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

TL;DR

This work generalizes Banach's matchbox problem to $k$ boxes under Knuth's big-chooser rule, analyzing the residue—the total number of matches in the $k-1$ nonempty boxes when the last box empties. A generating-function framework is developed, revealing that the residue's generating function has a denominator tied to a Raney-number generating function; the residue is expressed in terms of diagonal-state return probabilities and then analyzed via asymptotics of an asymmetric random walk. The authors establish that diagonal-state counts are given by Raney numbers, derive exact formulas for the residue and the order of first diagonal return, and uncover a rich combinatorial structure by linking diagonal-state probabilities to Manila-folder configurations, enabling closed-form expressions for diagonal-state statistics. The results exhibit phase transitions in growth rates with respect to $p$ (or $q$) and provide a detailed, interconnected picture of the stochastic process, diagonal-state dynamics, and combinatorial enumerations with potential broader relevance to multi-box occupancy problems and related random-walk models.

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.

Figures (7)

• Figure 1: The experimental value of $M^{(3)}_{100}(p)$. The horizontal axis represents the value of $p$ while the vertical axis represents $M^{(3)}_{100}(p)$.
• Figure 2: Paths counted by the numbers $d^{(k)}_{n,j}$: those paths from $((k-1)n,n)$ to $(j,0)$ that do not touch the boundary line or the $x$-axis en-route. This diagram relates to the proof of Lemma \ref{['expression:for:L']}. Here $k=3$.
• Figure 3: The value of $\lambda_{k,p}$ for $k \in \{2,3,4,5\}$. The lines for $\lambda_{2,p}$, $\lambda_{3,p}$, $\lambda_{4,p}$, and $\lambda_{5,p}$ are coloured red, green, blue, and brown, respectively.
• Figure 4: The experimental value of the random variable $Y$ where the first return to the diagonal state is at $(n-Y,n-Y,n-Y)$ for $n=100$.
• Figure 5: There are three ways to arrange two Manlia folders, each with two compartments. $|\mathsf{Manila}_3(2,0)|=2$ and $|\mathsf{Manila}_3(2,1)|=1$.

Theorems & Definitions (32)

• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.3
• proof
• Proposition 3.4
• proof
• Theorem 4.1
• Lemma 4.2