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

Eric Goles (), Michel Morvan () and Ha Duong Phan ()
Additional contact information
Eric Goles: Universidad de Chile, Departamento de Ingeniería Matematica, Escuela de Ingenierêa
Michel Morvan: LIAFA Université Denis Diderot Paris 7, Institut universitaire de France — Case
Ha Duong Phan: LIAFA Université Denis Diderot Paris 7 — Case

A chapter in Formal Power Series and Algebraic Combinatorics, 2000, pp 214-225 from Springer

Abstract: 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 (search for similar items in EconPapers)
Date: 2000
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-662-04166-6_19

Ordering information: This item can be ordered from
http://www.springer.com/9783662041666

DOI: 10.1007/978-3-662-04166-6_19

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().