Monomer-Dimer Tatami Tilings of Rectangular Regions

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 monome...

Monomer-dimer tatami tilings of square regions

We prove that the number of monomer-dimer tilings of an n × n square grid, with m < n monomers in which no four tiles meet at any point is m2 + (m + 1)2, when m and n have the same parity. In addition, we present a new proof of the result that there are n2 such tilings with n monomers, which divides the tilings into n classes of size 2. The sum of these tilings over all monomer counts has the c...

Explorations into Monomer-Dimer Tilings of Planar Regions

Counting Fixed-Height Tatami Tilings

A tatami tiling is an arrangement of 1 × 2 dominoes (or mats) in a rectangle with m rows and n columns, subject to the constraint that no four corners meet at a point. For fixed m we present and use Dean Hickerson’s combinatorial decomposition of the set of tatami tilings — a decomposition that allows them to be viewed as certain classes of restricted compositions when n ≥ m. Using this decompo...

Pfaffian solution of a dimer-monomer problem: Single monomer on the boundary.

We consider the dimer-monomer problem for the rectangular lattice. By mapping the problem into one of close-packed dimers on an extended lattice, we rederive the Tzeng-Wu solution for a single monomer on the boundary by evaluating a Pfaffian. We also clarify the mathematical content of the Tzeng-Wu solution by identifying it as the product of the nonzero eigenvalues of the Kasteleyn matrix.