50 research outputs found
The Combinatorics of Al-Salam-Chihara -Laguerre polynomials
We describe various aspects of the Al-Salam-Chihara -Laguerre polynomials.
These include combinatorial descriptions of the polynomials, the moments, the
orthogonality relation and a combinatorial interpretation of the linearization
coefficients. It is remarkable that the corresponding moment sequence appears
also in the recent work of Postnikov and Williams on enumeration of totally
positive Grassmann cells.Comment: 23 pages, to appear in Adv. in Appl. Math
Combinatorics of the three-parameter PASEP partition function
We consider a partially asymmetric exclusion process (PASEP) on a finite
number of sites with open and directed boundary conditions. Its partition
function was calculated by Blythe, Evans, Colaiori, and Essler. It is known to
be a generating function of permutation tableaux by the combinatorial
interpretation of Corteel and Williams.
We prove bijectively two new combinatorial interpretations. The first one is
in terms of weighted Motzkin paths called Laguerre histories and is obtained by
refining a bijection of Foata and Zeilberger. Secondly we show that this
partition function is the generating function of permutations with respect to
right-to-left minima, right-to-left maxima, ascents, and 31-2 patterns, by
refining a bijection of Francon and Viennot.
Then we give a new formula for the partition function which generalizes the
one of Blythe & al. It is proved in two combinatorial ways. The first proof is
an enumeration of lattice paths which are known to be a solution of the Matrix
Ansatz of Derrida & al. The second proof relies on a previous enumeration of
rook placements, which appear in the combinatorial interpretation of a related
normal ordering problem. We also obtain a closed formula for the moments of
Al-Salam-Chihara polynomials.Comment: 31 page
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
Separation of variables and combinatorics of linearization coefficients of orthogonal polynomials
We propose a new approach to the combinatorial interpretations of
linearization coefficient problem of orthogonal polynomials. We first establish
a difference system and then solve it combinatorially and analytically using
the method of separation of variables. We illustrate our approach by applying
it to determine the number of perfect matchings, derangements, and other
weighted permutation problems. The separation of variables technique naturally
leads to integral representations of combinatorial numbers where the integrand
contains a product of one or more types of orthogonal polynomials. This also
establishes the positivity of such integrals.Comment: Journal of Combinatorial Theory, Series A 120 (2013) 561--59
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
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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.Mathematic
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