  Inversions, also sometimes called reversals, are a major contributor to
variation among bacterial genomes, with studies suggesting that those involving
small numbers of regions are more likely than larger inversions. Deletions may
arise in bacterial genomes through the same biological mechanism as inversions,
and hence a model that incorporates both is desirable. However, while inversion
distances between genomes have been well studied, there has yet to be a model
which accounts for the combination of both deletions and inversions.
  To account for both of these operations, we introduce an algebraic model that
utilises partial permutations. This leads to an algorithm for calculating the
minimum distance to the most recent common ancestor of two bacterial genomes
evolving by inversions (of adjacent regions) and deletions. The algebraic model
makes the existing short inversion models more complete and realistic by
including deletions, and also introduces new algebraic tools into evolutionary
distance problems.
