Symmetric Functions in General

The theory of symmetric functions has many applications to enumerative combinatorics, as well as to such other branches of mathematics as group theory, Lie algebras, and algebraic geometry. Our aim in this chapter is to develop the basic combinatorial properties of symmetric functions; the connections with algebra will only be hinted at in Sections 7.18 and 7.24, Appendix 2, and in some exercises.

Let x = (x1, x2,…) be a set of indeterminates, and let n ∈ ℕ. A homogeneous symmetric function of degree n over a commutative ring R (with identity) is a formal power series

where (a) α ranges over all weak compositions α = (α1, α2, …) of n (of infinite length), (b) cα ∈ R, (c) xα stands for the monomial xα11, xα22,… and (d) f (xω(1)xω(2), …) = f (x1, x2,…) for every permutation ω of the positive integers ℙ. (A symmetric function of degree 0 is just an element of R.) Note that the term “symmetric function” is something of a misnomer; f(x) is not regarded as a function but rather as a formal power series. Nevertheless, for historical reasons we adhere to the above terminology.

The set of all homogeneous symmetric functions of degree n over R is denoted ΛnR. Clearly if f, g ∈ ΛnR and a, b ∈ R, then a f + b g ∈ ΛnR in other words, ΛnR is an R-module. For our purposes it will suffice to take R = ℚ (or sometimes ℚ with some indeterminates adjoined), so Λnℚ is a ℚ-vector space.

Algebraic Generating Functions

In this chapter we will investigate two classes of generating functions called algebraic and D-finite generating functions. We will also briefly discuss the theory of noncommutative generating functions, especially their connection with rational and algebraic generating functions. The algebraic functions are a natural generalization of rational functions, while D-flnite functions are a natural generalization of algebraic functions. Thus we have the hierarchy

Various other classes could be added to the hierarchy, but the three classes of (6.1) seem the most useful for enumerative combinatorics.

Definition. Let K be a field. A formal power series η ∈ K [[x]] is said to be algebraic if there exist polynomials P0(x),…, Pd(x) ∈ K[x], not all 0, such that

The smallest positive integer d for which (6.2) holds is called the degree of η.

Note that an algebraic series η has degree one if and only if η is rational. The set of all algebraic power series over K is denoted Kalg[[x]].

Example. Letη = ∑n≥0x n. By Exercise 1.4(a) wehave(l-4x)η2 - 1 = 0. Hence η is algebraic of degree one or two. If K has characteristic 2 then η = 1, which has degree one. Otherwise it is easy to see that η has degree two. Namely, if deg(η) = 1 then η = P(x)/Q(x) for some polynomials P(x), Q(x) ∈ K[x]. Thus

The degree (as a polynomial) of the left-hand side is odd while that of the right-hand side is even, a contradiction.

This is the second (and final) volume of a graduate-level introduction to enumerative combinatorics. To those who have been waiting twelve years since the publication of Volume 1,1 can only say that no one is more pleased to see Volume 2 finally completed than myself. I have tried to cover what I feel are the fundamental topics in enumerative combinatorics, and the ones that are the most useful in applications outside of combinatorics. Though the book is primarily intended to be a textbook for graduate students and a resource for professional mathematicians, I hope that undergraduates and even bright high-school students will find something of interest. For instance, many of the 66 combinatorial interpretations of Catalan numbers provided by Exercise 6.19 should be accessible to undergraduates with a little knowledge of combinatorics.

Much of the material in this book has never appeared before in textbook form. This is especially true of the treatment of symmetric functions in Chapter 7. Although the theory of symmetric functions and its connections with combinatorics is in my opinion one of the most beautiful topics in all of mathematics, it is a difficult subject for beginners to learn. The superb book by Macdonald on symmetric functions is highly algebraic and eschews the fundamental combinatorial tool in this subject, viz., the Robinson-Schensted-Knuth algorithm. I hope that Chapter 7 adequately fills this gap in the mathematical literature. Chapter 7 should be regarded as only an introduction to the theory of symmetric functions, and not as a comprehensive treatment.

Most textbooks written in our day have a short half-life. Published to meet the demands of a lucrative but volatile market, inspired by the table of contents of some out-of-print classic, garnished with multicolored tables, enhanced by nutshell summaries, enriched by exercises of dubious applicability, they decorate the shelves of college bookstores come September. The leftovers after Registration Day will be shredded by Christmas, unwanted even by remainder bookstores. The pageant is repeated every year, with new textbooks on the same shelves by other authors (or a new edition if the author is the same), as similar to the preceding as one can make them, short of running into copyright problems.

Every once in a long while, a textbook worthy of the name comes along; invariably, it is likely to prove aere perennius: Weber, Bertini, van der Waerden, Feller, Dunford and Schwartz, Ahlfors, Stanley.

The mathematical community professes a snobbish distaste for expository writing, but the facts are at variance with the words. In actual reality, the names of authors of the handful of successful textbooks written in this century are included in the list of the most celebrated mathematicians of our time.

Only another textbook writer knows the pains and the endless effort that goes into this kind of writing. The amount of time that goes into drafting a satisfactory exposition is always underestimated by the reader. The time required to complete one single chapter exceeds the time required to publish a research paper. But far from wasting his or her time, the author of a successful textbook will be amply rewarded by a renown that will spill into the distant future.