Permutations are my Samples

I have a question matching samples to a known distribution where my samples are actually permutations.

Let's suppose I have 100 people (p1, p2, ... p100) competing in a race. I get a single sample which is the top 15 finishers in order, maybe something like:
(p67, p34, p12, p90, ..., p2) <- finishing order for the top 15 people.

Let's suppose I re-run this race 20 times or so and get 20 different top-15 finishing orders. These are my 20 samples.

Now I can also analytically compute the probability of any 15-person permutation based on knowledge I have about the ability for each of the 100 people in the race. So I can theoretically compute a probability distribution over all possible 15-permutations (though there are too many to compute in practice).

I am looking for a goodness of fit type test where I can compare the likelihood of my 20 samples against the probability distribution of potential finishing orders. I suspect the race is "rigged", and the results I am observing do not match the known probability distribution over all permutations, i.e. I am "observing" a lot of very unlikely finishing orders. I have looked at trying some basics such as Chi-square and K-S testing. Anyone seen something like this before where the samples are permutations? Any help is very much appreciated.
I think it is 'rank ordered logits' for this, with 'partial rankings'. unfortunately for you, i have not looked at this type of data in recent memory, so is as far as i can take you on your journey. id bet some of the social science types around here would know.