  We study several aspects of the M\"{o}bius function, $\mu[\sigma,\pi]$, on
the poset of permutations under the pattern containment order.
  First, we consider cases where the lower bound of the poset is
indecomposable. We show that $\mu[\sigma,\pi]$ can be computed by considering
just the indecomposable permutations contained in the upper bound. We apply
this to the case where the upper bound is an increasing oscillation, and give a
method for computing the value of the M\"{o}bius function that only involves
evaluating simple inequalities.
  We then consider conditions on an interval which guarantee that the value of
the M\"{o}bius function is zero. In particular, we show that if a permutation
$\pi$ contains two intervals of length 2, which are not order-isomorphic to one
another, then $\mu[1,\pi] = 0$. This allows us to prove 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 $\pi$.
  Following this, we use ''2413-balloon'' permutations to show that the growth
of the principal M\"{o}bius function on the permutation poset is exponential.
This improves on previous work, which has shown that the growth is at least
polynomial.
  We then generalise 2413-balloon permutations, and find a recursion for the
value of the principal M\"{o}bius function of these generalisations.
