= Solution
The restriction form of the <restriction branching rule for a symmetric group> is
$$
\boxed{\operatorname{Res}^{S_n}_{S_{n-1}}S^\lambda
\cong\bigoplus_{\mu\in\lambda^-}S^\mu},
$$
where $\lambda^-$ contains the distinct partitions obtained by deleting one <Removable node of a Young diagram>. In particular, the restriction is multiplicity-free.
Restrict the alternating expression
$$
\psi^\lambda=\sum_{\pi\in S_N}\operatorname{sgn}(\pi)\xi^{\lambda-\mathrm{id}+\pi}.
$$
The supplied restriction formula for a Young permutation character, with $k=1$, says that each term restricts by subtracting one from each possible component. After collecting the alternating sums, this gives
$$
\operatorname{Res}^{S_n}_{S_{n-1}}\psi^\lambda
=\sum_i\psi^{\lambda-\epsilon_i}.
$$
If row $i$ has no removable node, part i straightens $\psi^{\lambda-\epsilon_i}$ against the adjacent term with the opposite sign, or makes it zero when two shifted entries coincide. The surviving terms are exactly $\psi^\mu$ for $\mu\in\lambda^-$. Since $\lambda$ and each surviving $\mu$ are partitions, $\psi^\lambda=\chi^\lambda$ and $\psi^\mu=\chi^\mu$. We obtain
$$
\operatorname{Res}^{S_n}_{S_{n-1}}\chi^\lambda
=\sum_{\mu\in\lambda^-}\chi^\mu.
$$
Complex representations of a <finite group> are semisimple, so equality of characters proves the asserted module decomposition.
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