Sequential selection of a monotone subsequence from a random permutation

Abstract

We find a two term asymptotic expansion for the optimal expected value of a sequentially selected monotone subsequence from a random permutation of length n n . A striking feature of this expansion is that it tells us that the expected value of optimal selection from a random permutation is quantifiably larger than optimal sequential selection from an independent sequence of uniformly distributed random variables; specifically, it is larger by at least ( 1 / 6 ) log ⁡ n + O ( 1 ) (1/6) \log n +O(1) .