Let $\pi \in NC(n)$ have blocks $V_1,\dots,V_k$. Then

\begin{align*} \mu_{NC(n)}(0_n,\pi)=\prod_{j=1}^{k}(-1)^{|V_j|-1}C_{|V_j|-1}. \end{align*}

More generally, let $\pi,\sigma\in NC(n)$ satisfy $\pi\le\sigma$. Form the quotient partition of the blocks of $\pi$ induced by $\sigma$: two blocks of $\pi$ lie in the same quotient block exactly when they are contained in the same block of $\sigma$. Each quotient block determines one planar region in the collapsed diagram. Let $\ell$ be the number of such regions, and let $m_r$ be the number of collapsed $\pi$-blocks lying in the $r$th region for $1\le r\le\ell$. With these integers, every interval $[\pi,\sigma] \subset NC(n)$ decomposes as a finite product of full noncrossing partition lattices $NC(m_1)\times\cdots\times NC(m_\ell)$. Under this standard interval decomposition,

\begin{align*} \mu_{NC(n)}(\pi,\sigma)=\prod_{r=1}^{\ell}\mu_{NC(m_r)}(0_{m_r},1_{m_r}). \end{align*}