# A Note on the Expected Length of the Longest Common Subsequences of two i.i.d. Random Permutations

## TL;DR

This paper investigates the expected length of the longest common subsequences of two independent random permutations, disproving a conjecture that it is at least proportional to the square root of n, and establishing a lower bound proportional to the cube root of n.

## Contribution

It resolves a question about the minimal expectation not occurring in the uniform case and provides a new lower bound of order n^{1/3} for the expected LCS length.

## Key findings

- Minimal expectation not attained in the uniform case
- Established a lower bound of order n^{1/3} for the expected LCS length
- Disproved the conjecture that the expectation is at least proportional to √n

## Abstract

We address a question and a conjecture on the expected length of the longest common subsequences of two i.i.d.$\ $random permutations of $[n]:=\{1,2,...,n\}$. The question is resolved by showing that the minimal expectation is not attained in the uniform case. The conjecture asserts that $\sqrt{n}$ is a lower bound on this expectation, but we only obtain $\sqrt[3]{n}$ for it.