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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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possible duplicate of Combinatorics problem based on Ferrers graph –  hardmath Feb 21 '13 at 12:11
    
There's a slight difference in these problems, in that here the question asks about partitions with largest part k, and there the question asks about partitions with parts at most size k. –  hardmath Feb 21 '13 at 12:15

2 Answers 2

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.

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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:

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