Frobenius alternant character formula (${\chi }^{\lambda }(\alpha )=[x^{\lambda +\delta }]p_{\alpha }(x)A_{\delta }(x)$)

For a partition of an integer $\lambda ⊢n$, padded to $m$ rows, let $\delta =(m-1,\dots ,0)$. If a permutation has ${\alpha }_q$ cycles of length $q$, put $p_{\alpha }={\prod }_q({\sum }_i{x}_i^q{)}^{{\alpha }_q}$. The character of the Specht module $S^{\lambda }$ is the displayed coefficient. Equivalently $p_{\alpha }{A}_{\delta }={\sum }_{\lambda }{\chi }^{\lambda }(\alpha )A_{\lambda +\delta }$, the alternant form of the Frobenius characteristic map. It gives finite coefficient computations without constructing representation matrices.