to a sign, equal to the number of normal occurrences of $\sigma$ in $\pi$ plus an extra term that seems to vanish for a significant proportion of intervals. For example, we know this extra term vanishes if $\sigma$ and $\pi$ have the same number of descents, which is a consequence of the result in [@Smith14]. Using *interval blocks*, which appear in [@SteTen10], we prove that if for all permutations $\lambda\in[\sigma,\pi)$ there is a singleton interval block, that is, a letter of $\pi$ which belongs to no occurrence of $\lambda$, the second term of the formula vanishes. The above mentioned cases are of zero proportion when the length of $\pi$ goes to infinity, but computer tests indicate that for a substantial proportion of intervals the second term of our formula vanishes. Why that is the case is still a mystery, but this suggests that many more families of intervals than are now known may turn out to have a tractable Möbius function.

It is shown in [@McSt13] that if $\pi$ is decomposable and has equal consecutive components then for any subpermutation $\sigma$ obtained by removing $k>1$ of the equal components, the interval $[\sigma,\pi]$ contains a disconnected subinterval. Many of the definitions 