  We calculate the large deviations for the length of the longest alternating
subsequence and for the length of the longest increasing subsequence in a
uniformly random permutation that avoids a pattern of length three. We treat
all six patterns in the case of alternating subsequences. In the case of
increasing subsequences, we treat two of the three patterns for which a
classical large deviations result is possible. The same rate function appears
in all six cases for alternating subsequences. This rate function is in fact
the rate function for the large deviations of the sum of IID symmetric
Bernoulli random variables. The same rate function appears in the two cases we
treat for increasing subsequences. This rate function is twice the rate
function for alternating subsequences.
