Dylan P. Thurston
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.
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Bivariate Generating Functions Enumerating Non-Bonding Dominoes on Rectangular Boards
本文研究了矩形棋盘上不重叠的非键合多米诺骨牌配置，利用转移矩阵法推导出有理生成函数，彻底解决了该枚举问题，并对最多六行或六列的棋盘进行了求值。
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.
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…
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.
We enumerate a certain class of monomino-domino coverings of square grids, which conform to the \emph{tatami} restriction; no four tiles meet. Let $\mathbf T_{n}$ be the set of monomino-domino tatami coverings of the $n\times n$ grid with the maximum number, $n$, of monominoes, oriented so that they have a monomino in each of the top left and top right corners. We give an algorithm for exhaustively generating the coverings in $\mathbf T_{n}$ with exactly $v$ vertical dominoes in constant amortized time, and an explicit formula for counting them. The polynomial that generates these counts has t…
A covering with dominoes of a rectilinear region is called \emph{tatami} if no four dominoes meet at any point. We describe a reduction from planar 3SAT to Domino Tatami Covering. As a consequence it is NP-complete to decide whether there is a perfect matching of a graph that meets every 4-cycle, even if the graph is restricted to be an induced subgraph of the grid-graph. The gadgets used in the reduction were discovered with the help of a SAT-solver.
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 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 s…
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.
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…
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 …
Components of domino tilings under flips in quadriculated tori
该研究针对方格环面的多米诺铺砌翻转图连通性问题，用图论方法证明非二分方格环面翻转图含两个同构分量且其完美匹配强迫数为整数区间，还用同调给出二分情形分量数的通用下界。
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 note on a flip-connected class of generalized domino tilings of the box $[0,2]^n$
本文研究了盒子 $[0,2]^n$ 的广义 $(n-d)$-多米诺平铺，定义了孪生对与翻转操作，并描述了一类具有翻转连通性质的平铺族。
We explore the connections between the well-studied Aztec Diamond graphs and a new family of graphs called the Half-Hexagons, discovered by Jonathan Novak. In particular, both families of graphs have very simple domino shuffling algorithms, which turn out to be intimately related. This connection allows us to prove an "arctic parabola" theorem for the Half-Hexagons as a corollary of the Arctic Circle theorem for the Aztec Diamond.
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 …
Domino tilings of three-dimensional cylinders: regularity of hamiltonian disks
研究了三维柱体多米诺平铺中圆盘的正则性，证明了具有狭窄小瓶颈的哈密顿圆盘是正则的，且无瓶颈即意味着正则性。
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 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…
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.
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.
In two papers, Little and Sellers introduced an exciting new combinatorial method for proving partition identities which is not directly bijective. Instead, they consider various sets of weighted tilings of a $1 \times \infty$ board with squares and dominoes, and for each type of tiling they construct a generating function in two different ways, which generates a $q$-series identity. Using this method, they recover quite a few classical $q$-series identities, but Euler's Pentagonal Number Theorem is not among them. In this paper, we introduce a key parameter when constructing the generating fu…
3D domino tilings: irregular disks and connected components under flips
本文研究了三维多米诺平铺在翻转操作下的连通分量，证明了存在瓶颈的盘通常是不正则的，并给出了强不正则盘最大连通分量大小的指数级上界。
Inspired by Propp's intruded Aztec diamond regions, we consider halved hexagons in which two aligned arrays of triangular holes have been removed from their boundaries. Unlike the intruded Aztec diamonds (whose numbers of domino tilings contain some large prime factors in their factorizations), the numbers of lozenge tilings of our doubly-intruded halved hexagons are given by simple product formulas in which all factors are linear in the parameters. In this paper, we present an extensive list of exact tiling enumerations of sixteen different types of doubly-intruded halved hexagons. We also pr…
Motivated in part by Propp's intruded Aztec diamond regions, we consider hexagonal regions out of which two horizontal chains of triangular holes (called ferns) are removed, so that the chains are at the same height, and are attached to the boundary. By contrast with the intruded Aztec diamonds (whose number of domino tilings contain some large prime factors in their factorization), the number of lozenge tilings of our doubly-intruded hexagons turns out to be given by simple product formulas in which all factors are linear in the parameters. We present in fact $q$-versions of these formulas, w…
Trimer covers in the triangular grid: twenty mostly open problems
本文将三角网格铺砌研究从二聚体推广到三聚体，引入苯环区域并提出二十个大多未解决的精确计数问题。
Colored vertex models and $k$-tilings of the Aztec diamond
本文利用彩色顶点模型计算 Aztec 钻石 $k$-铺砌的生成多项式，证明无相互作用铺砌与普通铺砌的双射，并求得相关北极曲线。
Boundary dents, the arctic circle and the arctic ellipse
本文研究带边界单位方格缺陷的阿兹特克钻石多米诺铺法计数，证明缺陷固定时铺法比值趋于Delannoy数，并利用缺陷位置趋近边界时的渐近行为推导出北极曲线为内切圆。
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