= Solution
Use the <Young–Jucys–Murphy elements> in the <group algebra> $\mathbb C S_n$. Their <joint spectrum> is
$$
\operatorname{Spec}(n)=\{(a_1,\ldots,a_n):X_rv=a_rv\ (1\leq r\leq n)\text{ for some }v\ne0\text{ in an irreducible module}\}.
$$
Equivalently one can use the <regular representation>, which contains every <irreducible representation>. The commuting elements are self-adjoint in a <unitary representation>, so there is a simultaneous <eigenbasis>.
A <standard Young tableau> has entries increasing along each row and down each column. If entry $r$ occupies cell $(i_r,j_r)$, its <Content of a Young-diagram cell> is $c_T(r)=j_r-i_r$. Define the set of <content vectors of standard Young tableaux> by
$$
\boxed{\operatorname{Cont}(n)=\{(c_T(1),\ldots,c_T(n)):T\text{ standard, of any shape of size }n\}.}
$$
The shape here is a <partition of an integer>; it is not itself the vector of <eigenvalues>. For $n=1$ both sets consist of $(0)$.
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