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

A composition π=π₁π₂⋯πₖ 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).