Law of large numbers for increasing subsequences of random permutations

Let 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 V ar(Zn,kn ) = o((EZn,kn ) ) as n → ∞ if and only if kn = o(n 2 5 ). In particular then, the weak law of large numbers holds for Zn,kn if kn = o(n 2 5 ); that is, lim n→∞ Zn,kn EZn,kn = 1, in probability. We also show the following approximation result for the uniform measure Un on Sn. Define the probability measure μn;kn on Sn by μn;kn = 1