A probabilistic approach to the asymptotics of the length of the longest alternating subsequence

TL;DR

This paper uses probabilistic methods to analyze the asymptotic behavior of the longest alternating subsequence in random permutations, extending to Markovian sequences and pattern-restricted permutations.

Contribution

It introduces a robust probabilistic approach to derive asymptotic properties for various sequence types, including finite alphabet words and Markovian sequences, with some original results.

Findings

Asymptotic mean and variance of LA_n( au) derived

Limiting distribution of LA_n( au) established

Method applicable to Markovian sequences and pattern-restricted permutations

Abstract

Let $LA_{n}(\tau)$ be the length of the longest alternating subsequence of a uniform random permutation $\tau\in[n]$. Classical probabilistic arguments are used to rederive the asymptotic mean, variance and limiting law of $LA_{n}(\tau)$. Our methodology is robust enough to tackle similar problems for finite alphabet random words or even Markovian sequences in which case our results are mainly original. A sketch of how some cases of pattern restricted permutations can also be tackled with probabilistic methods is finally presented.