关于单体-二聚体覆盖的一个互易定理
A Reciprocity Theorem for Monomer-Dimer Coverings
摘要：
格子的单-二聚体覆盖计数问题是统计力学中一个长期存在的问题。它只在二维二聚体覆盖的特殊情况下才被精确求解。在早期的工作中，斯坦利证明了一个互易原理，它控制着一个$m$的二聚体覆盖数$N(m,n)$通过$n$矩形网格（也称为完美匹配），其中$m$是固定的，而$n$是允许变化的。正如Propp重新解释的那样，Stanley的结果涉及到将$N(m,n)$扩展到$n < 0$的独特方式，使得所得到的双无限序列，对于$n \in {Z}$的$N(m,n)$，满足具有常系数的线性递归关系。特别地，Stanley表明$N(m,n)$总是满足关系式$N(m,-2-n) = ε_{m,n}N(m,n)$的整数，其中$ε_{m,n} = 1$除非$m\equiv$ 2（mod 4）和$n$是奇数，在这种情况下$ε_{m,n} = -1$。此外，Propp方法适用于高维情况。本文讨论了类似的调查数字$M(m,n)$，单体-二聚体覆盖，或等价（不一定完美）匹配的$m$的$n$矩形网格。我们发现，对于每个固定的$m$有一个唯一的方式扩展$M(m,n)$ $n < 0$，使所产生的双无限序列，$M(m,n)$ $n \in {Z}$，满足一个线性常系数递归关系。我们表明，$M(m,n)$，先验的有理数，总是一个整数，使用推广的组合模型提供的Propp。最后，我们给出了一个新的声明互惠多元生成函数的Stanley的结果如下。
英文摘要：
The problem of counting monomer-dimer coverings of a lattice is a longstanding problem in statistical mechanics. It has only been exactly solved for the special case of dimer coverings in two dimensions. In earlier work, Stanley proved a reciprocity principle governing the number $N(m,n)$ of dimer coverings of an $m$ by $n$ rectangular grid (also known as perfect matchings), where $m$ is fixed and $n$ is allowed to vary. As reinterpreted by Propp, Stanley's result concerns the unique way of extending $N(m,n)$ to $n < 0$ so that the resulting bi-infinite sequence, $N(m,n)$ for $n \in {Z}$, satisfies a linear recurrence relation with constant coefficients. In particular, Stanley shows that $N(m,n)$ is always an integer satisfying the relation $N(m,-2-n) = ε_{m,n}N(m,n)$ where $ε_{m,n} = 1$ unless $m\equiv$ 2(mod 4) and $n$ is odd, in which case $ε_{m,n} = -1$. Furthermore, Propp's method is applicable to higher-dimensional cases. This paper discusses similar investigations of the numbers $M(m,n)$, of monomer-dimer coverings, or equivalently (not necessarily perfect) matchings of an $m$ by $n$ rectangular grid. We show that for each fixed $m$ there is a unique way of extending $M(m,n)$ to $n < 0$ so that the resulting bi-infinite sequence, $M(m,n)$ for $n \in {Z}$, satisfies a linear recurrence relation with constant coefficients. We show that $M(m,n)$, a priori a rational number, is always an integer, using a generalization of the combinatorial model offered by Propp. Lastly, we give a new statement of reciprocity in terms of multivariate generating functions from which Stanley's result follows.
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