Intervals of permutations and the principal M\"{o}bius function

TL;DR

This paper investigates the distribution of the principal Möbius function in permutations, establishing a lower bound for permutations with a zero value and identifying structural conditions leading to this outcome.

Contribution

It provides new bounds on the proportion of permutations with zero principal Möbius function and characterizes structural permutation patterns that guarantee this property.

Findings

At least 39.95% of permutations have a principal Möbius function of zero.

Permutations containing specific interval patterns have a zero Möbius function.

Structural properties of permutations determine the value of the principal Möbius function.

Abstract

We show that the proportion of permutations of length $n$ with principal M\"{o}bius function equal to zero, $Z(n)$, is asymptotically bounded below by 0.3995. If a permutation $\pi$ contains two intervals of length 2, where one interval is an ascent and the other a descent, then we show that the value of the principal M\"{o}bius function $\mu [1, \pi]$ is zero, and we use this result to find the lower bound for $Z(n)$. We also show that if a permutation $\phi$ has certain properties, then any permutation $\pi$ which contains an interval order-isomorphic to $\phi$ has $\mu[1, \pi] = 0$.