Cyclic sieving is a well-known phenomenon where certain interesting polynomials, especially $q$-analogues, have useful interpretations related to actions and representations of the cyclic group. We propose a definition of sieving for an arbitrary group $G$ and study it for the dihedral group $I_2(n)$ of order $2n$. This requires understanding the generators of the representation ring of the dihedral group. For $n$ odd, we exhibit several instances of dihedral sieving which involve the generalized Fibonomial coefficients, recently studied by Amdeberhan, Chen, Moll, and Sagan. We also exhibit an instance of dihedral sieving involving Garsia and Haiman's $(q,t)$-Catalan numbers.

中文翻译：

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二面角筛分现象

循环筛分是一种众所周知的现象，其中某些有趣的多项式，尤其是 $q$-类似物，具有与循环群的动作和表示相关的有用解释。我们提出了一个任意群$G$ 的筛分定义，并针对阶$2n$ 的二面体群$I_2(n)$ 研究它。这需要了解二面体群的表示环的生成器。对于 $n$ 奇数，我们展示了一些涉及广义斐波项系数的二面角筛选实例，最近由 Amdeberhan、Chen、Moll 和 Sagan 研究。我们还展示了一个涉及 Garsia 和 Haiman 的 $(q,t)$-Catalan 数的二面角筛分实例。