Auspicious Tatami Mat Arrangements

The main purpose of this paper is to introduce the idea of tatami tilings, and to present some of the many interesting and fun questions that arise when studying them. Roughly speaking, we are considering are tilings of rectilinear regions with 1×2 dimer tiles and 1×1 monomer tiles, with the constraint that no four corners of the tiles meet. Typical problems are to minimize the number of monomers in a tiling, or to count the number of tilings in a particular shape. We determine the underlying structure of tatami tilings of rectangles and use this to prove that the number of tatami tilings of an n × n square with n monomers is n2n−1. We also prove that, for fixed-height, the number of tatami tilings of a rectangle is a rational function and outline an algorithm that produces the coefficients of the two polynomials of the numerator and the denominator. Many interesting and fun open problems remain to be solved. 1. What is a tatami tiling? Traditionally, a tatami mat is made from a rice straw core, with a covering of woven soft rush straw. Originally intended for nobility in Japan, they are now available in mass-market stores. The typical tatami mat occurs in a 1×2 aspect ratio and various configurations of them are used to cover floors in houses and temples. By parity considerations it may be necessary to introduce mats with a 1×1 aspect ratio in order to cover the floor of a room. Such a covering is said to be “auspicious” if no four corners of mats meet at a point. Hereafter, we only consider such auspicious arrangements, since without this constraint the problem is the classical and well-studied dimer tiling problem [3], citeStanley. Following Knuth [5], we will call the auspicious tatami arrangements, tatami tilings. The enumeration of tatami tilings that use only dimers (no monomers) was solved in [4]. Perhaps the most commonly occurring instance of tatami tilings is in paving stone layouts of driveways and sidewalks, where the most frequently used paver has a rectangular shape with a 1×2 aspect ratio. Two of the most common patterns, the “herringbone” and the “running bond”, have the tatami property (see Figure 1). Given a driveway to be paved, for example, with the shape shown in Figure 1, the question occurs of how to tatami tile it with the least number of monomers. The answer to this question could be interesting both because of aesthetic appeal, and because it could save work, since to make a monomer a worker typically cuts a 1× 2 paver in half. Before attempting to study tatami tilings in general orthogonal regions it is crucial to understand them in rectangles, and our results are about tatami tilings of rectangles. Here is an outline of the paper. In Section 2 we determine the structure of tatami tilings in a rectangle. In Section 3 we provide some counting results for tatami tilings in a rectangle: The number of tilings of an n× n square with n monomers is n2n−1 and for a fixed number of rows r, the ordinary generating function of the number of tilings of an r×n rectangle is a rational function. In Section 4 we return to the question of tatami tiling general orthogonal regions and introduce the “magnetic water strider problem”. Additional conjectures and open problems are also introduced in this section. 2. The structure of tatami tilings: T-diagrams We show that all tatami tilings have an underlying structure which partitions the grid into blocks. Each block is filled with either the vertical or horizontal bond pattern. We describe this structure precisely and prove some results for rectangular grids.