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On the rook polynomial of grid polyominoes

Rodica Dinu, Francesco Navarra

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.

On the rook polynomial of grid polyominoes

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.
Paper Structure (3 sections, 13 theorems, 5 equations, 13 figures, 1 table)

This paper contains 3 sections, 13 theorems, 5 equations, 13 figures, 1 table.

Table of Contents

  1. Simplicial complexes, polyominoes and their $K$-algebras
  2. Shellability of the simplicial complex attached to a grid polyomino
  3. Hilbert series and rook polynomial of a grid polyomino

Key Result

Proposition 1.1

Bruns_Herzog Let $\Delta$ be a shellable simplicial complex of dimension $d$ with shelling $F_1,\dots,F_m$. For $j\in \{2,\dots,m\}$ we denote by $r_j$ the number of facets of $\langle F_1,\dots,F_{j-1}\rangle\cap \langle F_j\rangle$ and we set $r_1=0$. Let $(h_0,\dots,h_{d+1})$ be the $h$-vector of

Figures (13)

  • Figure 1: A polyomino.
  • Figure 2: Examples of grid polyominoes.
  • Figure 3: Example of (generalized) steps in a facet of $\Delta_{\mathcal{P}}$.
  • Figure 4: Several arrangements of the cells of $\mathcal{P}$.
  • Figure 5: Sub-polyomino when $a=i-3$ and $b=j+1$.
  • ...and 8 more figures

Theorems & Definitions (31)

  • Proposition 1.1
  • Definition 1.2
  • Theorem 1.3
  • proof
  • Corollary 1.4
  • proof
  • Definition 2.1
  • Example 2.2
  • Lemma 2.3
  • proof
  • ...and 21 more