On Increasing Subsequences of Random Permutations

Let Ln be the length of a longest increasing subsequence in a random permutation of {1, ..., n}. It is known that the expected value of Ln is asymptotically equal to 2 √ n as n gets large. This note derives upper bound on the probability that Ln − 2 √ n exceeds certain quantities. In particular, we prove that Ln − 2 √ n has order at most n1/6 with high probability. Our main result is an isoperimetric upper bound of the probability that Ln − 2 √ n exceed θn1/6, which suggests that the variance V [Ln] might be n 1/3. We also find an explicit lower bound of the function β(c) := − lim n→∞ 1 √ n log Pr (Ln − 2 √ n > c √ n ) , c > 0 defined by Aldous and Diaconis [1].