Bounds on the Transposition Distance for Lonely Permutations

The problem of determining the transposition distance of permutations is a notoriously challenging one; to this date, neither there exists a polynomial algorithm for solving it, nor a proof that it is NPhard. Moreover, there are no tight bounds on the transposition distance of permutations in general. Our proposed approach merges two successful strategies: the classical reality and desire diagram proposed by Bafna and Pevzner and the more recent toric equivalence relation proposed by Eriksson et al. We focus on unitary toric equivalence classes and the corresponding lonely permutations. In a previous paper, we considered the case n+1 prime, proved that the reality and desire diagram of such lonely permutations has just one odd cycle and succeed in identifying in this subset of lonely permutations, new permutations for which the transposition distance is computed. The present paper extends regularity properties of the cycle structure for general n, yielding tight bounds for the transposition distance of lonely permutations. The subset of lonely permutations that are 3-permutations is characterized and consequently an upper bound is obtained for their transposition distances.