Séminaire Lotharingien de Combinatoire, vol. 47, B47a, 2001, 20 pages.
Dominique Foata, Guo Niu Han, Bodo Lass
Les nombres hyperharmoniques et
la fratrie du collectionneur de vignettes
The traditional coupon collector's problem is extended
to the case where the collector shares his harvest with his
own phratry. He remains the only coupon
provider, but gives his brothers the coupons that have already
appeared in his picture-book. When his book is completed, the
books of the other brothers have certain numbers of empty spots.
On the average, how many? We bring an answer to this question
and also derive the multivariable generating function for the
random variables involved. The problem gives rise to the
hyperharmonic numbers that are to be studied under various
aspects, in particular as solutions of finite-difference equations.
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