Two sided sand piles model and unimodal sequences
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 42 (2008) no. 3, pp. 631-646.

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.

DOI : 10.1051/ita:2008019
Classification : 68R05, 05A17
Mots clés : discrete dynamical system, sand piles model, partition, unimodal sequence, order, lattice, dominance ordering, fixed point
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     title = {Two sided sand piles model and unimodal sequences},
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     pages = {631--646},
     publisher = {EDP-Sciences},
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Thi Ha Duong Phan. Two sided sand piles model and unimodal sequences. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 42 (2008) no. 3, pp. 631-646. doi : 10.1051/ita:2008019. http://www.numdam.org/articles/10.1051/ita:2008019/

[1] R. Anderson, L. Lovász, P. Shor, J. Spencer, E. Tardos, and S. Winograd. Disks, ball, and walls: analysis of a combinatorial game. Amer. Math. Monthly 96 (1989) 481-493. | MR | Zbl

[2] P. Bak, C. Tang, and K. Wiesenfeld. Self-organized criticality. Phys. Rev. A 38 (1988) 364-374. | MR

[3] J. Bitar and E. Goles. Paralel chip firing games on graphs. Theoret. Comput. Sci. 92 (1992) 291-300. | MR | Zbl

[4] A. Bjorner, L. Lovász, and W. Shor. Chip-firing games on graphes. Eur .J. Combin. 12 (1991) 283-291. | MR | Zbl

[5] A. Bjorner and G. Ziegler. Introduction to greedoids. Matroid applications, N. White, Ed. Cambridge University Press (1991) 284-357. | MR | Zbl

[6] F. Brenti. Log-concave and unimodal sequences in algebra, combinatorics and geometry: an update. Contemporary Mathematics 178 (1994) 71-84. | MR | Zbl

[7] T. Brylawski. The lattice of interger partitions. Discrete Mathematics 6 (1973) 201-219. | MR | Zbl

[8] B.A. Davey and H.A. Priestley. Introduction to Lattices and Order. Cambridge University Press (1990). | MR | Zbl

[9] E. Duchi, R. Mantaci, D. Rossin, and H.D. Phan. Bidimensional sand pile and ice pile models. PUMA 17 (2006) 71-96. | MR

[10] E. Formenti, B. Masson, and T. Pisokas. Advances in symmetric sandpiles. Fundamenta Informaticae 20 (2006) 1-22. | MR | Zbl

[11] E. Goles and M.A. Kiwi. Games on line graphes and sand piles. Theoret. Comput. Sci. 115 (1993) 321-349. | MR | Zbl

[12] E. Goles, M. Morvan, and H.D. Phan. Lattice structure and convergence of a game of cards. Ann. Combin. 6 (2002) 327-335. | MR | Zbl

[13] E. Goles, M. Morvan, and H.D. Phan. Sandpiles and order structure of integer partitions. Discrete Appl. Math. 117 (2002) 51-64. | MR | Zbl

[14] C. Greene and D.J. Kleitman. Longest chains in the lattice of integer partitions ordered by majorization. Eur. J. Combin. 7 (1986) 1-10. | MR | Zbl

[15] M. Latapy, R. Mantaci, M. Morvan, and H.D. Phan. Structure of some sand piles model. Theoret. Comput. Sci, 262 (2001) 525-556. | MR | Zbl

[16] M. Latapy and H.D. Phan. The lattice of integer partitions and its infinite extension. To appear in Discrete Mathematics (2008).

[17] Ha Duong Phan. PhD thesis. Université Paris VII (1999).

[18] J. Spencer. Balancing vectors in the max norm. Combinatorica 6 (1986) 55-65. | MR | Zbl

[19] 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). | MR | Zbl

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