Longest alternating subsequences in pattern-restricted permutations

Inspired by the results of Stanley and Widom concerning the limiting distribution of the lengths of longest alternating subsequences in random permutations, and results of Deutsch, Hildebrand and Wilf on the limiting distribution of the longest increasing subsequence for pattern-restricted permutations, we find the limiting distribution of the longest alternating subsequence for pattern-restricted permutations in which the pattern is any one of the six patterns of length three. Our methodology uses recurrences, generating functions, and complex analysis, and also yields more detailed information. Several ideas for future research are listed.