This question is crossposted at math.stackexchange [here][1] and may be beyond the usual scope of the site. The question is located below. **In short, I am looking for an accessible explanation of the Murnagham Nakayama rule in relation to the following problem**. Pardon the long setup.

Let $Y$ be a standard Young tableau of shape $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_n)$. At the risk of being pedantic, standard here means $\lambda_1\geq\lambda_2\geq\ldots\lambda_n$ and the entries are strictly increasing rightward along rows and downard along columns (with each number occuring only once). Let $f_\lambda$ denote the number of standard Young tableau of shape $\lambda$. It is well known that $f_\lambda$ can be calculated via the Hook formula:

$$f_\lambda=\frac{|\lambda|!}{\prod_{i,j}h(i,j)}.$$ 

Now suppose I remove some boxes on the outer edge of the tableau (i'll call this a *pattern* from now on), giving a shape $\delta_i$ (I'll explain the $i$ in a moment). Here's an example: let $\lambda$ be the triangular tableau $(n-1,n-2,\ldots,1)$. Now I want to remove three adjacent boxes in an "L" shape:

![enter image description here][2]

Here I've indexed the location of the left box as $i$ which gives the $x$ coordinate of the box ($i=3$ in this example). Again, I'm calling the new tableau of shape $\delta_i$, and one easily finds $f_{\delta_i}/f_\lambda$ as the ratio of the particular hooks that change. I'm interested in the following. In the above example, I removed three adjacent boxes at location $i$. I want to evaluate sums of the form:

$$\sum_{i=1}^{n-2}\frac{f_{\delta_i}}{f_\lambda}$$.

In full generality, I am interested in removing *any* pattern of boxes indexed by $i$ (the sum's range changes appropriately depending on the pattern). Going along with my example, one can show that 

$$f_{\delta_i}/f_\lambda=\frac{1}{3N(N-1)(N-2)}a_ia_{n-i-1},$$

 where $N:=\binom{n}{2}$ and $a_i:=\frac{(2i+1)!!}{(2i-2)!!}$. One can then evaluate the sum using relatively straightforward generating functions for $a_i$, giving a tidy answer of $1/N$. 

>**Question:** I have heard that calculating $\sum_{i=1}^{n-2}\frac{f_{\delta_i}}{f_\lambda}$ can be done via the Murnaghan Nakayama rule. In particular, I've been told my $L$ example gives $-\frac{\chi^{\lambda}(\pi)}{\chi^\lambda(\mbox{id})}$ where $\pi$ is a 3-cycle. Can someone provide an accessible reference for this rule in the context above? I am not an expert in representation theory and the usual references I've seen (*Enumerative Combinatorics, Vol II* by Stanley) are somewhat beyond me at the moment (and in much greater generality). In short, I am looking for an accessible explanation of the Murnagham Nakayama rule for these types of questions. 


  [1]: http://math.stackexchange.com/questions/425256/on-applications-of-the-murnagham-nakayama-rule
  [2]: https://i.sstatic.net/xdyhE.png