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Two sided Sand Piles Model and unimodal sequences

Published online by Cambridge University Press:  03 June 2008

Thi Ha Duong Phan
Thi Ha Duong Phan*
Affiliation:
LIAFA Université Denis Diderot, Paris 7 - Case 7014-2, Place Jussieu- 75256 Paris Cedex 05-France and Institute of Mathematics, 18 Hoang Quoc Viet, Hanoi, Vietnam; [email protected] [email protected]
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Abstract

We introduce natural generalizations of two well-known dynamical systems, the Sand Piles Model and the Brylawski's model. We describe their order structure, their reachable configuration's characterization, their fixed points and their maximal and minimal length's chains. Finally, we present an induced model generating the set of unimodal sequences which amongst other corollaries, implies that this set is equipped with a lattice structure.

Keywords

Discrete dynamical systemSand Piles Modelpartitionunimodal sequenceorderlatticedominance orderingfixed point.
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 42 , Issue 3: JM'06 , July 2008 , pp. 631 - 646
Copyright
© EDP Sciences, 2008

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References

Anderson, R., Lovász, L., Shor, P., Spencer, J., Tardos, E., and Winograd, S.. Disks, ball, and walls: analysis of a combinatorial game. Amer. Math. Monthly 96 (1989) 481–493. CrossRef
Bak, P., Tang, C., and Wiesenfeld, K.. Self-organized criticality. Phys. Rev. A 38 (1988) 364–374. CrossRef
J. Bitar and E. Goles. Paralel chip firing games on graphs. Theoret. Comput. Sci. 92 (1992) 291–300.
Bjorner, A., Lovász, L., and Shor, W.. Chip-firing games on graphes. Eur .J. Combin. 12 (1991) 283–291. CrossRef
A. Bjorner and G. Ziegler. Introduction to greedoids. Matroid applications, N. White, Ed. Cambridge University Press (1991) 284–357.
Brenti, F.. Log-concave and unimodal sequences in algebra, combinatorics and geometry: an update. Contemporary Mathematics 178 (1994) 71–84. CrossRef
T. Brylawski. The lattice of interger partitions. Discrete Mathematics 6 (1973) 201–219.
B.A. Davey and H.A. Priestley. Introduction to Lattices and Order. Cambridge University Press (1990).
E. Duchi, R. Mantaci, D. Rossin, and H.D. Phan. Bidimensional sand pile and ice pile models. PUMA 17 (2006) 71–96.
Formenti, E., Masson, B., and Pisokas, T.. Advances in symmetric sandpiles. Fundamenta Informaticae 20 (2006) 1–22.
E. Goles and M.A. Kiwi. Games on line graphes and sand piles. Theoret. Comput. Sci. 115 (1993) 321–349.
Goles, E., Morvan, M., and Phan, H.D.. Lattice structure and convergence of a game of cards. Ann. Combin. 6 (2002) 327–335. CrossRef
Goles, E., Morvan, M., and Phan, H.D.. Sandpiles and order structure of integer partitions. Discrete Appl. Math. 117 (2002) 51–64. CrossRef
C. Greene and D.J. Kleitman. Longest chains in the lattice of integer partitions ordered by majorization. Eur. J. Combin. 7 (1986) 1–10.
M. Latapy, R. Mantaci, M. Morvan, and H.D. Phan. Structure of some sand piles model. Theoret. Comput. Sci, 262 (2001) 525–556.
M. Latapy and H.D. Phan. The lattice of integer partitions and its infinite extension. To appear in Discrete Mathematics (2008).
Ha Duong Phan. PhD thesis. Université Paris VII (1999).
J. Spencer. Balancing vectors in the max norm. Combinatorica 6 (1986) 55–65.
R. Stanley. Log-cocave and unimodal sequences in algebra, combinatorics and geometry. Graph theory and its applications: East and West (Jinan 1986). Ann. New York Acad. Sci. 576 (1989).