Abstract A classical problem in enumerative combinatorics is to count the number of derangements, i.e., permutations with no fixed point. In this paper, we study the cyclic derangement polynomials, which count the number of derangements in the wreath product C r ≀ S n . We first establish the relation between the cyclic Eulerian polynomials and the cyclic derangement polynomials. Then we show that the coefficients of the cyclic derangement polynomials have the asymptotic normality and the spiral property. Finally, we show the continued fraction expression of the generating function of the cyclic derangement polynomials.

中文翻译：

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环积Cr≀Sn的循环乱序多项式

摘要 枚举组合学中的一个经典问题是计算乱序的数量，即没有不动点的排列。在本文中，我们研究了循环乱序多项式，它计算花环积 C r ≀ S n 中的乱序数。我们首先建立循环欧拉多项式和循环乱序多项式之间的关系。然后我们证明循环乱序多项式的系数具有渐近正态性和螺旋性质。最后，我们展示了循环乱序多项式的生成函数的连分数表达式。