Auspicious Tatami Mat Arrangements
We introduce tatami tilings, and present some of the many interesting questions that arise when studying them. Roughly speaking, we are considering 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 n2 n − 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.
Keywordsdimer monomer tatami tilings combinatorics enumeration
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