  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 $\mathcal{P}$.
