Let $m,r\in \mathbb{N}, n=rm$. The character of symmetric group $\chi^{\lambda}$ where $\lambda$ is of the from $(k,1^{n-k})$ evaluated at the conjugacy class of $S_{n}$ of the from $(r,\ldots,r)$, where $r$ appearing $m--times$ is equal to \begin{equation} \chi_{r,\ldots,r}^{(k,1^{n-k})}= (-1)^{(r-1)(m-\Big[\frac{k}{r}\Big])+(k-1)mod\ r} \ \binom{m-1}{\Big[ \frac{(k-1)}{r} \Big]} \end{equation} \end{lemma} I want to prove the above statement using induction on $m$. Even I have a proof for the case $m=1$. After this step I want to use Murnaghan-Nakayama rule on $\chi_{r,\ldots,r}^{(k,1^{n-k})}$ to reduce the case to $m-1$. Here I don't know misinterpreting the rule. By the rule
\begin{equation} \chi_{r,\ldots,r}^{\lambda} =\sum_{\xi} (-1)^{l(\xi)}\chi_{r,\ldots,r}^{\lambda\setminus \xi} \end{equation}
The numbers of $r$ on the R.H.S is $(m-1)$. We are only interested when $\lambda$ are hooks. For a fixed $\lambda$
There is one or two possibility of $\xi$ that is the hook of length $r$? Am I having a confusion here from a hook we could delete two possible hooks of length $r$? Then the induction doesn't give the right formula. So what I am doing wrong here. Notice for $r=1$ it's the usual $dim\ \lambda$.