666 research outputs found
Polynomial Triangles Revisited
A polynomial triangle is an array whose inputs are the coefficients in
integral powers of a polynomial. Although polynomial coefficients have appeared
in several works, there is no systematic treatise on this topic. In this paper
we plan to fill this gap. We describe some aspects of these arrays, which
generalize similar properties of the binomial coefficients. Some combinatorial
models enumerated by polynomial coefficients, including lattice paths model,
spin chain model and scores in a drawing game, are introduced. Several known
binomial identities are then extended. In addition, we calculate recursively
generating functions of column sequences. Interesting corollaries follow from
these recurrence relations such as new formulae for the Fibonacci numbers and
Hermite polynomials in terms of trinomial coefficients. Finally, properties of
the entropy density function that characterizes polynomial coefficients in the
thermodynamical limit are studied in details.Comment: 24 pages with 1 figure eps include
Gibbs distributions for random partitions generated by a fragmentation process
In this paper we study random partitions of 1,...n, where every cluster of
size j can be in any of w\_j possible internal states. The Gibbs (n,k,w)
distribution is obtained by sampling uniformly among such partitions with k
clusters. We provide conditions on the weight sequence w allowing construction
of a partition valued random process where at step k the state has the Gibbs
(n,k,w) distribution, so the partition is subject to irreversible fragmentation
as time evolves. For a particular one-parameter family of weight sequences
w\_j, the time-reversed process is the discrete Marcus-Lushnikov coalescent
process with affine collision rate K\_{i,j}=a+b(i+j) for some real numbers a
and b. Under further restrictions on a and b, the fragmentation process can be
realized by conditioning a Galton-Watson tree with suitable offspring
distribution to have n nodes, and cutting the edges of this tree by random
sampling of edges without replacement, to partition the tree into a collection
of subtrees. Suitable offspring distributions include the binomial, negative
binomial and Poisson distributions.Comment: 38 pages, 2 figures, version considerably modified. To appear in the
Journal of Statistical Physic
Why Delannoy numbers?
This article is not a research paper, but a little note on the history of
combinatorics: We present here a tentative short biography of Henri Delannoy,
and a survey of his most notable works. This answers to the question raised in
the title, as these works are related to lattice paths enumeration, to the
so-called Delannoy numbers, and were the first general way to solve Ballot-like
problems. These numbers appear in probabilistic game theory, alignments of DNA
sequences, tiling problems, temporal representation models, analysis of
algorithms and combinatorial structures.Comment: Presented to the conference "Lattice Paths Combinatorics and Discrete
Distributions" (Athens, June 5-7, 2002) and to appear in the Journal of
Statistical Planning and Inference
A hybrid sampler for Poisson-Kingman mixture models
This paper concerns the introduction of a new Markov Chain Monte Carlo scheme
for posterior sampling in Bayesian nonparametric mixture models with priors
that belong to the general Poisson-Kingman class. We present a novel compact
way of representing the infinite dimensional component of the model such that
while explicitly representing this infinite component it has less memory and
storage requirements than previous MCMC schemes. We describe comparative
simulation results demonstrating the efficacy of the proposed MCMC algorithm
against existing marginal and conditional MCMC samplers
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