For future reference after the discussion https://codereview.stackexchange.com/questions/201725/disjoint-cycles-of-a-permutation
How to go from Cauchy notation to cyclic notation of a permutation with python
After insights from @Accumulation (working but with extra loop and with no inverse) and @Martin R (not working but with cool insight and inverse), here is what I have got: (it seems to work, but is far from the pythonic-elegance suggested by Martin R).
def permutation_from_cauchy_to_disjoints_cycles(cauchy_perm):