6 research outputs found
Generalized Integer Partitions, Tilings of Zonotopes and Lattices
In this paper, we study two kinds of combinatorial objects, generalized
integer partitions and tilings of two dimensional zonotopes, using dynamical
systems and order theory. We show that the sets of partitions ordered with a
simple dynamics, have the distributive lattice structure. Likewise, we show
that the set of tilings of zonotopes, ordered with a simple and classical
dynamics, is the disjoint union of distributive lattices which we describe. We
also discuss the special case of linear integer partitions, for which other
dynamical systems exist. These results give a better understanding of the
behaviour of tilings of zonotopes with flips and dynamical systems involving
partitions.Comment: See http://www.liafa.jussieu.fr/~latapy
The Lattice structure of Chip Firing Games and Related Models
In this paper, we study a famous discrete dynamical system, the Chip Firing
Game, used as a model in physics, economics and computer science. We use order
theory and show that the set of reachable states (i.e. the configuration space)
of such a system started in any configuration is a lattice, which implies
strong structural properties. The lattice structure of the configuration space
of a dynamical system is of great interest since it implies convergence (and
more) if the configuration space is finite. If it is infinite, this property
implies another kind of convergence: all the configurations reachable from two
given configurations are reachable from their infimum. In other words, there is
a unique first configuration which is reachable from two given configurations.
Moreover, the Chip Firing Game is a very general model, and we show how known
models can be encoded as Chip Firing Games, and how some results about them can
be deduced from this paper. Finally, we define a new model, which is a
generalization of the Chip Firing Game, and about which many interesting
questions arise.Comment: See http://www.liafa.jussieu.fr/~latap
Structure of some sand piles model
AbstractSand pile model (SPM) is a simple discrete dynamical system used in physics to represent granular objects. It is deeply related to integer partitions, and many other combinatorics problems, such as tilings or rewriting systems. The evolution of the system started with n stacked grains generates a lattice, denoted by SPM(n). We study here the structure of this lattice. We first explain how it can be constructed, by showing its strong self-similarity property. Then, we define SPM(∞), a natural extension of SPM when one starts with an infinite number of grains. Again, we give an efficient construction algorithm and a coding of this lattice using a self-similar tree. The two approaches give different recursive formulae for |SPM(n)|
Structure of some sand piles model
Abstract: spm (Sand Pile Model) is a simple discrete dynamical system used in physics to represent granular objects. It is deeply related to integer partitions, and many other combinatorics problems, such as tilings or rewriting systems. The evolution of the system started with n stacked grains generates a lattice, denoted by SPM(n). We study here the structure of this lattice. We first explain how it can be constructed, by showing its strong self-similarity property. Then, we define SPM(∞), a natural extension of spm when one starts with an infinite number of grains. Again, we give an efficient construction algorithm and a coding of this lattice using a self-similar tree. The two approaches give different recursive formulae for |SPM(n)|