Problem
COM-B2-M03-P011 Partitions and Transposition
#11
★★★☆☆ Level 3 of 5
Prove that the number of partitions of \(n\) into at most \(k\) parts equals the number of partitions of \(n\) in which each part is at most \(k\).
Draw the Ferrers diagram and transpose it.
A partition into at most \(k\) parts is represented by a diagram with at most \(k\) rows. Transpose the diagram: rows become columns. Now each row length is at most \(k\), so every part of the new partition is at most \(k\). The inverse transformation is transposition again, hence a bijection.
A diagram helps, but the statement is still precise without it.