  We construct an injection from the set of permutations of length $n$ that
contain exactly one copy of the decreasing pattern of length $k$ to the set of
permutations of length $n+2$ that avoid that pattern. We then prove that the
generating function counting the former is not rational, and in the case when
$k$ is even and $k\geq 4$, it is not even algebraic. We extend our injection
and our nonrationality result to a larger class of patterns.
