A combinatorial approach to jumping particles

In this paper we consider a model of particles jumping on a row of cells, called in physics the one dimensional totally asymmetric exclusion process (TASEP). More precisely we deal with the TASEP with open or periodic boundary conditions and with two or three types of particles. From the point of view of combinatorics a remarkable feature of this Markov chain is that it involves Catalan numbers in several entries of its stationary distribution. We give a combinatorial interpretation and a simple proof of these observations. In doing this we reveal a second row of cells, which is used by particles to travel backward. As a byproduct we also obtain an interpretation of the occurrence of the Brownian excursion in the description of the density of particles on a long row of cells.Comment: 24 figure

A combinatorial approach to jumping particles I: maximal flow regime

International audienceIn this paper we consider a model of particles jumping on a row of cells, called in physics the one dimensional totally asymmetric exclusion process (TASEP). More precisely we deal with the TASEP with two or three types of particles, with or without boundaries, in the maximal flow regime. From the point of view of combinatorics a remarkable feauture of these Markov chains is that they involve Catalan numbers in several entries of their stationary distribution. We give a combinatorial interpretation and a simple proof of these observations. In doing this we reveal a second row of cells, which is used by particles to travel backward. As a byproduct we also obtain an interpretation of the occurrence of the Brownian excursion in the description of the density of particles on a long row of cells

A combinatorial approach to jumping particles: the parallel TASEP

International audienceIn this paper we continue the combinatorial study of the TASEP. We consider here the parallel TASEP, in which particles jump simultaneously. We offer here an elementary derivation that extends the combinatorial approach we developed for the standard TASEP. In particular we show that this stationary distribution can be expressed in terms of refinements of Catalan numbers

On square permutations

Severini and Mansour introduced $\textit{square polygons}$, as graphical representations of $\textit{square permutations}$, that is, permutations such that all entries are records (left or right, minimum or maximum), and they obtained a nice formula for their number. In this paper we give a recursive construction for this class of permutations, that allows to simplify the derivation of their formula and to enumerate the subclass of square permutations with a simple record polygon. We also show that the generating function of these permutations with respect to the number of records of each type is algebraic, answering a question of Wilf in a particular case

On square permutations

Severini and Mansour introduced in [4]square polygons, as graphical representations of square permutations, that is, permutations such that all entries are records (left or right, minimum or maximum), and they obtained a nice formula for their number. In this paper we give a recursive construction for this class of permutations, that allows to simplify the derivation of their formula and to enumerate the subclass of square permutations with a simple record polygon. We also show that the generating function of these permutations with respect to the number of records of each type is algebraic, answering a question of Wilf in a particular case

A code for square permutations and convex permutominoes

In this article we consider square permutations, a natural subclass ofpermutations defined in terms of geometric conditions, that can also bedescribed in terms of pattern avoiding permutations, and convex permutoninoes,a related subclass of polyominoes. While these two classes of objects arisedindependently in various contexts, they play a natural role in the descriptionof certain random horizontally and vertically convex grid configurations. We propose a common approach to the enumeration of these two classes ofobjets that allows us to explain the known common form of their generatingfunctions, and to derive new refined formulas and linear time random generationalgorithms for these objects and the associated grid configurations.Comment: 18 pages, 10 figures. Revision according to referees' remark

A code for square permutations and convex permutominoes

In this article we consider square permutations, a natural subclass of permutations defined in terms of geometric conditions, that can also be described in terms of pattern avoiding permutations, and convex permutoninoes, a related subclass of polyominoes. While these two classes of objects arised independently in various contexts, they play a natural role in the description of certain random horizontally and vertically convex grid configurations. We propose a common approach to the enumeration of these two classes of objets that allows us to explain the known common form of their generating functions, and to derive new refined formulas and linear time random generation algorithms for these objects and the associated grid configurations