- 17
- 16 + 1
- 15 + 2
- 15 + 1 + 1
- 2 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
- 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
There are only five partitions of 17 that have distinct odd parts:
- 17
- 13 + 3 + 1
- 11 + 5 + 1
- 9 + 7 + 1
- 9 + 5 + 3
A Young Diagram is an arrangement of squares, that corresponds to a partition. You can create Young-like diagrams using the Relational Rods+ and the Colour Tiles mathies Learning Tools.
A conjugate is a partition that you get when you flip the rows and the columns of its Young Diagram. For example, the conjugate of
17 = 9 + 5 + 3
is
17 = 3 + 3 + 3 + 2 + 2 + 1 + 1 + 1 + 1
Sometimes, when you create a conjugate of a partition, you get the same partition. That partition is called a self-conjugate. For example,
17 = 5 + 4 + 4 + 3 + 1
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| Using Relational Rods+ |
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| Using Colour Tiles (the colours are unnecessary) |
It turns out that there are five partitions of 17 that are self-conjugates. Can you find the other 4? It is more fun if you use Colour Tiles, since you have the reflection and rotate buttons.
In our session, Greg and I stated the theorem that:
The number of partitions with distinct odd parts
is the same as the number of self-conjugates.
For 17, that number is 5. We gave a way to create a self-conjugate from a partition with distinct odd parts (and vice-versa) which establishes a one-to-one correspondence. Can you see how the two sets of five are related?
See the article on which our talk was based on the less-accessible wikipedia article that gave us the idea for more details.
