The Labeled Coupon Collector Problem

TL;DR

This paper extends the classic Coupon Collector Problem by considering random subsets and orderings of coupons, providing new theoretical insights and numerical methods for minimum and expected draws needed.

Contribution

It introduces two variations of the problem, characterizes the minimum number of draws, and develops a Markov chain model for expectation calculation.

Findings

Extended the CCP to subset and ordering scenarios

Provided a full characterization of the minimum number of draws

Developed a numerical Markov chain approach for expectations

Abstract

We generalize the well-known Coupon Collector Problem (CCP) in combinatorics. Our problem is to find the minimum and expected number of draws, with replacement, required to recover $n$ distinctly labeled coupons, with each draw consisting of a random subset of $k$ different coupons and a random ordering of their associated labels. We specify two variations of the problem, Type-I in which the set of labels is known at the start, and Type-II in which the set of labels is unknown at the start. We show that our problem can be viewed as an extension of the separating system problem introduced by R\'enyi and Katona, provide a full characterization of the minimum, and provide a numerical approach to finding the expectation using a Markov chain model, with special attention given to the case where two coupons are drawn at a time.