A parti at OAME 2017

Yesterday, Greg Clarke and I gave a session at OAME 2017, entitled "Reasoning and Proving with Relational Rods".  Toward the end, we talked about partitions of a number and Young Diagrams. There are 297 partitions of 17, including:

• 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

Using Relational Rods+

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.