通过Catalan对象枚举$k$ -Naples停车功能
Enumerating $k$-Naples Parking Functions Through Catalan Objects
摘要：
本文研究了停车场函数的一个推广，名为$k$ -Naples停车场函数，其中允许向后移动。向后移动的一个结果是，上升的$k$ -Naples的数量与下降的$k$ -Naples的数量不相同。本文主要研究了在升、降$k$ -Naples泊车函数的背景下，由Catalan数枚举的组合对象与升泊车函数的双射的推广.这些组合对象包括Dyck路径、二叉树、多边形三角剖分和非交叉分区。使用这些双射，我们列举了上升和下降的$k$那不勒斯停车功能。
英文摘要：
This paper studies a generalization of parking functions named $k$-Naples parking functions, where backward movement is allowed. One consequence of backward movement is that the number of ascending $k$-Naples is not the same as the number of descending $k$-Naples. This paper focuses on generalizing the bijections of ascending parking functions with combinatorial objects enumerated by the Catalan numbers in the setting of both ascending and descending $k$-Naples parking functions. These combinatorial objects include Dyck paths, binary trees, triangulations of polygons, and non-crossing partitions. Using these bijections, we enumerate both ascending and descending $k$-Naples parking functions.
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