Algebraic Properties of the Coordinate Ring of a Convex Polyomino

Abstract

We classify all convex polyominoes whose coordinate rings are Gorenstein. We also give an upper bound for the Castelnuovo-Mumford regularity of the coordinate ring of any convex polyomino in terms of the smallest interval which contains its vertices. We give a recursive formula for computing the multiplicity of a stack polyomino.

Hilbert-Poincaré series and Gorenstein property for some non-simple polyominoes

“…. , x n ] then HP S (t) = 1 (1−t) n . We will use frequently the following well known results (see for instance [24,Chapter 5] for a reference) Proposition 2.1.…”

“…We mention also other references, particularly inspiring for this work, for which we provide now some details. In [1] the author classifies all convex polyominoes whose coordinate rings are Gorenstein and compute the regularity of the coordinate ring of any stack polyomino in terms of the smallest interval which contains its vertices. In [8] the authors give a new combinatorial interpretation of the regularity of the coordinate ring attached to an L-convex polyomino, as the rook number of P, that is the maximum number of rooks which can be arranged in P in non-attacking positions.…”

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].…”

On Gröbner Basis 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].…”