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The Moebius function of separable and decomposable permutations

We give a recursive formula for the Moebius 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 Moebius 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. A consequence of the formula is that the Moebius function of such an interval $[σ,π]$ is bounded by the number of occurrences of $σ$ as a pattern in $π$. We also show that for any separable permutation $π$ the Moebius function of $(1,π)$ is either 0, 1 or -1.