  We investigate the structure and properties of an Artinian monomial complete
intersection quotient $A(n,d)=\mathbf{k} [x_{1}, \ldots, x_{n}] \big /
(x_{1}^{d}, \ldots, x_{n}^d)$. We construct explicit homogeneous bases of
$A(n,d)$ that are compatible with the $S_{n}$-module structure for $n=3$, all
exponents $d \ge 3$ and all homogeneous degrees $j \ge 0$. Moreover, we derive
the multiplicity formulas, both in recursive form and in closed form, for each
irreducible component appearing in the $S_{3}$-module decomposition of
homogeneous subspaces. 4, 5$.
