The Möbius function of separable and decomposable permutations

We give a recursive formula for the Möbius function of an interval [σ,π] in the poset of permutations ordered by pattern containment in the case where π is a decomposable permutation, that is, consists of two blocks where the first one contains all the letters 1,2,…,k for some k. This leads to many special cases of more explicit formulas. It also gives rise to a computationally efficient formula for the Möbius function in the case where σ and π are separable permutations. A permutation is separable if it can be generated from the permutation 1 by successive sums and skew sums or, equivalently, if it avoids the patterns 2413 and 3142.We also show that the Möbius function in the poset of separable permutations admits a combinatorial interpretation in terms of normal embeddings among permutations. A consequence of this interpretation is that the Möbius function of an interval [σ,π] of separable permutations is bounded by the number of occurrences of σ as a pattern in π. Another consequence is that for any separable permutation π the Möbius function of (1,π) is either 0, 1 or −1.