  We consider two related problems arising from a question of R. Graham on
quasirandom phenomena in permutation patterns. A ``pattern'' in a permutation
$\sigma$ is the order type of the restriction of $\sigma : [n] \to [n]$ to a
subset $S \subset [n]$. First, is it possible for the pattern counts in a
permutation to be exactly equal to their expected values under a uniform
distribution? Attempts to address this question lead naturally to an
interesting number theoretic problem: when does $k!$ divide $\binom{n}{k}$?
Second, if the tensor product of a permutation with large random permutations
is random-like in its pattern counts, what must the pattern counts of the
original permutation be? A recursive formula is proved which uses a certain
permutation ``contraction.''
