So I *think* it's possible to create an infinite sequence of transpositions `T` = { `t``i`, i ≥ 2 } satisfying ∀ `i` ∈ `[2,n!]` 


* ∃ `a,b` ∈ `[1,n]` s.t. `t`i = `(a b)`
* `n!|(j - i)` ⇒ `t``j` = `t``i`


such that if you consider the sequence of permutations Pn = { pi,n, i ∈ `[1,n!]` } defined by


* p1,n is the identity permutation
* pi+1,n = pi,n ti+1


that `P`n contains every permutation of `n` elements exactly once.




---


For example, if `T` starts with



```
{ (1 2), (1 3), (1 2), (1 3), (1 2), (1 4),
  (1 2), (1 3), (1 2), (1 3), (1 2), (2 4),
  (1 2), (1 3), (1 2), (1 3), (1 2), (3 4),
  (1 2), (1 3), (1 2), (1 3), (1 2), ....

```

then `P`4 is given by



```
{ (1 2 3 4), (2 1 3 4), (3 1 2 4), (1 3 2 4), (2 3 1 4), (3 2 1 4),
  (4 2 1 3), (2 4 1 3), (1 4 2 3), (4 1 2 3), (2 1 4 3), (1 2 4 3),
  (1 3 4 2), (3 1 4 2), (4 1 3 2), (1 4 3 2), (3 4 1 2), (4 3 1 2),
  (4 3 2 1), (3 4 2 1), (2 4 3 1), (4 2 3 1), (3 2 4 1), (2 3 4 1) }

```

and `P`4 contains every permutation of 4 elements exactly once.




---


What I'd like to find is a constructive proof for the existence of `T`.
I'm sure I'm retreading old ground here, I just don't seem to be searching for the right terms to turn up what I'm looking for.



Sequence of transpositions that generates the set of permutations
See for instance <http://en.wikipedia.org/wiki/Steinhaus-Johnson-Trotter_algorithm>.


