How would you use statistics to express the set of permutations on a set of integers? Not saying it isn't possible, I'm just asking how you would approach that.
You would describe a Poisson process... define a permutation that the set of them approaches Poisson distribution. The result is a randomized generalized sieve algorithm. (Also known as Gorosort.) You end up on a journey along combinatorial base of statistics.
It is a terribly complex description. Welcome to statistics. Permutation is a basic combinatorial concept that is assumed to not have to be explained.
