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 P_{n} = { p_{i,n}, i ∈ `[1,n!]`

} defined by

- p
_{1,n}is the identity permutation - p
_{i+1,n}= p_{i,n}t_{i+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.