The Length of the Longest Common Subsequence of Two Independent Mallows Permutations

TL;DR

This paper establishes a weak law of large numbers for the length of the longest common subsequence of two independent Mallows permutations, revealing asymptotic behavior depending on the parameter q as the permutation size grows.

Contribution

It provides the first asymptotic analysis of the longest common subsequence length for Mallows permutations under a varying parameter q.

Findings

Weak law of large numbers for LCS length

Asymptotic behavior depends on limit of n(1-q)

Results extend understanding of permutation similarity measures

Abstract

The Mallows measure is a probability measure on $S_n$ where the probability of a permutation $\pi$ is proportional to $q^{l(\pi)}$ with $q > 0$ being a parameter and $l(\pi)$ the number of inversions in $\pi$. We prove a weak law of large numbers for the length of the longest common subsequences of two independent permutations drawn from the Mallows measure, when $q$ is a function of $n$ and $n(1-q)$ has limit in $\mathbb{R}$ as $n \to \infty$.