# Lecture 12: Symbolic Method

SPOILER ALERT: This thread might give you hints to solve Homework 4 Problem 1. If you don’t want a hint (yet), stop reading!

Today in class I discussed “The Symbolic Method” for ordinary generating functions, and I promised you some references. The closest reference is P. Flajolet and R. Sedgewick. “Analytic Combinatorics” – an excellent book which also really goes into analytic methods with generating functions. It’s also freely available! http://algo.inria.fr/flajolet/Publications/book.pdf

Bergeron, Labelle, and Leroux. “Combinatorial species and tree-like structures” uses a slightly different language, but is closely related.

Also, I mentioned that you can use the symbolic method to “plainly see” that

$\displaystyle\sum_{n=0}^{\infty}S(n,k)x^n=\frac{x}{1-x}\frac{x}{1-2x}\cdots\frac{x}{1-kx}$

This is not so plain, it definitely requires some thought. Did anyone see how to do it?