Discrete Mathematics & Theoretical Computer Science 
Let $\textbf{as}_n$ denote the length of a longest alternating subsequence in a uniformly random permutation of order $n$. Stanley studied the distribution of $\textbf{as}_n$ using algebraic methods, and showed in particular that $\mathbb{E}(\textbf{as}_n) = (4n+1)/6$ and $\textrm{Var}(\textbf{as}_n) = (32n13)/180$. From Stanley's result it can be shown that after rescaling, $\textbf{as}_n$ converges in the limit to the Gaussian distribution. In this extended abstract we present a new approach to the study of $\textbf{as}_n$ by relating it to the sequence of local extrema of a random permutation, which is shown to form a "canonical'' longest alternating subsequence. Using this connection we reprove the abovementioned results in a more probabilistic and transparent way. We also study the distribution of the values of the local minima and maxima, and prove that in the limit the joint distribution of successive minimummaximum pairs converges to the twodimensional distribution whose density function is given by $f(s,t) = 3(1s)t e^{ts}$.
Source : ScholeXplorer
IsRelatedTo ARXIV 1303.2387 Source : ScholeXplorer IsRelatedTo DOI 10.3906/mat161059 Source : ScholeXplorer IsRelatedTo DOI 10.48550/arxiv.1303.2387
