On the M\"obius Function of Permutations With One Descent

TL;DR

This paper derives a formula for the M"obius function of intervals in the permutation poset for permutations with at most one descent, revealing unboundedness and specific zero values based on permutation structure.

Contribution

It provides a new explicit formula for the M"obius function in this class of permutations and explores its properties and conjectures.

Findings

M"obius function is unbounded on the permutation poset.

The function is zero for permutations with certain consecutive patterns.

Conjectures on M"obius function values for other permutation intervals.

Abstract

The set of all permutations, ordered by pattern containment, is a poset. We give a formula for the M\"obius function of intervals $[1,\pi]$ in this poset, for any permutation $\pi$ with at most one descent. We compute the M\"obius function as a function of the number and positions of pairs of consecutive letters in $\pi$ that are consecutive in value. As a result of this we show that the M\"obius function is unbounded on the poset of all permutations. We show that the M\"obius function is zero on any interval $[1,\pi]$ where $\pi$ has a triple of consecutive letters whose values are consecutive and monotone. We also conjecture values of the M\"obius function on some other intervals of permutations with at most one descent.