Hilbert series of parallelogram polyominoes

Abstract We present a conjecture about the reduced Hilbert series of coordinate ring simple polyomino in terms particular arrangements non-attacking rooks that can be placed on polyomino. By using computational approach, we prove above holds for all polyominoes up to rank 11. In addition, true class parallelogram polyominoes, by looking at those as planar distributive lattices. Finally, give combinatorial interpretation Gorensteinness polyominoes.