we found that $39\%$ of $48300$ random rank 10 intervals, where $|\sigma|=5$ and $|\pi|=15$, are single. We suspect that the proportion of intervals that are single approaches zero as the rank tends to infinity, because the likelihood that there exists some $\lambda\in[\sigma,\pi]$ such that $[\lambda,\pi]$ does not have a single block increases as the rank of the interval increases.

\[prop:single\] If $[\sigma,\pi]$ is single then $\mu(\sigma,\pi)=(-1)^{|\pi|-|\sigma|}\operatorname{\text{\rm NE}}(\sigma,\pi)$.

If $[\lambda,\pi]$ has a single block then $EZ^{\lambda,\pi}$ must be empty which implies $\sum_{S\in EZ^{\lambda,\pi}}(-1)^{|S|}=~0$. Therefore, if $[\sigma,\pi]$ is single then $\sum_{S\in EZ^{\lambda,\pi}}(-1)^{|S|}=0$ for all $\lambda\in[\sigma,\pi)$, combining this with Equation  completes the proof.

Intervals that contain a disconnected subinterval of rank at least 3 are non-shellable, as shown by Björner in [@Bjo80], and thus not amenable to some of the elegant methods of topological combinatorics, see [@McSt13] for further background. In the rest of the paper we consider a particular type of interval that is known to be disconnected and show that the sum in is nonzero for these intervals. Whether there is a topological “reason" for being nonzero in these cases we don’t know.

We consider decomposable permutations and write them in the form $\pi_1\oplus\cdots\oplus\pi_n$ where each $\pi_i$, which we call a 