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Let $\pi^\lambda$ be the <permutation character> of $M^\lambda$. <Young's rule> gives $\pi^\lambda=\chi^\lambda+\sum_{\mu>\lambda}K_{\mu\lambda}\chi^\mu$, with integer Kostka multiplicities. This dominance-triangular system has diagonal entries one, so recursive subtraction expresses every <irreducible character> as an integer linear combination of permutation characters. The latter count fixed tabloids, proving that all $\chi^\lambda(g)$ are integers.
For a $k$-cycle, the <Murnaghan–Nakayama rule> reduces to
$$
\chi^\lambda(g)=\sum_{\substack{\nu\subseteq\lambda,\ |\nu|=n-k\\\lambda/\nu\text{ a rim hook of length }k}}
(-1)^{\operatorname{ht}(\lambda/\nu)}f^\nu,
$$
where height is the number of occupied rows minus one, and $f^\nu$ is the degree of the remaining Specht <character> at the identity. In particular a full $n$-cycle has <character> zero unless $\lambda=(n-r,1^r)$ is a hook, in which case the value is $(-1)^r$.
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