permutations sampling by probability matrix
The problem has recently been actively studied:
https://arxiv.org/abs/2007.15043
https://arxiv.org/abs/1904.03836
https://projecteuclid.org/euclid.ejs/1586419218
https://academic.oup.com/comnet/article-abstract/6/6/833/4772759
P.S. It is true, that problem is not fully or correctly defined, but Raphael's comments are at least a bit confusing.