to $\sigma$ after the length of $\pi$ is added to each element of $\sigma$. In [@BJJS11] a formula for the Möbius function is presented for intervals of *decomposable* permutations, that is, permutations that can be written as the direct sum of two or more non-empty permutations. This formula, however, is recursive and bottoms out in intervals bounded by indecomposable permutations, for which there is no general formula for the Möbius function.

Furthermore, in [@BJJS11] a formula is presented for intervals of *separable* permutations, that is, permutations that avoid 2413 and 3142, or equivalently, permutations that can be written using only direct sums, skew sums and the singleton permutation 1. A formula for the Möbius function of intervals of permutations with a fixed number of descents is given in [@Smith14], where a *descent* occurs at position $i$ in a permutation $\pi=\pi_1\ldots\pi_n$ if $\pi_i>\pi_{i+1}$. Further results have been presented in [@McSt13; @Smith13; @SteTen10]. However, the proportion of intervals $[\sigma,\pi]$ which satisfy any of these properties approaches zero as the length of $\pi$ increases. There are indications that the formula we present here reduces the computation of the Möbius function to polynomial time for a significant proportion of intervals.

Many of the results 