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

TL;DR

This paper introduces and analyzes new statistics called symmetric peaks and valleys in compositions of integers, providing explicit generating functions that count compositions based on these features.

Contribution

It develops explicit formulas for generating functions that count compositions with symmetric peaks and valleys according to their heights and depths.

Findings

Derived explicit formulas for generating functions

Counted compositions with symmetric peaks and valleys

Analyzed the distribution of heights and depths

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).