because our induction assumption is that $P_{<p}$ has been retracted and our condition of the proposition is $f^*(P_{<p})=Q_{<q}$. Therefore, $-\mu(W(q))=\mu(Q_{<q})\mu(f^{-1}(Q_{\ge q}))$ and summing over all $q\in Q$ completes the proof.

An interesting question is whether Proposition \[prop:mob2.5\] can be generalised to show homotopy equivalence. Also, is there a more general condition that encompasses the conditions of both Proposition \[prop:mob2.5\] and Theorem 2.5 of [@Bjo05]?

Applications {#sec:app}
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By Theorem \[thm:main\] we know the Möbius function is linked to the number of normal embeddings, which depend on the adjacencies.

\[lem:adj\] The average total number of letters in the tails of adjacencies in a permutation of length $n$ is $2(\frac{n-1}{n})$. In particular, when $n$ tends to infinity the average number of letters in the tails of adjacencies tends to 2.

Note first that $k=1$ cannot be in the tail of an increasing adjacency and $n$ cannot be in the tail of a decreasing adjacency. For $k>1$ the number of permutations of length $n$ in which $k$ is in the tail of an increasing adjacency is $(n-1)!$, because these are precisely all permutations of the letters $1,2,\ldots,n$ where $(k-1)k$ is regarded as a single letter. So the probability that a letter $k>1$ is 