= Solution
If the cycle <integer partition>[partition] contains a part $r$, the <Murnaghan–Nakayama rule> is
$$
\boxed{\chi^\lambda((r,\nu))=\sum_{\substack{\mu\subseteq\lambda\\\lambda/\mu\text{ a border strip of size }r}}(-1)^{\operatorname{ht}(\lambda/\mu)}\chi^\mu(\nu).}
$$
A <border strip> is a connected skew <Young diagram> with no $2\times2$ block; its height is its number of occupied rows minus one. The remaining boxes must be a <integer partition>[partition] diagram. The recursion starts with <character> one for the empty diagram at the empty <cycle type>, and an empty sum is zero. Repeated removal for all cycle lengths computes the <character> value.
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