Combinatorics Through Guided Discovery, 2004a

Chapter 4
Generating Functions
4.1 The Idea of Generating Functions
4.1.1 Visualizing Counting with Pictures
Suppose you are going to choose three pieces of fruit from among apples, pears
and bananas for a snack. We can symbolically represent all your choices as
+ + + + + + + + + .
Here we are using a picture of a piece of fruit to stand for taking a piece of that fruit.
Thus stands for taking an apple, for taking an apple and a pear, and
for taking two apples. You can think of the plus sign as standing for the “exclusive
or,” that is, + would stand for “I take an apple or a banana but not both.” To
say “I take both an apple and a banana,” we would write . We can extend the
analogy to mathematical notation by condensing our statement that we take three
pieces of fruit to
3 + 3 + 3 + 2 + 2 + 2 + 2 + 2 + 2 + .
In this notation 3 stands for taking a multiset of three apples, while 2
stands for taking a multiset of two apples and a banana, and so on. What our
notation is really doing is giving us a convenient way to list all three element
multisets chosen from the set {, , }.1
Suppose now that we plan to choose between one and three apples, between
one and two pears, and between one and two bananas. In a somewhat clumsy way
we could describe our fruit selections as
+ 2 + ···+ 2 2 + ···+ 2 2 2
+ 3 + ···+ 3 2 + ···+ 3 2 2 . (4.1)
1This approach was inspired by George Pólya’s paper “Picture Writing,” in the December, 1956 issue
of the American Mathematical Monthly, page 689. While we are taking a somewhat more formal approach
than Pólya, it is still completely in the spirit of his work.
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