45 research outputs found
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
Stammering tableaux
The PASEP (Partially Asymmetric Simple Exclusion Process) is a probabilistic
model of moving particles, which is of great interest in combinatorics, since
it appeared that its partition function counts some tableaux. These tableaux
have several variants such as permutations tableaux, alternative tableaux,
tree- like tableaux, Dyck tableaux, etc. We introduce in this context certain
excursions in Young's lattice, that we call stammering tableaux (by analogy
with oscillating tableaux, vacillating tableaux, hesitating tableaux). Some
natural bijections make a link with rook placements in a double staircase,
chains of Dyck paths obtained by successive addition of ribbons, Laguerre
histories, Dyck tableaux, etc.Comment: Clarification and better exposition thanks reviewer's report
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
Combinatorial Hopf algebras, noncommutative Hall-Littlewood functions, and permutation tableaux
We introduce a new family of noncommutative analogues of the Hall-Littlewood
symmetric functions. Our construction relies upon Tevlin's bases and simple
q-deformations of the classical combinatorial Hopf algebras. We connect our new
Hall-Littlewood functions to permutation tableaux, and also give an exact
formula for the q-enumeration of permutation tableaux of a fixed shape. This
gives an explicit formula for: the steady state probability of each state in
the partially asymmetric exclusion process (PASEP); the polynomial enumerating
permutations with a fixed set of weak excedances according to crossings; the
polynomial enumerating permutations with a fixed set of descent bottoms
according to occurrences of the generalized pattern 2-31.Comment: 37 pages, 4 figures, new references adde
Combinatorial mappings of exclusion processes
We review various combinatorial interpretations and mappings of
stationary-state probabilities of the totally asymmetric, partially asymmetric
and symmetric simple exclusion processes (TASEP, PASEP, SSEP respectively). In
these steady states, the statistical weight of a configuration is determined
from a matrix product, which can be written explicitly in terms of generalised
ladder operators. This lends a natural association to the enumeration of random
walks with certain properties.
Specifically, there is a one-to-many mapping of steady-state configurations
to a larger state space of discrete paths, which themselves map to an even
larger state space of number permutations. It is often the case that the
configuration weights in the extended space are of a relatively simple form
(e.g., a Boltzmann-like distribution). Meanwhile, various physical properties
of the nonequilibrium steady state - such as the entropy - can be interpreted
in terms of how this larger state space has been partitioned.
These mappings sometimes allow physical results to be derived very simply,
and conversely the physical approach allows some new combinatorial problems to
be solved. This work brings together results and observations scattered in the
combinatorics and statistical physics literature, and also presents new
results. The review is pitched at statistical physicists who, though not
professional combinatorialists, are competent and enthusiastic amateurs.Comment: 56 pages, 21 figure