Given an ordered string T of unique elements, consider an operation that takes k elements and permutes them randomly.

If you consider all the permutations of T, and you categorize them according to the *minimum* number of operations required to reach T, how many permutations are in each grouping?

For example, ABC, where an operation is swapping 2 elements:

0 operations: ABC (1 permutation)

1 operation: BAC, CBA, ACB (3 permutations)

2 operations: CAB, BCA (2 permutations)

So we could write this as

### Swap 2
- n= 2: 1,3,2

So what do these groupings look like?

### Swap 2
- n= 2: 1,1
- n= 3: 1,3,2
- n= 4: 1,6,11,6
- n= 5: 1,10,35,50,24
- n= 6: 1,15,85,225,274,120
- n= 7: 1,21,175,735,1624,1764,720
- n= 8: 1,28,322,1960,6769,13132,13068,5040

h3.Swap 3

- n= 3: 1,5
- n= 4: 1,14,9
- n= 5: 1,30,89
- n= 6: 1,55,439,225
- n= 7: 1,91,1519,3429
- n= 8: 1,140,4214,24940,11025
- n= 9: 1,204,10038,122156,230481

h3.Swap 4

- n= 4: 1,23
- n= 5: 1,75,44
- n= 6: 1,190,529
- n= 7: 1,406,4633
- n= 8: 1,770,27341,12208
- n= 9: 1,1338,118173,243368