Switching Rook Polynomials of Collections of Cells: Palindromicity and Domino-Stability
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| Format: | Preprint |
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2025
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| _version_ | 1866911301140217856 |
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| author | Navarra, Francesco Qureshi, Ayesha Asloob Rinaldo, Giancarlo |
| author_facet | Navarra, Francesco Qureshi, Ayesha Asloob Rinaldo, Giancarlo |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_13982 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Switching Rook Polynomials of Collections of Cells: Palindromicity and Domino-Stability Navarra, Francesco Qureshi, Ayesha Asloob Rinaldo, Giancarlo Combinatorics Commutative Algebra 05A15, 05B50, 05E40 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. |
| title | Switching Rook Polynomials of Collections of Cells: Palindromicity and Domino-Stability |
| topic | Combinatorics Commutative Algebra 05A15, 05B50, 05E40 |
| url | https://arxiv.org/abs/2511.13982 |