Zeros of the Möbius function of permutations

Brignall, Robert; Jelínek, Vít; Kynčl, Jan and Marchant, David (2019). Zeros of the Möbius function of permutations. Mathematika, 65(4) pp. 1074–1092.

DOI: https://doi.org/10.1112/S0025579319000251

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 Möbius function $\mu$ [1, $\pi$ ] of the interval [1, $\pi$ ] is zero. As a consequence, we prove that the proportion of permutations of length $\textit{n}$ with principal Möbius function equal to zero is asymptotically bounded below by (1 $\$ - $\ \sfrac{1}{e)^2}$ $\geq$ 0.3995. This is the first result determining the value of $\mu\left$ [1, $\pi\right$ ] for an asymptotically positive proportion of permutations $\pi$ . We further establish other general conditions on a permutation $\pi$ that ensure $\mu\left$ [1, $\pi\right$ ] $\$ = $\$ 0, including the occurrence in $\pi$ of any interval of the form $\alpha\oplus\$ 1 $\ \oplus\ \beta$ .