The Length of the Longest Increasing Subsequence of a Random Mallows Permutation

TL;DR

This paper establishes a weak law of large numbers for the length of the longest increasing subsequence in Mallows distributed permutations as the permutation size grows and the parameter approaches 1, with a specific scaling.

Contribution

It provides the first asymptotic result for the LIS length of Mallows permutations under a particular scaling regime.

Findings

Weak law of large numbers for LIS length

Asymptotic behavior as n→∞ and q→1

Limit depends on the limit of n(1-q)

Abstract

The Mallows measure on the symmetric group $S_n$ is the probability measure such that each permutation has probability proportional to $q$ raised to the power of the number of inversions, where $q$ is a positive parameter and the number of inversions of $\pi$ is equal to the number of pairs $i<j$ such that $\pi_i > \pi_j$. We prove a weak law of large numbers for the length of the longest increasing subsequence for Mallows distributed random permutations, in the limit that $n$ tends to infinity and $q$ tends to 1 in such a way that $n(1-q)$ has a limit in $\R$.