507 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
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
Symmetric unimodal expansions of excedances in colored permutations
We consider several generalizations of the classical -positivity of
Eulerian polynomials (and their derangement analogues) using generating
functions and combinatorial theory of continued fractions. For the symmetric
group, we prove an expansion formula for inversions and excedances as well as a
similar expansion for derangements. We also prove the -positivity for
Eulerian polynomials for derangements of type . More general expansion
formulae are also given for Eulerian polynomials for -colored derangements.
Our results answer and generalize several recent open problems in the
literature.Comment: 27 pages, 10 figure
Growth models, random matrices and Painleve transcendents
The Hammersley process relates to the statistical properties of the maximum
length of all up/right paths connecting random points of a given density in the
unit square from (0,0) to (1,1). This process can also be interpreted in terms
of the height of the polynuclear growth model, or the length of the longest
increasing subsequence in a random permutation. The cumulative distribution of
the longest path length can be written in terms of an average over the unitary
group. Versions of the Hammersley process in which the points are constrained
to have certain symmetries of the square allow similar formulas. The derivation
of these formulas is reviewed. Generalizing the original model to have point
sources along two boundaries of the square, and appropriately scaling the
parameters gives a model in the KPZ universality class. Following works of Baik
and Rains, and Pr\"ahofer and Spohn, we review the calculation of the scaled
cumulative distribution, in which a particular Painlev\'e II transcendent plays
a prominent role.Comment: 27 pages, 5 figure
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
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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
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
Counting derangements and Nash equilibria
The maximal number of totally mixed Nash equilibria in games of several
players equals the number of block derangements, as proved by McKelvey and
McLennan.On the other hand, counting the derangements is a well studied
problem. The numbers are identified as linearization coefficients for Laguerre
polynomials. MacMahon derived a generating function for them as an application
of his master theorem. This article relates the algebraic, combinatorial and
game-theoretic problems that were not connected before. New recurrence
relations, hypergeometric formulas and asymptotics for the derangement counts
are derived. An upper bound for the total number of all Nash equilibria is
given.Comment: 22 pages, 1 table; Theorem 3.3 adde
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