Hilbert series of parallelogram polyominoes

Abstract

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 Gorensteinness of parallelogram polyominoes.

The Stanley-Reisner ideal of the rook complex of polyominoes

“…Among the most popular topics in combinatorics related to polyominoes one finds enumerating polyominoes of given size, including the asymptotic growth of the numbers of polyominoes, tiling problems, and reconstruction of polyominoes. The actual research on polyominoes under an algebraic point of view focuses on the study of the polyomino ideal, a quadratic binomial ideal associated to the geometry of polyominoes (see [12,14,10,11,2,15,13,3]). In the last three papers, the authors compute some algebraic invariants of the polyomino ideal by studying the rook polynomial n i=1 r i t i , i.e.…”