A tight upper bound on the number of cyclically adjacent transpositions to sort a permutation

We consider the problem of upper bounding the number of cyclically adjacent transpositions needed to sort a permutation. It is well known that any permutation can be sorted using at most n ( n - 1 ) 2 adjacent transpositions. We show that, if we allow all adjacent transpositions, as well as the transposition that interchanges the element in position 1 with the element in the last position, then the number of transpositions needed is at most ź n 2 4 ź . Upper bound on number of cyclically adjacent transpositions needed to sort a permutation of length n of ź n 2 4 ź .This upper bound matches the known lower bound.Answers open question in Feng, Chitturi and Sudborough 4.Relevant quantity in the design of interconnection networks, and the evolutionary history of the genome.