Integer partitions, tilings of 2D-gons and lattices
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 36 (2002) no. 4, pp. 389-399.

In this paper, we study two kinds of combinatorial objects, generalized integer partitions and tilings of 2D-gons (hexagons, octagons, decagons, etc.). We show that the sets of partitions, ordered with a simple dynamics, have the distributive lattice structure. Likewise, we show that the set of tilings of a 2D-gon is the disjoint union of distributive lattices which we describe. We also discuss the special case of linear integer partitions, for which other dynamical models exist.

DOI : https://doi.org/10.1051/ita:2003004
Classification : 05A17,  11P81,  05B45,  06B99,  06D99,  68R05,  52C20,  52C23,  52C40
Mots clés : integer partitions, tilings of 2D-gons, lattices, sand pile model, discrete dynamical models
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     title = {Integer partitions, tilings of $2D$-gons and lattices},
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Latapy, Matthieu. Integer partitions, tilings of $2D$-gons and lattices. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 36 (2002) no. 4, pp. 389-399. doi : 10.1051/ita:2003004. http://www.numdam.org/articles/10.1051/ita:2003004/

[1] G.E. Andrews, The Theory of Partitions. Addison-Wesley Publishing Company, Encyclopedia Math. Appl. 2 (1976). | MR 557013 | Zbl 0655.10001

[2] G.D. Bailey, Coherence and enumeration of tilings of 3-zonotopes. Discrete Comput. Geom. 22 (1999) 119-147. | MR 1692682 | Zbl 0992.52006

[3] G.D. Bailey, Tilings of zonotopes: Discriminantal arrangements, oriented matroids, and enumeration, Ph.D. Thesis. University of Minnesota (1997).

[4] T. Brylawski, The lattice of integer partitions. Discrete Math. 6 (1973) 210-219. | MR 325405 | Zbl 0283.06003

[5] B.A. Davey and H.A. Priestley, Introduction to Lattices and Orders. Cambridge University Press (1990). | MR 1058437 | Zbl 0701.06001

[6] N.G. De Bruijn, Algebraic theory of penrose's non-periodic tilings of the plane. Konink. Nederl. Akad. Wetensch. Proc. Ser. A 43 (1981). | Zbl 0457.05021

[7] N.G. De Bruijn, Dualization of multigrids. J. Phys. France Coloq (1981) 3-9. | MR 866319

[8] N. Destainville, R. Mosseri and F. Bailly, Configurational entropy of codimension-one tilings and directed membranes. J. Statist. Phys. 87 (1997) 697. | MR 1459040 | Zbl 0952.52500

[9] N. Destainville, R. Mosseri and F. Bailly, Fixed-boundary octogonal random tilings: A combinatorial approach. Preprint (1999). | Zbl 0984.52007

[10] N. Destainville, Entropie configurationnelle des pavages aléatoires et des membranes dirigées, Ph.D. Thesis. University Paris VI (1997).

[11] S. Elnitsky, Rhombic tilings of polygons and classes of reduced words in coxeter groups. J. Combin. Theory 77 (1997) 193-221. | MR 1429077 | Zbl 0867.05019

[12] E. Goles and M.A. Kiwi, Games on line graphs and sand piles. Theoret. Comput. Sci. 115 (1993) 321-349. | MR 1224440 | Zbl 0785.90120

[13] R. Kenyon, Tilings of polygons with parallelograms. Algorithmica 9 (1993) 382-397. | MR 1208568 | Zbl 0778.52012

[14] M. Latapy and H.D. Phan, The lattice of integer partitions and its infinite extension, in DMTCS, Special Issue, Proc. of ORDAL'99. Preprint (to appear) available at http://www.liafa.jussieu.fr/~latapy/

[15] M. Latapy, Generalized integer partitions, tilings of zonotopes and lattices, in Proc. of the 12-th international conference Formal Power Series and Algebraic Combinatorics (FPSAC'00), edited by A.A. Mikhalev, D. Krob and E.V. Mikhalev. Springer (2000) 256-267. Preprint available at http://www.liafa.jussieu.fr/~latapy/ | Zbl 0960.05005

[16] M. Latapy, R. Mantaci, M. Morvan and Ha Duong Phan, Structure of some sand piles model. Theoret. Comput. Sci. 262 (2001) 525-556. Preprint available at http://www.liafa.jussieu.fr/~latapy/ | MR 1836234 | Zbl 0983.68085

[17] R.P. Stanley, Ordered structures and partitions. Mem. ACM 119 (1972). | MR 332509 | Zbl 0246.05007

[18] G. Ziegler, Lectures on Polytopes. Springer-Verlag, Grad. Texts in Math. (1995). | MR 1311028 | Zbl 0823.52002

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