1,337 research outputs found
Restricting Dyck Paths and 312-avoiding Permutations
Dyck paths having height at most and without valleys at height are
combinatorially interpreted by means of 312-avoding permutations with some
restrictions on their \emph{left-to-right maxima}. The results are obtained by
analyzing a restriction of a well-known bijection between the sets of Dyck
paths and 312-avoding permutations. We also provide a recursive formula
enumerating these two structures using ECO method and the theory of production
matrices. As a further result we obtain a family of combinatorial identities
involving Catalan numbers
Singularity analysis, Hadamard products, and tree recurrences
We present a toolbox for extracting asymptotic information on the
coefficients of combinatorial generating functions. This toolbox notably
includes a treatment of the effect of Hadamard products on singularities in the
context of the complex Tauberian technique known as singularity analysis. As a
consequence, it becomes possible to unify the analysis of a number of
divide-and-conquer algorithms, or equivalently random tree models, including
several classical methods for sorting, searching, and dynamically managing
equivalence relationsComment: 47 pages. Submitted for publicatio
The normal distribution is -infinitely divisible
We prove that the classical normal distribution is infinitely divisible with
respect to the free additive convolution. We study the Voiculescu transform
first by giving a survey of its combinatorial implications and then
analytically, including a proof of free infinite divisibility. In fact we prove
that a subfamily Askey-Wimp-Kerov distributions are freely infinitely
divisible, of which the normal distribution is a special case. At the time of
this writing this is only the third example known to us of a nontrivial
distribution that is infinitely divisible with respect to both classical and
free convolution, the others being the Cauchy distribution and the free
1/2-stable distribution.Comment: AMS LaTeX, 29 pages, using tikz and 3 eps figures; new proof
including infinite divisibility of certain Askey-Wilson-Kerov distibution
Q-systems, Heaps, Paths and Cluster Positivity
We consider the cluster algebra associated to the -system for as a
tool for relating -system solutions to all possible sets of initial data. We
show that the conserved quantities of the -system are partition functions
for hard particles on particular target graphs with weights, which are
determined by the choice of initial data. This allows us to interpret the
simplest solutions of the Q-system as generating functions for Viennot's heaps
on these target graphs, and equivalently as generating functions of weighted
paths on suitable dual target graphs. The generating functions take the form of
finite continued fractions. In this setting, the cluster mutations correspond
to local rearrangements of the fractions which leave their final value
unchanged. Finally, the general solutions of the -system are interpreted as
partition functions for strongly non-intersecting families of lattice paths on
target lattices. This expresses all cluster variables as manifestly positive
Laurent polynomials of any initial data, thus proving the cluster positivity
conjecture for the -system. We also give an alternative formulation in
terms of domino tilings of deformed Aztec diamonds with defects.Comment: 106 pages, 38 figure
Interactions between Business Cycles, stock Market Cycles and Interest Rates: the Stylised Facts.
In this paper, we study the co-movements between stock market indices and real economic activity over the business cycle in France, Germany, Italy, the United Kingdom and the United States, using two complementary approaches in our analysis. First, we identify the turning points in real economy indicators and stock market indices and determine the extent to which these series co-move. Second, we calculate the correlations between the cyclical components of real economy indicators and excess returns, on the one hand, and the correlations between the structural components and these indicators, on the other. We then analyse the co-movements between three-month interest rates and the cyclical and structural components of the real economy and stock market indices.Stock returns ; Comovements ; Turning points ; Spectral analysis.
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