2,496 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
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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
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
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
Tableaux combinatorics for the asymmetric exclusion process
The partially asymmetric exclusion process (PASEP) is an important model from
statistical mechanics which describes a system of interacting particles hopping
left and right on a one-dimensional lattice of sites. It is partially
asymmetric in the sense that the probability of hopping left is times the
probability of hopping right. Additionally, particles may enter from the left
with probability and exit from the right with probability .
In this paper we prove a close connection between the PASEP and the
combinatorics of permutation tableaux. (These tableaux come indirectly from the
totally nonnegative part of the Grassmannian, via work of Postnikov, and were
studied in a paper of Steingrimsson and the second author.) Namely, we prove
that in the long time limit, the probability that the PASEP is in a particular
configuration is essentially the generating function for permutation
tableaux of shape enumerated according to three statistics. The
proof of this result uses a result of Derrida, Evans, Hakim, and Pasquier on
the {\it matrix ansatz} for the PASEP model.
As an application, we prove some monotonicity results for the PASEP. We also
derive some enumerative consequences for permutations enumerated according to
various statistics such as weak excedence set, descent set, crossings, and
occurences of generalized patterns.Comment: Clarified exposition, more general result, new author (SC), 19 pages,
6 figure
Generalized Dumont-Foata polynomials and alternative tableaux
Dumont and Foata introduced in 1976 a three-variable symmetric refinement of
Genocchi numbers, which satisfies a simple recurrence relation. A six-variable
generalization with many similar properties was later considered by Dumont.
They generalize a lot of known integer sequences, and their ordinary generating
function can be expanded as a Jacobi continued fraction.
We give here a new combinatorial interpretation of the six-variable
polynomials in terms of the alternative tableaux introduced by Viennot. A
powerful tool to enumerate alternative tableaux is the so-called "matrix
Ansatz", and using this we show that our combinatorial interpretation naturally
leads to a new proof of the continued fraction expansion.Comment: 17 page
The Heine-Stieltjes correspondence and a new angular momentum projection for many-particle systems
A new angular momentum projection for systems of particles with arbitrary
spins is formulated based on the Heine-Stieltjes correspondence, which can be
regarded as the solutions of the mean-field plus pairing model in the strong
pairing interaction G ->Infinity limit. Properties of the Stieltjes zeros of
the extended Heine-Stieltjes polynomials, of which the roots determine the
projected states, and the related Van Vleck zeros are discussed. The
electrostatic interpretation of these zeros is presented. As examples,
applications to n nonidentical particles of spin-1/2 and to identical bosons or
fermions are made to elucidate the procedure and properties of the Stieltjes
zeros and the related Van Vleck zeros. It is shown that the new angular
momentum projection for n identical bosons or fermions can be simplified with
the branching multiplicity formula of U(N) supset O(3) and the special choices
of the parameters used in the projection. Especially, it is shown that the
solutions for identical bosons can always be expressed in terms of zeros of
Jacobi polynomials. However, unlike non-identical particle systems, the
n-coupled states of identical particles are non-orthogonal with respect to the
multiplicity label after the projection.Comment: 14 pages LaTeX with no figur
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