(i) Let $G$ be a group, and $X$ and $Y$ finite $G$-sets. Define the permutation representation $\mathbf{C}[X]$ and compute its character. Show that

$$\operatorname{dim}_{\operatorname{Hom}_{G}}(\mathbf{C}[X], \mathbf{C}[Y])$$

is equal to the number of $G$-orbits in $X \times Y$.

(ii) Let $G=S_{n}(n \geqslant 4), X=\{1, \ldots, n\}$, and

$$Z=\{\{i, j\} \subseteq X \mid i \neq j\}$$

be the set of 2 -element subsets of $X$. Decompose $\mathbf{C}[Z]$ into irreducibles, and determine the dimension of each irreducible constituent.