Formal Power Series and Algebraic Combinatorics pp 214-225 | Cite as

# About the Dynamics of Some Systems Based on Integer Partitions and Compositions

## 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.

## Keywords

Maximal Element Transition Rule Maximal Chain Sand Pile Finite Lattice## Preview

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## References

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