The Moebius function of separable and decomposable permutations

Abstract

We give a recursive formula for the Moebius function of an interval $[\sigma ,\pi ]$ in the poset of permutations ordered by pattern containment in the case where $\pi$ 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 $\sigma$ and $\pi$ 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 $[\sigma ,\pi ]$ is bounded by the number of occurrences of $\sigma$ as a pattern in $\pi$. We also show that for any separable permutation $\pi$ the Moebius function of $(1,\pi )$ is either 0, 1 or -1.

20 pages, 2 figures

References in corpus (1)

• The enumeration of simple permutations

Cited by in corpus (1)

• The Möbius function of the consecutive pattern poset