I know two formulas by the name of Frobenius.
The first one computes the number $$\mathcal{N}(G;C_1,\dotsc,C_k):=|\{(c_1,\dotsc,c_k)\in C_1\times\cdots C_k\:|\:c_1\cdots c_k=1\}|,$$$$\mathcal{N}(G;C_1,\dotsc,C_k):=|\{(c_1,\dotsc,c_k)\in C_1 \times \cdots \times C_k\:|\:c_1\cdots c_k=1\}|,$$ where $G$ is a finite group and $C_1,\dotsc,C_k$ conjugacy classes in $G$. The Frobenius' formula says that $$\mathcal{N}(G;C_1,\dotsc,C_k)=\frac{|C_1|\cdots |C_k|}{|G|}\sum_{\chi}\frac{\chi(C_1)\cdots\chi(C_k)}{\chi(1)^{k-2}},$$ where the sum is over all characters of irreducible representations of $G$.
The other says that, if $\lambda=(\lambda_1,\dotsc,\lambda_\ell)$ and $\mu=(\mu_1,\dotsc,\mu_m)$ are partitions of $n$, the coefficient of $x_1^{\lambda_1+\ell-1}x_2^{\lambda_2+\ell-2}\cdots x_{\ell}^{\lambda_\ell}$ in $$\prod_{1\leq i<j\leq \ell}(x_i-x_j)\prod_{i=1}^m(x_1^{\mu_i}+x_2^{\mu_i}+\cdots+x_\ell^{\mu_i})$$ is the character of $S^\lambda$ (the Speecht module) evaluated at an element of $S_n$ with cycle type $\mu$.
It would be lovely if we could compute the characters of the Speecht modules utilizing the first formula. If it is possible, how could I do it?
