A note on tiling with integer-sided rectangles

On tiling rectangles via the Frobenius number

In this paper, we give some sufficient conditions for a $n$-dimensional rectangle to be tiled with a set of bricks. These conditions are obtained by using the so-called Frobenius number.

A special tiling of the rectangle

We count tilings of a rectangle of integer sides m-1 and n-1 by a special set of tiles. The result is obtained fron the study of the kernel of the adjacency matrix of an n x n rectangular graph of Z x Z.

Tiling n X m rectangles with 1 X 1 and s X s squares

We consider tilings of a rectangle which is n units wide and m units long by non-overlapping 1 X 1 squares and s X s squares. Bivariate generating functions are computed with the Transfer Matrix Method for moderately large but fixed widths n as a function of the parameter m and of the number of s X s squares in the rectangle.

Tromino tilings of Domino-Deficient Rectangles

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…

Generating functions for straight polyomino tilings of narrow rectangles

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.

Tiling of rectangles with squares via Diophantine approximation

This article shines new light on the classical problem of tiling rectangles with squares efficiently with a novel method. With a twist on the traditional approach of resistor networks, we provide new and improved results on the matter using the theory of Diophantine Approximation, hence overcoming long-established difficulties, such as generalizations to higher-dimensional analogues. The universality of the method is demonstrated through its applications to different tiling problems. These include tiling rectangles with other rectangles, with their respective higher-dimensional counterparts, a…

Transformations of 2-port networks and tiling by rectangles

In this paper, we study 2-port networks and introduce new concepts of voltage drop and $Π$-equivalence. The main result is that each planar network is $Π$-equivalent to a network with no more than 5 edges. This implies that if an octagon in the shape of the letter $Π$ can be tiled by squares then it can be tiled by no more than 5 rectangles with rational aspect ratios. Kenyon's theorem from 1998 proves this only for 6 rectangles.

平坦 торus 的轴对齐矩形铺砌的最小长度

The minimum length of an axis-aligned rectangular tiling of a flat torus

该研究确定了平坦 torus 轴对齐矩形铺砌的最小周长总和，并证明该最小值可通过单个或两个矩形实现。

矩形与特殊边钩表的直线多联骨牌铺砌

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。

Tiling simply connected regions with rectangles

In [BNRR], it was shown that tiling of general regions with two rectangles is NP-complete, except for a few trivial special cases. In a different direction, Rémila showed that for simply connected regions by two rectangles, the tileability can be solved in quadratic time (in the area). We prove that there is a finite set of at most 10^6 rectangles for which the tileability problem of simply connected regions is NP-complete, closing the gap between positive and negative results in the field. We also prove that counting such rectangular tilings is #P-complete, a first result of this kind.

Generalizing the divisibility property of rectangle domino tilings

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 by rectangles and alternating current

This paper is on tilings of polygons by rectangles. A celebrated physical interpretation of such tilings due to R.L. Brooks, C.A.B. Smith, A.H. Stone and W.T. Tutte uses direct-current circuits. The new approach of the paper is an application of alternating-current circuits. The following results are obtained: - a necessary condition for a rectangle to be tilable by rectangles of given shapes; - a criterion for a rectangle to be tilable by rectangles similar to it but not all homothetic to it; - a criterion for a generic polygon to be tilable by squares. These results generalize the ones of C.…

Double Aztec Rectangles

We investigate the connection between lozenge tilings and domino tilings by introducing a new family of regions obtained by attaching two different Aztec rectangles. We prove a simple product formula for the generating functions of the tilings of the new regions, which involves the statistics as in the Aztec diamond theorem (Elkies, Kuperberg, Larsen, and Propp, J. Algebraic Combin. 1992). Moreover, we consider the connection between the generating function and MacMahon's $q$-enumeration of plane partitions fitting in a given box

关于正方形与矩形单孔多米诺铺砖的一个注记

A Note on One-Hole Domino Tilings of Squares and Rectangles

本文研究奇数乘奇数矩形网格中单孔多米诺铺砖（即近完美匹配）的计数问题，证明了正方形网格任意特定空缺位置的近完美匹配数均为$2^k$的倍数，从而证实Kong猜想，并给出矩形情形的奇偶性判定。

A rigidity property of ribbon L-shaped n-ominoes and generalizations

Let n integer greater or equal to 4 and even and let T_n be the set of ribbon L-shaped n-ominoes. We study tiling problems for regions in a square lattice by T_n. Our main result shows a remarkable rigidity property: a tiling of the first quadrant by T_n is possible if and only if it reduces to a tiling by 2 x n and n x 2 rectangles. An application is the classification of all rectangles that can be tiled by T_n: a rectangle can be tiled by T_n if and only if both of its sides are even and at least one side is divisible by n. Another application is the existence of the local move property for …

On Subtilings of Polyomino Tilings

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…

On the number of tilings of a square by rectangles

We develop a recursive formula for counting the number of rectangulations of a square, i.e the number of combinatorially distinct tilings of a square by rectangles. Our formula specializes to give a formula counting generic rectangulations, as analyzed by Reading in [5]. Our computations agree with [5] as far as was calculated and extend to the non-generic case. An interesting feature of the number of rectangulations is that it appears to have an 8-fold periodicity modulo 2. We verify this periodicity for small values of n, but the general result remains elusive, perhaps hinting at some unseen…

The tilings of deficient squares by ribbon L-tetrominoes are diagonally cracked

We consider tilings of deficient rectangles by the set $\mathcal{T}_4$ of ribbon $L$-tetrominoes. A tiling exists iff the rectangle is a square of odd side. The missing cell is on the main NW--SE diagonal, in an odd position if the square is $(4m+1)\times (4m+1)$ and in an even position for $(4m+3)\times (4m+3)$. The majority of the tiles in a tiling are paired and each pair tiles a $2\times 4$ rectangle. The tiles in an irregular position and the missing cell form a NW--SE diagonal crack, located in a thin region symmetric about the diagonal, made out of $3\times 3$ squares that overlap over …

Signed tilings by ribbon L n-ominoes, n even, via Groebner bases

Let $\mathcal{T}_n$ be the set of ribbon $L$-shaped $n$-ominoes for some $n\ge 4$ even, and let $\mathcal{T}_n^+$ be $\mathcal{T}_n$ with an extra $2\times 2$ square. We investigate signed tilings of rectangles by $\mathcal{T}_n$ and $\mathcal{T}_n^+$. We show that a rectangle has a signed tiling by $\mathcal{T}_n$ if and only if both sides of the rectangle are even and one of them is divisible by $n$, or if one of the sides is odd and the other side is divisible by $n\left (\frac{n}{2}-2\right ).$ We also show that a rectangle has a signed tiling by $\mathcal{T}_n^+, n\ge 6$ even, if and only…

Fault-Free Tileability of Rectangles, Cylinders, Tori, and Möbius Strips with Dominoes

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…

Signed tilings by ribbon L n-ominoes, n odd, via Groebner bases

We show that a rectangle can be signed tiled by ribbon L n-ominoes, n odd, if and only if it has a side divisible by n. A consequence of our technique, based on the exhibition of an explicit Groebner basis, is that any k-inflated copy of the skewed L n-omino has a signed tiling by skewed L n-ominoes. We also discuss regular tilings by ribbon L n-ominoes, n odd, for rectangles and more general regions. We show that in this case obstructions appear that are not detected by signed tilings.

算术级数中的T型四格骨牌（T-tetrominos）

T-Tetrominos in Arithmetic Progression

本文针对T-tetrominos铺砌矩形的经典结论，研究所有合法铺砌中必然存在的有序性程度，以骨牌算术级数的长度作为有序性度量标准。

Dissection of a rectangle into rectangles with given side ratios

Given a collection of N rectangles such that the side ratio of each one is a quadratic irrationality, we find all rectangles which can be tiled by rectangles similar to one of the given ones. It means that each possible shape can be used several times or not used at all, so that the number of rectangles in the tiling is not necessarily equal to N.

Polynomial mixing of the edge-flip Markov chain for unbiased dyadic tilings

We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall, and Spencer in 2002. A dyadic tiling of size n is a tiling of the unit square by n non-overlapping dyadic rectangles, each of area 1/n, where a dyadic rectangle is any rectangle that can be written in the form [a2^{-s}, (a+1)2^{-s}] \times [b2^{-t}, (b+1)2^{-t}] for non-negative integers a,b,s,t. The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doing …

New Combinatorial Interpretations of the Fibonacci Numbers Squared, Golden Rectangle Numbers, and Jacobsthal Numbers Using Two Types of Tile

We consider the tiling of an $n$-board (a board of size $n\times1$) with squares of unit width and $(1,1)$-fence tiles. A $(1,1)$-fence tile is composed of two unit-width square subtiles separated by a gap of unit width. We show that the number of ways to tile an $n$-board using unit-width squares and $(1,1)$-fence tiles is equal to a Fibonacci number squared when $n$ is even and a golden rectangle number (the product of two consecutive Fibonacci numbers) when $n$ is odd. We also show that the number of tilings of boards using $n$ such square and fence tiles is a Jacobsthal number. Using combi…

Tetromino tilings and the Tutte polynomial

We consider tiling rectangles of size 4m x 4n by T-shaped tetrominoes. Each tile is assigned a weight that depends on its orientation and position on the lattice. For a particular choice of the weights, the generating function of tilings is shown to be the evaluation of the multivariate Tutte polynomial Z\_G(Q,v) (known also to physicists as the partition function of the Q-state Potts model) on an (m-1) x (n-1) rectangle G, where the parameter Q and the edge weights v can take arbitrary values depending on the tile weights.

A note on a 2-enumeration of antisymmetric monotone triangles

In their unpublished work, Jockusch and Propp showed that a 2-enumeration of antisymmetric monotone triangles is given by a simple product formula. On the other hand, the author proved that the same formula counts the domino tilings of the quartered Aztec rectangle. In this paper, we explain this phenomenon directly by building a correspondence between the antisymmetric monotone triangles and domino tilings of the quartered Aztec rectangle.

可由两两不可比较的整数矩形铺砌的最小正方形

The smallest square tileable by pairwise incomparable integer rectangles

本文证明 27 是可被两两不可比较的整数矩形铺砌的最小正方形边长，通过结构约简与穷举搜索验证了所有 n≤26 的情况。

带状铺砌的数量

The Number of Ribbon Tilings for Strips

本文研究了矩形和带状的 $n$ 阶带状铺砌数量，证明了增长率的存在性并给出上下界，同时构建递归系统枚举了 $n \leq 8$ 的带状铺砌及其生成函数。

Auspicious tatami mat arrangements

An \emph{auspicious tatami mat arrangement} is a tiling of a rectilinear region with two types of tiles, $1 \times 2$ tiles (dimers) and $1 \times 1$ tiles (monomers). The tiles must cover the region and satisfy the constraint that no four corners of the tiles meet; such tilings are called \emph{tatami tilings}. The main focus of this paper is when the rectilinear region is a rectangle. We provide a structural characterization of rectangular tatami tilings and use it to prove that the tiling is completely determined by the tiles that are on its border. We prove that the number of tatami tiling…

方形平铺算术的代数方法

An Algebraic Approach to the Arithmetic of Square Tilings

本文提出一种代数方法，通过块矩阵编码平铺几何，利用区间矩阵的全幺模性证明二维方形平铺分母最小公倍数以2^n为界，并推广到d维得到d^n界，同时简洁证明Kenyon定理。

Singular polynomials of generalized Kasteleyn matrices

Kasteleyn counted the number of domino tilings of a rectangle by considering a mutation of the adjacency matrix: a Kasteleyn matrix K. In this paper we present a generalization of Kasteleyn matrices and a combinatorial interpretation for the coefficients of the characteristic polynomial of KK^\ast (which we call the singular polynomial), where K is a generalized Kasteleyn matrix for a planar bipartite graph. We also present a q-version of these ideas and a few results concerning tilings of special regions such as rectangles.

Distances in domino flip graphs

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.

Rectangular tileability and complementary tileability are undecidable

Does a given a set of polyominoes tile some rectangle? We show that this problem is undecidable. In a different direction, we also consider tiling a cofinite subset of the plane. The tileability is undecidable for many variants of this problem. However, we present an algorithm for testing whether the complement of a finite region is tileable by a set of rectangles.

何时可以用整数正方形铺满整数矩形？

When Can You Tile an Integer Rectangle with Integer Squares?

本文刻画了整数矩形可被边长至少为2的整数正方形铺满的条件：当两维均至少为10时总是可行，小维度时行为呈周期性或由穷举确定。

Counting With Irrational Tiles

We introduce and study the number of tilings of unit height rectangles with irrational tiles. We prove that the class of sequences of these numbers coincides with the class of diagonals of N-rational generating functions and a class of certain binomial multisums. We then give asymptotic applications and establish connections to hypergeometric functions and Catalan numbers.

通用矩形划分

Generic rectangulations

本文研究了通用矩形划分的计数问题，通过建立与避免特定模式的排列集之间的双射关系，提出了新的计数方法。

矩形分拆函数$p(m,n)$的渐近行为

The asymptotic behavior of the rectangle partition function $p(m,n)$

本文研究矩形分拆函数$p(m,n)$的渐近行为，通过初等方法推导其对数渐近公式，证实相关猜想并推广整数分拆的Hardy-Ramanujan公式。

圆形Aztec窗口，Aztec钻石定理的对偶以及斜对角裂缝关联的一个奇特对称性

Round Aztec windows, a dual of the Aztec diamond theorem and a curious symmetry of the correlation of diagonal slits

本文研究了Aztec矩形和环面Aztec钻石中特定孔洞形状的多米诺平铺数量，证明了其平铺数和完美匹配数可由简单乘积公式表示，并揭示了斜对角裂缝关联的对称性。

论将正多边形剖分为矩形

On Dissecting Polygons into Rectangles

本文研究正 n 边形重排为矩形的最少剖分块数 r(n)，给出 n=3 至 12 的构造上界，并在多数情形改进正方形剖分界。

用于 Meir-Moser 装填问题的 Slack-Pack 算法

Slack-Pack algorithm for Meir-Moser packing problem

针对 Meir-Moser 矩形铺砌问题，本文提出允许零件堆间留隙的 Slack-Pack 算法，以论证 Tao 关于大索引矩形可铺满等面积正方形的结果可推广至 t=1。