I am going through the lecture note How to get character tables of symmetric groups.
On page 2, it computes the character table of $S_4$. The procedure starts with building the table of the characters of permutation representations for $S_4$. The table is given below.
The note says the following about the table.
The top row lists the different conjugacy classes by way of listing partitions; the left-hand edge lists the different representations we get for different shapes of tabloid. The top line is the trivial representation, while the bottom one is the regular representation.
I am trying to understand what is going on here. It is clear that the columns are the corresponding partitions, hence, the conjugacy classes. $\sigma_\lambda$ for a given shaped $\lambda$ is defined as follows.
So, for example, how $\sigma_{3, 1}$ gives $2$ for the partition $2, 1, 1$?
Let's go over how these representations are defined:
We have a certain partition of the set $\{1,2,\ldots,n\}$; this partition is into an ordered collection of unordered sets. Such a partition might look like $[\{1,3\}, \{2,4\}]$; note this is exactly the same partition as $[\{3,1\}, \{2,4\}]$, but not the same as $[\{2,4\}, \{1,3\}]$. The problem with that last one is the first element of the array is not the same. All of these partitions have the same shape: $2,2$ means an array, with a set of size $2$ in the first spot, and another set of size $2$ in the second spot.
For each such shape, we can define a representation (over $\mathbb{C}$), and the character is what you are calling $\sigma_{\text{shape}}$. That character can be determined for each "permutation type" by simply asking how many partitions (of that shape) it fixes.
Here's a concrete example: for the "permutation type" $2,1,1$, and the shape $2,2$, we can choose any permutation in $S_4$ of that type: let's use $(1,2)$. The question is then: how many partitions of shape $2,2$ does $(1,2)$ fix? The answer is two: $[\{1,2\}, \{3,4\}]$ and $[\{3,4\}, \{1,2\}]$. Thus the entry in your table, under column $2,1,1$ and row $\sigma_{2,2}$, is $2$.