= Two-subset incidence realization of a two-row Specht module
{title2=$S^{(n-2,2)}=\ker\bigl(e_{ij}\mapsto e_i+e_j\bigr)$}
Over a <field> of characteristic zero with $n\ge4$, let $V_2$ be the permutation representation on unordered two-element subsets and $V_1$ the representation on singletons. The equivariant incidence map $D:V_2\to V_1$ sends $e_{ij}$ to $e_i+e_j$. It is surjective: if coefficients $c_i$ annihilate its image, then $c_i+c_j=0$ for all distinct $i,j$, forcing all coefficients to vanish. Thus its kernel has dimension $\binom n2-n$.
A polytabloid of shape $(n-2,2)$, identified by its bottom-row pair, is an alternating four-edge vector $e_{cd}-e_{ad}-e_{cb}+e_{ab}$. Its incidence is zero. The <hook-length formula> gives the same dimension $\binom n2-n$ for this <Specht module>, proving the identity. <Maschke's theorem> yields $V_2\cong S^{(n-2,2)}\oplus V_1$.
For a Young subgroup with block sizes $\mu_1,\ldots,\mu_l$, there are $l$ singleton orbits and $\binom l2+\#\{i:\mu_i\ge2\}$ pair orbits. Taking fixed vectors and using <Frobenius reciprocity> therefore gives the multiplicity of $S^{(n-2,2)}$ in $M^\mu$ as $\binom l2+\#\{i:\mu_i\ge2\}-l$. This derives the multiplicity without <Young's rule>; block sizes $(3,2,1)$ give exactly two.
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