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

TL;DR

This paper explores the properties of switching rook polynomials, establishing a new combinatorial criterion called domino-stability that characterizes when these polynomials are palindromic, which relates to algebraic Gorensteinness.

Contribution

It introduces domino-stability as a new combinatorial property and proves its equivalence to palindromicity of switching rook polynomials, linking combinatorics to algebraic properties.

Findings

Switching rook polynomial is palindromic if and only if the collection is domino-stable.

Domino-stability characterizes Gorenstein $K$-algebras from collections of cells.

New insights into algebraic properties of polyominoes and cell collections.

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.