标题： 在$(2p + 1) \times (2q + 1) $晶格上包装二聚体 显示英文标题

标题： Packing dimers on $(2p + 1) \times (2q + 1) $ lattices

摘要： 我们使用计算方法来研究在\emph{奇数乘奇数}晶格上排列二聚体的方式数目。 在这种情况下，晶格中总是有一个空位。 我们表明，在$(2k+1) \times (2k+1)$ \emph{奇数} 平方晶格上的二聚体配置数目具有一些显著的数论性质，与在$2k \times 2k$\emph{甚至}平方晶格上紧密排列的二聚体的数目类似，后者存在精确解。 此外，我们证明了在任意宽度为$n \ge 1$的奇数宽晶格条带的自由能有限尺寸修正中，存在一个明确的对数项。 这个对数项决定了奇数正方形晶格自由能的不同行为。 这些发现揭示了统计物理模型与数论之间深层次且之前未被探索的联系，并表明单体-二聚体问题可能是可解的。 显示英文摘要

摘要： We use computational method to investigate the number of ways to pack dimers on \emph{odd-by-odd} lattices. In this case, there is always a single vacancy in the lattices. We show that the dimer configuration numbers on $(2k+1) \times (2k+1)$ \emph{odd} square lattices have some remarkable number-theoretical properties in parallel to those of close-packed dimers on $2k \times 2k$ \emph{even} square lattices, for which exact solution exists. Furthermore, we demonstrate that there is an unambiguous logarithm term in the finite size correction of free energy of odd-by-odd lattice strips with any width $n \ge 1$. This logarithm term determines the distinct behavior of the free energy of odd square lattices. These findings reveal a deep and previously unexplored connection between statistical physics models and number theory, and indicate the possibility that the monomer-dimer problem might be solvable.