= Solution
The <permutation module> $M^\lambda$ has a <basis> consisting of row <tabloids> of shape $\lambda$: tableaux are identified when each row has the same set of entries. A permutation acts by applying it to every entry, and the action is extended linearly over $F$. Its <dimension> is $n!/\prod_i\lambda_i!$.
Here the two singleton rows form an ordered pair of distinct entries, so
$$
\boxed{\dim M^{(n-2,1,1)}=n(n-1).}
$$
A <standard tableau> of the same shape must put one in its top-left corner. Choose two entries from $2,\ldots,n$ for the two remaining boxes of the first column, in increasing order; all other entries then fill the top row in increasing order. Thus the standard-<polytabloid> <basis> gives
$$
\boxed{\dim S^{(n-2,1,1)}=\binom{n-1}2=\frac{(n-1)(n-2)}2.}
$$
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