Solution
= Solution
The two <integer partition>[partition] diagrams are transposes, so their characteristic-zero <Specht modules> differ by tensoring with the <sign representation>. A <transposition> has sign minus one. Consequently
$$
\boxed{\chi^{(2,1^{n-2})}((i,j))=-\chi^{(n-1,1)}((i,j))=3-n.}
$$
The middle <character> is $n-3$, since the standard <representation> is the point-permutation <representation> minus its trivial line, and a <transposition> fixes $n-2$ points.