Hilbert Series of simple thin polyominoes

TL;DR

This paper establishes a connection between the Hilbert series of simple thin polyominoes and rook polynomials, providing a characterization of Gorenstein cases and advancing combinatorial algebra understanding.

Contribution

It proves that the reduced Hilbert series of simple thin polyominoes equals their rook polynomial, offering a new combinatorial interpretation and classification of Gorenstein polyominoes.

Findings

Hilbert series equals rook polynomial for simple thin polyominoes

Characterization of Gorenstein simple thin polyominoes

Provides a combinatorial approach to algebraic properties

Abstract

Let P be a simple thin polyomino, roughly speaking a polyomino that has no holes and does not contain a square tetromino as a subpolyomino. In this paper, we determine the reduced Hilbert 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.