  In this paper we will give a M\"obius number of $NC^{k}(W) \setminus
\bf{mins} \cup \{\hat{0} \}$ for a Coxeter group $W$ which contains an
affirmative answer for the conjecture 3.7.9 in Armstrong's paper [ Generalized
noncrossing partitions and combinatorics of Coxeter groups.
arXiv:math/0611106].
