Chapter 8.4, Problem 57E

Generating functions are useful in studying the number of different types of partitions of an integer n. A partition of a positive integer is a way to write this integer as the sum of positive integers where repetition is allowed and the order of the integers in the sum does not matter. For example, the partitions of 5 (with no restrictions) are $1+1+1+1+1,1+1+1+2,1+1+3,$

$1+2+2,1+4,2+3$ , and 5. Exercises 53-58 illustrate some of these uses.

Show that if n is a positive integer, then the number of partitions of n into distinct parts equals the number of partitions of n into odd parts with repetitions allowed; that is, $p_o\left(n\right)=p_d\left(n\right)$[Hint: Show that the generating functions for $p_o\left(n\right)$ and $p_d\left(n\right)$ are equal.]