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.