ABSTRACT
Whenever we encounter the number n! playing a natural role in some mathematical context, it is almost certain that some interesting combinatorial objects are lurking. This is a recurrent theme in this book, which has a mostly algebraic emphasis. Until very recently a conjecture, known simply as the n! conjecture, was still open. Although it is now settled, many important questions surrounding this conjecture remain unanswered. The actual statement1 of the conjecture makes it appear deceptively easy. Simply stated, the dimension of a certain space of n-variable polynomials had to be equal to n!. The only known proof of this (found after at least a decade of intense research by many top-level mathematicians) is rather intricate and makes use of algebraic geometry notions that lie beyond the intended scope of this book.
