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Using Ferrer's diagrams, show that the number of partitions of $n$ into parts of size 1 or 2 is equal to the number of partitions of $n$ into two parts.

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2 Answers 2

It isn’t: it’s equal to the number of partitions of $n$ into at most two parts. The concept that you want is the conjugate of a partition.

Added: Here’s the Ferrers diagram of a partition whose parts all have sizes $1$ or $2$:

$$\begin{align*} &\bullet\bullet\\ &\bullet\bullet\\ &\bullet\\ &\bullet\\ &\bullet \end{align*}$$

And here is the Ferrers diagram of its conjugate partition:

$$\begin{align*} &\bullet\bullet\bullet\bullet\bullet\\ &\bullet\bullet \end{align*}$$

Notice that it has two parts. Here’s another example:

$$\begin{align*} &\bullet\bullet\\ &\bullet\bullet\\ &\bullet\bullet\\ &\bullet\bullet\\ &\bullet\bullet \end{align*}$$

And its conjugate:

$$\begin{align*} &\bullet\bullet\bullet\bullet\bullet\\ &\bullet\bullet\bullet\bullet\bullet\end{align*}$$

And one more example:

$$\begin{align*} &\bullet\\ &\bullet\\ &\bullet\\ &\bullet\\ &\bullet \end{align*}$$

And its conjugate

$$\bullet\bullet\bullet\bullet\bullet$$

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And how can I do this? –  tobyyy May 12 '13 at 17:46
    
@aaaaa: Did you read the material at the link? What does the conjugate of a partition of $n$ into parts of size $1$ and $2$ look like? –  Brian M. Scott May 12 '13 at 17:48
    
I did, but I don't know how to prove it. I don't have any idea. –  tobyyy May 13 '13 at 14:09
1  
@aaaaa: Do you see that the conjugate of a partition whose parts have sizes $1$ or $2$ is always a partition with $1$ or $2$ parts, and vice versa? Use that relation to define a bijection between the set of partitions with parts of size $1$ or $2$ and the set of partitions into at most two parts. –  Brian M. Scott May 13 '13 at 16:54

Hint: Change your direction of viewing the diagram.

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