Imagine a race with $n$ runners, all equally skilled so that the outcome is just based on luck. If the race is run $m$ times, what are the chances that two runners end up with the same exact average rank? I want to know if there is a closed-form solution.
Here is a more formal statement of the problem. Consider permutations $\sigma_1, \ldots, \sigma_m$ of $\mathbb{N}_n$ drawn uniformly at random. What is the probability that there exist $i,j \in \mathbb{N}_n$ such that $i \neq j$ and $\sum_{k=1}^{m}\sigma_k(i) = \sum_{k=1}^{m}\sigma_k(j)$?