# Show that the number of partitions of $n$ into $k$ parts that are all $m$ or less is

Full problem: Show that the number of partitions of $$n$$ into $$k$$ parts that are all $$m$$ or less is the number of partitions of $$km-n$$ into fewer than $$m$$ parts that are all $$k$$ or less.

I'm a little confused about how to show this, I tried drawing a Ferrer's diagram, because it's reminding me a little but of identifying self-conjugate partitions but I'm not sure... Can anyone help with this?

• I think it should be "partitions of $n$ into $k$ OR LESS parts"? Commented Oct 14, 2019 at 1:15
• @Dzoooks no, it's just into k parts!
Commented Oct 14, 2019 at 1:16

Hint: Consider the $$k \times m$$ rectangle into which the partition diagrams of both types fit and find a bijection between the two types:

$$\begin{matrix} {\color{red} \bullet} &{\color{red} \bullet} & {\color{red} \bullet} & {\color{blue} \bullet} \\ {\color{red} \bullet} &{\color{red} \bullet} & {\color{blue} \bullet}&{\color{blue} \bullet} \\ {\color{red} \bullet} & {\color{red} \bullet} &{\color{blue} \bullet} &{\color{blue} \bullet} \\ {\color{red} \bullet}& {\color{blue} \bullet}& {\color{blue} \bullet} &{\color{blue} \bullet}\end{matrix}$$

• I'm just a bit unclear on how this rectangle counts both of the number of necessary partitions.. What is the relation supposed to be between km and n?
• $n$ is the size of the red partition. $km-n$ is the size of the blue partition. Just do it for a few examples. Commented Oct 14, 2019 at 1:38
• I see it with examples, but to word it properly- If we take a $k•m$ rectangle, and remove the a partition of size $n$ that is $k$ parts, size $m$ or less, then the remaining area is exactly partitioned into fewer than $m$ parts that are k or less.