Longest Increasing Subsequences and Log Concavity

摘要

Let be a permutation of [n] = {1, . . . , n} and denote by the length of a longest increasing subsequence of . Let be the number of permutations of [n] with . Chen conjectured that the sequence is log concave for every fixed positive integer n. We conjecture that the same is true if one is restricted to considering involutions and we show that these two conjectures are closely related. We also prove various analogues of these conjectures concerning permutations whose output tableaux under the Robinson-Schensted algorithm have certain shapes. In addition, we present a proof of Deift that part of the limiting distribution is log concave. Various other conjectures are discussed.