The following identity involving Eulerian numbers is well-known: \begin{equation} A(n,m)=\sum_{k=0}^{m}(-1)^k \binom{n+1}{k} (m+1-k)^n. \end{equation} where $A(n,m)$ is the number of permutations $(\pi_1~\pi_2~\cdots~\pi_n)$ of $\{1,2,\ldots,n\}$ having $m$ ascents, namely, $m$ places where $\pi_j < \pi_{j+1}$.
Does anyone know how to prove the above identity? (I found this identity on a research paper which contains no proof.)