# Analysis of the gift exchange problem

## TL;DR

This paper investigates the combinatorial structure of the gift exchange game, deriving formulas and asymptotic behaviors for the number of possible game sequences based on parameters n and sigma.

## Contribution

It provides new formulas and asymptotic expansions for counting the game sequences, extending previous work inspired by integer sequence analysis.

## Key findings

- Derived formulas for the number of game sequences.
- Provided asymptotic expansions for large n.
- Connected the problem to integer sequence analysis.

## Abstract

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.