I am trying to calculate the following sum of harmonics: $$ \sum_{\sigma}\exp\left(i\frac{2\pi}{N}\sum_{k=1}^Nk\cdot\sigma(k)\right), $$ over all permutations $\sigma$ with $N$ being the size of the task. Inside the exponents, the phases represent sums over indices $k$ multiplied by permutation $\sigma(k)$. Because of this product, it is unclear for me how to handle the task.
I wonder, if there exist an analytical solution for that? I would appreciate any help with that.