Average length of the longest $k$-alternating subsequence

TL;DR

This paper proves a conjecture regarding the average maximum length of $k$-alternating subsequences in permutations, extending known results for the case $k=1$ to general $k$.

Contribution

It establishes a proof for Armstrong's conjecture on the average length of $k$-alternating subsequences, generalizing Stanley's classic result.

Findings

Proves Armstrong's conjecture on average $k$-alternating subsequence length.

Extends known results from $k=1$ to arbitrary $k$.

Provides a mathematical foundation for understanding permutation subsequences.

Abstract

We prove a conjecture of Drew Armstrong on the average maximal length of $k$-alternating subsequence of permutations. The $k=1$ case is a well-known result of Richard Stanley.