$\dim \operatorname{Hom}_{S_n}(\mathbb C [X_l], \mathbb C [X_k])$ is equal to the number of double cosets $P_l \backslash S_n/P_k$

Question

I have the following problem:

$S_n$ is acting on the set of $k$ element subsets of $\{1,...,n\}$ ($X_k$). Show that $\dim\operatorname{Hom}_{S_n}(\mathbb C [X_l], \mathbb C [X_k])$ is equal to the number of double cosets $P_l \backslash S_n/P_k$, $1\leq l \leq k \leq n$, where $P_i$ is the stabilizer of $\{1,...,i\}$

I’m asking for any hints. Thank you