A Probabilistic Approach To The Asymptotics Of The Length Of The Longest Alternating Subsequence

Let LA(n)(tau) be the length of the longest alternating subsequence of a uniform random permutation tau is an element of [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.