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

3D domino tilings: irregular disks and connected components under flips

We consider three-dimensional domino tilings of cylinders $R_N = D \times [0,N]$ where $D \subset \mathbb{R}^2$ is a fixed quadriculated disk and $N \in \mathbb{N}$. A flip is a local move in the space of tilings $\mathcal{T}(R_N)$: remove two adjacent dominoes and place them back after a rotation. The twist is a flip invariant which associates an integer number to each tiling. For some disks $D$, called regular, two tilings of $R_N$ with the same twist can be joined by a sequence of flips once we add vertical space to the cylinder. We have that if $D$ is regular then the size of the largest connected component under flips of $\mathcal{T}(R_N)$ is $\Theta(N^{-\frac{1}{2}}|\mathcal{T}(R_N)|)$. The domino group $G_D$ captures information of the space of tilings. It is known that $D$ is regular if and only if $G_{D}$ is isomorphic to $\mathbb{Z} \oplus \mathbb{Z}/(2)$; sufficiently large rectangles are regular. We prove that certain families of disks are irregular. We show that the existence of a bottleneck in a disk $D$ often implies irregularity. In many, but not all, of these cases, we also prove that $D$ is strongly irregular, i.e., that there exists a surjective homomorphism from $G_{D}^+$ (a subgroup of index two of $G_{D}$) to the free group of rank two. Moreover, we show that if $D$ is strongly irregular then the cardinality of the largest connected component under flips of $\mathcal{T}(R_N)$ is $O(c^N |\mathcal{T}(R_N)|)$ for some $c \in (0,1)$.Comment: 35 pages, 29 figure

Domino tilings and flips in dimensions 4 and higher

In this paper we consider domino tilings of bounded regions in dimension $n \geq 4$. We define the twist of such a tiling, an elements of ${\mathbb{Z}}/(2)$, and prove it is invariant under flips, a simple local move in the space of tilings. We investigate which regions $D$ are regular, i.e. whenever two tilings $t_0$ and $t_1$ of $D \times [0,N]$ have the same twist then $t_0$ and $t_1$ can be joined by a sequence of flips provided some extra vertical space is allowed. We prove that all boxes are regular except $D = [0,2]^3$. Furthermore, given a regular region $D$, we show that there exists a value $M$ (depending only on $D$) such that if $t_0$ and $t_1$ are tilings of equal twist of $D \times [0,N]$ then the corresponding tilings can be joined by a finite sequence of flips in $D \times [0,N+M]$. As a corollary we deduce that, for regular $D$ and large $N$, the set of tilings of $D \times [0,N]$ has two twin giant components under flips, one for each value of the twist.Comment: 28 pages, 14 figure

Domino tilings of three-dimensional regions: flips, trits and twists

In this paper, we consider domino tilings of regions of the form $\mathcal{D} \times [0,n]$, where $\mathcal{D}$ is a simply connected planar region and $n \in \mathbb{N}$. It turns out that, in nontrivial examples, the set of such tilings is not connected by flips, i.e., the local move performed by removing two adjacent dominoes and placing them back in another position. We define an algebraic invariant, the twist, which partially characterizes the connected components by flips of the space of tilings of such a region. Another local move, the trit, consists of removing three adjacent dominoes, no two of them parallel, and placing them back in the only other possible position: performing a trit alters the twist by $\pm 1$. We give a simple combinatorial formula for the twist, as well as an interpretation via knot theory. We prove several results about the twist, such as the fact that it is an integer and that it has additive properties for suitable decompositions of a region.Comment: 38 pages, 17 figures. Most of this material is also covered in the first author's Ph.D. thesis (arXiv:1503.04617

Local dimer dynamics in higher dimensions

We consider local dynamics of the dimer model (perfect matchings) on hypercubic boxes $[n]^d$. These consist of successively switching the dimers along alternating cycles of prescribed (small) lengths. We study the connectivity properties of the dimer configuration space equipped with these transitions. Answering a question of Freire, Klivans, Milet and Saldanha, we show that in three dimensions any configuration admits an alternating cycle of length at most 6. We further establish that any configuration on $[n]^d$ features order $n^{d-2}$ alternating cycles of length at most $4d-2$. We also prove that the dynamics of dimer configurations on the unit hypercube of dimension $d$ is ergodic when switching alternating cycles of length at most $4d-4$. Finally, in the planar but non-bipartite case, we show that parallelogram-shaped boxes in the triangular lattice are ergodic for switching alternating cycles of lengths 4 and 6 only, thus improving a result of Kenyon and R\'emila, which also uses 8-cycles. None of our proofs make reference to height functions.Comment: 14 pages, 4 figure

Enumerative Combinatorics

Enumerative Combinatorics focusses on the exact and asymptotic counting of combinatorial objects. It is strongly connected to the probabilistic analysis of large combinatorial structures and has fruitful connections to several disciplines, including statistical physics, algebraic combinatorics, graph theory and computer science. This workshop brought together experts from all these various fields, including also computer algebra, with the goal of promoting cooperation and interaction among researchers with largely varying backgrounds