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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.

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DOI (Digital Object Identifier) Link: https://doi.org/10.1112/S0025579319000251
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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.

Item Type: Journal Item
Copyright Holders: 2019 University College London
ISSN: 0025-5793
Academic Unit/School: Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics
Faculty of Science, Technology, Engineering and Mathematics (STEM)
Item ID: 66369
Depositing User: ORO Import
Date Deposited: 03 Sep 2019 08:20
Last Modified: 05 Oct 2019 02:44
URI: http://oro.open.ac.uk/id/eprint/66369
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