636 research outputs found
The sorting index
We consider a bivariate polynomial that generalizes both the length and
reflection length generating functions in a finite Coxeter group. In seeking a
combinatorial description of the coefficients, we are led to the study of a new
Mahonian statistic, which we call the sorting index. The sorting index of a
permutation and its type B and type D analogues have natural combinatorial
descriptions which we describe in detail.Comment: 14 pages, minor changes, new references adde
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 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
A barrier for further approximating Sorting By Transpositions
The Transposition Distance Problem (TDP) is a classical problem in genome
rearrangements which seeks to determine the minimum number of transpositions
needed to transform a linear chromosome into another represented by the
permutations and , respectively. This paper focuses on the
equivalent problem of Sorting By Transpositions (SBT), where is the
identity permutation . Specifically, we investigate palisades, a family
of permutations that are "hard" to sort, as they require numerous
transpositions above the celebrated lower bound devised by Bafna and Pevzner.
By determining the transposition distance of palisades, we were able to provide
the exact transposition diameter for -permutations (TD3), a special subset
of the Symmetric Group , essential for the study of approximate solutions
for SBT using the simplification technique. The exact value for TD3 has
remained unknown since Elias and Hartman showed an upper bound for it. Another
consequence of determining the transposition distance of palisades is that,
using as lower bound the one by Bafna and Pevzner, it is impossible to
guarantee approximation ratios lower than when approximating SBT. This
finding has significant implications for the study of SBT, as this problem has
been subject of intense research efforts for the past 25 years
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