The distribution of cycles in breakpoint graphs of signed permutations

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

Two enumerative results on cycles of permutations

Answering a question of Bona, it is shown that for n≥2 the probability that 1 and 2 are in the same cycle of a product of two n-cycles on the set {1,2,…,n} is 1/2 if n is odd and 1/2 - 2/(n-1)(n+2) if n is even. Another result concerns the polynomial P[subscript λ](q) = ∑[subscript w]q[superscript κ]((1,2,…,n)⋅w), where w ranges over all permutations in the symmetric group S[subscript n] of cycle type λ, (1,2,…,n) denotes the n-cycle 1→2→⋯→n→1, and κ(v) denotes the number of cycles of the permutation v. A formula is obtained for P[subscript λ](q) from which it is deduced that all zeros of P[subscript λ](q) have real part 0.National Science Foundation (U.S.) (Grant 0604423