Asymptotic behavior of permutation records

TL;DR

This paper investigates the asymptotic properties of permutation statistics related to records in the symmetric group, revealing simple functions that describe their scaled behavior as the group size grows.

Contribution

It introduces a probabilistic approach to analyze the asymptotic behavior of record-related statistics in permutations, providing new insights into their limiting distributions.

Findings

Asymptotic formulas for the number of permutations with k records

Asymptotic behavior of the sum of record positions in permutations

Simple functions describe the scaled asymptotics of these statistics

Abstract

We study the asymptotic behavior of two statistics defined on the symmetric group S_n when n tends to infinity: the number of elements of S_n having k records, and the number of elements of S_n for which the sum of the positions of their records is k. We use a probabilistic argument to show that the scaled asymptotic behavior of these statistics can be described by remarkably simple functions.