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We show that the proportion of permutations of length $n$ with principalM\"{o}bius function equal to zero, $Z(n)$, is asymptotically bounded below by0.3995. If a permutation $\pi$ contains two intervals of length 2, where oneinterval is an ascent and the other a descent, then we show that the value ofthe principal M\"{o}bius function $\mu [1, \pi]$ is zero, and we use thisresult to find the lower bound for $Z(n)$. We also show that if a permutation$\phi$ has certain properties, then any permutation $\pi$ which contains aninterval order-isomorphic to $\phi$ has $\mu[1, \pi] = 0$.

Intervals of permutations and the principal M\"{o}bius function