The Variance and the Asymptotic Distribution of the Length of Longest $k$-alternating Subsequences

Abstract

We obtain an explicit formula for the variance of the number of $k$-peaks in a uniformly random permutation. This is then used to obtain an asymptotic formula for the variance of the length of longest $k$-alternating subsequence in random permutations. Also a central limit is proved for the latter statistic.

Typos and errors corrected