Bounds on the Location of the Maximum Stirling Numbers of the Second Kind

TL;DR

This paper establishes bounds on the position of the maximum Stirling numbers of the second kind using probabilistic methods and Lambert's W-function, providing precise estimates for the mode of these numbers.

Contribution

It introduces new bounds on the mode of Stirling numbers of the second kind, utilizing a probabilistic approach and Lambert's W-function for the first time in this context.

Findings

Bounds on K_n are tight and precise for all n>=2.

The mode K_n is closely approximated by exp(w(n)) with small integer adjustments.

The approach links combinatorial properties with probabilistic and special functions techniques.

Abstract

Let K_n denote the smaller mode of the nth row of Stirling numbers of the second kind S(n, k). Using a probablistic argument, it is shown that for all n>=2, [exp(w(n))]-2<=K_n<=[exp(w(n))]+1, where [x] denotes the integer part of x, and w(n) is Lambert's W-function.