A probabilistic approach to the asymptotics of the length of the longest alternating subsequence (2010)
ABSTRACT: Let $LA_{n}(τ)$ be the length of the longest alternating subsequence of a uniform random permutation $τ\in[n]$. Classical probabilistic arguments are used to rederive the asymptotic mean, variance and limiting law of $LA_{n}(τ)$. 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.