The $h$-polynomial and the rook polynomial of some polyominoes

Abstract

Let $X$ be a convex polyomino such that its vertex set is a sublattice of ${\mathbb{N}}^2$. Let $\mathbb{k}[X]$ be the toric ring (over a field $\mathbb{k}$) associated to $X$ in the sense of Qureshi, \emph{J. Algebra}, 2012. Write the Hilbert series of $\mathbb{k}[X]$ as $(1+h_{1}t+h_2{t}^2+\cdots )/(1-t{)}^{\dim (\mathbb{k}[X])}$. For $k\in \mathbb{N}$, let $r_k$ be the number of configurations in $X$ with $k$ pairwise non-attacking rooks. We show that $h_2<r_2$ if $X$ is not a thin polyomino. This partially confirms a conjectured characterization of thin polyominoes by Rinaldo and Romeo, \emph{J. Algebraic Combin.}, 2021.

5 pages