In this paper we consider tilings of rectangular regions with two types of tiles, $1 imes 2$ tiles (dimers) and $1 imes 1$ tiles (monomers). The tiles must cover the region and satisfy the constraint that no four corners of the tiles meet; such tilings are called tatami tilings. We provide a structural characterization and use it to prove that the tiling is completely determined by the tiles that are on its border. We prove that the number of tatami tilings of an $n imes n$ square with $n$ monomers is $n2^{n-1}$. We also show that, for fixed-height, the generating function for the number of tatami tilings of a rectangle is a rational function, and outline an algorithm that produces the generating function.
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机译：在本文中，我们考虑矩形区域的平铺，其中包含两种类型的图块：$ 1 x 2 $瓷砖（二聚体）和$ 1 x 1 $瓷砖（单体）。瓷砖必须覆盖该区域并满足瓷砖四个角不相交的约束;这种瓷砖称为榻榻米瓷砖。我们提供了结构特征，并用它来证明平铺完全由其边界上的平铺确定。我们证明，具有$ n $个单体的$ n times n $正方形的榻榻米平铺数为$ n2 ^ {n-1} $。我们还表明，对于固定高度，矩形的榻榻米平铺数目的生成函数是一个有理函数，并概述了生成该生成函数的算法。
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