The Stanley–Reisner ideal of the rook complex of polyominoes

We study the properties of the rook complex [Formula: see text] of a polyomino [Formula: see text] seen as independence complex of a graph [Formula: see text], and the associated Stanley–Reisner ideal [Formula: see text]. In particular, we characterize the polyominoes [Formula: see text] having a pure rook complex, and the ones whose Stanley–Reisner ideal has linear resolution. Furthermore, we prove that for a class of polyominoes the Castelnuovo–Mumford regularity of [Formula: see text] coincides with the induced matching number of [Formula: see text].