Enumerating symmetric and non-symmetric peaks in words

Let \([k] = \{1, 2, \ldots, k\}\) be an alphabet over \(k\) letters. A word \(\omega\) of length \(n\) over alphabet \([k]\) is an element of \([k]^n\) and is also called \(k\)-ary word of length \(n\). We say that \(\omega\) contains a peak, if exists \(2 \leq i \leq n-1\) such that \(\omega_{i-1} \omega_{i+1}\). We say that \(\omega\) contains a symmetric peak, if exists \(2 \leq i \leq n-1\) such that \(\omega_{i-1} = \omega_{i+1} < \omega_i\), and contains a non-symmetric peak, otherwise. In this paper, we find an explicit formula for the generating functions for the number of \(k\)-ary words of length \(n\) according to the number of symmetric peaks and non-symmetric peaks in terms of Chebyshev polynomials of the second kind. Moreover, we find the number of symmetric and non-symmetric peaks in \(k\)-ary word of length \(n\) in two ways by using generating functions techniques, and by applying probabilistic methods.