What is a good way of defining a non-uniform probability distribution on a permutation of k objects?

For example, suppose the parameter was an ideal ordering, and the probability of an ordering was proportional to its Levenshtein distance from the ideal ordering. But too me this seems hacky.

Another approach might to specify a transition matrix...

I guess I'm wondering if there is a more canonical probability distribution over permutations that you can skew in one way or another with a basic set of parameters