  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.
