A generalization of Chebyshev polynomials and non rooted posets

Bjorner was the first to determine the Mobius functions of factor orders and subword orders. To determine the Mobius functions, he used involutions, shellability, and generating functions. [2][3][4] Bjorner and Stanley found an interesting relation among the subword order derived from a two point set {a, b} , symmetric groups and composition orders. [6] Factor orders, subword orders, and generalized subword orders were studied in the context of Mobius functions derived from word orders. In [10] Sagan and Vatter gave a description of the Mobius function of the generalized subword order derived from positive integers in two ways, namely the sign reversing involution and the discrete Morse theory. More generally they gave a combinatorial description of the Mobius functions derived from rooted forests. And in [5][10] they gave a very interesting conjecture which connects with the relation between a non-rooted forest P2 as in Notation.1 and Chebyshev polynomials.