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Integer Partitions, Tilings of 2D-gons and Lattices

Matthieu Latapy (2010)

RAIRO - Theoretical Informatics and Applications

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.

Isohedrally compatible tilings

Philip M. Maynard (2002)

Visual Mathematics

Methods of Perfect Coloring

R. Pérez-Gómez, Ceferino Ruiz (2000)

Visual Mathematics

Moments of inertia associated with the lozenge tilings of a hexagon.

Fischer, Ilse (2001)

Séminaire Lotharingien de Combinatoire [electronic only]

Monomer-dimer tatami tilings of rectangular regions.

Erickson, Alejandro, Ruskey, Frank, Woodcock, Jennifer, Schurch, Mark (2011)

The Electronic Journal of Combinatorics [electronic only]

Notions of denseness.

Kuperberg, Greg (2000)

Geometry & Topology

On deformations of spherical isometric foldings

Ana M. Breda, Altino F. Santos (2010)

Czechoslovak Mathematical Journal

The behavior of special classes of isometric foldings of the Riemannian sphere $S^{2}$ under the action of angular conformal deformations is considered. It is shown that within these classes any isometric folding is continuously deformable into the standard spherical isometric folding $f_{s}$ defined by $f_{s} (x, y, z) = (x, y, | z |)$ .

On the counting of fully packed loop configurations: some new conjectures.

Zuber, J.-B. (2004)

The Electronic Journal of Combinatorics [electronic only]

On the Fundamental Group of self-affine plane Tiles

Jun Luo, Jörg M. Thuswaldner (2006)

Annales de l’institut Fourier

Let $A \in ℤ^{2} \times ℤ^{2}$ be an expanding matrix, $𝒟 \subset ℤ^{2}$ a set with $| det (A) |$ elements and define $𝒯$ via the set equation $A 𝒯 = 𝒯 + 𝒟$ . If the two-dimensional Lebesgue measure of $𝒯$ is positive we call $𝒯$ a self-affine plane tile. In the present paper we are concerned with topological properties of $𝒯$ . We show that the fundamental group $π_{1} (𝒯)$ of $𝒯$ is either trivial or uncountable and provide criteria for the triviality as well as the uncountability of $π_{1} (𝒯)$ . Furthermore, we give a short proof of the fact that the closure of each component of $int (𝒯)$ is a locally...