This question is a follow-up to that found on this page. How can I write the solution to this summation?
$$\sum_\sigma \exp\left[\alpha \sum_{j=1}^N j \sigma(j)\right],$$
where $\alpha$ is a real constant and $\sigma$ represents the sum over all $N!$ elements of the permutation group. This question differs from the linked page since the solution suggested there can not be used here (the summation cannot be separated and the summations switched).