Balls in Boxes: Variations on a Theme of Warren Ewens and Herbert Wilf

TL;DR

This paper critiques and simplifies the analysis of the balls-and-boxes problem, advocating for the Poisson Approximation over complex methods, with practical implications for real-world applications.

Contribution

It demonstrates that the Poisson Approximation is preferable to Ewens and Wilf's method for analyzing maximums in balls-and-boxes problems, simplifying the approach.

Findings

Poisson Approximation outperforms complex methods in real-world scenarios.

Extremely unlikely events are negligible in practical applications.

The paper provides a Maple package for implementation.

Abstract

We comment on, elaborate, and extend the work of Warren Ewens and Herbert Wilf, described in their http://www.pnas.org/content/104/27/11189.full.pdf about the maximum in balls-and-boxes problem. In particular we meta-apply their ingenious method to show that it is not really needed, and that one is better off using the so-called Poisson Approximation, at least in applications to the real world, because extremely unlikely events mever happen in real life. This article is accompanied by the Maple package http://www.math.rutgers.edu/~zeilberg/tokhniot/BallsInBoxes">BallsInBoxes.