About the Dynamics of Some Systems Based on Integer Partitions and Compositions
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  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 . We prove that this new model has a structure of lattice.
KeywordsMaximal Element Transition Rule Maximal Chain Sand Pile Finite Lattice
Unable to display preview. Download preview PDF.
- 1.R. Anderson, L. Lovâsz, P. Shor, J. Spencer, E. Tardos, and S. Winograd. Disks, ball, and walls: analysis of a combinatorial game. Amer. math. Monthly, (96):481493, 1989.Google Scholar
- 2.G.E. Andrews. The Theory of Partitions. Addison-Wesley Publishing Company, 1976.Google Scholar
- 3.P. Bak, C. Tang, and K. Wiesenfeld. Self-organized criticality. Phys. rev. A, (38): 364 - 374, 1988.Google Scholar
- 4.Garrett Birkhoff. Lattice Theory American Mathematical Society, 1967.Google Scholar
- 5.T. Brylawski. The lattice of interger partitions. Discrete Mathematics, (6): 201 - 219, 1973.Google Scholar
- 6.B.A. Davey and H.A. Priestley. Introduction to Lattices and Order. Cambridge University Press, 1990.Google Scholar
- 7.K. Eriksson. Strongly Convergent Games and Coexeter Groups. PhD thesis, Departement of Mathematics, KTH, S-100 44 Stockholm, Sweden, 1993.Google Scholar
- 8.E. Goles and M.A. Kiwi. Games on line graphes and sand piles. Theoret. Comput. Sci., (115): 321 - 349, 1993.Google Scholar
- 9.E. Goles, M. Morvan, and H.D. Phan. Sand piles and order structure of integer partitions. submitted, 1997.Google Scholar
- 10.C. Greene and D.J. Kleiman. Longest chains in the lattice of integer partitions ordered by majorization. European J.Combin., (7): 1 - 10, 1986.Google Scholar
- 11.J. Spencer. Balancing vectors in the max norm. Combinatorica, (6): 55 - 65, 1986.Google Scholar