Structure of some sand pile model

Details Structure of some sand pile model Structure of some sand pile model

2000-08-02

arxiv.org/abs/cs/0008002v1

Computer Science

Discrete Mathematics

To appear in Theoretical Computer Science

Scientific paper

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.

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