15,922 research outputs found
Generalized permutation patterns - a short survey
An occurrence of a classical pattern p in a permutation Ļ is a subsequence of Ļ whose letters are in the same relative order (of size) as those in p. In an occurrence of a generalized pattern, some letters of that subsequence may be required to be adjacent in the permutation. Subsets of permutations characterized by the avoidanceāor the prescribed number of occurrencesā of generalized patterns exhibit connections to an enormous variety of other combinatorial structures, some of them apparently deep. We give a short overview of the state of the art for generalized patterns
Transition probabilities for general birth-death processes with applications in ecology, genetics, and evolution
A birth-death process is a continuous-time Markov chain that counts the
number of particles in a system over time. In the general process with
current particles, a new particle is born with instantaneous rate
and a particle dies with instantaneous rate . Currently no robust and
efficient method exists to evaluate the finite-time transition probabilities in
a general birth-death process with arbitrary birth and death rates. In this
paper, we first revisit the theory of continued fractions to obtain expressions
for the Laplace transforms of these transition probabilities and make explicit
an important derivation connecting transition probabilities and continued
fractions. We then develop an efficient algorithm for computing these
probabilities that analyzes the error associated with approximations in the
method. We demonstrate that this error-controlled method agrees with known
solutions and outperforms previous approaches to computing these probabilities.
Finally, we apply our novel method to several important problems in ecology,
evolution, and genetics
Restricted Motzkin permutations, Motzkin paths, continued fractions, and Chebyshev polynomials
We say that a permutation is a Motzkin permutation if it avoids 132 and
there do not exist such that . We study the
distribution of several statistics in Motzkin permutations, including the
length of the longest increasing and decreasing subsequences and the number of
rises and descents. We also enumerate Motzkin permutations with additional
restrictions, and study the distribution of occurrences of fairly general
patterns in this class of permutations.Comment: 18 pages, 2 figure
Introduction to Partially Ordered Patterns
We review selected known results on partially ordered patterns (POPs) that
include co-unimodal, multi- and shuffle patterns, peaks and valleys ((modified)
maxima and minima) in permutations, the Horse permutations and others. We
provide several (new) results on a class of POPs built on an arbitrary flat
poset, obtaining, as corollaries, the bivariate generating function for the
distribution of peaks (valleys) in permutations, links to Catalan, Narayna, and
Pell numbers, as well as generalizations of few results in the literature
including the descent distribution. Moreover, we discuss q-analogue for a
result on non-overlapping segmented POPs. Finally, we suggest several open
problems for further research.Comment: 23 pages; Discrete Applied Mathematics, to appea
Generalized permutation patterns -- a short survey
An occurrence of a classical pattern p in a permutation \pi is a subsequence
of \pi whose letters are in the same relative order (of size) as those in p. In
an occurrence of a generalized pattern, some letters of that subsequence may be
required to be adjacent in the permutation. Subsets of permutations
characterized by the avoidance--or the prescribed number of occurrences--of
generalized patterns exhibit connections to an enormous variety of other
combinatorial structures, some of them apparently deep. We give a short
overview of the state of the art for generalized patterns.Comment: 11 pages. Added a section on asymptotics (Section 8), added more
examples of barred patterns equal to generalized patterns (Section 7) and
made a few other minor additions. To appear in ``Permutation Patterns, St
Andrews 2007'', S.A. Linton, N. Ruskuc, V. Vatter (eds.), LMS Lecture Note
Series, Cambridge University Pres
On rational approximation of algebraic functions
We construct a new scheme of approximation of any multivalued algebraic
function by a sequence of rational
functions. The latter sequence is generated by a recurrence relation which is
completely determined by the algebraic equation satisfied by . Compared
to the usual Pad\'e approximation our scheme has a number of advantages, such
as simple computational procedures that allow us to prove natural analogs of
the Pad\'e Conjecture and Nuttall's Conjecture for the sequence
in the complement \mathbb{CP}^1\setminus
\D_{f}, where \D_{f} is the union of a finite number of segments of real
algebraic curves and finitely many isolated points. In particular, our
construction makes it possible to control the behavior of spurious poles and to
describe the asymptotic ratio distribution of the family . As an application we settle the so-called 3-conjecture of
Egecioglu {\em et al} dealing with a 4-term recursion related to a polynomial
Riemann Hypothesis.Comment: 25 pages, 8 figures, LaTeX2e, revised version to appear in Advances
in Mathematic
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