This is part 3 of our series on generating functions. Part 1 handled the quicksort recurrence. Part 2 used ordinary generating functions to count marbles in a can.

In this post we introduce exponential generating functions, which are what you reach for when the order of the objects you are counting matters. As with the earlier posts, our working notes live on the Generating Functions page of our wiki.

When Order Matters

The marbles-in-a-can problem from part 2 was order-independent. A can containing three reds, two blues, and one yellow is the same can no matter what order you put …