The M\"obius function of the consecutive pattern poset

An occurrence of a consecutive permutation pattern $p$ in a permutation $\pi$ is a segment of consecutive letters of $\pi$ whose values appear in the same order of size as the letters in $p$. The set of all permutations forms a poset with respect to such pattern containment. We compute the M\"obius function of intervals in this poset, providing what may be called a complete solution to the problem. For most intervals our results give an immediate answer to the question. In the remaining cases, we give a polynomial time algorithm to compute the M\"obius function. In particular, we show that the M\"obius function only takes the values -1, 0 and 1.