Murnaghan-Nakayama rule

Statement
Suppose $$n$$ is a natural number. Suppose $$\lambda$$ is an unordered integer partition of $$n$$. Suppose $$g$$ is an element of the symmetric group $$S_n$$ and $$\chi_\lambda$$ is the character of the irreducible representation corresponding to $$\lambda$$. Then, if we can write $$g = g_1g_2$$, where $$g_1$$ is a $$m$$-cycle for $$m \le n$$ and $$g_2$$ is a permutation of the remaining $$n-m$$ elements, we have:

$$\chi_\lambda(g) = \sum (-1)^{r(\mu)} \chi_\mu(h)$$

Here, $$\mu$$ ranges over all unordered integer partitions of $$n - m$$, and $$r(\mu)$$ is one less than the number of parts in $$\mu$$, i.e., one less than the number of rows in the Young diagram of $$\mu$$.