= Solution
For an integer tuple $\ell=(\ell_1,\ldots,\ell_m)$, the <monomial alternant> is
$$
A_\ell(x)=|x^{\ell_1},\ldots,x^{\ell_m}|=\det(x_i^{\ell_j})_{1\le i,j\le m}.
$$
Negative exponents require nonzero coordinates; all exponents in the <character> expansion below are nonnegative. The <power-sum symmetric polynomial> is $s_k(x)=\sum_{i=1}^m x_i^k$. Put $\delta=(m-1,m-2,\ldots,0)$, so $A_\delta(x)=\prod_{i<j}(x_i-x_j)$ is the <Vandermonde determinant>.
Let a <conjugacy class> of $S_n$ have $\alpha_k$ cycles of length $k$, with $\sum k\alpha_k=n$, and put $p_\alpha(x)=\prod_k s_k(x)^{\alpha_k}$. The product $p_\alpha A_\delta$ is an <alternating polynomial> homogeneous of degree $n+\binom m2$. In an alternating <polynomial>, a monomial with two equal exponents has zero coefficient, since interchanging those variables fixes the monomial and reverses its sign. Grouping the remaining monomials by their permutation orbits gives a unique expansion in alternants $A_\ell$ with $\ell_1>\cdots>\ell_m\ge0$.
Such tuples of the indicated total degree are exactly $\ell_i=\lambda_i+m-i$ for <partitions of an integer> $\lambda$ of $n$ with at most $m$ parts. Define the <class function>
$$
\boxed{\omega_\lambda(\alpha)=[x_1^{\lambda_1+m-1}\cdots x_m^{\lambda_m}]\,p_\alpha(x)A_\delta(x).}
$$
The displayed monomial occurs with coefficient $1$ in $A_{\lambda+\delta}$ and in no other ordered alternant, so
$$
p_\alpha(x)A_\delta(x)=\sum_{\lambda\vdash n,\ \ell(\lambda)\le m}\omega_\lambda(\alpha)A_{\lambda+\delta}(x).
$$
This proves the expansion and explicitly defines its coefficients. They depend only on the cycle counts and hence are <class functions>. Identifying these coefficients with <Specht module> <characters> is the <Frobenius alternant character formula>, which the remaining parts allow us to assume.
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