Hilbert Series of simple thin polyominoes

Abstract

Let 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-Poincaré 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.

On Gröbner bases and Cohen-Macaulay property of closed path polyominoes

“…In [22] the authors study the reduced Gröbner basis of polyomino ideals and introduce some conditions in order to the generators of I P form the reduced Gröbner basis with respect to some suitable degree reverse lexicographic monomial orders. Eventually, for further references about several algebraic properties of polyomino ideals we report [1], [9], [10] and [26].…”