Hilbert series of simple thin polyominoes

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