= Solution
For a tableau $t$, let $C_t$ be its column stabilizer and $e_t=\sum_{\sigma\in C_t}\operatorname{sgn}(\sigma)\{\sigma t\}$ its <polytabloid>. The <Specht module> is the span of all such <polytabloids> inside the <tabloid> permutation module.
A <tabloid> with second row of size $k$ is determined by that subset, so $\dim M^{(n-k,k)}=\binom nk$. The two-row hook formula gives
$$
\dim S^{(n-k,k)}=\frac{n!(n-2k+1)}{(n-k+1)!k!}=\binom nk-\binom n{k-1}.
$$
At $k=3$ this proves
$$
\boxed{\dim M^{(n-3,3)}=\dim M^{(n-2,2)}+\dim S^{(n-3,3)}.}
$$
By <Maschke's theorem>, characteristic-zero permutation modules are completely reducible, so their Specht submodules are summands. The nonzero sum of all three-subset tabloids is fixed by every permutation; its span is a copy of the trivial module $S^{(n)}$.
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