The Gift Exchange Problem

The aim of this paper is to solve the "gift exchange" problem: you are one of n players, and there are n wrapped gifts on display; when your turn comes, you can either choose any of the remaining wrapped gifts, or you 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 S times. The problem is to determine the number of ways that the game can be played out, for given values of S and n. Several recurrences and explicit formulas are given for these numbers, although some open questions remain.