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Some new aspects of the coupon collectors’s problem. (English) Zbl 1111.05005

Summary: We extend the classical coupon collector’s problem to one in which two collectors are simultaneously and independently seeking collections of \(d\) coupons. We find, in finite terms, the probability that the two collectors finish at the same trial, and we find, using the methods of Gessel and Viennot, the probability that the game has the following “ballot-like” character: the two collectors are tied with each other for some initial number of steps, and after that the player who first gains the lead remains ahead throughout the game. As a by-product we obtain the evaluation in finite terms of certain infinite series whose coefficients are powers and products of Stirling numbers of the second kind.
We study the variant of the original coupon collector’s problem in which a single collector wants to obtain at least \(h\) copies of each coupon. Here we give a simpler derivation of results of D. J. Newman and L. Shepp [Am. Math. Mon. 67, 58–61 (1960; Zbl 0092.35502)] and extend those results. Finally, we obtain the distribution of the number of coupons that have been obtained exactly once (“singletons”) at the conclusion of a successful coupon collecting sequence.
[This paper originally appeared in SIAM J. Discrete Math. 17, 1–17 (2003; Zbl 1038.05002).]

MSC:

05A15 Exact enumeration problems, generating functions
05A05 Permutations, words, matrices
60C05 Combinatorial probability
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