Odds of two runners ending up with the same average rank across multiple races?

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 $$\sum_{k=1}^{m}\sigma_k(i) = \sum_{k=1}^{m}\sigma_k(j)$$?

• For two runners, it's the same as the probability of having the same number of heads and tails after $m$ flips of a fair coin, so $0$ if $m$ is odd and ${m \choose m/2}(.5)^m$ if $m$ is even. – user113102 Apr 23 at 14:43
• For 2 races, the probability for $n$ runners is $1-\frac{a(n)}{n!}$, where $a(n)$ is the sequence oeis.org/A099152. This makes me think it's pretty hard in general. – user113102 Apr 23 at 16:41