Hilbert series of Parallelogram Polyominoes

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