  Using techniques from Poisson approximation, we prove explicit error bounds
on the number of permutations that avoid any pattern. Most generally, we bound
the total variation distance between the joint distribution of pattern
occurrences and a corresponding joint distribution of independent Bernoulli
random variables, which as a corollary yields a Poisson approximation for the
distribution of the number of occurrences of any pattern. We also investigate
occurrences of consecutive patterns in random Mallows permutations, of which
uniform random permutations are a special case. These bounds allow us to
estimate the probability that a pattern occurs any number of times and, in
particular, the probability that a random permutation avoids a given pattern.
