# On Applications of the Murnaghan-Nakayama rule

The question is located below. In short, I am looking for an accessible explanation of the Murnaghan-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:

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 Murnaghan-Nakayama rule.

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