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Introduction
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Let $\sigma$ and $\pi$ be permutations of positive integers. We define an *occurrence* of $\sigma$ in $\pi$ to be a subsequence of $\pi$ with the same relative order of size as the letters in $\sigma$. For example, $132$ occurs twice in $23541$, as the subsequences 254 and 354. The *permutation poset* $\mathcal{P}$ consists of all permutations with the partial order $\sigma\le\pi$ if there is an occurrence of $\sigma$ in $\pi$. An *interval* $[\sigma,\pi]$ in $\mathcal{P}$ is the subposet $\{z\in\mathcal{P}\,|\,\sigma\le z\le\pi\}$. The *Möbius function* for a poset is defined recursively as: $\mu(a,b)=0$ if $a\not\le b$, $\mu(a,a)=1$ for all $a$ and, for $a<b$: $$\mu(a,b)=-\sum_{a\le z< b}\mu(a,z).$$

The first systematic study of the Möbius function of general posets appeared in [@Rota64] and the first result pertaining to the Möbius function of intervals of $\mathcal{P}$ appeared in [@SagVat06], where a formula for intervals of *layered* permutations was presented. A layered permutation is the direct sum of decreasing permutations, where the *direct sum* $\sigma\oplus\pi$ of two permutations $\sigma$ and $\pi$ is obtained by appending $\pi$ to $\sigma$ after adding the length of $\sigma$ to each letter of $\pi$. For example, $312\oplus213=312546$. There is an analogous *skew sum* $\sigma\ominus\pi$ where $\pi$ is appended 