Let $\sigma=\sigma_1 \sigma_2 \cdot \cdot \cdot \sigma_n \in S_n$ which means a permutation of the elements $1,2,...,n$. $\sigma_j$ is called a left-right maximum of $ \sigma$ if $\sigma_k <\sigma_j$ for all $k<j$. $a_{n,k}$ is the number of permutations in $S_n$ with exactly $k$ left right maxima. I have to show that $a_{n,k}=\frac{n!}{k!}[z^n](\log\frac{1}{1-z})^k$.
I know that these are the Stirling numbers of first kind, but I have no idea how to proove it. Do you have ideas? Please help me.