Abstract

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 \textit{T. Brylawski} [Discrete Math. 6, 201-219 (1973; Zbl 0283.06003)] 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 [\textit{E. Goles} and \textit{M. A. Kiwi}, Theor. Comput. Sci. 115, No. 2, 321-349 (1993; Zbl 0785.90120)]. 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 \(n\).