Switching Rook Polynomials of Collections of Cells: Palindromicity and Domino-Stability

TL;DR

This work provides a complete combinatorial characterization of when the switching rook polynomial of a collection of cells is palindromic, equating palindromicity with a geometric condition called domino-stability. It establishes a bidirectional link: domino-stability implies palindromicity via a canonical maximum rook configuration and a constructive bijection, and palindromicity implies domino-stability through a case analysis. Consequently, the authors connect palindromicity to the Gorenstein property of coordinate rings $K[\mathcal{P}]$, offering conjectural equivalences among Gorensteinness, palindromic $h$-polynomials, palindromic switching rook polynomials, and domino-stability. The results extend known phenomena for skew diagrams and polyominoes and provide both theoretical and computational evidence toward a unified criterion for Gorenstein coordinate rings in this combinatorial-algebraic setting.

Abstract

The rook polynomial is a generating function that enumerates the number of ways to place rooks, with no two in the same row or column, on a collection of cells regarded as a pruned chessboard. In combinatorial commutative algebra, special attention is devoted to its variant, the switching rook polynomial, which is conjectured to coincide with the $h$-polynomial of the $K$-algebra associated with the given collection of cells. In this context, palindromicity plays a crucial role, as it reflects the algebraic property of Gorensteinness. In this paper, we introduce a new combinatorial property, called domino-stability, and we prove that the switching rook polynomial of a collection of cells $\mathcal{P}$ is palindromic if and only if $\mathcal{P}$ is domino-stable. Building upon this result, we derive new insights into the characterization of Gorenstein $K$-algebras arising from polyominoes or, more generally, from collections of cells.

Figures (12)

• Figure 1: A polyomino, a weakly connected collection of cells and a bounding rectangle of a polyomino.
• Figure 2: Examples of some arrangements of non-attacking rooks.
• Figure 3: Three $3$-rook configurations in $\mathcal{P}$.
• Figure 4: A family of axis-aligned rectangles and its gluing.
• Figure 5: A polyomino and a gluing of $\overline{\mathcal{B}_1}$.

Theorems & Definitions (37)

• Theorem 1
• Corollary 2
• Proposition 3
• Conjecture 4
• Definition 1.1
• Remark 1.2
• Definition 1.4
• Remark 1.5
• Definition 2.1: Gluing of axis-aligned rectangles
• Proposition 2.2