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

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


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.


Maximal Element Transition Rule Maximal Chain Sand Pile Finite Lattice 
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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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