On a Conjecture of Schilling Regarding the Coupon Collector's Problem

Abstract: This article addresses a conjecture by Schilling concerning the optimality of the uniform distribution in the generalized Coupon Collector's Problem (CCP) where, in each round, a subset (package) of $s$ coupons is drawn from a total of $n$ distinct coupons. While the classical CCP (with single-coupon draws) is well understood, the group-draw variant - where packages of size $s$ are drawn - presents new challenges and has applications in areas such as biological network models. Schilling conjectured that, for $2 \leq s \leq n-1$, the uniform distribution over all possible packages minimizes the expected number of rounds needed to collect all coupons if and only if $s = n-1$. We prove Schilling's conjecture in full by presenting, for all other values of $s$, "natural" non-uniform distributions yielding strictly lower expected collection times. Explicit formulas and asymptotic analyses are provided for the expected number of rounds under these and related distributions. The article further explores the behavior of the expected collection time as $s$ varies under the uniform distribution, including the cases where $s$ is constant, proportional to $n$, or nearly $n$. Keywords: Coupon Collector's Problem (CCP), Group Drawings, Uniform Distribution, Expected Collection Time, Schilling's Conjecture, Optimal Distribution.