Title: Genus Permutations and Genus Partitions Show Chinese title

Title: 基因排列与基因分割

Abstract: For a given permutation or set partition there is a natural way to assign a genus. Counting all permutations or partitions of a fixed genus according to cycle lengths or block sizes, respectively, is the main content of this article. After a variable transformation, the generating series are rational functions with poles located at the ramification points in the new variable. The generating series for any genus is given explicitly for permutations and up to genus 2 for set partitions. Extending the topological structure not just by the genus but also by adding more boundaries, we derive the generating series of non-crossing partitions on the cylinder from known results of non-crossing permutations on the cylinder. Most, but not all, outcomes of this article are special cases of already known results, however they are not represented in this way in the literature, which however seems to be the canonical way. To make the article as accessible as possible, we avoid going into details into the explicit connections to Topological Recursion and Free Probability Theory, where the original motivation came from. Show Chinese abstract

Abstract: 对于给定的排列或集合划分，有一种自然的方式来分配一个亏格。 根据循环长度或块大小，分别统计固定亏格的排列或划分的数量是本文的主要内容。 经过变量变换后，生成级数是具有极点位于新变量中的分支点的有理函数。 对于排列，任何亏格的生成级数都明确给出，而对于集合划分，仅给出到亏格2的生成级数。 通过不仅扩展亏格的拓扑结构，还通过添加更多的边界，我们从圆柱面上已知的非交叉排列的结果中推导出圆柱面上非交叉划分的生成级数。 本文的大多数但不是全部结果都是已知结果的特例，然而这些结果在文献中并未以这种方式表示，尽管这似乎是标准的方式。 为了使文章尽可能易于理解，我们避免深入探讨与拓扑递归和自由概率论的显式联系，这些是原始动机的来源。