Longest monotone subsequences and rare regions of pattern-avoiding permutations

TL;DR

This paper investigates the distribution of longest monotone subsequences in pattern-avoiding permutations, revealing linear expected lengths and analyzing the structure of rare regions in scaled permutations.

Contribution

It provides exact results for classes avoiding two length-3 patterns and explores the geometric structure of rare regions in pattern-avoiding permutations.

Findings

Longest monotone subsequences have expected length proportional to n.

The complement of the rare region is a closed set containing the main diagonal.

The boundary of the rare region above the diagonal is a Lipschitz continuous curve.

Abstract

We consider the distributions of the lengths of the longest monotone and alternating subsequences in classes of permutations of size $n$ that avoid a specific pattern or set of patterns, with respect to the uniform distribution on each such class. We obtain exact results for any class that avoids two patterns of length 3, as well as results for some classes that avoid one pattern of length 4 or more. In our results, the longest monotone subsequences have expected length proportional to $n$ for pattern-avoiding classes, in contrast with the $\sqrt n$ behaviour that holds for unrestricted permutations. In addition, for a pattern $\tau$ of length $k$, we scale the plot of a random $\tau$-avoiding permutation down to the unit square and study the "rare region," which is the part of the square that is exponentially unlikely to contain any points. We prove that when $\tau_1>\tau_k$, the complement of the rare region is a closed set that contains the main diagonal of the unit square. For the case $\tau_1=k,$ we also show that the lower boundary of the part of the rare region above the main diagonal is a curve that is Lipschitz continuous and strictly increasing on $[0,1]$.