A lower bound on the reversal and transposition diameter

One possible model to study genome evolution is to represent genomes as permutations of genes and compute distances based on the minimum number of certain operations (re-arrangements) needed to transform one permutation into another. Under this model, the shorter the distance, the closer the genomes are. Two operations that have been extensively studied are the reversal and the transposition. A reversal is an operation that reverses the order of the genes on a certain portion of the permutation. A transposition is an operation that "cuts" a certain portion of the permutation and "pastes" it elsewhere in the same permutation. In this note, we show that the reversal and transposition distance of the signed permutation pi(n) = (-1 -2... -(n - 1) -n) with respect to the identity is [n/2] +2 for all n greater than or equal to 3. We conjecture that this value is the diameter of the permutation group under these operations.