The heights of symmetric peaks and the depth of symmetric valleys over compositions of an integer

Abstract

A composition $\pi=\pi_1\pi_2\cdots\pi_k$ 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 $\pi$. 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).