= Specht filtration of the ordered-pair permutation module
For $n\geq4$, identify $M^{(n-2,1^2)}$ with the permutation module on ordered pairs $(i,j)$ with $i\ne j$. Let $p_1(i,j)=e_i$, $p_2(i,j)=e_j$, and $q(i,j)=\{i,j\}$. With $U=\ker p_1$ and $V=U\cap\ker p_2$, one has a Specht filtration
$$
0<S^{(n-2,1^2)}=\ker(q|_V)<V<U<\ker\varepsilon<M^{(n-2,1^2)}
$$
whose successive quotients are
$$
S^{(n-2,1^2)},\quad S^{(n-2,2)},\quad
S^{(n-1,1)},\quad S^{(n-1,1)},\quad S^{(n)}.
$$
The same construction for $n=3$ omits the zero $S^{(n-2,2)}$ factor.
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