The structure of a linear chip firing game and related models

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 worst classical one, introduced by Brylawski (Discrete Math. 6 (1973) 201) 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 is associated to a fixed point for the SPM rule. In the second part, we extend the SPM rule in a natural way and obtain a model called Linear Chip Firing Game (Theoret. Comput. Sci. 115 (1993) 321). We prove that this new model has interesting properties: the induced order is a lattice, a natural greedoid can be associated to the model and it also defines a strongly convergent game. In the last section, we generalize the SPM rule in another way and obtain other lattice structure parametrized by some theta, denoted by L(n, theta), which form a decreasing sequence of lattices when theta varies in [ - n + 2, n]. For each theta, we characterize the fixed point of L(n, theta) and give the value of its maximal sized chain's length. We also note that L(n, -n + 2) is the lattice of all compositions of n. (C) 2002 Elsevier Science B.V. All rights reserved.