  Inspired by the results of Baik, Deift and Johansson on the limiting
distribution of the lengths of the longest increasing subsequences in random
permutations, we find those limiting distributions for pattern-restricted
permutations in which the pattern is any one of the six patterns of length 3.
We show that the (132)-avoiding case is identical to the distribution of
heights of ordered trees, and that the (321)-avoiding case has interesting
connections with a well known theorem of Erd\H os-Szekeres.
