Hilbert Series of simple thin polyominoes

Abstract

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.