1,009 research outputs found
Pattern-avoiding alternating words
A word is alternating if either
(when the word is up-down) or (when the word is
down-up). In this paper, we initiate the study of (pattern-avoiding)
alternating words. We enumerate up-down (equivalently, down-up) words via
finding a bijection with order ideals of a certain poset. Further, we show that
the number of 123-avoiding up-down words of even length is given by the
Narayana numbers, which is also the case, shown by us bijectively, with
132-avoiding up-down words of even length. We also give formulas for
enumerating all other cases of avoidance of a permutation pattern of length 3
on alternating words
Avoiding vincular patterns on alternating words
A word is alternating if either
(when the word is up-down) or (when the word is
down-up). The study of alternating words avoiding classical permutation
patterns was initiated by the authors in~\cite{GKZ}, where, in particular, it
was shown that 123-avoiding up-down words of even length are counted by the
Narayana numbers.
However, not much was understood on the structure of 123-avoiding up-down
words. In this paper, we fill in this gap by introducing the notion of a
cut-pair that allows us to subdivide the set of words in question into
equivalence classes. We provide a combinatorial argument to show that the
number of equivalence classes is given by the Catalan numbers, which induces an
alternative (combinatorial) proof of the corresponding result in~\cite{GKZ}.
Further, we extend the enumerative results in~\cite{GKZ} to the case of
alternating words avoiding a vincular pattern of length 3. We show that it is
sufficient to enumerate up-down words of even length avoiding the consecutive
pattern and up-down words of odd length avoiding the
consecutive pattern to answer all of our enumerative
questions. The former of the two key cases is enumerated by the Stirling
numbers of the second kind.Comment: 25 pages; To appear in Discrete Mathematic
An Isomonodromy Cluster of Two Regular Singularities
We consider a linear matrix ODE with two coalescing regular
singularities. This coalescence is restricted with an isomonodromy condition
with respect to the distance between the merging singularities in a way
consistent with the ODE. In particular, a zero-distance limit for the ODE
exists. The monodromy group of the limiting ODE is calculated in terms of the
original one. This coalescing process generates a limit for the corresponding
nonlinear systems of isomonodromy deformations. In our main example the latter
limit reads as , where is the -th Painlev\'e equation. We
also discuss some general problems which arise while studying the
above-mentioned limits for the Painlev\'e equations.Comment: 44 pages, 8 figure
Experimental Quantum Process Discrimination
Discrimination between unknown processes chosen from a finite set is
experimentally shown to be possible even in the case of non-orthogonal
processes. We demonstrate unambiguous deterministic quantum process
discrimination (QPD) of non-orthogonal processes using properties of
entanglement, additional known unitaries, or higher dimensional systems. Single
qubit measurement and unitary processes and multipartite unitaries (where the
unitary acts non-separably across two distant locations) acting on photons are
discriminated with a confidence of in all cases.Comment: 4 pages, 3 figures, comments welcome. Revised version includes
multi-partite QP
Non-Schlesinger Deformations of Ordinary Differential Equations with Rational Coefficients
We consider deformations of and matrix linear ODEs with
rational coefficients with respect to singular points of Fuchsian type which
don't satisfy the well-known system of Schlesinger equations (or its natural
generalization). Some general statements concerning reducibility of such
deformations for ODEs are proved. An explicit example of the general
non-Schlesinger deformation of -matrix ODE of the Fuchsian type with
4 singular points is constructed and application of such deformations to the
construction of special solutions of the corresponding Schlesinger systems is
discussed. Some examples of isomonodromy and non-isomonodromy deformations of
matrix ODEs are considered. The latter arise as the compatibility
conditions with linear ODEs with non-singlevalued coefficients.Comment: 15 pages, to appear in J. Phys.
Symmetry reductions of a particular set of equations of associativity in twodimensional topological field theory
The WDVV equations of associativity arising in twodimensional topological
field theory can be represented, in the simplest nontrivial case, by a single
third order equation of the Monge-Ampe`re type. By investigating its Lie point
symmetries, we reduce it to various nonlinear ordinary differential equations,
and we obtain several new explicit solutions.Comment: 10 pages, Latex, to appear in J. Phys. A: Math. Gen. 200
Shot Noise in Anyonic Mach-Zehnder Interferometer
We show how shot noise in an electronic Mach-Zehnder interferometer in the
fractional quantum Hall regime probes the charge and statistics of quantum Hall
quasiparticles. The dependence of the noise on the magnetic flux through the
interferometer allows for a simple way to distinguish Abelian from non-Abelian
quasiparticle statistics. In the Abelian case, the Fano factor (in units of the
electron charge) is always lower than unity. In the non-Abelian case, the
maximal Fano factor as a function of the magnetic flux exceeds one.Comment: references adde
On pattern avoiding indecomposable permutations
Comtet introduced the notion of indecomposable permutations in 1972. A permutation is indecomposable if and only if it has no proper prefix which is itself a permutation. Indecomposable permutations were studied in the literature in various contexts. In particular, this notion has been proven to be useful in obtaining non-trivial enumeration and equidistribution results on permutations. In this paper, we give a complete classification of indecomposable permutations avoiding a classical pattern of length 3 or 4, and of indecomposable permutations avoiding a non-consecutive vincular pattern of length 3. Further, we provide a recursive formula for enumerating 12 ••• k-avoiding indecomposable permutations for k ≥ 3. Several of our results involve the descent statistic. We also provide a bijective proof of a fact relevant to our studies
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