Structure of some sand piles model

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 rst explain how it can be constructed, by showing its strong self-similarity property. Then, we de ne SPM(1), a natural extension of spm when one starts with an in nite number of grains. Again, we give an e cient construction algorithm and a coding of this lattice using a self-similar tree. The two approaches give di erent recursive formulae for jSPM(n)j.