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 contains two intervals of length 2, where one interval is an ascent and the other a descent, then the Möbius function [1,] of the interval [1,] is zero. As a consequence, we prove that the proportion of permutations of length with principal Möbius function equal to zero is asymptotically bounded below by (1- 0.3995. This is the first result determining the value of [1,] for an asymptotically positive proportion of permutations . We further establish other general conditions on a permutation that ensure [1,]=0, including the occurrence in of any interval of the form 1.

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