On the rook polynomial of grid polyominoes

TL;DR

This paper establishes a mathematical equivalence between the rook polynomial of grid polyominoes and the h-polynomial of their coordinate rings, using simplicial complex theory, extending prior results for simpler polyominoes.

Contribution

It proves the equivalence for grid polyominoes, generalizing previous findings from frame polyominoes with a single hole.

Findings

Rook polynomial equals the h-polynomial of the coordinate ring.

Extension of results from frame polyominoes to grid polyominoes.

Application of simplicial complex theory to polyomino analysis.

Abstract

Grid polyominoes form a class of thin polyominoes with one or more holes arranged in a grid-like pattern in the plane. In this paper, we prove that the rook polynomial of grid polyominoes coincides with the h-polynomial of their corresponding coordinate ring. Our approach is based on the theory of simplicial complexes and extends previous results for frame polyominoes, which are special cases of polyominoes with exactly one hole.