整数复合上对称峰的高度和对称谷的深度
The heights of symmetric peaks and the depth of symmetric valleys over compositions of an integer
摘要：
正整数$n$的组合$π=π_1π_2\cdotsπ_k$是一个或多个正整数的有序集合，其和为$n$。被加数，即$k$，称为$π$的部分数。本文介绍了整数$n$的完全$k$部分组成的两个统计量：对称峰的高度和对称谷的深度在$n$的所有组成。根据对称峰（谷）的个数和峰（谷）的总高（深），导出了$n$与恰好$k$部分合成的母函数的显式表达式.
英文摘要：
A composition $π=π_1π_2\cdotsπ_k$ of a positive integer $n$ is an ordered collection of one or more positive integers whose sum is $n$ . The number of summands, namely $k$, is called the number of parts of $π$. In this paper, we introduce two statistics over compositions of an integer $n$ with exactly $k$ parts: heights of symmetric peaks and depths of symmetric valleys over all compositions of $n$. We derive an explicit formula for the generating functions of compositions of $n$ with exactly $k$ parts according to the number of symmetric peaks (valleys) and the total heights (depths) of peaks (valleys).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
