of $N=O(t^5)$ for $b_{t,2}(n)\ge b_{t,1}(n)$ and show that $b_{t,2}(n)\ge b_{t,3}(n)$ for all $n\ge 4$.