  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(infini), 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 the cardinal of
SPM(n), where no closed formula have ever been found.

Discrete Mathematics;Statistical Mechanics;Data Structures and Algorithms;Combinatorics