The <Murnaghan–Nakayama rule> states that if a permutation has a $k$-cycle and remaining cycle type $\rho$, then
$$
\chi^\lambda(k,\rho)
=\sum_H(-1)^{\operatorname{leg}(H)}
\chi^{\lambda\setminus H}(\rho),
$$
where $H$ ranges over removable <rim hook>[rim hooks] of length $k$ and $\operatorname{leg}(H)$ is one less than the number of rows occupied by $H$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 160 / 2 / b / i / Solution

The <Murnaghan–Nakayama rule> states that if a permutation has a $k$-cycle and remaining cycle type $\rho$, then
$$
\chi^\lambda(k,\rho)
=\sum_H(-1)^{\operatorname{leg}(H)}\chi^{\lambda\setminus H}(\rho),
$$
where $H$ runs over the removable <rim hook>[rim hooks] of length $k$. Iterating removes rim hooks whose lengths are the cycle lengths, and the character value is the signed sum over all complete removal sequences.

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 160 / 2 / a / i / Solution

{wiki}

A rim hook, or border strip, is a connected skew Young diagram containing no $2\times2$ square. Its leg length is one less than its number of occupied rows.


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 Incoming links: Rim hook