Results and conjectures on the role of the uniform distribution in the coupon collector's problem with group drawings

Abstract

In the coupon collector's problem with group drawings, a collector buys independent, identically distributed subsets of fixed size s≥2 out of a totality of n coupons. Let Wn,s denote the number of such subsets necessary to obtain each coupon at least once. We prove that, among all distributions that act on the class of all (ns) subsets of size s of the coupons, the uniform distribution minimizes the expectation of Wn,s if s=n−1. However, if 3≤s≤100 and s+2≤n≤500, computer algebra shows that the uniform distribution does not minimize E(Wn,s). We conjecture that the latter property holds for each s≥3 and each n≥s+2.