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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.

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