<a href="http://www.wiki.canisiusmath.net/index.php?title=Well-Known_Generators_of_the_Symmetric_Group">It is well known that all symmetric group can be generated using two generators</a>

The two generators are:  
1) $(1,2)$  
2) $(1,2,3,\dots ,n)$

>Question: Is there a deterministic algorithm to generate all permutations without repetition using only these two generators?  
(Bonus 1: The algorithm generates the permutations in a cycle. Bonus 2: Not requiring the inverse of generator 2)

It is easy to generate all of them without repetition using $n-1$ generators, by the
<a href="http://en.wikipedia.org/wiki/Steinhaus%E2%80%93Johnson%E2%80%93Trotter_algorithm">Steinhaus-Johnson-Trotter algorithm</a>.  
It is easy to generate all of them, with repetition, using two generators.  
However I was unable to find a way to generate all without repetition and using only two generators.

As this approach seems natural, I suspect someone should have worked on it but I was unable to find any references online.  

Does anyone knows the status of this problem?