Transposition Diameter and Lonely Permutations

Determining the transposition distance of permutations was proven recently to be \(\textup{NP}\)-hard. However, the problem of the transposition diameter is still open. The known lower bounds for the diameter were given by Meidanis, Walter and Dias when the lengths of the permutations are even and by Elias and Hartman when the lengths are odd. A better lower bound for the transposition diameter was proposed using the new definition of super-bad permutations, that would be a particular family of the lonely permutations. We show that there are no super-bad permutations, by computing the exact transposition distance of the union of two copies of particular lonely permutations that we call knot permutations. Meidanis, Walter, Dias, Elias and Hartman, therefore, still hold the current best lower bound. Moreover, we consider the union of distinct lonely permutations and manage to define an alternative family of permutations that meets the current lower bound.