of normal occurrences have an extra condition for the case when $\pi$ has this property. We prove a result that indicates the second term of our formula for the Möbius function is often non-zero in this case. Exactly what the connection is between this second term and the topology of such intervals is another mystery.

Computing the Möbius function using the original recursive formula has exponential complexity, whereas our formula splits the computation into two parts. The first part, that is, computing the number of normal occurrences, can be done in polynomial time and the second part has exponential complexity in the general case, but computational evidence suggests that in a significant proportion of cases this second term vanishes. Our formula here is the first formula for arbitrary intervals of permutations that seems to have polynomial time complexity for a significant proportion of intervals.

In Section \[sec:defn\] we introduce some definitions, give a brief introduction to the topology of posets and present a poset fibration of $[\sigma,\pi]$ that we later use to compute $\mu(\sigma,\pi)$. In Section \[sec:result\] we present and prove our main result, that the Möbius function of intervals of $\mathcal{P}$ equals the number of normal occurrences plus an 