The Wikipedia article on the Fisher Yates shuffle has a subsection on Potential Sources of Bias (Implementation Errors). It contains the following observation related to the possible outcomes when shuffling a three element array [1, 2, 3] using an incorrect implementation:

```
There are 6 possible permutations of this array (3! = 6), but the algorithm
produces 27 possible shuffles (3^3 = 27). In this case, [1, 2, 3], [3, 1, 2],
and [3, 2, 1] each result from 4 of the 27 shuffles, while each of the
remaining 3 permutations occurs in 5 of the 27 shuffles.
```

I can easily enumerate the 27 possibilities using a probability tree and confirm that `123, 312, 321`

occur four times each while `132, 213, 231`

occur five times each.

This over representation is also confirmed by the first image in this blog post.

Here's my question: what is so special about `132, 213, 231`

that results in them occuring more often than `123, 312, 321`

? Intuitively, I'd expect some sort of vague "symmetry" between the bias vs. non-bias cases, but that's clearly not true here.

Also, for the case of four elements, the same blog post indicates that `2341, 2314, 2143, 1342`

occur far more frequently than the other permutations. Is there any particular reason for it?

Some related questions: