Sorting with fixed-length reversals

A popular puzzle TOP-SPIN consists of a permutation of 20 numbered disks on an oval track, with a turnstile capable of reversing a string of 4 consecutive disks. The goal is to sort the disks into identity permutation using reversals. We consider the more general case of sorting n element permutations with a turnstile of size k. The problem of computing the reversal distances is of considerable importance in reconstructing the evolutionary history of the genome. The minimum reversal sequence sorting the one genome to another corresponds to the most likely evolutionary path between them. We give a complete solution for all n and k, of the number of equivalence classes of n-permutations under k:-reversal, for both permutations and circular permutations. We also prove the upper and lower bounds on the reversal distances between two permutations.