Generating function for partitions with bounded parts
= Generating function for partitions with bounded parts
The generating function for partitions whose parts are at most $e$ is
$$
\prod_{i=1}^e\frac1{1-x^i}.
$$
Conjugating Young diagrams shows that it also counts partitions with at most $e$ parts. Multiplication by $x^e$ gives the generating function for partitions with exactly $e$ positive parts.