矩形网格的邻接矩阵的核
The kernel of the adjacency matrix of a rectangular mesh
摘要：
给定一个m × n的矩形网格，它的邻接矩阵A，只有整数元素，可以解释为任意域K上的向量空间之间的映射。我们描述A的核：它是两个自然子空间的直和，它们的维数等于$\lceil c/2 \rceil$和$\lfloor c/2 \rfloor$，其中c = gcd（m+1，n+1）- 1.我们证明了这两个向量空间都有基，其中元素等于0，1和-1。当K = Z/（2）时，这些子空间的核元素由一种特殊的矩形平铺来描述。作为一个推论，我们计算一个整数边的矩形与一组指定的瓷砖的瓷砖的数量。
英文摘要：
Given an m x n rectangular mesh, its adjacency matrix A, having only integer entries, may be interpreted as a map between vector spaces over an arbitrary field K. We describe the kernel of A: it is a direct sum of two natural subspaces whose dimensions are equal to $\lceil c/2 \rceil$ and $\lfloor c/2 \rfloor$, where c = gcd (m+1,n+1) - 1. We show that there are bases to both vector spaces, with entries equal to 0, 1 and -1. When K = Z/(2), the kernel elements of these subspaces are described by rectangular tilings of a special kind. As a corollary, we count the number of tilings of a rectangle of integer sides with a specified set of tiles.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
