Formal power series are central to enumerative combinatorics. A formal power series is just an alternative way of representing an infinite sequence of numbers. However, the use of formal power series introduces new techniques, especially from analysis, into the subject, as we will see.

This chapter provides an introduction to formal power series and the operations we can do on them. Examples are taken from elementary combinatorics of subsets, partitions and permutations, and will be discussed in more detail in the next chapter. We also take a first look at the use of analysis for finding (exactly or asymptotically) the coefficients of a formal power series; we return to this in the last two chapters. The exponential, logarithmic and binomial series are of crucial importance; we discuss these, but leave their combinatorial content until the next chapter.

Fibonacci numbers

Leonardo of Pisa, also known as Fibonacci, published a book in 1202 on the use of Arabic numerals, then only recently introduced to Europe. He contended that they made calculation much easier than existing methods (the use of an abacus, or the clumsy Roman numerals used for records). As an exercise, he gave the following problem:

Let Fn be the number of pairs of rabbits at the end of the nth month. Then F0 = 1 (given), and F1 = 1 (since the rabbits do not breed in the first month of life). Also, for n ≥ 2, we have

since at the end of the nth month, we have all the pairs who were alive a month earlier, and also new pairs produced by all pairs at least two months old (that is, those which were alive two months earlier).

In this chapter I list a few books, papers and websites which may be useful if you would like to follow up some of the things I have discussed.

The On-line Encyclopedia of Integer Sequences

The On-line Encyclopedia of Integer Sequences, available at the URL https://oeis.org/,

is an essential resource for anyone doing research in combinatorics. For example, suppose you are trying to count the number of arrangements of n zeros and ones around a circle in which no two ones are consecutive (Exercise 2.8(b)). You might reasonably assume that n ≥ 3, and calculate that for n = 3,4,5 there are respectively 4, 7 and 11 such arrangements. If you type these three numbers into the Encyclopedia, you find many matches (I found 521 when I tried it on 24 October 2016), but near the top (indeed, at the top when I did the experiment) is an entry for the Lucas numbers. The entry gives a recurrence relation (identical to that for Fibonacci numbers), congruences modulo primes, representation in terms of hyperbolic functions, and much more, including (most importantly) ten references to the literature, short programs for computing the numbers in various programming languages, further web links, open problems, and cross-references to related sequences. Now you can either prove directly that the numbers you are interested in satisfy the Fibonacci recurrence (and hence coincide with the Lucas numbers), or check in the literature for further information which will help you make the identification.

On the Encyclopedia's website, you will find different ways of viewing the sequence and information about it, pointers to interesting or mysterious sequences,

a formula for the terms of the sequence or its generating function if known, and several articles by the editor Neil Sloane and others describing uses of the Encyclopedia in research. I have used it myself on a number of occasions.

This book is about counting. Of course this doesn't mean just counting a single finite set. Usually, we have a family of finite sets indexed by a natural number n, and we want to find F(n), the cardinality of the nth set in the family. For example, we might want to count the subsets or permutations of a set of size n, lattice paths of length n, words of length n in the alphabet {0, 1} with no two consecutive 1s, and so on.

What is counting?

There are several kinds of answer to this question:

1. • An explicit formula (which may be more or less complicated, and in particular may involve a number of summations). In general, we regard a simple formula as preferable; replacing a formula with two summations by one with only one is usually a good thing.

2. • A recurrence relation expressing F(n) in terms of n and the values of F(m) for m < n. This allows us to compute F(0),F(1), … in turn, up to any desired value.

3. • A closed form for a generating function for F. We will have much more to say about generating functions later on. Roughly speaking, a generating function represents a sequence of numbers by a power series, which in some cases converges to an analytic function in some domain in the complex plane. An explicit formula for the generating function for a sequence of numbers is regarded as almost as good as a formula for the numbers themselves.

If a generating function converges, it is possible to find the coefficients by analytic methods (differentiation or contour integration).

In the examples below, we use two forms of generating function for a sequence (a0,a1, a2, …) of natural numbers: the ordinary generating function, given by

and the exponential generating function, given by

We will study these further in the next chapter, and meet them many times during later chapters. In Chapter 10, we will see a sort of explanation of why some problems need one kind of generating function and some need the other.