Lengths of Monotone Subsequences in a Mallows Permutation

TL;DR

This paper investigates the typical lengths and deviations of the longest increasing and decreasing subsequences in Mallows permutations, revealing their order of magnitude and geometric structure, with implications for related probabilistic models.

Contribution

It provides the first detailed analysis of subsequence lengths in Mallows permutations for a broad parameter range, including large deviation bounds and a law of large numbers.

Findings

Longest increasing subsequence length is of order n(1-q)^(1/2).

Most points in the permutation are in a symmetric strip of width 1/(1-q).

Results connect the model to last passage percolation in a strip.

Abstract

We study the length of the longest increasing and longest decreasing subsequences of random permutations drawn from the Mallows measure. Under this measure, the probability of a permutation pi in S_n is proportional to q^{inv(pi)} where q is a real parameter and inv(pi) is the number of inversions in pi. The case q=1 corresponds to uniformly random permutations. The Mallows measure was introduced by Mallows in connection with ranking problems in statistics. We determine the typical order of magnitude of the lengths of the longest increasing and decreasing subsequences, as well as large deviation bounds for them. We also provide a simple bound on the variance of these lengths, and prove a law of large numbers for the length of the longest increasing subsequence. Assuming without loss of generality that q<1, our results apply when q is a function of n satisfying n(1-q) -> infty. The case that n(1-q)=O(1) was considered previously by Mueller and Starr. In our parameter range, the typical length of the longest increasing subsequence is of order n(1-q)^(1/2), whereas the typical length of the longest decreasing subsequence has four possible behaviors according to the precise dependence of n and q. We show also that in the graphical representation of a Mallows-distributed permutation, most points are found in a symmetric strip around the diagonal whose width is of order 1/(1-q). This suggests a connection between the longest increasing subsequence in the Mallows model and the model of last passage percolation in a strip.