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ZEROS OF THE MÖBIUS FUNCTION OF PERMUTATIONS

  • Robert Brignall (a1), Vít Jelínek (a2), Jan Kynčl (a3) and David Marchant (a4)

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 $\unicode[STIX]{x1D6FC}\oplus 1\oplus \unicode[STIX]{x1D6FD}$ .

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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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References

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ZEROS OF THE MÖBIUS FUNCTION OF PERMUTATIONS

  • Robert Brignall (a1), Vít Jelínek (a2), Jan Kynčl (a3) and David Marchant (a4)

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