We want to come up with some nice sets for our symmetric group to act on. Our first step in this direction is to define a “Young tableau”.
If is 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 Young tableaux of shape if .
For example, if , the Ferrers diagram is
We see that , and so there are Young tableaux of shape . They are
We write for the entry in the place. For example, the last tableau above has , , and .
We also call a Young tableau of shape a “-tableau”, and we write . We can write a generic -tableau as .