Homology of polyomino tilings on flat surfaces

由n-列多项式和Groebner基构成的符号四角拼接

Signed polyomino tilings by n-in-line polyominoes and Groebner bases

Conway和Lagarias观察到，六边形晶格中的三角形区域T（m）允许由三线四角形（三角形）有符号镶嵌当且仅当m=9d-1或m=9d，对于某个整数d。我们应用整数上的Groebner基理论证明了T（m）允许由n-列多项式（n-骨架）进行符号镶嵌当且仅当m=dn^2-1或m=dn^2对于某个整数d。显式描述的Groebner基础，使我们能够计算“Groebner离散体积”的晶格区域，通过应用带余除法到它的“牛顿多项式”。直接的结果是描述了$n$ -串联多角体的tile同调群。

对称的多角镶嵌、三角形、理想和Groebner基

Symmetric polyomino tilings, tribones, ideals, and Groebner bases

本文应用Groebner基理论研究平面域上的符号对称多角分块。作为对Conway和Lagarias结果的补充，我们证明了六角晶格中的三角形区域T_N= T_{3k-1}和T_N= T_{3k}允许由关于三角形的120度旋转对称的三线四角形（三角形）有符号镶嵌当且仅当对于某个整数r，N= 27 r-1或N= 27 r.

Polyomino卷积和平铺问题

Polyomino convolutions and tiling problems

我们定义了一个卷积运算集的polyominoes和使用它来获得一个标准，一个给定的polyomino不平铺平面（旋转和平移允许）。我们应用该标准的几个家庭的polyominoes，并表明该标准检测到一些情况下，是不能检测的广义着色参数。

平面上的几种多项式镶嵌

Some Polyomino Tilings of the Plane

我们计算了各种宽度的矩形由右tromino，$L$ tetromino，和$T$ tetromino平铺的数量的生成函数。这使我们能够放置下界上的熵平铺的平面由每一个这些。对于$T$ tetromino，我们也从二维Ising模型的解导出了一个下界.

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…

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.