Hilbert-Poincaré series and Gorenstein property for some non-simple polyominoes

Abstract

In this paper we compute the reduced Poincaré-Hilbert series of the coordinate ring attached to a closed path P having no zig-zag walks, as a combination of the Poincaré-Hilbert series of convenient simple thin polyominoes. As a consequence we find the Krull dimension and the regularity of K[P] and we prove that the h-polynomial is exactly the rook polynomial of P. Finally we characterize the Gorenstein prime closed paths using the S-property.

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.…”