Knuth's big-chooser matchbox process: the case of many matchboxes

TL;DR

This paper generalizes Knuth's matchbox problem to multiple boxes, deriving generating functions for expected matches remaining, analyzing asymptotic behavior, and connecting the problem to combinatorial structures like Raney numbers and manila folder configurations.

Contribution

It extends Knuth's matchbox problem to many boxes, providing new generating functions, asymptotic analysis, and combinatorial interpretations involving Raney numbers and folder configurations.

Findings

Derived the generating function for the expected residue in multiple matchboxes.

Analyzed the asymptotic behavior of the expected residue for all probabilities p.

Connected the problem to combinatorial structures, enabling closed-form expressions for certain probabilities.

Abstract

Banach's matchbox problem considers the setting of two matchboxes that each initially contain the same number of matches. Boxes are chosen with equal probability and a match removed each time. The problem concerns the law of the number of matches remaining in one box once the other box empties. Knuth considered a generalization of this problem whereby `big-choosers' arrive with probability $p$ and remove a match from the box with the most number remaining, and `little-choosers' arrive with probability $1-p$ and remove a match from the box with the least number remaining. In this paper we consider Knuth's generalization for the case of $k$ matchboxes. We determine the generating function for the expected number of matches remaining in $k-1$ matchboxes once a box first empties, a quantity we refer to as the `residue'. Interestingly, this generating function is a quotient whose denominator contains a generating function for a special case of the Raney numbers. The form for this generating function allows us to give an expression for the expected residue in terms of a sum that involves diagonal state return probabilities, where a diagonal state is a configuration in which all matchboxes each contain the same number of matches. We use analytic techniques to determine the asymptotic behaviour of this expected value for all values of $p$, which involves the study of an asymmetric random walk. We also consider the expected value of the order of the first return to a diagonal state and determine its asymptotic behaviour. The coefficients of the diagonal state probability generating function are shown to be related to `manila folder configurations in a filing cabinet', and we make this connection precise. This allows us to use known results for the enumeration of such manila folder configurations to give a closed form expression for the diagonal state return probabilities.