[3.2] Exercise 22
Show that \( p(n \mid \text{long rectangle}) \) equals the number of divisors of \( n \) that are less than or equal to \( \sqrt{n} \).
Solution
Suppose we start with a partition of \(n\) with distinct parts meaning that each row has a distinct length and observe what happens if we conjugate the partition:
With conjugation, rows becomes columns. Since each row had a distinct length, then each column will have distinct length after conjugation. This means, each row has a maximum length of \(m\) which means that each part is at most \(m\).
Conversely, conjugation maps partitions whose parts are all at most \(m\) to partitions with at most \(m\) parts.