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

  • Eric Goles
  • Michel Morvan
  • Ha Duong Phan
Conference paper

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Eric Goles
    • 1
  • Michel Morvan
    • 2
  • Ha Duong Phan
    • 3
  1. 1.Departamento de Ingeniería Matematica, Escuela de IngenierêaUniversidad de ChileSantiagoChile
  2. 2.Institut universitaire de France — CaseLIAFA Université Denis Diderot Paris 7Paris Cedex 05France
  3. 3.LIAFA Université Denis Diderot Paris 7 — CaseParis Cedex 05France

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