Shellable simplicial complex and switching rook polynomial of frame polyominoes

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 $\mathcal{P}$.

To appear in Journal of Pure and Applied Algebra. 26 pages, 19 figures

References in corpus (3)

• Ideals generated by 2-minors, collection of cells and stack polyominoes
• Hilbert-Poincaré series and Gorenstein property for some non-simple polyominoes
• Primality of weakly connected collections of cells and weakly closed path polyominoes

Cited by in corpus (3)

• On Cohen-Macaulay non-prime collections of cells
• Non-simple polyominoes of Kőnig type and their canonical module
• Shellable flag simplicial complexes of non-simple polyominoes