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Analysis of the gift exchange problem

Moa Apagodu
David Applegate
Neil J.A. Sloane
Doron Zeilberger
The Electronic Journal of Combinatorics, vol. 24 (2017), P3.9


In the gift exchange game there are n players and n wrapped gifts. When a player’s number is called, that person can either choose one of the remaining wrapped gifts, or can “steal” a gift from someone who has already unwrapped it, subject to the restriction that no gift can be stolen more than a total of σ times. The problem is to determine the number of ways that the game can be played out, for given values of σ and n. Formulas and asymptotic expansions are given for these numbers. This work was inspired in part by a 2005 remark by Robert A. Proctor in the On-Line Encyclopedia of Integer Sequences.