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

2024/08/17 by Walaa Asakly Noor Kezil, Kezil, Walaa Asakly Noor
Mathematics · #Advanced Differential Equations and Dynamical Systems #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2408.09166

openalex publication_date 2024/08/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A composition π=π₁π₂⋯π_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).

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