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A fairly old problem in modular representation theory is to determine the vanishing behavior of the $\mathrm{H}om$ groups and higher $\mathrm{E}xt$ groups of Weyl modules and to compute the dimension of the $\mathbb{Z}/\left(p\right)$-vector space ${\mathrm{H}om}_{{\overline{A}}_r}({\overline{K}}_{\lambda },{\overline{K}}_{\mu })$ for any partitions $\lambda$, $\mu$ of $r$, which is the intertwining number. K. Akin, D. A. Buchsbaum, and D. Flores solved this problem in the cases of partitions of length two and three. In this paper, we describe the vanishing behavior of the groups ${\mathrm{H}om}_{{\overline{A}}_r}({\overline{K}}_{\lambda },{\overline{K}}_{\mu })$ and provide a new formula for the intertwining number for any...