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

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.