The distribution of cycles in breakpoint graphs of signed permutations

TL;DR

This paper extends the enumeration and statistical analysis of cycle distributions in breakpoint graphs from unsigned to signed permutations, providing explicit formulas for expectations and variances, and comparing these distributions to other genomic distance measures.

Contribution

It introduces explicit formulas for the expected value and variance of cycle distributions in breakpoint graphs of signed permutations, expanding prior unsigned permutation results.

Findings

Derived formulas for expected cycle counts in signed permutations

Compared cycle distribution with other genomic distance measures

Provided simpler proofs for existing results

Abstract

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.