Analysis of the gift exchange problem

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. This is a sequel to the earlier article [arXiv:0907.0513] by the second and third authors, differing from it in that there are two additional authors and several new theorems, including the resolution of most of the conjectures, and the extensive tables have been omitted.