= Solution
Conjugation in the <symmetric group> sends $(1\ 2)$ to $(\sigma(1)\ \sigma(2))$. Hence $\sigma$ commutes with $g$ exactly when it preserves the unordered pair $\{1,2\}$. It can independently swap that pair and arbitrarily permute the remaining points. Therefore the <centraliser of a transposition> is
$$
\boxed{C_{S_n}(g)=\langle(1\ 2)\rangle\times S_{\{3,\ldots,n\}},\qquad |C_{S_n}(g)|=2(n-2)!.}
$$
Let $m=n/2$. The <orbits of a group action> generated by $h$ are its $m$ two-point blocks. Any <centraliser> element must permute those blocks, and can choose independently whether to swap the two points in each image block. Conversely every such choice commutes with $h$, since it preserves the partner relation. This <centraliser of a fixed-point-free involution> is the <permutation wreath product> $C_2\wr S_m$, with
$$
\boxed{|C_{S_n}(h)|=2^m m! = 2^{n/2}(n/2)!.}
$$
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