  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 M\"{o}bius 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 M\"{o}bius 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 $\phi$ 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 $\phi$ 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 $\mu[1, \pi] = 0$.
