  We consider the problem of counting the occurrences of patterns of the form
$xy-z$ within flattened permutations of a given length. Using symmetric
functions, we find recurrence relations satisfied by the distributions on
$\mathcal{S}_n$ for the patterns 12-3, 21-3, 23-1 and 32-1, and develop a
unified approach to obtain explicit formulas. By these recurrences, we are able
to determine simple closed form expressions for the number of permutations
that, when flattened, avoid one of these patterns as well as expressions for
the average number of occurrences. In particular, we find that the average
number of 23-1 patterns and the average number of 32-1 patterns in
$\text{Flatten}(\pi)$, taken over all permutations $\pi$ of the same length,
are equal, as are the number of permutations avoiding either of these patterns.
We also find that the average number of 21-3 patterns in $\text{Flatten}(\pi)$
over all $\pi$ is the same as it is for 31-2 patterns.
