On Increasing Subsequences of Random Permutations

LetLnbe the length of a longest increasing subsequence in a random permutation of {1, …, n}. It is known that the expected value ofLnis asymptotically equal to[formula]asngets large. This note derives upper bound on the probability that[formula]exceeds certain quantities. In particular, we prove that[formula]is at most ordern1/6with high probability. Our main result is an isoperimetric upper bound of the probability that[formula]exceedθn1/6, which suggests that the varianceV[Ln] might ben1/3. We also find an explicit lower bound of the function[formula], defined by D. Aldous and P. Diaconis.