\[\begin{aligned}
&\sum_{x=0}^{k} \binom{m}{x}\binom{n-m}{k-x}x^L \\
=&\sum_{i=0}^{L} S_2(L,i)i!\sum_{x=0}^{k} \binom{m}{x}\binom{n-m}{k-x}\binom{x}{i} \\
=&\sum_{i=0}^{L} S_2(L,i)i!\sum_{x=0}^{k} \binom{m}{i}\binom{n-m}{k-x}\binom{m-i}{x-i} \\
=&\sum_{i=0}^{L} S_2(L,i)i!\binom{m}{i} \binom{n-i}{k-i} \\
=&\sum_{i=0}^{L} \left( \sum_{j=0}^{i} \frac{(-1)^{i-j}j^L}{j!(i-j)!} \right)i!\binom{m}{i} \binom{n-i}{k-i} \\
\end{aligned}
\]
卷积处理斯特林数。