Thank you all for the great ideas and help. I got it to work on my machine!

@Carl Love 

Oh I forgot to put another condition that in that case, when the bijections are viewed as permutations, it must be prodcut of two cycles with no fixed points. In that case, it is juts pair partitions, so setpartition works. Thanks for pointing it out.

@Carl Love 

thanks a lot. It works as I wanted. Also, if we take two sets to be the same, I just used the function "setpartition" with argument 2. Would this function presented here be more efficient than this? 

@Carl Love Sorry for the late response.It worked for me greatly. Thank you!

@Carl Love 

Thank you so much for your answer!

It seems great for permuting permutations.

If I were to modify it to make it work for any lists of positive integers, where elements are not necessarily distinct,

would the implementation be totally different?