Zeros of the M\"{o}bius function of permutations

TL;DR

This paper investigates conditions under which the M"obius function of permutation intervals is zero, providing asymptotic bounds and identifying structural patterns that lead to a zero value.

Contribution

It introduces new structural criteria for permutations that guarantee a zero M"obius function, including asymptotic bounds and specific interval configurations.

Findings

Proves permutations with certain interval patterns have zero M"obius function.

Establishes a lower bound of (1-1/e)^2 for permutations with zero M"obius function.

Identifies structural conditions involving direct sums and specific interval types leading to zero M"obius function.

Abstract

We show that if a permutation $\pi$ contains two intervals of length 2, where one interval is an ascent and the other a descent, then the M\"{o}bius function $\mu[\pi]$ of the interval $[1,\pi]$ is zero. As a consequence, we show that the proportion of permutations of length $n$ with principal M\"{o}bius function equal to zero is asymptotically bounded below by $(1-1/e)^2\ge 0.3995$. This is the first result determining the value of $\mu[1,\pi]$ for an asymptotically positive proportion of permutations $\pi$. We also show that if a permutation $\phi$ can be expressed as a direct sum of the form $\alpha \oplus 1 \oplus \beta$, then any permutation $\pi$ containing an interval order-isomorphic to $\phi$ has $\mu[1, \pi]=0$; we deduce this from a more general result showing that $\mu [\sigma, \pi]=0$ whenever $\pi$ contains an interval of a certain form. Finally, we show that if a permutation $\pi$ contains intervals isomorphic to certain pairs of permutations, or to certain permutations of length six, then $\mu[1, \pi] = 0$.