We study edit distances between permutations of n elements, defined as

the minimum number of operations (from some given set of allowed

transformations, fixed in advance) needed to transform a given

permutation into the identity permutation (1 2 … n). Some well-studied

examples of such transformations include reversals, which reverse the

order of elements in a given interval; element exchanges, which swap any

two elements of the permutation; and block-transpositions, which swap

adjacent, non-intersecting intervals in the permutation.

We

consider the distributions of those distances, i.e. the sequences formed

by the number of permutations at distance 0, 1, 2, … from the

identity permutation. We have observed that many of those distributions

are unimodal (i.e. increasing then decreasing, non-strictly), and that

some of them are log-concave, which means that the terms of the sequence

S satisfy the relation s(k-1) * s(k+1) <= s(k)^2 for every value of

k. This property is well-known to imply unimodality.

Given that

all those distances are defined in a similar way, albeit using a

different set of operations for each of them, we are working on

developing a unified approach to proving our observations on their

distributions. Another reason for developing that approach is that many

of the techniques usually developed for proving such results fail to

generalise to other distributions than the ones they were successfully

applied to.