On the length of the longest subsequence avoiding an arbitrary pattern in a random permutation

We consider the distribution of the length of the longest subsequence avoiding an arbitrary pattern, pi, in a random permutation of length n. The well-studied case of a longest increasing subsequence corresponds to pi = 21. We show that there is some constant c, such that as n -> infinity the mean value of this length is asymptotic to 2/c(pi)n and that the distribution of the length is tightly concentrated around its mean. We observe some apparent connections between c, and the Stanley-Wilf limit of the class of permutations avoiding the pattern pi.