Sorting in a probabilistic order
One of the popular models for biased permutations is the Mallows model, dating to a paper of Mallows from 1957. Lu and Boutilier, quoting Doignon et al., give the following recipe for sampling a permutation according to the Mallows distribution, given a parameter $0 &lt; \phi \leq 1$:


Start with the permutation 1.
Insert 2 into position 1 with probability $\frac{\phi-\phi^2}{1-\phi^2}$, and into position 2 with probability $\frac{1-\phi}{1-\phi^2}$.
Insert 3 into positions 1,2,3 with probabilities 
$\frac{\phi^2-\phi^3}{1-\phi^3},\frac{\phi-\phi^2}{1-\phi^3},\frac{1-\phi}{1-\phi^3}$, respectively.
Insert $4,\ldots,n$ in an analogous manner.


When inserting $x$ into position $i$, what you do is shift positions $i,\ldots,x-1$ one step forward, and insert $x$ in the resulting empty spot.

The probability to obtain a permutation $\pi$ is proportional to $\phi$ raised to the number of inversions in $\pi$ (the Kendall $\tau$ distance between $\pi$ and the identity permutation).

Another popular model is the Plackett–Luce model from 1959. There are other models, for example Tallis–Dansie.