Let $X$ be a convex polyomino such that its vertex set is a sublattice of $\mathbb{N}^2$. Let $\Bbbk[X]$ be the toric ring (over a field $\Bbbk$) associated to $X$ in the sense of Qureshi, J. Algebra, 2012. Write the Hilbert series of $\Bbbk[X]$ as $(1 + h_1 t + h_2 t^2 + \cdots )/(1-t)^{\dim(\Bbbk[X])}$. 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, J. Algebraic Combin., 2021.
@article{10_37236_10902,
author = {Manoj Kummini and Dharm Veer},
title = {The \(h\)-polynomial and the rook polynomial of some polyominoes},
journal = {The electronic journal of combinatorics},
year = {2023},
volume = {30},
number = {2},
doi = {10.37236/10902},
zbl = {1521.13035},
url = {http://geodesic.mathdoc.fr/articles/10.37236/10902/}
}
TY - JOUR
AU - Manoj Kummini
AU - Dharm Veer
TI - The \(h\)-polynomial and the rook polynomial of some polyominoes
JO - The electronic journal of combinatorics
PY - 2023
VL - 30
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/10902/
DO - 10.37236/10902
ID - 10_37236_10902
ER -
%0 Journal Article
%A Manoj Kummini
%A Dharm Veer
%T The \(h\)-polynomial and the rook polynomial of some polyominoes
%J The electronic journal of combinatorics
%D 2023
%V 30
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/10902/
%R 10.37236/10902
%F 10_37236_10902
Manoj Kummini; Dharm Veer. The \(h\)-polynomial and the rook polynomial of some polyominoes. The electronic journal of combinatorics, Tome 30 (2023) no. 2. doi: 10.37236/10902
