Analysis of the gift exchange problem

Moa Apagodu, David Applegate, N. J.A. Sloane, Doron Zeilberger

Research output: Contribution to journalArticle

1 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)1-15
Number of pages15
JournalElectronic Journal of Combinatorics
Issue number3
StatePublished - Jul 14 2017

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics
  • Applied Mathematics


  • Almkvist-Zeilberger algorithm
  • Bessel polynomials
  • Gift swapping
  • Hypergeometric functions
  • Restricted stirling numbers
  • Set partitions
  • Wilf-Zeilberger summation

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