Hilbert–Poincaré Series and Gorenstein Property for Some Non-simple Polyominoes

Let P be a closed path having no zig-zag walks, a kind of non-simple thin polyomino. In this paper, we give a combinatorial interpretation of the h-polynomial of K[P] , showing that it is the rook polynomial of P. It is known by Rinaldo and Romeo (J Algebr Comb 54:607–624, 2021), that if P is a simple thin polyomino, then the h-polynomial is equal to the rook polynomial of P and it is conjectured that this property characterizes all thin polyominoes. Our main demonstrative strategy is to compute the reduced Hilbert–Poincaré series of the coordinate ring attached to a closed path P having no zig-zag walks, as a combination of the Hilbert–Poincaré series of convenient simple thin polyominoes. As a consequence, we prove that the Krull dimension is equal to |V(P)|-rankP and the regularity of K[P] is the rook number of P. Finally, we characterize the Gorenstein prime closed paths, proving that K[P] is Gorenstein if and only if P consists of maximal blocks of length three.