a complicated Sum

Question

I don't know how i can calculate this complicated multivariate sum : $$ S(l,k)=\sum_{|m|=k} s_{l,m}=\sum_{|m|=k} l(l-1)(l-2)\dots(l-m+1) $$ Where $m=(m_1,\dots,m_n)$, $l=(l_1,\dots,l_n)$, and $k$ a non-negative integer and $|m|=m_1+\dots+m_n$. when a term of this sum is explicitly for a $m=(m_1,\dots,m_n)$ such that $\sum_{i=1}^n |m_i|=k$ : $$ s_{l,m}=\prod_{j=1}^{n}l_j \times\prod_{j=1}^{n}(l_j-1)\times\dots\times \prod_{j=1}^{n}(l_j-m_j+1) $$