We introduce a family of planar regions, called Aztec diamonds, and study the ways in which these regions can be tiled by dominoes. Our main result is a generating function that not only gives the number of domino tilings of the Aztec diamond of order $n$ but also provides information about the orientation of the dominoes (vertical versus horizontal) and the accessibility of one tiling from another by means of local modifications. Several proofs of the formula are given. The problem turns out to have connections with the alternating sign matrices of Mills, Robbins, and Rumsey, as well as the square ice model studied by Lieb.
The number of domino tilings of a region with reflective symmetry across a line is combinatorially shown to depend on the number of domino tilings of particular subregions, modulo 4. This expands upon previous congruency results for domino tilings, modulo 2, and leads to a variety of corollaries, including that the number of domino tilings of a k x 2k rectangle is congruent to 1 mod 4.
We consider tilings of quadriculated regions by dominoes and of triangulated regions by lozenges. We present an overview of results concerning tileability, enumeration and the structure of the space of tilings.
Helfgott and Gessel gave the number of domino tilings of an Aztec Rectangle with defects of size one on the boundary of one side. In this paper we extend this to the case of domino tilings of an Aztec Rectangle with defects on all boundary sides.
The expanded Aztec diamond is a generalized version of the Aztec diamond, with an arbitrary number of long columns and long rows in the middle. In this paper, we count the number of domino tilings of the expanded Aztec diamond. The exact number of domino tilings is given by recurrence relations of state matrices by virtue of the state matrix recursion algorithm, recently developed by the author to solve various two-dimensional regular lattice model enumeration problems.
In this paper, we continue the study of domino-tilings of Aztec diamonds. In particular, we look at certain ways of placing ``barriers'' in the Aztec diamond, with the constraint that no domino may cross a barrier. Remarkably, the number of constrained tilings is independent of the placement of the barriers. We do not know of a combinatorial explanation of this fact; our proof uses the Jacobi-Trudi identity.
We use the subgraph replacement method to prove a simple product formula for the tilings of an 8-vertex counterpart of Propp's quasi-hexagons (Problem 16 in New Perspectives in Geometric Combinatorics, Cambridge University Press, 1999), called quasi-octagon.
We consider the problem of counting and classifying domino tilings of a quadriculated torus. The counting problem for rectangles was studied by Kasteleyn and we use many of his ideas. Domino tilings of planar regions can be represented by height functions; for a torus given by a lattice L, these functions exhibit arithmetic L-quasiperiodicity. The additive constants determine the flux of the tiling, which can be interpreted as a vector in the dual lattice (2L)*. We give a characterization of the actual flux values, and of how corresponding tilings behave. We also consider domino tilings of the…
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).
This paper studies the dimer model on the dual graph of the square-octagon lattice, which can be viewed as the domino tilings with impurities in some sense. In particular, under a certain boundary condition, we give an exact formula representing the probability of finding an impurity at a given site in a uniformly random dimer configuration in terms of simple random walks on the square lattice.
In a region $R$ consisting of unit squares, a domino is the union of two adjacent squares and a (domino) tiling is a collection of dominoes with disjoint interior whose union is the region. The flip graph $\mathcal{T}(R)$ is defined on the set of all tilings of $R$ such that two tilings are adjacent if we change one to another by a flip (a $90^{\circ}$ rotation of a pair of side-by-side dominoes). It is well-known that $\mathcal{T}(R)$ is connected when $R$ is simply connected. By using graph theoretical approach, we show that the flip graph of $2m\times(2n+1)$ quadriculated cylinder is still …
We consider domino tilings of three-dimensional cubiculated regions. A flip is a local move: two neighboring parallel dominoes are removed and placed back in a different position. The twist is an integer associated to each tiling, which is invariant under flips. A balanced quadriculated disk $D$ is regular if 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 define the domino group of a quadriculated disk and prove that $D$ is regular if and only if its domino …
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.
We use the subgraph replacement method to investigate new properties of the tilings of regions on the square lattice with diagonals drawn in. In particular, we show that the centrally symmetric tilings of a generalization of the Aztec diamond are always enumerated by a simple product formula. This result generalizes the previous work of Ciucu (1997) and Yang (1992) about symmetric tilings of the Aztec diamond. We also use our method to prove a closed-form product formula for the number of centrally symmetric tilings of a quasi-hexagon.
University of Nebraska – Lincoln（内布拉斯加大学林肯分校）；Faculty of Mathematics and Computer Science, University of Science, Ho Chi Minh City（胡志明市科技大学数学与计算机科学学院）；Vietnam National University, Ho Chi Minh City（胡志明市越南国立大学）
We solve and generalize an open problem posted by James Propp (Problem 16 in New Perspectives in Geometric Combinatorics, Cambridge University Press, 1999) on the number of tilings of quasi-hexagonal regions on the square lattice with every third diagonal drawn in. We also obtain a generalization of Douglas' Theorem on the number of tilings of a family of regions of the square lattice with every second diagonal drawn in.
A Töplitz determinant whose entries are described by a q-analogue of the Narayana polynomials is evaluated by means of Laurent biorthogonal polynomials which allow of a combinatorial interpretation in terms of Schröder paths. As an application, a new proof is given to the Aztec diamond theorem by Elkies, Kuperberg, Larsen and Propp concerning domino tilings of the Aztec diamonds. The proof is based on the correspondence with non-intersecting Schröder paths developed by Eu and Fu.
We study the 2-adic behavior of the number of domino tilings of a 2n-by-2n square as nvaries. It was previously known that this number was of the form 2^n f(n)^2, where f(n) is an odd, positive integer. We show that the function f is uniformly continuous under the 2-adic metric, and thus extends to a function on all of Z. The extension satisfies the functional equation f(-1-n) = +- f(n), where +- sign is + if n is congruent to 0 or 3 modulo 4 and - otherwise.
We prove combinatorially that the parity of the number of domino tilings of a region is equal to the parity of the number of domino tilings of a particular subregion. Using this result we can resolve the holey square conjecture. We furthermore give combinatorial proofs of several other tiling parity results, including that the number of domino tilings of a particular family of rectangles is always odd.
This article is about measuring and visualizing distances between domino tilings. Given two tilings of a simply connected square tiled surface, we're interested in the minimum number of flips between two tilings. Given a certain shape, we're interested in computing the diameters of the flip graphs, meaning the maximal distance between any two of its tilings. Building on work of Thurston and others, we give geometric interpretations of distances which result in formulas for the diameters of the flip graphs of rectangles or Aztec diamonds.
We consider a generating function of the domino tilings of an Aztec rectangle with several boundary unit squares removed. Our generating function involves two statistics: the rank of the tiling and half number of vertical dominoes as in the Aztec diamond theorem by Elkies, Kuperberg, Larsen and Propp. In addition, our work deduces a combinatorial explanation for an interesting connection between the number of lozenge tilings of a semihexagon and the number of domino tilings of an Aztec rectangle.
A domino covering of a board is saturated if no domino is redundant. We introduce the concept of a fragment tiling and show that a minimal fragment tiling always corresponds to a maximal saturated domino covering. The size of a minimal fragment tiling is the domination number of the board. We define a class of regular boards and show that for these boards the domination number gives the size of a minimal X-pentomino covering. Natural sequences that count maximal saturated domino coverings of square and rectangular boards are obtained. These include the new sequences A193764, A193765, A193766, …
We consider domino tilings of $3$-dimensional cubiculated regions. A three-dimensional domino is a 2x2x1 rectangular cuboid. We are particularly interested in regions of the form $R_N = D \times [0,N]$ where $D$ is a fixed quadriculated disk. In dimension 3, the twist associates to each tiling $t$ an integer $Tw(t)$. We prove that, when $N$ goes to infinity, the twist follows a normal distribution. A flip is a local move: two neighboring parallel dominoes are removed and placed back in a different position. The twist is invariant under flips. A quadriculated disk $D$ is regular if, whenever tw…
As a continuation to our previous work [9, 10], we consider the domino tiling problem with impurities. (1) if we have more than two impurities on the boundary, we can compute the number of corresponding perfect matchings by using the hitting matrix method[4]. (2) we have an alternative proof of the main result in [9] and result in (1) above using the formula by Kenyon-Wilson [6, 7] of counting the number of groves on the circular planar graph. (3) we study the behavior of the probability of finding the impurity at a given site when the size of the graph tends to infinity, as well as the scalin…
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 …
Several articles deal with tilings with squares and dominoes of the well-known regular square mosaic in Euclidean plane, but not any with the hyperbolic regular square mosaics. In this article, we examine the tiling problem with colored squares and dominoes of one type of the possible hyperbolic generalization of $(2\times n)$-board.
Let T(m,n) denote the number of ways to tile an m-by-n rectangle with dominos. For any fixed m, the numbers T(m,n) satisfy a linear recurrence relation, and so may be extrapolated to negative values of n; these extrapolated values satisfy the relation T(m,-2-n) = epsilon_{m,n} T(m,n), where epsilon_{m,n} is -1 if m is congruent to 2 (mod 4) and n is odd, and is +1 is otherwise. This is equivalent to a fact demonstrated by Stanley using algebraic methods. Here I give a proof that provides, among other things, a uniform combinatorial interpretation of T(m,n) that applies regardless of the sign o…
We consider the subgroup of the abelian sandpile group of the grid graph consisting of configurations of sand that are symmetric with respect to central vertical and horizontal axes. We show that the size of this group is (i) the number of domino tilings of a corresponding weighted rectangular checkerboard; (ii) a product of special values of Chebyshev polynomials; and (iii) a double-product whose factors are sums of squares of values of trigonometric functions. We provide a new derivation of the formula due to Kasteleyn and to Temperley and Fisher for counting the number of domino tilings of …
We consider tromino tilings of $m\times n$ domino-deficient rectangles, where $3|(mn-2)$ and $m,n\geq0$, and characterize all cases of domino removal that admit such tilings, thereby settling the open problem posed by J. M. Ash and S. Golomb in \cite {marshall}. Based on this characterization, we design a procedure for constructing such a tiling if one exists. We also consider the problem of counting such tilings and derive the exact formula for the number of tilings for $2\times(3t+1)$ rectangles, the exact generating function for $4\times(3t+2)$ rectangles, where $t\geq0$, and an upper bound…
Consider the $2n$-by-$2n$ matrix $M=(m_{i,j})_{i,j=1}^{2n}$ with $m_{i,j} = 1$ for $i,j$ satisfying $|2i-2n-1|+|2j-2n-1| \leq 2n$ and $m_{i,j} = 0$ for all other $i,j$, consisting of a central diamond of 1's surrounded by 0's. When $n \geq 4$, the $λ$-determinant of the matrix $M$ (as introduced by Robbins and Rumsey) is not well-defined. However, if we replace the 0's by $t$'s, we get a matrix whose $λ$-determinant is well-defined and is a polynomial in $λ$ and $t$. The limit of this polynomial as $t \to 0$ is a polynomial in $λ$ whose value at $λ=1$ is the number of domino tilings of a $2n$-…
We introduce a class of graphs called compound graphs, generalizing rectangles, which are constructed out of copies of a planar bipartite base graph. The main result is that the number of perfect matchings of every compound graph is divisible by the number of matchings of its base graph. Our approach is to use Kasteleyn's theorem to prove a key lemma, from which the divisibility theorem follows combinatorially. This theorem is then applied to provide a proof of Problem 21 of Propp's Enumeration of Matchings, a divisibility property of rectangles. Finally, we present a new proof, in the same sp…
Tiling planar regions with dominoes is a classical problem in which the decision and counting problems are polynomial. We prove a variety of hardness results (both NP- and #P-completeness) for different generalizations of dominoes in three and higher dimensions.
In this paper, we consider the set of all domino tilings of a cubiculated region. The primary question we explore is: How can we move from one tiling to another? Tiling spaces can be viewed as spaces of subgraphs of a fixed graph with a fixed degree sequence. Moves to connect such spaces have been explored in algebraic statistics. Thus, we approach this question from an applied algebra viewpoint, making new connections between domino tilings, algebraic statistics, and toric algebra. Using results from toric ideals of graphs, we are able to describe moves that connect the tiling space of a give…
We prove an asymptotic formula for the probability that, if one chooses a domino tiling of a large Aztec diamond at random according to the uniform distribution on such tilings, the tiling will contain a domino covering a given pair of adjacent lattice squares. This formula quantifies the effect of the diamond's boundary conditions on the behavior of typical tilings; in addition, it yields a new proof of the arctic circle theorem of Jockusch, Propp, and Shor. Our approach is to use the saddle point method to estimate certain weighted sums of squares of Krawtchouk polynomials (whose relevance t…
In this thesis, we consider domino tilings of three-dimensional regions, especially those of the form $\mathcal{D} \times [0,N]$. In particular, we investigate the connected components of the space of tilings of such regions by flips, the local move performed by removing two adjacent dominoes and placing them back in the only other possible position. For regions of the form $\mathcal{D} \times [0,2]$, we define a polynomial invariant $P_t(q)$ that characterizes tilings that are "almost in the same connected component", in a sense discussed in the thesis. We also prove that the space of domino …
We formulate and prove a variational principle (in the sense of thermodynamics) for random domino tilings, or equivalently for the dimer model on a square grid. This principle states that a typical tiling of an arbitrary finite region can be described by a function that maximizes an entropy integral. We associate an entropy to every sort of local behavior domino tilings can exhibit, and prove that almost all tilings lie within epsilon (for an appropriate metric) of the unique entropy-maximizing solution. This gives a solution to the dimer problem with fully general boundary conditions, thereby…
The arctic circle theorem of Jockusch, Propp, and Shor asserts that uniformly random domino tilings of an Aztec diamond of high order are frozen with asymptotically high probability outside the "arctic circle" inscribed within the diamond. A similar arctic circle phenomenon has been observed in the limiting behavior of random square Young tableaux. In this paper, we show that random domino tilings of the Aztec diamond are asymptotically related to random square Young tableaux in a more refined sense that looks also at the behavior inside the arctic circle. This is done by giving a new derivati…
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…
The inverse Kasteleyn matrix of a bipartite graph holds much information about the perfect matchings of the system such as local statistics which can be used to compute local and global asymptotics. In this paper, we consider three different weightings of domino tilings of the Aztec diamond and show using recurrence relations, we can compute the inverse Kasteleyn matrix. These weights are the one-periodic weighting where the horizontal edges have one weight and the vertical edges have another weight, the q^{vol} weighting which corresponds to multiplying the product of tile weights by q if we …
There is a natural generalization of domino tilings to tilings of a polygon by hexagons, or, dually, configurations of oriented curves that meet in triples. We show exactly when two such tilings can be connected by a series of moves analogous to the domino flip move. The triple diagrams that result have connections to Legendrian knots, cluster algebras, and planar algebras.
Let U be a multiply-connected region in R^2 with smooth boundary. Let P_epsilon be a polyomino in epsilon Z^2 approximating U as epsilon tends to 0. We show that, for certain boundary conditions on P_epsilon, the height distribution on a random domino tiling (dimer covering) of P_epsilon is conformally invariant in the limit as epsilon tends to 0, in the sense that the distribution of heights of boundary components only depends on the conformal type of U. The mean height and all the moments are explicitly evaluated.
Tilings are around us everywhere, and our curiosity draws us to study their properties. A tiling is a way of arranging pieces on a board, such that there is no space left uncovered, nor any space covered by more than one tile. In particular, we study fault-free tilings of boards with dominoes. To be fault-free every line that intersects the tiling must also intersect the interior of at least one of the tiles. Fault-free rectangular boards have been well studied, however we look at boards that are cylinders, tori, and Möbius strips. Using dominoes we study the various shapes of boards and see h…
We generalize Aztec diamond theorem (N. Elkies, G. Kuperberg, M. Larsen, and J. Propp, Alternating-sign matrices and domino tilings, Journal Algebraic Combinatoric, 1992) by showing that the numbers of tilings of a certain family of regions in the square lattice with southwest-to-northeast diagonals drawn in are given by powers of 2. We present a proof for the generalization by using a bijection between domino tilings and non-intersecting lattice paths.
We show that the number of configurations of the 20 Vertex model on certain domains with domain wall type boundary conditions is equal to the number of domino tilings of Aztec-like triangles, proving a conjecture from [P. Di Francesco and E. Guitter, Twenty-Vertex Model with Domain Wall Boundaries and Domino Tilings, Elec. Jour. of Combinatorics 27 (2020), no. 2, P2.13]. The result is based on the integrability of the 20 Vertex model and uses a connection to the U-turn boundary 6 Vertex model to re-express the number of 20 Vertex configurations as a simple determinant, which is then related to…
