  A partition $\alpha$ is said to contain another partition (or pattern) $\mu$
if the Ferrers board for $\mu$ is attainable from $\alpha$ under removal of
rows and columns. We say $\alpha$ avoids $\mu$ if it does not contain $\mu$. In
this paper we count the number of partitions of $n$ avoiding a fixed pattern
$\mu$, in terms of generating functions and their asymptotic growth rates.
  We find that the generating function for this count is rational whenever
$\mu$ is (rook equivalent to) a partition in which any two part sizes differ by
at least two. In doing so, we find a surprising connection to metacyclic
$p$-groups. We further obtain asymptotics for the number of partitions of $n$
avoiding a pattern $\mu$. Using these asymptotics we conclude that the
generating function for $\mu$ is not algebraic whenever $\mu$ is rook
equivalent to a partition with distinct parts whose first two parts are
positive and differ by 1.
