  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?
