Intervals of permutations and the principal Möbius function

Abstract

We show that the proportion of permutations of length $n$ with principal Möbius function equal to zero, $Z(n)$, is asymptotically bounded below by 0.3995. If a permutation $\pi$ contains two intervals of length 2, where one interval is an ascent and the other a descent, then we show that the value of the principal Möbius function $\mu [1,\pi ]$ is zero, and we use this result to find the lower bound for $Z(n)$. We also show that if a permutation $\varphi$ has certain properties, then any permutation $\pi$ which contains an interval order-isomorphic to $\varphi$ has $\mu [1,\pi ]=0$.

18 pages, 5 figures, 4 tables. An expanded version of this paper, with two additional authors, is available at arXiv:1810.05449