Law of large numbers for increasing subsequences of random permutations

Abstract

AbstractLet the random variable Zn,k denote the number of increasing subsequences of length k in a random permutation from Sn, the symmetric group of permutations of {1,…,n}. We show that Var(Z) = o((EZ)2) as n → ∞ if and only if $k_n=o(n^{{2\over 5}})$. In particular then, the weak law of large numbers holds for Z if $k_n=o(n^{{2\over 5}})$; that is, We also show the following approximation result for the uniform measure Un on Sn. Define the probability measure μ on Sn by where U denotes the uniform measure on the subset of permutations that contain the increasing subsequence {x1,x2,…,x}. Then the weak law of large numbers holds for Z if and only if where ∣∣˙∣∣ denotes the total variation norm. In particular then, (*) holds if $k_n=o(n^{{2\over 5}})$.In order to evaluate the asymptotic behavior of the second moment, we need to analyze occupation times of certain conditioned two‐dimensional random walks. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006