Example 2. Show that the number of

Example 2. Show that the number of partitions of a number into parts which have at most one
of each distinct even part (e.g. 1 + 1 + 1 + 2 + 3 + 4) equals the number of partitions of the number
in which each part can appear at most three times (e.g. 1 + 1 + 1 + 2 + 2 + 4 + 4 + 4).
Solution. We shall approach this problem with generating functions. To find the generating function
for the number of ways to partition a number into parts which have at most one of each distinct even
part let us look at the generating functions for even and odd numbers separately. Two contributes
a factor of 1 + x
2 because you can either have zero or one two’s. Four contributes a factor of 1 + x
4
because you can either have zero or one fours. Continuing this logic we find that even numbers