# The Cali Garmo

does Math

## Ferrers Diagram

By Cali G , Published on Tue 07 January 2020
Category: math / symmetric functions

## Background

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 $n$s) 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.

Theorem:

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 $n^k$.

Alternatively:

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