Abstract

In this paper, we study the dynamics of sand grains falling in sand piles. Usually sand piles are characterized by a decreasing integer partition and grain moves are described in terms of transitions between such partitions. We study here four main transition rules. The more classical one, introduced by Brylawski [5] induces a lattice structure L_{B} (n) (called dominance ordering) between decreasing partitions of a given integer n. We prove that a more restrictive transition rule, called SPM rule, induces a natural partition of L_{B} (n) in suborders, each one associated to a fixed point for SPM rule. In the second part, we generalize the SPM rule and obtain other lattice structure parametrized by some θ: L(n, θ), which form for θ ∈ [n, −n + 2, n] a decreasing sequence of lattices. For each θ, we characterize the fixed point of L(n, θ) and give the value of its maximal sized chain’s lenght. We also note that L(n, −n + 2) is the lattice of all compositions of n. In the last section, we extend the SPM rule in another way and obtain a model called Chip Firing Game [8]. We prove that this new model has a structure of lattice.