 Solution

Since $X_1=0$, every joint weight has $a_1=0$. In $S_2$, $X_2=(12)$ acts by $1$ on the trivial representation and by $-1$ on the sign representation, which exhaust the irreducible representations. ThusThe content vector of a standard Young tableau conditions similarly require the second coordinate to be a neighbour of $0$, soThese vectors correspond respectively to the single-row tableau of shape $(2)$ and the single-column tableau of shape $(1,1)$.