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, π, in a random permutation of length n. The well‐studied case of a longest increasing subsequence corresponds to π = 21. We show that there is some constant c_π such that as n →∞ the mean value of this length is asymptotic to and that the distribution of the length is tightly concentrated around its mean. We observe some apparent connections between c_π and the Stanley–Wilf limit of the class of permutations avoiding the pattern π. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007