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).