Hilbert-Poincar\'e series and Gorenstein property for some non-simple polyominoes

TL;DR

This paper explores the algebraic properties of certain non-simple polyominoes, establishing a combinatorial interpretation of their $h$-polynomial, computing their Hilbert-Poincaré series, and characterizing when their coordinate rings are Gorenstein.

Contribution

It provides a new combinatorial interpretation of the $h$-polynomial for non-simple polyominoes and characterizes Gorenstein properties in this context.

Findings

The $h$-polynomial equals the rook polynomial for the studied polyominoes.

The Krull dimension equals the number of vertices minus the rank of the polyomino.

The Gorenstein property holds if and only if the polyominoes consist of maximal blocks of length three.

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-Poincar\'e series of the coordinate ring attached to a closed path $\mathcal{P}$ having no zig-zag walks, as a combination of the Hilbert-Poincar\'e series of convenient simple thin polyominoes. As a consequence we prove that the Krull dimension is equal to $\vert V(\mathcal{P})\vert -\mathrm{rank}\, \mathcal{P}$ 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.