306 research outputs found
The Matrix Ansatz, Orthogonal Polynomials, and Permutations
In this paper we outline a Matrix Ansatz approach to some problems of
combinatorial enumeration. The idea is that many interesting quantities can be
expressed in terms of products of matrices, where the matrices obey certain
relations. We illustrate this approach with applications to moments of
orthogonal polynomials, permutations, signed permutations, and tableaux.Comment: to appear in Advances in Applied Mathematics, special issue for
Dennis Stanto
Crossings, Motzkin paths and Moments
Kasraoui, Stanton and Zeng, and Kim, Stanton and Zeng introduced certain
-analogues of Laguerre and Charlier polynomials. The moments of these
orthogonal polynomials have combinatorial models in terms of crossings in
permutations and set partitions. The aim of this article is to prove simple
formulas for the moments of the -Laguerre and the -Charlier polynomials,
in the style of the Touchard-Riordan formula (which gives the moments of some
-Hermite polynomials, and also the distribution of crossings in matchings).
Our method mainly consists in the enumeration of weighted Motzkin paths, which
are naturally associated with the moments. Some steps are bijective, in
particular we describe a decomposition of paths which generalises a previous
construction of Penaud for the case of the Touchard-Riordan formula. There are
also some non-bijective steps using basic hypergeometric series, and continued
fractions or, alternatively, functional equations.Comment: 21 page
Tableaux combinatorics for the asymmetric exclusion process and Askey-Wilson polynomials
Introduced in the late 1960's, the asymmetric exclusion process (ASEP) is an
important model from statistical mechanics which describes a system of
interacting particles hopping left and right on a one-dimensional lattice of n
sites with open boundaries. It has been cited as a model for traffic flow and
protein synthesis. In the most general form of the ASEP with open boundaries,
particles may enter and exit at the left with probabilities alpha and gamma,
and they may exit and enter at the right with probabilities beta and delta. In
the bulk, the probability of hopping left is q times the probability of hopping
right. The first main result of this paper is a combinatorial formula for the
stationary distribution of the ASEP with all parameters general, in terms of a
new class of tableaux which we call staircase tableaux. This generalizes our
previous work for the ASEP with parameters gamma=delta=0. Using our first
result and also results of Uchiyama-Sasamoto-Wadati, we derive our second main
result: a combinatorial formula for the moments of Askey-Wilson polynomials.
Since the early 1980's there has been a great deal of work giving combinatorial
formulas for moments of various other classical orthogonal polynomials (e.g.
Hermite, Charlier, Laguerre, Meixner). However, this is the first such formula
for the Askey-Wilson polynomials, which are at the top of the hierarchy of
classical orthogonal polynomials.Comment: An announcement of these results appeared here:
http://www.pnas.org/content/early/2010/03/25/0909915107.abstract This version
of the paper has updated references and corrects a gap in the proof of
Proposition 6.11 which was in the published versio
Linearization coefficients for orthogonal polynomials using stochastic processes
Given a basis for a polynomial ring, the coefficients in the expansion of a
product of some of its elements in terms of this basis are called linearization
coefficients. These coefficients have combinatorial significance for many
classical families of orthogonal polynomials. Starting with a stochastic
process and using the stochastic measures machinery introduced by Rota and
Wallstrom, we calculate and give an interpretation of linearization
coefficients for a number of polynomial families. The processes involved may
have independent, freely independent or q-independent increments. The use of
noncommutative stochastic processes extends the range of applications
significantly, allowing us to treat Hermite, Charlier, Chebyshev, free Charlier
and Rogers and continuous big q-Hermite polynomials. We also show that the
q-Poisson process is a Markov process.Comment: Published at http://dx.doi.org/10.1214/009117904000000757 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
The combinatorics of associated Hermite polynomials
We develop a combinatorial model of the associated Hermite polynomials and
their moments, and prove their orthogonality with a sign-reversing involution.
We find combinatorial interpretations of the moments as complete matchings,
connected complete matchings, oscillating tableaux, and rooted maps and show
weight-preserving bijections between these objects. Several identities,
linearization formulas, the moment generating function, and a second
combinatorial model are also derived.Comment: [v1]: 18 pages, 16 figures; presented at FPSAC 2007 [v2]: Some minor
errors fixed (thanks Bill Chen, Jang Soo Kim) and text rearranged and cleaned
up; no real content changes [v3]: fixed typos, to appear in European J.
Combinatoric
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