  We show that the growth of the principal M\"obius function on the permutation
poset is exponential. This improves on previous work, which has shown that the
growth is at least polynomial. We define a method of constructing a permutation
from a smaller permutation which we call "ballooning". We show that if $\beta$
is a 2413-balloon, and $\pi$ is the 2413-balloon of $\beta$, then $\mu[1, \pi]
= 2 \mu[1, \beta]$. This allows us to construct a sequence of permutations
$\pi_1, \pi_2, \pi_3\ldots$ with lengths $n, n+4, n+8, \ldots$ such that $\mu
[1, \pi_{i+1}] = 2 \mu [1, \pi_{i}]$, and this gives us exponential growth.
Further, our construction method gives permutations that lie within a
hereditary class with finitely many simple permutations. We also find an
expression for the value of $\mu[1, \pi]$, where $\pi$ is a 2413-balloon, with
no restriction on the permutation being ballooned.
