The M\"obius function of permutations with an indecomposable lower bound

TL;DR

This paper investigates the M"obius function in permutation posets with indecomposable bounds, providing formulas and recursions that simplify calculations based on the structure of the permutations involved.

Contribution

It introduces a method to compute the M"obius function relying solely on indecomposable permutations within the bounds, simplifying previous complex calculations.

Findings

The M"obius function depends only on indecomposable permutations in the bounds.

A recursion for increasing oscillations is established, involving simple inequalities.

The approach simplifies the calculation of M"obius sums in permutation posets.

Abstract

We show that the M\"obius function of an interval in a permutation poset where the lower bound is sum (resp. skew) indecomposable depends solely on the sum (resp. skew) indecomposable permutations contained in the upper bound, and that this can simplify the calculation of the M\"obius sum. For increasing oscillations, we give a recursion for the M\"obius sum which only involves evaluating simple inequalities.