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arXiv · 2306.11467

Shellable simplicial complex and switching rook polynomial of frame polyominoes

Rizwan Jahangir·Francesco Navarra
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Abstract

Let $\mathcal{P}$ be a frame polyomino, a new kind of non-simple polyomino. In this paper we study the $h$-polynomial of $K[\mathcal{P}]$ in terms of the switching rook polynomial of $\mathcal{P}$ using the shellable simplicial complex $\Delta(\mathcal{P})$ attached to $\mathcal{P}$. We provide a suitable shelling order for $\Delta(\mathcal{P})$ and we define a bijection between the set of the canonical configurations of $j$ rooks in $\mathcal{P}$ and the facets of $\Delta(\mathcal{P})$ with $j$ steps. Finally we use a well-known combinatorial result, due to McMullen and Walkup, about the $h$-vector of a shellable simplicial complex to interpret the $h$-polynomial of $K[\mathcal{P}]$ as the switching rook polynomial of $\mathcal{P}$.

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Rizwan Jahangir, Francesco Navarra. 2023-06-20. Shellable simplicial complex and switching rook polynomial of frame polyominoes. https://doi.org/10.1016/j.jpaa.2023.107576

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