The Cali Garmo

does Math

Ferrers Diagram

By Cali G , Published on Tue 07 January 2020
Category: math / 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 \lambda = (\lambda_1, \lambda_2, \ldots) is a partition, then its Ferrers diagram has \lambda_1 dots in the first row, \lambda_2 dots in the second row, etc. Likewise, its Young diagram has \lambda_1 boxes in the first row, \lambda_2 boxes in the second row, etc.

As an example, let \lambda = (4,2,1) be a partition of 7. 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 (4,2,1) 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 \lambda be a partition of n. The order \abs{\lambda} of \lambda is n. Alternatively, it is the number of boxes in its Young diagram.

Recall that n^k = (n,n, \ldots, n) (with k number of ns) is a partition of n \cdot k. Recall also that for two partitions \lambda and \mu then \lambda \subseteq \mu if \lambda_i \leq \mu_i for every i. An alternative way to see this is through Young diagrams: \lambda \subseteq \mu if the Young diagram of \lambda is contained in the Young diagram of \mu.

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


We show this in an example. Let n = 3 and k = 2. Then n^k = (3,3) and is represented by the Young diagram:

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

Partitions of 5:

Partitions of 4:

Partitions of 3:

Partitions of 2:

Partitions of 1:

Partitions of 0: \emptyset

Therefore we have Note that the coefficient c_i is just the number of Young diagram of a partition of i that fits inside n^k.


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