  Breakpoint graphs are ubiquitous structures in the field of genome
rearrangements. Their cycle decomposition has proved useful in computing and
bounding many measures of (dis)similarity between genomes, and studying the
distribution of those cycles is therefore critical to gaining insight on the
distributions of the genomic distances that rely on it. We extend here the work
initiated by Doignon and Labarre, who enumerated unsigned permutations whose
breakpoint graph contains $k$ cycles, to signed permutations, and prove
explicit formulas for computing the expected value and the variance of the
corresponding distributions, both in the unsigned case and in the signed case.
We also compare these distributions to those of several well-studied distances,
emphasising the cases where approximations obtained in this way stand out.
Finally, we show how our results can be used to derive simpler proofs of other
previously known results.
