A Generalization of the Chebyshev Polynomials and Nonrooted Posets

In this article we give a generalization of the Chebyshev polynomials of the first kind. Then we describe a Möbius function of the generalized subword order over Ps $s\in\mathbb N$. These results give the affirmative answer for the conjecture proposed in [A. Björner and B. Sagan, “Rationality of the Möbius function of the composition poset,” Theoretical Computer Science 359, no. 1–3 (2006): 282–98.] and [B. Sagan and V. Vatter, “The Möbius function of the composition poset,” Journal of Algebraic Combinatorics 24, no. 2 (2006): 117–36].