On the Limiting Distribution for the Longest Alternating Sequence in a Random Permutation

TL;DR

This paper proves a conjecture about the limiting distribution of the longest alternating subsequence length in random permutations, showing it converges to a Gaussian distribution after proper normalization.

Contribution

It provides a proof of Stanley's conjecture that the distribution converges to a Gaussian, based on the explicit generating function.

Findings

Distribution of longest alternating subsequence converges to Gaussian

Explicit generating function for the probability distribution

Variance of the limiting distribution is 8/45

Abstract

Recently Richard Stanley initiated a study of the distribution of the length as(w) of the longest alternating subsequence in a random permutation w from the symmetric group $S_n$. Among other things he found an explicit formula for the generating function (on n and k) for the probability that as(w) is at most k and conjectured that the distribution, suitably centered and normalized, tended to a Gaussian with variance 8/45. In this note we present a proof of the conjecture based on the generating function.