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

Hilbert series of simple thin polyominoes

In this paper we introduce some monomial orders for the class of closed path polyominoes and we prove that the set of the generators of the polyomino ideal attached to a closed path forms the reduced Gröbner basis with respect to these monomial orders. It is known that the polyomino ideal attached to a closed path containing an $L$-configuration or a ladder of at least three steps, equivalently having no zig-zag walks, is prime. As a consequence, we obtain that the coordinate ring of a closed path having no zig-zag walks is a normal Cohen-Macaulay domain.

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On Gröbner Basis and Cohen-Macaulay Property of Closed Path Polyominoes

Let $$\mathcal {P}$$ be a closed path having no zig-zag walks, a kind of non-simple thin polyomino. In this paper, we give a combinatorial interpretation of the h-polynomial of $$K[\mathcal {P}]$$ , showing that it is the rook polynomial of $$\mathcal {P}$$ . It is known by Rinaldo and Romeo (J Algebr Comb 54:607–624, 2021), that if $$\mathcal {P}$$ is a simple thin polyomino, then the h-polynomial is equal to the rook polynomial of $$\mathcal {P}$$ and it is conjectured that this property characterizes all thin polyominoes. Our main demonstrative strategy is to compute the reduced Hilbert–Poincaré series of the coordinate ring attached to a closed path $$\mathcal {P}$$ having no zig-zag walks, as a combination of the Hilbert–Poincaré series of convenient simple thin polyominoes. As a consequence, we prove that the Krull dimension is equal to $$\vert V(\mathcal {P})\vert -\textrm{rank}\, \mathcal {P}$$ and the regularity of $$K[\mathcal {P}]$$ is the rook number of $$\mathcal {P}$$ . Finally, we characterize the Gorenstein prime closed paths, proving that $$K[\mathcal {P}]$$ is Gorenstein if and only if $$\mathcal {P}$$ consists of maximal blocks of length three. 

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Hilbert–Poincaré Series and Gorenstein Property for Some Non-simple Polyominoes

. In this paper we compute the reduced Poincar´e-Hilbert series of the coordinate ring attached to a closed path P having no zig-zag walks, as a combination of the Poincar´e-Hilbert series of convenient simple thin polyominoes. As a consequence we ﬁnd the Krull dimension and the regularity of K [ P ] and we prove that the h -polynomial is exactly the rook polynomial of P . Finally we characterize the Gorenstein prime closed paths using the S -property.

Hilbert-poincar´e series and gorenstein property for closed path polyominoes

In this article, we study the primary decomposition of some binomial ideals. In particular, we introduce the concept of polyocollection, a combinatorial object that generalizes the definitions of collection of cells and polyomino, that can be used to compute a primary decomposition of non-prime polyomino ideals. Furthermore, we give a description of the minimal primary decomposition of non-prime closed path polyominoes. In particular, for such a class of polyominoes, we characterize the set of all zig-zag walks and show that the minimal prime ideals have a very nice combinatorial description.

Polyocollection ideals and primary decomposition of polyomino ideals

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Claudia Andrei

Papers

Algebraic Properties of the Coordinate Ring of a Convex Polyomino