Let $G$ be a finite group, $H<G$ be a subgroup, and let $h \in H$. Denote by $C_H(h)=\{x\in H: xh=hx\}$ and $C_G(h)=\{y\in G: yh=hy\}$ the centralizers of $h$ in groups $H$ and $G$, respectively.
Question: Is the following inequality always true: $$ \frac{|C_G(h)|}{|C_H(h)|} \le \frac{|G|}{|H|} $$ In our calculation we noticed that this holds for $G=S_n$, $H=S_k \times S_{n-k}$, so I was curious if this holds in general.
Yes. This is a special case of the following:
If $H$ and $K$ are subgroups of $G$, then $|G|\geq |HK|=|H||K|/|H\cap K|$, so $|G|/|H|\geq |K|/|H\cap K|$.
(Note that $HK$ might not be a subgroup of $G$, but the result still holds.)
In you case, you have $K=C_G(h)$.