Sequential Selection of a Monotone Subsequence from a Random Permutation

TL;DR

This paper derives an asymptotic expansion for the expected value of the optimal monotone subsequence selected sequentially from a random permutation, revealing it exceeds that from independent uniform variables by at least (1/6)log n plus lower order terms.

Contribution

It provides the first two-term asymptotic expansion for the expected value of the optimal sequential monotone subsequence from a permutation, highlighting a quantifiable difference from independent sequences.

Findings

Expected value exceeds that from independent sequences by at least (1/6)log n + O(1)

Asymptotic expansion for the optimal expected value is established

Quantifies the advantage of permutation-based selection over i.i.d. sequences

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. A striking feature of this expansion is that tells us that the expected value of optimal selection from a random permutation is quantifiably larger than optimal sequential selection from an independent sequences of uniformly distributed random variables; specifically, it is larger by at least (1/6)log n +O(1).