[Paper Review] On the Average Maximal Number of Balls in a Bin Resulting from Throwing r Balls into n Bins T times

Amir Behrouzi-Far, Doron Zeilberger|arXiv (Cornell University)|May 19, 2019

Analytic Number Theory Research2 references4 citations

TL;DR

This paper investigates the asymptotic behavior of the average maximal number of balls in a bin when throwing r balls into n bins T times, using the holonomic ansatz and recurrence relations to derive that the excess over the mean load grows as $ C_{n,r} \sqrt{T} + O(1/\sqrt{T}) $. It provides exact recurrences and estimates for $ C_{n,r} $ for small $ n $ and $ r $, and offers a $150 donation to the OEIS for a closed-form expression in terms of $ n $, $ r $, and $ \pi $.

ABSTRACT

We use the holonomic ansatz to estimate the asymptotic behavior, in $T$, of the average maximal number of balls in a bin that is obtained when one throws uniformly at random (without replacement) $r$ balls into $n$ bins, $T$ times. Our approach works, in principle, for any fixed $n$ and $r$. We were able to do the cases $(n,r)$ = $(2,1),(3,1),(4,1), (4,2)$, but things get too complicated for larger values of $n$ and $r$. We are pledging a \$150 donation to the OEIS for an explicit expression, (in terms of $n$, $r$, and $π$) for the constant $C_{n,r}$ such that that average equals $\frac{n}{r}\,T+C_{n,r} \sqrt{T}+O(1/\sqrt{T})$. In this version we announce that the problem has been solved (to the extent possible) by Marcus Michelen.

Motivation & Objective

• To determine the asymptotic behavior of the expected excess load in the most heavily loaded bin after T rounds of throwing r balls into n bins.
• To compute exact linear recurrence relations for the excess load $ A(n,r;T) $ for small values of $ n $ and $ r $, enabling computation for large $ T $.
• To estimate the constant $ C_{n,r} $ in the asymptotic expansion $ A(n,r;T) = C_{n,r} \sqrt{T} + O(1/\sqrt{T}) $, and to challenge the mathematical community to find a closed-form expression for $ C_{n,r} $.

Proposed method

• Using the holonomic ansatz and Wilf-Zeilberger algorithmic proof theory to derive linear recurrence relations with polynomial coefficients for $ A(n,r;T) $, valid for any fixed $ n $ and $ r $.
• Computing initial values of $ A(n,r;T) $ via brute-force enumeration of multinomial coefficients and symmetric functions.
• Fitting the recurrence relations using symbolic computation, then verifying them a posteriori using theoretical guarantees of existence.
• Leveraging the recurrence to compute $ A(n,r;T) $ for very large $ T $, bypassing the infeasibility of direct enumeration.
• Applying Stirling's approximation to the recurrence solutions to extract asymptotic constants $ C_{n,r} $.
• Using contour integral and multi-sum representations of $ A(n,r;T) $ as potential pathways to closed-form expressions for $ C_{n,r} $.

Experimental results

Research questions

• RQ1 For fixed $ n $ and $ r $, what is the asymptotic behavior of the expected maximum bin load minus the mean load as $ T \to \infty $?
• RQ2 Can a linear recurrence with polynomial coefficients be derived for $ A(n,r;T) $, and can it be used to compute values for arbitrarily large $ T $?
• RQ3 What is the exact value of the constant $ C_{n,r} $ in the asymptotic expansion $ A(n,r;T) = C_{n,r} \sqrt{T} + O(1/\sqrt{T}) $?
• RQ4 Is there a closed-form expression for $ C_{n,r} $ in terms of $ n $, $ r $, and $ \pi $?
• RQ5 How do heuristic estimates for $ C_{n,r} $ compare with the numerically computed values for small $ n $ and $ r $?

Key findings

• For $ (n,r) = (2,1) $, the recurrence $ A(T) = \frac{1}{T-1} A(T-1) + A(T-2) $ holds with initial conditions $ A(1) = 1/2 $, $ A(2) = 1/2 $, and the asymptotic constant is $ C_{2,1} = (2\pi)^{-1/2} \approx 0.398942 $.
• For $ (n,r) = (3,1) $, a fifth-order linear recurrence was derived, yielding $ C_{3,1} \approx 0.489 $, and the recurrence was verified a posteriori.
• For $ (n,r) = (4,1) $, a ninth-order linear recurrence was computed and stored in a public file, with $ C_{4,1} \approx 0.516 $.
• For $ (n,r) = (4,2) $, an eighth-order linear recurrence was derived, yielding $ C_{4,2} \approx 0.59430 $.
• The authors estimate $ C_{n,r} \approx (r/n) \sqrt{\pi \log n} \log(n/r) $ as a heuristic, though it diverges from exact values for small $ n $ and $ r $.
• A $150 donation to the OEIS was pledged for a closed-form expression of $ C_{n,r} $, with $50 for $ r=1 $ and $100 for the general case, and a donation was made in honor of Marcus Michelen and Brendan McKay for partial progress.