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

Abstract

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 (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 $|V(\mathcal{P})|-\mathrm{rank}\mathcal{P}$ 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.

21 pages, 8 figures

References in corpus (2)

• Ideals generated by 2-minors, collection of cells and stack polyominoes
• On Gröbner bases and Cohen-Macaulay property of closed path polyominoes

Cited by in corpus (5)

• Polyocollection ideals and primary decomposition of polyomino ideals
• On Cohen-Macaulay non-prime collections of cells
• Shellable simplicial complex and switching rook polynomial of frame polyominoes
• Non-simple polyominoes of Kőnig type and their canonical module
• Shellable flag simplicial complexes of non-simple polyominoes