A fairly old problem in modular representation theory is to determine the vanishing behavior of the $\mathop {\mathrm Hom}\nolimits $ groups and higher $\mathop {\mathrm Ext}\nolimits $ groups of Weyl modules and to compute the dimension of the $\mathbb{Z} /(p)$-vector space $\mathop {\mathrm Hom}\nolimits _{\bar{A}_r}(\bar{K}_\lambda ,\bar{K}_\mu )$ 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 $\mathop {\mathrm Hom}\nolimits _{\bar{A}_r}(\bar{K}_\lambda ,\bar{K}_\mu )$ and provide a new formula for the intertwining number for any $n$-rowed partition.