Combinatorics Through Guided Discovery, 2004a

82 4. Generating Functions
• Problem 202. In Problem 201.b you gave the generating function for the
number of partitions of an integer into parts of size at most m. Explain
why this is also the generating function for partitions of an integer into at
most m parts. Notice that this is the generating function for the number of
partitions whose Young diagram fits into the space between the line y =0
and the line y = m. (h)
• Problem 203. When studying partitions of integers, it is inconvenient to
restrict ourselves to partitions with at most m parts or partitions with maximum
part size m.
(a) Give the generating function for the number of partitions of an integer
into parts of any size. Don’t forget to use q rather than x as your
variable. (h)
(b) Find the coefficient of q 4 in this generating function. (h)
(c) find the coefficient of q 5 in this generating function.
(d) This generating function involves an infinite product. Describe the
process you would use to expand this product into as many terms of
a power series as you choose. (h)
(e) Rewrite any power series that appear in your product as quotients of
polynomials or as integers divided by polynomials.
⇒
Problem 204. In Problem 203, we multiplied together infinitely many power
series. Here are two notations for infinite products that look rather similar:
∞∏
1+x + x 2 + ···+ x i and
i=1
∞∏
1+x i + x 2i + ···+ x i2 .
i=1
However, one makes sense and one doesn’t. Figure out which one makes
sense and explain why it makes sense and the other one doesn’t. If we want
a product of the form
∞∏
1+p i (x),
i=1
where each p i (x) is a nonzero polynomial in x to make sense, describe a
relatively simple assumption about the polynomials p i (x) that will make the
product make sense. If we assumed the terms p i (x) were nonzero power
series, is there a relatively simple assumption we could make about them
in order to make the product make sense? (Describe such a condition or
explain why you think there couldn’t be one.) (h)