= Solution
Since $X_1=0$, every joint weight has \b[$a_1=0$]. In $S_2$, $X_2=(1\ 2)$ acts by $1$ on the <trivial representation> and by $-1$ on the <sign representation>, which exhaust the <irreducible representations>. Thus
$$
\boxed{\operatorname{Spec}(2)=\{(0,1),(0,-1)\}}.
$$
The <content vector of a standard Young tableau> conditions similarly require the second coordinate to be a neighbour of $0$, so
$$
\boxed{\operatorname{Cont}(2)=\{(0,1),(0,-1)\}}.
$$
These vectors correspond respectively to the single-row tableau of shape $(2)$ and the single-column tableau of shape $(1,1)$.
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