Zeros of the Möbius function of permutations

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 $\mu [\pi ]$ of the interval $[1,\pi ]$ 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 $\mu [1,\pi ]$ for an asymptotically positive proportion of permutations $\pi$. We also show that if a permutation $\varphi$ can be expressed as a direct sum of the form $\alpha \oplus 1\oplus \beta$, then any permutation $\pi$ containing an interval order-isomorphic to $\varphi$ has $\mu [1,\pi ]=0$; we deduce this from a more general result showing that $\mu [\sigma ,\pi ]=0$ whenever $\pi$ contains an interval of a certain form. Finally, we show that if a permutation $\pi$ contains intervals isomorphic to certain pairs of permutations, or to certain permutations of length six, then $\mu [1,\pi ]=0$.

21 pages, 7 figures, 1 tables. This is an expanded version of the preprint "Intervals of permutations and the principal Möbius function", available at arXiv:1806.10362, with two additional authors