1,026 research outputs found
Odd permutations are nicer than even ones
International audienceWe give simple combinatorial proofs of some formulas for the number of factorizations of permutations in S n as a product of two n-cycles, or of an n-cycle and an (nâ1)-cycle. ... The parameter number of cycles plays a central role in the algebraic theory of the symmetric group, however there are very few results giving a relationship between the number of cycles of two permutations and that of their product. ... The first results on the subject go back to Ore, Bertram, Stanley (see [13], [1] and [15]), who proved some existence theorems. These results allowed to obtain ..
Getting more flavour out of one-flavour QCD
We argue that no notion of flavour is necessary when performing amplitude
calculations in perturbative QCD with massless quarks. We show this explicitly
at tree-level, using a flavour recursion relation to obtain multi-flavoured QCD
from one-flavour QCD. The method relies on performing a colour decomposition,
under which the one-flavour primitive amplitudes have a structure which is
restricted by planarity and cyclic ordering. An understanding of SU(3)_c group
theory relations between QCD primitive amplitudes and their organisation around
the concept of a Dyck tree is also necessary. The one-flavour primitive
amplitudes are effectively N=1 supersymmetric, and a simple consequence is that
all of tree-level massless QCD can be obtained from Drummond and Henn's closed
form solution to tree-level N=4 super Yang-Mills theory.Comment: 27 pages, 6 figure
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
Mixing times of lozenge tiling and card shuffling Markov chains
We show how to combine Fourier analysis with coupling arguments to bound the
mixing times of a variety of Markov chains. The mixing time is the number of
steps a Markov chain takes to approach its equilibrium distribution. One
application is to a class of Markov chains introduced by Luby, Randall, and
Sinclair to generate random tilings of regions by lozenges. For an L X L region
we bound the mixing time by O(L^4 log L), which improves on the previous bound
of O(L^7), and we show the new bound to be essentially tight. In another
application we resolve a few questions raised by Diaconis and Saloff-Coste, by
lower bounding the mixing time of various card-shuffling Markov chains. Our
lower bounds are within a constant factor of their upper bounds. When we use
our methods to modify a path-coupling analysis of Bubley and Dyer, we obtain an
O(n^3 log n) upper bound on the mixing time of the Karzanov-Khachiyan Markov
chain for linear extensions.Comment: 39 pages, 8 figure
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