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