When the Law of Large Numbers Fails 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}. In a recent paper [4] we showed that 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. The method of proof employed there used the second moment method and demonstrated that this method cannot work if the condition kn = o(n 2 5 ) does not hold. It follows from results concerning the longest increasing subsequence of a random permutation that the law of large numbers cannot hold for Zn,kn if kn ≥ cn 1 2 , with c > 2. Presumably there is a critical exponent l0 such that the law of large numbers holds if kn = O(n ), with l < l0, and does not hold if lim supn→∞ kn nl > 0, for some l > l0. Several phase transitions concerning increasing subsequences occur at l = 1 2 , and these would suggest that l0 = 1 2 . However, in this paper, we show that the law of large numbers fails for Zn,kn if lim supn→∞ kn n 4 9 = ∞. Thus the critical exponent, if it exists, must satisfy l0 ∈ [ 2 5 , 4 9 ].