The Möbius–Chebyshev conjecture for subwords over the poset Λ

Let $\Lambda$ be the poset with elements $a,b,c$ and relations $a<c$ and $b<c$, and let $\Lambda^*$ carry subword order. For words $u\leq w$ in this poset, write $\mu(u,w)$ for the Möbius function of the resulting poset. Let $T_n(x)$ be the unique polynomial satisfying

$$T_n(\cos\theta)=\cos(n\theta).$$

Möbius–Chebyshev conjecture. For all integers $i\leq j$, $\mu(a^i,c^j)$ is the coefficient of $x^{j-i}$ in $T_{i+j}(x)$.

Numerical evidence suggests a connection between the Möbius function of $\Lambda^*$ and the Chebyshev polynomials of the first kind; the general Möbius function for subwords over $\Lambda$ remains to be determined.