on the Möbius function of intervals of $\mathcal{P}$, and also of some posets of words, are linked to the number of what have been termed *normal occurrences*, or *normal embeddings*, in the literature, see [@Bjo90; @Bjo93; @BJJS11; @SagVat06; @Smith14]. The first appearance of normal occurrences is in Björner’s paper [@Bjo90] where a formula for the Möbius function of intervals of words with subword order is presented. The definition of a normal occurrence has varied in these papers, but all follow a similar theme.

Our definition of normal occurrences, which is simpler than most previous ones, is based upon the *adjacencies* of a permutation, where an adjacency in a permutation is a maximal sequence of increasing or decreasing consecutively valued letters in consecutive positions and the *tail* of an adjacency is all but its first letter. A *normal occurrence* of $\sigma$ in $\pi$, in our definition, is any occurrence that includes all the tails of all the adjacencies of $\pi$. This definition of normal occurrences based on adjacencies does not seem to have been considered previously, but in [@Smith14] we presented a slightly different version.

We present a formula, in Theorem \[thm:main\], that shows the Möbius function of $[\sigma,\pi]$ is, up 