Palindromic switching rook polynomials and domino-stable collections of cells

Abstract

Let $\mathcal{P}$ be a finite collection of cells. Two non-attacking rook configurations on $\mathcal{P}$ are switching-equivalent if one can be obtained from the other by a sequence of switches inside rectangles contained in $\mathcal{P}$. The switching rook polynomial of $\mathcal{P}$ enumerates the resulting equivalence classes according to the number of rooks. We characterize the collections of cells whose switching rook polynomial is palindromic. More precisely, we introduce a new combinatorial property, called domino-stability, and we prove that the switching rook polynomial of $\mathcal{P}$ is palindromic if and only if $\mathcal{P}$ is domino-stable.

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