For a permutation $\pi: [n]\rightarrow [n]$, we define the displacement of $\pi$ to be $\sum_{i\in [n]} |i-\pi(i)|$.
For given $k$, prove that the number of even permutations of $[n]$ with displacement $2k$ minus the number of odd permutations of $[n]$ with displacement $2k$ is $(-1)^{k}\binom{n-1}{k}$.