Cristopher Moore
We calculate the generating functions for the number of tilings of rectangles of various widths by the right tromino, the $L$ tetromino, and the $T$ tetromino. This allows us to place lower bounds on the entropy of tilings of the plane by each of these. For the $T$ tetromino, we also derive a lower bound from the solution of the Ising model in two dimensions.
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Conway and Lagarias observed that a triangular region T(m) in a hexagonal lattice admits signed tiling by three-in-line polyominoes (tribones) if and only if m=9d-1 or m=9d for some integer d. We apply the theory of Groebner bases over integers to show that T(m) admits a signed tiling by n-in-line polyominoes (n-bones) if and only if m=dn^2-1 or m=dn^2 for some integer d. Explicit description of the Groebner basis allows us to calculate the "Groebner discrete volume" of a lattice region by applying the division algorithm to its `Newton polynomial'. Among immediate consequences is a description…
We apply the theory of Groebner bases to the study of signed, symmetric polyomino tilings of planar domains. Complementing the results of Conway and Lagarias we show that the triangular regions T_N=T_{3k-1} and T_N=T_{3k} in a hexagonal lattice admit a signed tiling by three-in-line polyominoes (tribones) symmetric with respect to the 120 degrees rotation of the triangle if and only if either N=27r-1 or N=27r for some integer r.
We define a convolution operation on the set of polyominoes and use it to obtain a criterion for a given polyomino not to tile the plane (rotations and translations allowed). We apply the criterion to several families of polyominoes, and show that the criterion detects some cases that are not detectable by generalized coloring arguments.
Undecidability of Tiling the Plane with a Set of 5 Polyominoes
本文证明五个多联骨牌能否平移平铺平面是不可判定的，通过新边标记法并专用一个多联骨牌实现匹配判定。
Automatic Enumeration of Tilings by Polyominoes
该研究基于Zeilberger的计算框架，复现L-四格骨牌铺砌的生成函数，还得到宽度5-9的显式生成函数及OEIS未收录的新序列，拓展了多联骨牌铺砌的枚举研究。
Let $m,k$ be fixed positive integers. Determining the generating function for the number of tilings of an $m\times n$ rectangle by $k\times 1$ rectangles is a long-standing open problem to which the answer is only known in certain special cases. We give an explicit formula for this generating function in the case where $m<2k$. This result is used to obtain the generating function for the number of tilings of an $m\times n \times k$ box with $k\times k\times 1$ bricks.
Straight polyomino tilings of rectangles and special rim-hook tableaux
通过故障线分解与Hadamard积框架，结合Littlewood恒等式和Graham定理，推导了$2k\ imes n$矩形被$k\ imes 1$直线瓷砖覆盖的加权生成函数显式表达式。
We consider a problem concerning tilings of rectangular regions by a finite library of polyominoes. We specifically look at rectangular regions of dimension $n\times m$ and ask whether or not a tiling of this region can be rearranged so that tiling of the $n\times m$ rectangle can be realized as a tiling of an $n\times m'$ rectangle and an $n\times m"$ rectangle, $m=m'+m"$. We call this a subtiling. We show that the associated decision problem is $\mathsf{NP}$-complete when restricted to rectangular polyominoes. We also show that for certain finite libraries of polyominoes, if $m$ is sufficien…
The homology group of a tiling introduced by M. Reid is studied for certain topological tilings. As in the planar case, for finite square grids on topological surfaces, the method of homology groups, namely the non-triviality of some specific element in the group allows a `coloring proof' of impossibility of a tiling. Several results about the non-existence of polyomino tilings on certain square-tiled surfaces are proved in the paper.
Some 2-adic conjectures concerning polyomino tilings of Aztec diamonds
本文研究不同骨牌集合铺砌阶为$n$的阿兹特克钻石的方式数，并关注该计数函数在模2的幂下的特殊规律。
A proof of Ollinger's conjecture: undecidability of tiling the plane with a set of $8$ polyominoes
本文证明 Ollinger 猜想，通过改变多联骨牌模拟 Wang tiles 的朝向并提出颜色编码新方法，确立用 8 个多联骨牌平移铺砌平面的不可判定性。
Undecidability of Translational Tiling of the Plane with Orthogonally Convex Polyominoes
针对仅由几何形状决定的平面平移铺砌问题，本文摆脱以往极凹铺砌块构造，证明用一组 7 个正交凸多联骨牌铺砌平面仍不可判定。
Undecidability of Translational Tiling of the Plane with Four Tiles
本文研究平面平移平铺的不可判定性，将所需多联骨牌数量从5个降至4个，证明了用4个不连通多联骨牌进行平面平移平铺不可判定。
Locked Polyomino Tilings
该研究构造了t≥3的锁定t联骨牌铺砌，证明了任意大有限正方形网格上3、4联骨牌及无限网格上任意大t的锁定铺砌存在，还给出加权网格的相关构造，为ReCom选区重划算法的不可约性猜想提供反例。
Translational Aperiodic Sets of 7 Polyominoes
本文证明了用7个多边形进行平面翻译铺砌是不可判定的，并构建了大小为7的非周期集，打破了Ammann的30年记录。
Enumeration of Colored Tilings on Graphs via Generating Functions
本文针对图划分为连通着色块的问题，利用二元生成函数与组合技术计算特定图族中块的期望数量，并推广了着色多联骨牌网格铺砌枚举。
We fix $n$ and say a square in the two-dimensional grid indexed by $(x,y)$ has color $c$ if $x+y \equiv c \pmod{n}$. A {\it ribbon tile} of order $n$ is a connected polyomino containing exactly one square of each color. We show that the set of order-$n$ ribbon tilings of a simply connected region $R$ is in one-to-one correspondence with a set of {\it height functions} from the vertices of $R$ to $\mathbb Z^{n}$ satisfying certain difference restrictions. It is also in one-to-one correspondence with the set of acyclic orientations of a certain partially oriented graph. Using these facts, we des…
A polyomino graph $H$ is a connected finite subgraph of the infinite plane grid such that each finite face is surrounded by a regular square of side length one and each edge belongs to at least one square. In this paper, we show that if $K$ is a maximum resonant set of $H$, then $H-K$ has a unique perfect matching. We further prove that the maximum forcing number of a polyomino graph is equal to its Clar number. Based on this result, we have that the maximum forcing number of a polyomino graph can be computed in polynomial time. We also show that if $K$ is a maximal alternating set of $H$, the…
A polyomino is a finite, edge-connected set of cells in the plane. At the present time, an enumeration of all polyominoes is nowhere in sight. On the other hand, there are several subsets of polyominoes for which generating functions are known. For example, there exists extensive knowledge about column-convex polyominoes, a model introduced by Temperley in 1956. While studying column-convex polyominoes, researchers also gave a look at diagonally convex polyominoes (DCPs), but noticed an awkward feature: when the last diagonal of a DCP is deleted, the remaining object is not always a polyomino.…
In this paper, we introduce a generalization of a class of tilings which appear in the literature: the tilings over which a height function can be defined (for example, the famous tilings of polyominoes with dominoes). We show that many properties of these tilings can be seen as the consequences of properties of the generalized tilings we introduce. In particular, we show that any tiling problem which can be modelized in our generalized framework has the following properties: the tilability of a region can be constructively decided in polynomial time, the number of connected components in the …
We consider here square tilings of the plane. By extending the formalism introduced in [3] we build a correspondence between plane maps endowed with an harmonic vector and square tilings satisfying a condition of regularity. In the case of periodic plane square tiling the relevant space of harmonic vectors is actually isomorphic to the first homology group of a torus. So, periodic plane square tilings are described by two parameters and the set of parameters is split into angular sectors. The correspondence between symmetry of the square tiling and symmetry of the plane maps and harmonic vecto…
This article addresses the problem of enumerating the tilings of a plane by lozenges, under the restriction that these tilings be doubly periodic. Kasteleyn's Pfaffian method is applied to compute the generating function of those permutations. The monomials of this function represent the different types of tilings, grouping them according to the number of lozenges in each orientation. We present an alternative approach to compute these types. Finally, two additional classes of tilings are proposed as open enumeration problems.
It is shown that the descending plane partitions of Andrews can be geometrically realized as cyclically symmetric rhombus tilings of a certain hexagon where an equilateral triangle of side length 2 has been removed from its centre. Thus, the lattice structure for descending plane partitions, as introduced by Mills, Robbins and Rumsey, allows for an elegant visualization.
Let $\cal T$ be a tiling of the plane with equilateral triangles no two of which share a side. We prove that if the side lengths of the triangles are bounded from below by a positive constant, then $\cal T$ is periodic and it consists of translates of only at most three different triangles. As a corollary, we prove a theorem of Scherer and answer a question of Nandakumar. The same result has been obtained independently by Richter and Wirth.
The deductive method ruled mathematics for the last 2500 years, now it is the turn of the inductive method. Here we make a modest start by using the inductive method to discover and prove (rigorously) explicit generating functions for the number of dimer (and monomer-dimer) tilings of large families of "skinny" plane regions.
It has been common knowledge since 1950 that seven colours can be assigned to tiles of an infinite honeycomb with cells of unit diameter such that no two tiles of the same colour are closer than $d(7)=\frac{\sqrt{7}}{2}$ apart. Various authors have described tilings using $k>7$ colours, giving corresponding values for $d(k)$, but it is generally unknown whether these are the largest possible for a given $k$. Here, for many $k$, we describe tilings with larger values of $d(k)$ than previously reported.
Tilings of Flat Tori by Congruent Hexagons
本文在所有顶点度数为3的条件下，分类三类一般六边形及两种特殊情形的平面铺砌，并据此刻画平坦环面的六边形铺砌与模空间。
Perfect precise colorings of plane semiregular tilings
本文研究平面半正镶嵌的完美精确着色问题，为$k\leq 6$的某些$k$价半正镶嵌族构造了满足对称群置换与顶点邻接颜色约束的着色方案。
We study properties of (bi-infinite) arrays having all adjacent $k\times k$ adjacent minors equal to one. If we further add the condition that all adjacent $(k-1)\times (k-1)$ minors be nonzero, then these arrays are necessarily of rank $k$. It follows that we can explicit construct all of them. Several nice properties are made apparent. In particular, we revisit, with this perspective, the notion of frieze patterns of Coxeter. This shed new light on their properties. A connexion is also established with the notion of $T$-systems of Statistical Physics.
Edge-covering plane-filling curves on grid colorings: a pedestrian approach
本文研究正多边形平面铺嵌上的边覆盖平面填充曲线，通过将铺嵌对称群的边类划分为对应于有限子群的精化着色，构造出称为曲线集的平面填充曲线族，并以图示示例直观展示。
We derive a simple bijection between geometric plane perfect matchings on $2n$ points in convex position and triangulations on $n+2$ points in convex position. We then extend this bijection to monochromatic plane perfect matchings on periodically $k$-colored vertices and $(k+2)$-gonal tilings of convex point sets. These structures are related to a generalization of Temperley-Lieb algebras and our bijections provide explicit one-to-one relations between matchings and tilings. Moreover, for a given element of one class, the corresponding element of the other class can be computed in linear time.
For a given tiling of the euclidean plane ${\bf E}^2$, we call the degree of freedom of perturbed edges of prototiles {\it escher degree}. In this paper we consider non-periodic L-tilings by 2 prototiles and obtain the escher degree of them.
A survey of tilings in the plane for a general audience.
Lozenge Tilings of Hexagons with Intrusions I: Generalized Intrusion
本文通过从六边形区域移除广义侵入结构构造新区域，证明其菱形铺砌数量及带权生成函数具有简单乘积公式，从而推广 MacMahon 定理并给出 $q$-模拟。
Motivated by the enumeration of a class of plane partitions studied by Proctor and by considerations about symmetry classes of plane partitions, we consider the problem of enumerating lozenge tilings of a hexagon with ``maximal staircases'' removed from some of its vertices. The case of one vertex corresponds to Proctor's problem. For two vertices there are several cases to consider, and most of them lead to nice enumeration formulas. For three or more vertices there do not seem to exist nice product formulas in general, but in one special situation a lot of factorization occurs, and we pose t…
It has been proven that the lozenge tilings of a quartered hexagon on the triangular lattice are enumerated by a simple product formula. In this paper we give a new proof for the tiling formula by using Kuo's graphical condensation. Our result generalizes a Proctor's theorem on enumeration of plane partitions contained in a "maximal staircase".
Substitution discrete plane tilings with $2n$-fold rotational symmetry for odd n
本文研究替换离散平面平铺，证明Sub Rosa平铺不是离散平面，并提出新的Planar Rosa替换平铺，证明其满足离散平面条件，提供了10重情况的显式构造和一般奇数n的构造方法。
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。
A semi-regular tiling of the hyperbolic plane is a tessellation by regular geodesic polygons with the property that each vertex has the same vertex-type, which is a cyclic tuple of integers that determine the number of sides of the polygons surrounding the vertex. We determine combinatorial criteria for the existence, and uniqueness, of a semi-regular tiling with a given vertex-type, and pose some open questions.
We study empirical statistical and gap distributions of several important tilings of the plane. In particular, we consider the slope distributions, the angle distributions, pair correlation, squared-distance pair correlation, angle gap distributions, and slope gap distributions for the Ammann Chair tiling, the recently discovered fifteenth pentagonal tiling, and a few pertinent tilings related to these famous examples. We also consider the spatial statistics of generalized Ulam sets in two dimensions. Additionally, we carefully prove a tight asymptotic formula for the time steps in which Ulam …
Frieze patterns and aperiodic tilings of the plane
研究平面非周期花边图案，基于菱形彭罗斯平铺和戈德雷什 - 兰松 - 比拉尔平铺给出例子，通过用正整数装饰顶点使其满足菱形规则，类似康威和考克斯特定义的图案。
There exist tilings of the plane with pairwise noncongruent triangles of equal area and bounded perimeter. Analogously, there exist tilings with triangles of equal perimeter, the areas of which are bounded from below by a positive constant. This solves a problem of Nandakumar.
A first step in investigating colour symmetries of periodic and nonperiodic patterns is determining the number of colours which allow perfect colourings of the pattern under consideration. A perfect colouring is one where each symmetry of the uncoloured pattern induces a global permutation of the colours. Two cases are distinguished: Either perfect colourings with respect to all symmetries, or with respect to orientation preserving symmetries only (no reflections). For the important class of colourings of regular tilings (and some Laves tilings) of the Euclidean or hyperbolic plane, this mainl…
Planar Rosa : a family of quasiperiodic substitution discrete plane tilings with $2n$-fold rotational symmetry
提出 Planar Rosa 替换平铺族，证明其具有 $2n$ 阶旋转对称性且满足离散平面条件，并指出 Sub Rosa 平铺不满足该条件，完善了奇偶 $n$ 的结果。
An irregular vertex in a tiling by polygons is a vertex of one tile and belongs to the interior of an edge of another tile. In this paper we show that for any integer $k\geq 3$, there exists a normal tiling of the Euclidean plane by convex hexagons of unit area with exactly $k$ irregular vertices. Using the same approach we show that there are normal edge-to-edge tilings of the plane by hexagons of unit area and exactly $k$ many $n$-gons ($n>6$) of unit area. A result of Akopyan yields an upper bound for $k$ depending on the maximal diameter and minimum area of the tiles. Our result complement…
The Honeycomb Conjecture in normed planes and an alpha-convex variant of a theorem of Dowker
本文研究了normed平面中的蜂窝猜想，证明了正凸平铺的平均平方周长最小，并探讨了与Dowker定理α-凸变种相关的联系，进一步给出了关于平铺单元等周比的问题解答。
The article 'A "regular" pentagonal tiling of the plane' by P. L. Bowers and K. Stephenson defines a conformal pentagonal tiling. This is a tiling of the plane with remarkable combinatorial and geometric properties. However, it doesn't have finite local complexity in any usual sense, and therefore we cannot study it with the usual tiling theory. The appeal of the tiling is that all the tiles are conformally regular pentagons. But conformal maps are not allowable under finite local complexity. On the other hand, the tiling can be described completely by its combinatorial data, which rather auto…
Undecidability of tiling the plane with a fixed number of Wang bars
本文证明了使用29个Wang条铺满平面的问题是不可判定的，并进一步推导出颜色缺失25的Wang瓷砖集的铺砌问题同样不可判定。
Periodic tilings play a role in the decorative arts, in construction and in crystal structures. Combinatorial tiling theory allows the systematic generation, visualization and exploration of such tilings of the plane, sphere and hyperbolic plane, using advanced algorithms and software.Here we present a "galaxy" of tilings that consists of the set of all 2.4 billion different types of periodic tilings that have Dress complexity up to 24. We make these available in a database and provide a new program called Tegula that can be used to search and visualize such tilings. Availability: All tilings …
In \emph{$k$-bootstrap percolation}, we fix $p\in (0,1)$, an integer $k$, and a plane graph $G$. Initially, we infect each face of $G$ independently with probability $p$. Infected faces remain infected forever, and if a healthy (uninfected) face has at least $k$ infected neighbors, then it becomes infected. For fixed $G$ and $p$, the \emph{percolation threshold} is the largest $k$ such that eventually all faces become infected, with probability at least $1/2$. For a large class of infinite graphs, we show that this threshold is independent of $p$. We consider bootstrap percolation in tilings o…
Hexagonal Tiling of the Plane
本文在弱于凸性的假设下证明了平面中恰好存在三种能铺砌的凸六边形。
We prove that a Schur function of rectangular shape $(M^n)$ whose variables are specialized to $x_1,x_1^{-1},...,x_n,x_n^{-1}$ factorizes into a product of two odd orthogonal characters of rectangular shape, one of which is evaluated at $-x_1,...,-x_n$, if $M$ is even, while it factorizes into a product of a symplectic character and an even orthogonal character, both of rectangular shape, if $M$ is odd. It is furthermore shown that the first factorization implies a factorization theorem for rhombus tilings of a hexagon, which has an equivalent formulation in terms of plane partitions. A simila…
Conformal Tilings, Combinatorial Curvature, and the Type Problem
本文研究共形铺砌，证明对n≥7不存在复平面的边对边共形正则n边形铺砌，并揭示组合曲率与底层黎曼曲面万有覆盖类型的关系，基于Stone和Oh的工作给出新证明与应用。
Suppose $f\in L^1(\mathbb{R}^d)$, $Λ\subset\mathbb{R}^d$ is a finite union of translated lattices such that $f+Λ$ tiles with a weight. We prove that there exists a lattice $L\subset{\mathbb{R}}^d$ such that $f+L$ also tiles, with a possibly different weight. As a corollary, together with a result of Kolountzakis, it implies that any convex polygon that multi-tiles the plane by translations admits a lattice multi-tiling, of a possibly different multiplicity. Our second result is a new characterization of convex polygons that multi-tile the plane by translations. It also provides a very efficien…
A tiling (edge-to-edge) of the plane is a family of tiles that cover the plane without gaps or overlaps. Vertex figure of a vertex in a tiling to be the union of all edges incident to that vertex. A tiling is $k$-vertex-homogeneous if any two vertices with congruent vertex figures are symmetric with each other and the vertices form precisely $k$ transitivity classes with respect to the group of all symmetries of the tiling. In this article, we discuss that if a map is the quotient of a plane's $k$-vertex-homogeneous lattice ($k \ge 4$) then what would be the sharp bounds of the number of verte…
Given a flag in each of the vertex-transitive tessellations of the Euclidean plane by regular polygons, we determine the flag stabilizer under the action of the automorphism group of a regular cover. In so doing we give a presentation of these tilings as quotients of regular (infinite) polyhedra.
A family of non-periodic tilings of the plane by right golden triangles
本文研究了一种由两种大小右三角形构成的替换铺砌家族，证明其可通过局部规则用装饰瓷砖生成，即为sofic铺砌。
In this paper we present a combinatorial generalization of the fact that the number of plane partitions that fit in a $2a\times b\times b$ box is equal to the number of such plane partitions that are symmetric, times the number of such plane partitions for which the transpose is the same as the complement. We use the equivalent phrasing of this identity in terms of symmetry classes of lozenge tilings of a hexagon on the triangular lattice. Our generalization consists of allowing the hexagon have certain symmetrically placed holes along its horizontal symmetry axis. The special case when there …
Suppose $P$ is a symmetric convex polygon in the plane. We give a polynomial time algorithm that decides if $P$ can tile the plane by transations at some level (not necessarily at level one; this is multiple tiling). The main technical contribution is a polynomial time algorithm that selects, if this is possible, for each $j=1,2,\ldots,n$ one of two given vectors $e_j$ or $τ_j$ so that the selection spans a discrete additive subgroup.
Boltzmann samplers and the recursive method are prominent algorithmic frameworks for the approximate-size and exact-size random generation of large combinatorial structures, such as maps, tilings, RNA sequences or various tree-like structures. In their multiparametric variants, these samplers allow to control the profile of expected values corresponding to multiple combinatorial parameters. One can control, for instance, the number of leaves, profile of node degrees in trees or the number of certain subpatterns in strings. However, such a flexible control requires an additional non-trivial tun…
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