Counting peaks and valleys in a partition of a set

A partition π of the set [n] = {1,2,...,n} is a collection {B₁,B₂,...,B_k} of nonempty disjoint subsets of [n] (called blocks) whose union equals [n]. In this paper, we find an explicit formula for the generating function for the number of partitions of [n] with exactly k blocks according to the number of peaks (valleys) in terms of Chebyshev polynomials of the second kind. Furthermore, we calculate explicit formulas for the total number of peaks and valleys in all the partitions of [n] with exactly k blocks, providing both algebraic and combinatorial proofs.