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Ferrers Diagram

By Cali G, Published on Tue 07 January 2020
Category: Symmetric functions

Both Young diagrams and Ferrers diagrams represent the same thing: a partition. The main difference between the two is what they use to represent the diagram:

Definitions: A Ferrers diagram is a way to represent a partition using dots. A Young diagram is a way to represent a partition using square boxes. If is a partition, then its Ferrers diagram has dots in the first row, dots in the second row, etc. Likewise, its Young diagram has boxes in the first row, boxes in the second row, etc.

As an example, let be a partition of . Then its Ferrers diagram is given by and its Young diagram is given by

There are 3 types of ways we can draw a Young diagram. The way given is known as the "French notation". There are two other notations: English and Russian.

The English notation flips the Young diagram upside down so that the rows go from top to bottom. The Russian notation puts everything on a diagonal so that the boxes are going towards the north-east.

In our example for the English notation is given by: and the Russian notation is given by:

Distribution of Young diagram

We next consider how the Young diagrams are distributed. For this we'll need the following definition.

Definition 1: Let be a partition of . The order of is . Alternatively, it is the number of boxes in its Young diagram.

Recall that (with number of s) is a partition of . Recall also that for two partitions and then if for every . An alternative way to see this is through Young diagrams: if the Young diagram of is contained in the Young diagram of .

To see the distribution, we will also recall some notations.

Theorem:

We show this in an example. Let and . Then and is represented by the Young diagram:

As our sum is over every which is contained in this Young diagram, we list every possible Young diagram that is contained in the above one:

Partitions of :

Partitions of :

Partitions of :

Partitions of :

Partitions of :

Partitions of :

Therefore we have Note that the coefficient is just the number of Young diagram of a partition of that fits inside .

Alternatively:

So we see that the two are equal in our example.

References

  • To be added