Combinatorics Through Guided Discovery, 2004a

4.2. Generating functions for integer partitions 83
• Problem 205. What is the generating function (using q for the variable) for
the number of partitions of an integer in which each part is even? (h)
• Problem 206. What is the generating function (using q as the variable) for
the number of partitions of an integer into distinct parts, that is, in which
each part is used at most once? (h)
• Problem 207. Use generating functions to explain why the number of partitions
of an integer in which each part is used an even number of times
equals the generating function for the number of partitions of an integer in
which each part is even. (h)
Problem 208. Use the fact that
1 − q 2i
1 − q i =1+q i
and the generating function for the number of partitions of an integer into
distinct parts to show how the number of partitions of an integer k into
distinct parts is related to the number of partitions of an integer k into odd
parts. (h)
Problem 209. Write down the generating function for the number of ways
to partition an integer into parts of size no more than m, each used an
odd number of times. Write down the generating function for the number
of partitions of an integer into parts of size no more than m, each used
an even number of times. Use these two generating functions to get a
relationship between the two sequences for which you wrote down the
generating functions. (h)
⇒
Problem 210. In Problem 201.b and Problem 202 you gave the generating
functions for, respectively, the number of partitions of k into parts the largest
of which is at most m and for the number of partitions of k into at most m
parts. In this problem we will give the generating function for the number
of partitions of k into at most n parts, the largest of which is at most m.
That is we will analyze ∑ ∞
i=0 a kq k where a k is the number of partitions of k
into at most n parts, the largest of which is at most m. Geometrically, it is