Large deviations for the longest alternating and the longest increasing subsequence in a random permutation avoiding a pattern of length three

TL;DR

This paper derives large deviation principles for the longest alternating and increasing subsequences in pattern-avoiding permutations, revealing universal rate functions linked to Bernoulli sums.

Contribution

It provides the first large deviations analysis for these subsequences in pattern-avoiding permutations, identifying universal rate functions across different pattern classes.

Findings

The same rate function applies to all six alternating subsequence cases.

For increasing subsequences, the rate function is twice that of alternating subsequences.

Explicit large deviation results are obtained for specific pattern-avoiding classes.

Abstract

We calculate the large deviations for the length of the longest alternating subsequence and for the length of the longest increasing subsequence in a uniformly random permutation that avoids a pattern of length three. We treat all six patterns in the case of alternating subsequences. In the case of increasing subsequences, we treat two of the three patterns for which a classical large deviations result is possible. The same rate function appears in all six cases for alternating subsequences. This rate function is in fact the rate function for the large deviations of the sum of IID symmetric Bernoulli random variables. The same rate function appears in the two cases we treat for increasing subsequences. This rate function is twice the rate function for alternating subsequences.