In this paper, we enumerate the pairs of permutations that are long cycles and whose product has a given cycle-type. Our main result is a simple relation concerning the desired numbers for a few related cycle-types. The relation refines a formula of the number of pairs of long cycles whose product has $k$ cycles independently obtained by Zagier and Stanley relying on group characters, and was previously obtained by Feray and Vassilieva by counting some colored permutations first and then relying on some algebraic computations in the ring of symmetric functions. Our approach here is simpler and combinatorial.

中文翻译：

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对排列乘积的 Zagier-Stanley 结果进行组合改进

在本文中，我们列举了长周期的排列对，其乘积具有给定的周期类型。我们的主要结果是关于一些相关循环类型的所需数量的简单关系。该关系式提炼了一个乘积具有 $k$ 个循环的长循环对数的公式，由 Zagier 和 Stanley 依靠群特征独立获得，之前 Feray 和 Vassilieva 通过先计算一些颜色排列，然后依靠一些得到对称函数环中的代数计算。我们这里的方法更简单和组合。