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ZEROS OF THE MÖBIUS FUNCTION OF PERMUTATIONS

Part of: Combinatorics

Published online by Cambridge University Press:  14 August 2019

Robert Brignall ,
Vít Jelínek ,
Jan Kynčl  and
David Marchant
Robert Brignall
Affiliation:
School of Mathematics and Statistics, The Open University, Milton Keynes, MK7 6AA, U.K. email robert.brignall@open.ac.uk
Vít Jelínek
Affiliation:
Computer Science Institute, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic email jelinek@iuuk.mff.cuni.cz
Jan Kynčl
Affiliation:
Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic email kyncl@kam.mff.cuni.cz
David Marchant
Affiliation:
School of Mathematics and Statistics, The Open University, Milton Keynes, MK7 6AA, U.K. email david.marchant@open.ac.uk
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Abstract

We show that if a permutation $\unicode[STIX]{x1D70B}$ contains two intervals of length 2, where one interval is an ascent and the other a descent, then the Möbius function $\unicode[STIX]{x1D707}[1,\unicode[STIX]{x1D70B}]$ of the interval $[1,\unicode[STIX]{x1D70B}]$ is zero. As a consequence, we prove that the proportion of permutations of length $n$ with principal Möbius function equal to zero is asymptotically bounded below by $(1-1/e)^{2}\geqslant 0.3995$. This is the first result determining the value of $\unicode[STIX]{x1D707}[1,\unicode[STIX]{x1D70B}]$ for an asymptotically positive proportion of permutations $\unicode[STIX]{x1D70B}$. We further establish other general conditions on a permutation $\unicode[STIX]{x1D70B}$ that ensure $\unicode[STIX]{x1D707}[1,\unicode[STIX]{x1D70B}]=0$, including the occurrence in $\unicode[STIX]{x1D70B}$ of any interval of the form $\unicode[STIX]{x1D6FC}\oplus 1\oplus \unicode[STIX]{x1D6FD}$.

MSC classification

Primary: 05A05: Permutations, words, matrices
Type
Research Article
Information
Mathematika , Volume 65 , Issue 4 , 2019 , pp. 1074 - 1092
Copyright
Copyright © University College London 2019 

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Footnotes

V. Jelínek and J. Kynčl were supported by project 16-01602Y of the Czech Science Foundation (GAČR). J. Kynčl was also supported by Charles University project UNCE/SCI/004.

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