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Brignall, Robert; Jelínek, Vít; Kynčl, Jan and Marchant, David
(2019).
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
.