The Moebius function of separable and decomposable permutations

TL;DR

This paper presents recursive and explicit formulas for the Moebius function in the permutation pattern poset, especially for decomposable and separable permutations, enabling efficient computation and bounding of the function.

Contribution

It introduces a recursive formula for the Moebius function in decomposable permutations and an explicit, efficient formula for separable permutations, with bounds and specific values.

Findings

Moebius function bounded by pattern occurrences in separable permutations

Explicit formulas enable efficient computation of the Moebius function

Moebius function of intervals in separable permutations is limited to 0, 1, or -1

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.