Auspicious tatami mat arrangements

TL;DR

This paper studies a special tiling problem called tatami tilings, characterizes their structure on rectangles, and provides formulas and algorithms for counting such tilings.

Contribution

It offers a structural characterization of rectangular tatami tilings and derives explicit counting formulas and generating functions.

Findings

Number of n x n tatami tilings with n monomers is n*2^{n-1}.

Tiling is determined by border tiles.

Generating function for fixed-height rectangles is rational.

Abstract

An \emph{auspicious tatami mat arrangement} is a tiling of a rectilinear region with two types of tiles, $1 \times 2$ tiles (dimers) and $1 \times 1$ tiles (monomers). The tiles must cover the region and satisfy the constraint that no four corners of the tiles meet; such tilings are called \emph{tatami tilings}. The main focus of this paper is when the rectilinear region is a rectangle. We provide a structural characterization of rectangular tatami tilings 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 $n \times n$ 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.