Domino tilings and related models: space of configurations of domains with holes

We first prove that the set of domino tilings of a fixed finite figure is a distributive lattice, even in the case when the figure has holes. We then give a geometrical interpretation of the order given by this lattice, using (not necessarily local) transformations called {\em flips}. This study allows us to formulate an exhaustive generation algorithm and a uniform random sampling algorithm. We finally extend these results to other types of tilings (calisson tilings, tilings with bicolored Wang tiles).Comment: 17 pages, 11 figure

Fast domino tileability

Domino tileability is a classical problem in Discrete Geometry, famously solved by Thurston for simply connected regions in nearly linear time in the area. In this paper, we improve upon Thurston's height function approach to a nearly linear time in the perimeter.Comment: Appeared in Discrete Comput. Geom. 56 (2016), 377-39

Flip invariance for domino tilings of three-dimensional regions with two floors

We investigate tilings of cubiculated regions with two simply connected floors by 2 x 1 x 1 bricks. More precisely, we study the flip connected component for such tilings, and provide an algebraic invariant that "almost" characterizes the flip connected components of such regions, in a sense that we discuss in the paper. We also introduce a new local move, the trit, which, together with the flip, connects the space of domino tilings when the two floors are identical.Comment: 33 pages, 34 figures, 2 tables. We updated the reference lis

Tilings of quadriculated annuli

Tilings of a quadriculated annulus A are counted according to volume (in the formal variable q) and flux (in p). We consider algebraic properties of the resulting generating function Phi_A(p,q). For q = -1, the non-zero roots in p must be roots of unity and for q > 0, real negative.Comment: 33 pages, 12 figures; Minor changes were made to make some passages cleare

A note on a flip-connected class of generalized domino tilings of the box [0,2]n[0,2]^n

Let $n,d\in \mathbb{N}$ and $n>d$. An $(n-d)$-domino is a box $I_1\times \cdots \times I_n$ such that $I_j\in \{[0,1],[1,2]\}$ for all $j\in N\subset [n]$ with $|N|=d$ and $I_i=[0,2]$ for every $i\in [n]\setminus N$. If $A$ and $B$ are two $(n-d)$-dominoes such that $A\cup B$ is an $(n-(d-1))$-domino, then $A,B$ is called a twin pair. If $C,D$ are two $(n-d)$-dominoes which form a twin pair such that $A\cup B=C\cup D$ and $\{C,D\}\neq \{A,B\}$, then the pair $C,D$ is called a flip of $A,B$. A family $\mathscr{D}$ of $(n-d)$-dominoes is a tiling of the box $[0,2]^n$ if interiors of every two members of $\mathscr{D}$ are disjoint and $\bigcup_{B\in \mathscr{D}}B=[0,2]^n$. An $(n-d)$-domino tiling $\mathscr{D}'$ is obtained from an $(n-d)$-domino tiling $\mathscr{D}$ by a flip, if there is a twin pair $A,B\in \mathscr{D}$ such that $\mathscr{D}'=(\mathscr{D}\setminus \{A,B\})\cup \{C,D\}$, where $C,D$ is a flip of $A,B$. A family of $(n-d)$-domino tilings of the box $[0,2]^n$ is flip-connected, if for every two members $\mathscr{D},\mathscr{E}$ of this family the tiling $\mathscr{E}$ can be obtained from $\mathscr{D}$ by a sequence of flips. In the paper some flip-connected class of $(n-d)$-domino tilings of the box $[0,2]^n$ is described

Cut-and-paste of quadriculated disks and arithmetic properties of the adjacency matrix

We define cut-and-paste, a construction which, given a quadriculated disk obtains a disjoint union of quadriculated disks of smaller total area. We provide two examples of the use of this procedure as a recursive step. Tilings of a disk $\Delta$ receive a parity: we construct a perfect or near-perfect matching of tilings of opposite parities. Let $B_\Delta$ be the black-to-white adjacency matrix: we factor $B_\Delta = L\tilde DU$, where $L$ and $U$ are lower and upper triangular matrices, $\tilde D$ is obtained from a larger identity matrix by removing rows and columns and all entries of $L$, $\tilde D$ and $U$ are equal to 0, 1 or -1.Comment: 20 pages, 17 figure