二维二聚体问题的一种新解法
A new solution for the two dimensional dimer problem
摘要：
经典的1961年解决问题的数量确定完美匹配（或二聚体覆盖）的矩形网格图-由于独立Kasteleyn和Temperley和费舍尔-包括改变符号的一些项目的邻接矩阵，使Pfweian的新矩阵给出了完美匹配的数量，然后评估这个Pfweian。另一个经典的方法是使用不相交格路上的Lindström-Gessel-Viennot定理将完美匹配的数量表示为行列式，然后计算这个行列式。本文利用柯西-比奈定理，提出了一种求解二维二聚体问题的新方法。它只涉及在20世纪30年代中期二聚体问题被提出时已知的事实，所以它可能在二聚体问题仍然悬而未决时就被发现了。我们提供明确的产品公式的正方形和六角晶格。与Kasteleyn、Temperley和Fisher的原始公式相比，我们的正方形晶格公式的一个优点是，我们的公式具有线性因子数，而前者的因子数是二次的。我们的六角晶格的结果产生一个公式的数量的周期性的阶梯表面，适合在一个无限管给定的横截面，这可以被看作是一个对应的MacMahon的盒装平面分割定理。
英文摘要：
The classical 1961 solution to the problem of determining the number of perfect matchings (or dimer coverings) of a rectangular grid graph -- due independently to Kasteleyn and to Temperley and Fisher -- consists of changing the sign of some of the entries in the adjacency matrix so that the Pfaffian of the new matrix gives the number of perfect matchings, and then evaluating this Pfaffian. Another classical method is to use the Lindström-Gessel-Viennot theorem on non-intersecting lattice paths to express the number of perfect matchings as a determinant, and then evaluate this determinant. In this paper we present a new method for solving the two dimensional dimer problem, which relies on the Cauchy-Binet theorem. It only involves facts that were known in the mid 1930's when the dimer problem was phrased, so it could have been discovered while the dimer problem was still open. We provide explicit product formulas for both the square and the hexagonal lattice. One advantage of our formula for the square lattice compared to the original formula of Kasteleyn, Temperley and Fisher is that ours has a linear number of factors, while the number of factors in the former is quadratic. Our result for the hexagonal lattice yields a formula for the number of periodic stepped surfaces that fit in an infinite tube of given cross-section, which can be regarded as a counterpart of MacMahon's boxed plane partition theorem.
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。
