Zeros of the Möbius function of permutations
We show that if a permutation $π$ contains two intervals of length 2, where one interval is an ascent and the other a descent, then the Möbius function $μ[π]$ of the interval $[1,π]$ is zero. As a consequence, we show that the proportion of permutations of length $n$ with principal Möbius function equal to zero is asymptotically bounded below by $(1-1/e)^2\ge 0.3995$. This is the first result determining the value of $μ[1,π]$ for an asymptotically positive proportion of permutations $π$. We also show that if a permutation $ϕ$ can be expressed as a direct sum of the form $α\oplus 1 \oplus β$, then any permutation $π$ containing an interval order-isomorphic to $ϕ$ has $μ[1, π]=0$; we deduce this from a more general result showing that $μ[σ, π]=0$ whenever $π$ contains an interval of a certain form. Finally, we show that if a permutation $π$ contains intervals isomorphic to certain pairs of permutations, or to certain permutations of length six, then $μ[1, π] = 0$.