$n$ -三角形网格中凸五边形和凸六边形的个数

Enumeration，菱形镶嵌六边形，中央有三角形孔

Enumeration of lozenge tilings of hexagons with a central triangular hole

本文研究边长为$a,b+m,c,a+m,b,c+m$的六边形菱形镶嵌的未加权和加权枚举问题，其中边长为$m$的等边三角形已从中心移去.我们给出了封闭的公式为平原枚举和一定的$(-1)$ -枚举这些菱形平铺。在这种情况下，$a=b=c$，我们还提供了封闭的公式，某些加权枚举的菱形镶嵌是循环对称的。对于$m=0$，后面的公式专用于关于循环对称平面划分的加权枚举的语句。一个这样的专业化提供了一个证明的猜想Stembridge对一定的加权计数的循环对称平面分区。在我们的证明中采用的工具是不相交格路径和行列式评价理论的非标准应用。特别是，我们评估的决定因素$\det_{0\le i,j\le n-1}\big(\om δ_{ij}+\binom {m+i+j}j\big)$，其中$\om$是任何6次根的单位。这些行列式的评价是一个著名的结果，由于安德鲁斯（发明。53（1979），193- 225），其对应于$\om=1$。

On the chromatic numbers of signed triangular and hexagonal grids

A signed graph is a simple graph with two types of edges. Switching a vertex $v$ of a signed graph corresponds to changing the type of each edge incident to $v$. A homomorphism from a signed graph $G$ to another signed graph $H$ is a mapping $φ: V(G) \rightarrow V(H)$ such that, after switching any number of the vertices of $G$, $φ$ maps every edge of $G$ to an edge of the same type in $H$. The chromatic number $χ_s(G)$ of a signed graph $G$ is the order of a smallest signed graph $H$ such that there is a homomorphism from $G$ to $H$. We show that the chromatic number of signed triangular grid…

A factorization theorem for lozenge tilings of a hexagon with triangular holes

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 …

A new proof for the number of lozenge tilings of quartered hexagons

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".

Lozenge Tiling Function Ratios for Hexagons with Dents on Two Sides

We give a formula for the number of lozenge tilings of a hexagon on the triangular lattice with unit triangles removed from arbitrary positions along two non-adjacent, non-opposite sides. Our formula implies that for certain families of such regions, the ratios of their numbers of tilings are given by simple product formulas.

具有六边形和三角形面且每个顶点周围有三个面的多面体

Polyhedra with hexagonal and triangular faces and three faces around each vertex

本文研究了由六边形和三角形组成且每个顶点有三个面的3-正则平面图（trihexes），通过signature标记法建立了其与等价类的双射，并据此限制了给定顶点数下的trihexes数量及证明了无六边形“belts”的猜想。

关于Spectre铺砌六边形簇的观察

Observations on the hex clusters of the Spectre tilings

该研究通过用六边形装饰Spectre瓦片得到三角形六边形簇并探究其结构，重新证明Spectre铺砌存在且具唯一层级结构，且证明非计算机辅助。