extra term that we define. In Section \[sec:fibration\] we present a result that links the Möbius function of two posets connected by a poset fibration satisfying a certain condition. This indicates there is possibly a more general condition for the main result of Björner, Wachs and Welker in [@Bjo05]. In Section \[sec:app\] we apply our formula to show that the Möbius function of $[\sigma,\pi]$ equals the number of normal occurrences of $\sigma$ in $\pi$ if for each $\lambda\in[\sigma,\pi)$ there is at least one letter of $\pi$ which is not in any occurrence of $\lambda$. We also show that the value of the second term of our formula for the Möbius function of $[\sigma,\pi]$ is often nonzero when $\pi$ has a decomposition into a direct sum with consecutive equal components. Furthermore, we consider for which permutations all occurrences are normal.

Definitions and Preliminaries {#sec:defn}
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In this section we introduce some definitions required to present our main result. We begin with an important property of permutations that is fundamental to our results:

\[defn:emb\] An *adjacency* in a permutation is a maximal sequence, of length $\ell\ge1$, of increasing or decreasing consecutively valued letters in consecutive order. The *tail* of an adjacency of 