We conclude our foray into Hurwitz theory by introducing some mathematical machinery which is useful to “organize things”. Admittedly, there are lots of Hurwitz numbers… as in, infinitely many! But we have seen that they are not just some random collection of numbers unrelated to each other: we saw in Theorem 7.5.3 that they are all determined by base curve genus 0, three-branch-point Hurwitz numbers via recursive formulas.
It is sometimes convenient to consider infinite sets of numbers with some kind of recursive structure as coefficients of a power series, which is called a generating function. When the encoding is appropriate, operations on generating functions correspond to recursions on the collection of numbers.
We begin by introducing the notion of generating functions through some simple examples, which include the mind-boggling statement that there are “e” isomorphism classes of finite sets if, when we count them, we divide by the order of automorphism groups. Once we are warmed up, we introduce the Hurwitz potential, one ginormous power series that contains all Hurwitz numbers as coefficients of its monomials. We then derive two interesting applications of this point of view. The first is that the relationship between connected and disconnected Hurwitz numbers is controlled by one simple functional equation relating the connected and disconnected Hurwitz potentials. The second is that all (infinitely many!) recursions coming from a specific type of degeneration formula are encoded in a unique differential operator, called the cut-and-join operator, which vanishes when applied to the Hurwitz potential.
The book (Wilf, 2006) introduces generating functions with this sentence:
“A generating function is a clothesline on which we hang up a sequence of numbers for display.”
Behind the humorous character of this statement lies the philosophy that encoding sequences of numbers as coefficients of power series is a convenient way to encode and manipulate combinatorial information. Let us make some precise definitions.
Definition 10.1.1. Given a sequence of numbers, the ordinary generating function for A is the formal power series:
The exponential generating function for A is defined to be
If either of the above power series converges in a neighborhood of x = 0, then we also refer to the analytic function that the power series converges to as the generating function for A.