Hilbert series of simple thin polyominoes

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