Conway's napkin problem

Abstract

The napkin problem was first posed by John H. Conway, and written up as a `toughie' in "Mathematical Puzzles: A Connoisseur's Collection," by Peter Winkler. To paraphrase Winkler's book, there is a banquet dinner to be served at a mathematics conference. At a particular table, $n$ men are to be seated around a circular table. There are $n$ napkins, exactly one between each of the place settings. Being doubly cursed as both men and mathematicians, they are all assumed to be ignorant of table etiquette. The men come to sit at the table one at a time and in random order. When a guest sits down, he will prefer the left napkin with probability $p$ and the right napkin with probability $q=1-p$. If there are napkins on both sides of the place setting, he will choose the napkin he prefers. If he finds only one napkin available, he will take that napkin (though it may not be the napkin he wants). The third possibility is that no napkin is available, and the unfortunate guest is faced with the prospect of going through dinner without any napkin! We think of the question of how many people don't get napkins as a statistic for signed permutations, where the permutation gives the order in which people sit and the sign tells us whether they initially reach left or right. We also keep track of the number of guests who get a napkin, but not the napkin they prefer. We build a generating function for the joint distribution of these statistics, and use it to answer questions like: What is the probability that every guest receives a napkin? How many guests do we expect to be without a napkin? How many guests are happy with the napkin they receive?