Let \(\mathcal {P}\) be a simple thin polyomino, namely a polyomino that has no holes and does not contain a square tetromino as a subpolyomino. In this paper, we determine the reduced Hilbert–Poincaré series \(h(t)/(1-t)^d\) of \(K[\mathcal {P}]\) by proving that h(t) is the rook polynomial of \(\mathcal {P}\). As an application, we characterize the Gorenstein simple thin polyominoes.

中文翻译：

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希尔伯特系列的简单薄多胺

令\（\ mathcal {P} \）是一个简单的薄型多米诺骨牌，即没有孔且不包含方形四聚体作为亚多米诺骨牌的多米诺骨牌。在本文中，我们通过证明h（t）是H（t）是H（t）的简化希尔伯特-庞加莱级数\（h（t）/（1-t）^ d \）的\（K [\ mathcal {P}] \）来确定。\（\ mathcal {P} \）的rook多项式。作为一种应用，我们表征了戈伦施泰因简单的稀聚氨基酸。