  Let P be a simple thin polyomino, roughly speaking a polyomino that has no
holes and does not contain a square tetromino as a subpolyomino. In this paper,
we determine the reduced Hilbert series $h(t)/(1-t)^d$ of K[P] by proving that
$h(t)$ is the rook polynomial of P. As an application, we characterize the
Gorenstein simple thin polyominoes.
