Partition bijections

Question

How do I prove bijectively that the number of partitions of $n$ with largest part $k$ equals the number of partitions of n with exactly $k$ parts.

Answer 1

Ferrers diagrams.

Write the partition $n=a_1+a_2+\cdots+a_r+k$ as a row of $k$ dots, below it a row of $a_r$ dots, and so on, down to a row of $a_1$ dots, all left-justified. Then the columns give a partition of $n$ into exactly $k$ parts.

Answer 2

Imagine a random partition of a number n into 5 distinct parts:

24 = 10+6+3+2+1

|*|* * * * * * * * *
|*|* * * * *
|*|* *
|*|*
|*|

the bar piece will always be in such a partition: it shows that such partitions are in correspondence with partitions of n whose largest part is 5:

* * * * *
* * * *
* * *
* *
* *
*
*
*
*