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

TL;DR

This paper derives formulas for the variance and distribution of the length of the longest $k$-alternating subsequence in random permutations, establishing a central limit theorem.

Contribution

It provides an explicit variance formula for $k$-peaks and introduces an asymptotic distribution result for the longest $k$-alternating subsequence.

Findings

Explicit variance formula for the number of $k$-peaks

Asymptotic formula for the variance of the longest $k$-alternating subsequence

Central limit theorem for the length of the longest $k$-alternating subsequence

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.