Shellable simplicial complex and switching rook polynomial of frame polyominoes

Abstract

Let P be a frame polyomino, a new kind of non-simple polyomino. In this paper we study the h-polynomial of K[P] in terms of the switching rook polynomial of P using the shellable simplicial complex Δ(P) attached to P. We provide a suitable shelling order for Δ(P) and we define a bijection between the set of the canonical configurations of j rooks in P and the facets of Δ(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[P] as the switching rook polynomial of P.