# The Unapologetic Mathematician

## Partitions and Ferrers Diagrams

We’ve discussed partitions before, but they’re about to become very significant. Let $\lambda=(\lambda_1,\dots,\lambda_k)$ be a sequence of positive integers with $\lambda_1\geq\dots\geq\lambda_k$. We write

$\displaystyle\lvert\lambda\rvert=\sum\limits_{i=1}^k\lambda_i$

If $\lvert\lambda\rvert=n$ we say $\lambda$ is a partition of $n$, and we write $\lambda\vdash n$. A partition, then, is a way of breaking a positive integer $n$ into a bunch of smaller positive integers, and sorting them in (the unique) decreasing order.

We visualize partitions with Ferrers diagrams. The best way to explain this is with an example: if $\lambda=(3,3,2,1)$, the Ferrers diagram of $\lambda$ is

$\displaystyle\begin{array}{ccc}\bullet&\bullet&\bullet\\\bullet&\bullet&\bullet\\\bullet&\bullet&\\\bullet&&\end{array}$

The diagram consists of left-justified rows, one for each part in the partition $\lambda$, and arranged from top to bottom in decreasing order. We can also draw the Ferrers diagram as boxes

$\displaystyle\begin{array}{|c|c|c|}\cline{1-3}\hphantom{X}&\hphantom{X}&\\\cline{1-3}&&X\\\cline{1-3}&&\\\cline{1-2}&&\\\cline{1-1}\end{array}$

The dangling vertical lines aren’t supposed to be there, but I’m having a hell of a time getting WordPress’ $\LaTeX$ processor to recognize an \hfill command so I can place \vline elements at the edges of columns. This should work but.. well, see for yourself:

$\displaystyle\begin{array}{ccc}\cline{1-3}\vline\hfill\hphantom{X}\hfill\vline&\hphantom{X}\hfill\vline&\hfill\vline\\\cline{1-3}\vline\hfill\hfill\vline&\hfill\vline&X\hfill\vline\\\cline{1-3}\vline\hfill\hfill\vline&\hfill\vline&\\\cline{1-2}\vline\hfill\hfill\vline&&\\\cline{1-1}\end{array}$

So, if anyone knows how to make this look like the above diagram, but without the dangling vertical lines, I’d appreciate the help.

Anyway, in both of those ugly, ugly Ferrers diagrams, the $X$ is placed in the $(2,3)$ position; we see this by counting down two boxes and across three boxes. We will have plenty of call to identify which positions in a Ferrers diagram are which in the future.

December 8, 2010 -

1. Hello,
Thank you very much for your posts on representations.

I would suggest something like this for the Ferrers’ diagrams, hoping that the multicolumn command is recognized !

$\begin{array}{|c|c|c|}\cline{1-3}\hphantom{X}&\hphantom{X}&\hphantom{X}\\\cline{1-3}&&\\\cline{1-3}&&\multicolumn{1}{c}{}\\\cline{1-2}&\multicolumn{2}{c}{}\\\cline{1-1}\end{array}$

Comment by FelixCQ | December 9, 2010 | Reply

• Thanks, Felix, I’ll examine that one more closely.

Comment by John Armstrong | December 9, 2010 | Reply

2. […] a partition of , we define a Young tableau of shape to be an array of numbers. We start with the Ferrers diagram of the partition , and we replace the dots with the numbers to in any order. Clearly, there are […]

Pingback by Young Tableaux « The Unapologetic Mathematician | December 9, 2010 | Reply

3. Hello John,

Firstly I just wanted to say thank you for this blog- I’ve just started a PhD in representation theory and this has been an invaluable resource and good way to idle away some hours usefully!

As for drawing young tableux, try the youngtab package (just google it- should be the first to come up). I used it in a dissertation and was very easy to use with good results.

Thanks again!

Comment by Andrew Poulton | December 13, 2010 | Reply

• Glad you find it useful, Andrew. Unfortunately, WordPress doesn’t let me use packages here, so I’m stuck with base $\LaTeX$. So it goes.

Comment by John Armstrong | December 13, 2010 | Reply

4. […] let . This is a pretty trivial “partition”, consisting of one piece of length . The Ferrers diagram of looks […]

Pingback by Permutation Representations from Partitions « The Unapologetic Mathematician | December 14, 2010 | Reply

5. […] can interpret this by looking at their Ferrers diagrams. First look at the first rows of the diagrams. Are there more dots in the diagram for than in that […]

Pingback by The Dominance Order on Partitions « The Unapologetic Mathematician | December 17, 2010 | Reply

6. […] “composition” is sort of like a partition, except the parts are allowed to come in any specified order. That is, a composition of is an […]

Pingback by Compositions « The Unapologetic Mathematician | January 6, 2011 | Reply

7. […] for row tabloids. Of course, in doing so we have to alter our definition of the dominance order on Ferrers diagrams to take columns into account instead of […]

Pingback by The Column Dominance Order « The Unapologetic Mathematician | January 20, 2011 | Reply

8. […] to take note of it the idea of an “inner corner” and an “outer corner” of a Ferrers diagram, and thus of a […]

Pingback by Inner and Outer Corners « The Unapologetic Mathematician | January 26, 2011 | Reply