[3.2] Exercise 21
Let us say that a partition is a long rectangle if its Ferrers graph is rectangular with length at least as great as height. Use the above idea of merging rows with corresponding columns to show that
\[
p(n \mid \text{consecutive parts differ by } 2)\;=\;p(n \mid \text{long rectangle}).\]
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.