Generalized Descents and Normality

TL;DR

This paper proves that the distribution of d-descents in permutations approaches a normal distribution as permutation size grows, even when the parameter d increases with n, using Janson's dependency criterion.

Contribution

It extends the normality result of d-descents to cases where d grows with n, broadening previous understanding of permutation statistics.

Findings

Distribution of d-descents converges to normal as n increases

Normality holds even when d grows with n up to a certain limit

Uses Janson's dependency criterion for proof

Abstract

We use Janson's dependency criterion to prove that the distribution of $d$-descents of permutations of length $n$ converge to a normal distribution as $n$ goes to infinity. We show that this remains true even if $d$ is allowed to grow with $n$, up to a certain degree.