  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
sigma times. The problem is to determine the number of ways that the game can
be played out, for given values of sigma 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.
